The Development of Year 3 Students’
Place-Value Understanding:
Representations and Concepts
Peter Stanley Price
Dip.Teach., B.Ed., M.Ed., A.C.P.
Centre for Mathematics and Science Education
School of Mathematics, Science and Technology Education
Faculty of Education
Queensland University of Technology
A Thesis
submitted in partial fulfilment of the requirements
for the award of the degree of
Doctor of Philosophy
March, 2001
Keywords
Place value, base-ten blocks, Year 3, mathematical understanding, place-value
software, representations of number, conceptions of number, electronic base-ten
blocks, conceptual structures for multidigit numbers, feedback, misconceptions of
number, independent-place construct, face-value construct, mathematics teaching
with technology, number models, Payne-Rathmell model for teaching number topics.
i
Abstract
Understanding base-ten numbers is one of the most important mathematics
topics taught in the primary school, and yet also one of the most difficult to teach and
to learn. Research shows that many children have inaccurate or faulty number
conceptions, and use rote-learned procedures with little regard for quantities
represented by mathematical symbols. Base-ten blocks are widely used to teach
place-value concepts, but children often do not perceive the links between numbers,
symbols, and models. Software has also been suggested as a means of improving
children’s development of these links but there is little research on its efficacy.
Sixteen Queensland Year 3 students worked cooperatively with the researcher
for 10 daily sessions, in 4 groups of 4 students of either high or low mathematical
achievement level, on tasks introducing the hundreds place. Two groups used
physical base-ten blocks and two used place-value software incorporating electronic
base-ten blocks. Individual interviews assessed participants’ place-value
understanding before and after teaching sessions. Data sources were videotapes of
interviews and teaching sessions, field notes, workbooks, and software audit trails,
analysed using a grounded theory method.
There was little difference evident in learning by students using either
physical or electronic blocks. Many errors related to the “face-value” construct,
counting and handling errors, and a lack of knowledge of base-ten rules were
evident. Several students trusted the counting of blocks to reveal number
relationships. The study failed to confirm several reported schemes describing
children’s conceptual structures for multidigit numbers. Many participants
demonstrated a preference for grouping or counting approaches, but not stable
mental models characterising their thinking about numbers generally. The
independent-place construct is proposed to explain evidence in both the study and
the literature that shows students making single-dimensional associations between a
place, a set of number words, and a digit, rather than taking account of groups of 10.
Feedback received in the two conditions differed greatly. Electronic feedback was
more positive and accurate than feedback from blocks, and reduced the need for
human-based feedback. Primary teachers are urged to monitor students’ use of baseten blocks closely, and to challenge faulty number conceptions by asking appropriate
questions.
iii
Table of Contents
Keywords .........................................................................................................................i
Abstract......................................................................................................................... iii
Table of Contents ...........................................................................................................v
List of Tables .................................................................................................................ix
List of Figures.................................................................................................................x
Supplementary Material ...............................................................................................x
Statement of Original Authorship ..............................................................................xi
Acknowledgments ...................................................................................................... xiii
Chapter 1:
The Problem........................................................................................ 1
1.1 Recommendations for Changes in Mathematics Education............................1
1.2 The Learning of Place-Value Concepts..............................................................3
1.2.1
1.2.2
1.3
1.4
1.5
1.6
Conceptual Structures and Difficulties With Place-Value Concepts ...................3
Use of Number Representations............................................................................3
The Research Question.........................................................................................5
Overview of Research Methodology...................................................................5
Significance of the study.......................................................................................6
Outline of the Thesis .............................................................................................7
Chapter 2:
Review of Literature........................................................................... 9
2.1 Chapter Overview.................................................................................................9
2.2 Issues in Mathematics Education......................................................................10
2.2.1
2.2.2
2.2.3
Students’ Active Involvement in Mathematics Learning...................................10
Number Sense......................................................................................................13
Use of Technological Devices ............................................................................15
2.3 Place-value Understanding ................................................................................17
2.3.1
2.3.2
Place Value ..........................................................................................................18
Place-value Understanding..................................................................................20
2.4 The Contribution of Cognitive Science to Mathematics Education .............21
2.4.1
2.4.2
2.4.3
Understanding Mathematics................................................................................22
Mental Models.....................................................................................................23
Analogical Reasoning..........................................................................................37
2.5 Teaching Place-value Understanding ...............................................................43
2.5.1
2.5.2
2.5.3
Teaching Approaches ..........................................................................................43
Building Place-Value Connections .....................................................................45
Use of Concrete Materials...................................................................................51
2.6 Computers and Mathematics Education .........................................................55
2.6.1
2.6.2
Claimed Benefits of Computers ..........................................................................55
Cognitive Aspects of Computer Use...................................................................57
2.7 Chapter Summary; Statement of the Problem ...............................................59
Chapter 3:
Methods ............................................................................................. 61
3.1 Chapter Overview...............................................................................................61
3.2 Aims of the Study................................................................................................61
3.3 Variables ..............................................................................................................62
v
3.3.1
3.3.2
Mathematical Achievement Level...................................................................... 62
Number Representation Format ......................................................................... 63
3.4 Data collection and analysis. ............................................................................. 63
3.5 Design Issues........................................................................................................ 63
3.5.1
3.5.2
Assumptions........................................................................................................ 63
Theoretical and Methodological Stance............................................................. 64
3.6 Pilot Study ........................................................................................................... 68
3.6.1
3.6.2
3.6.3
3.6.4
3.6.5
Purposes of the Pilot Study................................................................................. 68
Selection of Pilot Study Participants.................................................................. 69
Pilot Study Procedures........................................................................................ 70
Pilot Study Data Collection and Analysis.......................................................... 70
Changes Made to Study Design After Pilot Study............................................. 70
3.7 Main Study .......................................................................................................... 75
3.7.1
3.7.2
3.7.3
3.7.4
3.7.5
Selection of Participants ..................................................................................... 76
Teaching Program............................................................................................... 77
Instruments - First and Second Interviews......................................................... 83
Administration Procedures ................................................................................. 85
Data Collection and Analysis ............................................................................. 87
3.8 Validity and Reliability ...................................................................................... 92
3.9 Limitations........................................................................................................... 93
3.10 Chapter Summary .............................................................................................. 94
Chapter 4:
Results................................................................................................ 97
4.1 Chapter Overview............................................................................................... 97
4.1.1
Restatement of the Research Question............................................................... 97
4.2 Transcript Conventions Used in this Thesis.................................................... 98
4.3 Place-Value Task Performance Revealed in Interview Results .................... 99
4.3.1
4.3.2
Methods used to Analyse Interview Data .......................................................... 99
Overview of Interview Results......................................................................... 100
4.4 Students’ Conceptions of Numbers ................................................................ 107
4.4.1
4.4.2
4.4.3
4.4.4
4.4.5
Grouping Approaches....................................................................................... 107
Counting Approaches ....................................................................................... 115
Face-Value Interpretation of Symbols ............................................................. 123
Summary of Approaches to Interview Questions ............................................ 132
Changeability of Participants’ Number Conceptions ...................................... 134
4.5 Digit Correspondence Tasks: Four Categories of Response ....................... 136
4.5.1
4.5.2
4.5.3
4.5.4
4.5.5
Category I: Face-Value Interpretation of Digits .............................................. 137
Category II: No Referents For Individual Digits ............................................. 137
Category III: Correct Total Represented by Each Digit, but Tens not Explained
........................................................................................................................... 140
Category IV: Correct Number of Referents, Tens Place Mentioned............... 141
Summary of Responses to Digit Correspondence Tasks ................................. 142
4.6 Errors, Misconceptions, and Limited Conceptions ......................................143
4.6.1
4.6.2
4.6.3
4.6.4
Counting Errors ................................................................................................ 143
Blocks Handling Errors .................................................................................... 145
Errors in Naming and Writing Symbols for Numbers..................................... 149
Errors in Applying Values to Blocks ............................................................... 152
4.7 Use of Materials to Represent Numbers ........................................................158
vi
4.7.1
4.7.2
4.7.3
4.7.4
4.7.5
4.7.6
4.7.7
4.7.8
Counting of Representational Materials ...........................................................158
Use of Trial-and-Error Methods........................................................................162
Handling Larger Numbers.................................................................................164
Interpreting Non-Canonical Block Arrangements............................................167
Face-value Interpretations of Symbols .............................................................169
Predictions About Trading ................................................................................172
Feedback ............................................................................................................176
Using Blocks To Discover Number Relationships...........................................186
4.8 Chapter Summary ............................................................................................192
Chapter 5:
Discussion ........................................................................................ 193
5.1 Chapter Overview.............................................................................................193
5.2 Participants’ Ideas About Multidigit Numbers ............................................193
5.2.1
5.2.2
5.2.3
5.2.4
Participants’ Preferences for Grouping or Counting Approaches....................195
Comparison of Grouping and Counting Approaches .......................................198
Difficulties With Existing Conceptual Structure Schemes ..............................204
Face-value Interpretations of Symbols .............................................................208
5.3 Independent-Place Construct ..........................................................................213
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
5.3.6
Description & Definition of the Independent-Place Construct ........................213
Comparison of the Independent-Place Construct and the Face-Value Construct
............................................................................................................................214
Evidence for the Independent-Place Construct in This Study..........................214
Evidence of the Independent-Place Construct in the Literature.......................217
Written Computation and the Independent-Place Construct............................221
Place-Value Tasks and the Independent-Place Construct ................................222
5.4 Participants’ Construction of Meaning..........................................................223
5.4.1
5.4.2
5.4.3
‘Organic’ Understanding...................................................................................224
Participants’ “Invented” Answers .....................................................................225
Teaching, Learning, and Constructing Meaning ..............................................227
5.5 Effects of Physical or Electronic Base-Ten Blocks on Place-Value
Understanding ............................................................................................................227
5.5.1
5.5.2
5.5.3
5.5.4
5.5.5
Differences in Learning of Participants Using Physical or Electronic Blocks 228
Sensory Impact of Physical or Electronic Blocks.............................................228
How Numbers Are Represented by Physical or Electronic Blocks .................230
Development of Links Among Blocks, Symbols, and Numbers .....................232
Support for the Development of Number Concepts .........................................234
5.6 Place-Value Understanding Demonstrated by High- and Low-AchievementLevel Participants ......................................................................................................235
5.6.1
Similarities in Place-Value Understanding of High- and Low-AchievementLevel Participants ............................................................................................................235
5.6.2
Differences in Place-Value Understanding of High- and Low-AchievementLevel Participants ............................................................................................................236
Chapter 6:
Conclusions ..................................................................................... 239
6.1 Chapter Overview.............................................................................................239
6.2 Conclusions About Answers to Research Questions ....................................239
6.2.1
Conceptual Structures for Multidigit Numbers Evident in Participants’
Responses.........................................................................................................................239
6.2.2
Misconceptions, Errors, or Limited Conceptions Evident In Participants’
Responses.........................................................................................................................240
vii
6.2.3
Effects of the Two Materials on Students’ Learning of Place-Value Concepts
........................................................................................................................... 243
6.2.4
Differences Between Place-Value Understanding of High- and LowAchievement-Level Participants..................................................................................... 248
6.3 Implications for Teaching................................................................................249
6.3.1
Implications of Using Physical Base-Ten Blocks to Teach Place-Value
Concepts .......................................................................................................................... 249
6.3.2
Implications of Using Electronic Base-Ten Blocks to Teach Place-Value
Concepts .......................................................................................................................... 253
6.3.3
Implications of the Independent-Place Construct for Teaching Mathematics 255
6.3.4
Implications of Construction of Meaning for Teaching Mathematics ............ 256
6.4 Recommendations for Further Research....................................................... 258
Appendix A – Design of Software used in the Study.............................................261
Appendix B - Overview of Teaching Session Content for Interviews and
Teaching Phase of Pilot Study..................................................................................277
Appendix C – Summary of Pilot Study Teaching Program .................................279
Appendix D - Excerpt of Teaching Script of Pilot Study: Session 1.................... 281
Appendix E – Audit Trail Example.........................................................................283
Appendix F – Results of The Year Two Diagnostic Net, Used to Select
Participants for the Main Study .............................................................................. 287
Appendix G – List of Participants ........................................................................... 289
Appendix H - Main Study Teaching Program ....................................................... 291
Appendix I - Main Study Interview 1 Instrument................................................. 299
Appendix J - Main Study Interview 2 Instrument ................................................301
Appendix K – Letter Requesting Consent by Parents or Guardians of
Prospective Participants ...........................................................................................303
Appendix L – Coding Teaching Session Transcripts for Feedback .................... 305
Appendix M – Descriptions of Numeration Skills Targeted by Interview
Questions and Criteria for Their Assessment ........................................................311
Appendix N – Transcript of Interview 1 Question 6 (a) with Terry ...................315
Appendix O – Transcript of Interview 2 Question 6 (a) with Hayden................319
Appendix P – Transcript of Low/Blocks Group Answering Task 28 (a)............321
Appendix Q – Transcripts of Task 4 (a) from 4 groups........................................ 325
Appendix R – Transcript Excerpts Showing Participants Predicting Equivalence
of Traded Blocks........................................................................................................355
Appendix S – Transcript of Task 4 (d) from Low/Blocks Group........................365
Appendix T – Comparison Between Ross’s (1989) Model and a Proposed Model
for Categories of Responses to Digit Correspondence Tasks...............................369
Appendix U – Sample Coding of Transcript for Feedback .................................. 371
References...................................................................................................................373
Supplementary Material – Hi-Flyer Maths Installation Files [CD-ROM] .........385
viii
List of Tables
TABLE 2.1.
TABLE 2.2.
TABLE 3.1.
TABLE 3.2.
TABLE 4.1.
TABLE 4.2.
TABLE 4.3.
TABLE 4.4.
TABLE 4.5.
TABLE 4.6.
TABLE 4.7.
TABLE 4.8.
TABLE 4.9.
TABLE 4.10.
TABLE 4.11.
TABLE 4.12.
TABLE 4.13.
TABLE 4.14.
TABLE 4.15.
TABLE 4.16.
TABLE 4.17.
TABLE 4.18.
TABLE 4.19.
TABLE 4.20.
TABLE 4.21.
TABLE 5.1.
TABLE H.1.
TABLE L.1.
TABLE L.2.
TABLE L.3.
Aspects of Place-value Understanding Described in the Literature... 26
Task Performance Illustrating Limited Conceptions in Place-value
Understanding..................................................................................... 31
Phases of the Research Design ........................................................... 75
Participant Groups for the Main Study............................................... 76
Transcript Notations ........................................................................... 98
Summary of Participants’ Numeration Skills Identified in two
Interviews ......................................................................................... 102
Summary of Numeration Skills Demonstrated by Each Participant and
by Each Group.................................................................................. 103
Summary of Place-value Understanding Criteria Achieved by Highachievement-level and Low-Achievement-Level Participants......... 105
Summary of Place-value Understanding Criteria Achieved by
Participants in Computer and Blocks Groups .................................. 106
Use of Grouping Approaches for Selected Interview Questions...... 113
Use of Grouping Approaches by Each Group.................................. 113
Use of a Counting Approach for Selected Interview Questions....... 121
Use of Counting Approaches by Each Group .................................. 122
Incidence of Face-value Interpretations for Written Symbols after
Selected Interview Questions ........................................................... 130
Use of Face-Value Interpretations of Symbols by Each Group ....... 131
Incidence of Approaches Adopted for Selected Interview Questions....
......................................................................................................... 133
Response Categories for Interview Digit Correspondence Questions ...
......................................................................................................... 142
Summary of Digit Correspondence Response Categories................ 143
Participants’ Written Responses to Task 27 (b) ............................... 171
Incidents of Feedback of Each Source per Group ............................ 177
Percentage of Feedback Compared With Answer Status ................. 180
Quality of Feedback Provided for Correct or Incorrect Answers..... 181
Percent of Feedback for Correct Answers from Each Source.......... 182
Percent of Feedback for Incorrect Answers from Each Source ....... 183
Feedback Providing Answers from Each Source for Each Group ... 187
Comparison of Results of Digit Correspondence Tasks Between This
Study and Ross (1989)...................................................................... 209
Overview of Teaching Program Tasks ............................................. 284
Source of Feedback .......................................................................... 307
Effects of Feedback .......................................................................... 307
Responses to Feedback..................................................................... 308
ix
List of Figures
Figure 2.1.
The face value of each individual numerical symbol, together with its
position relative to the ones place, determines the value it represents.
............................................................................................................ 18
Figure 2.2. Relationships inherent in base-ten blocks. ......................................... 41
Figure 2.3. Relationships among numbers, written symbols, and concrete
materials.............................................................................................. 46
Figure 2.4. Conceptual gap between written symbols and concrete materials. .... 48
Figure 2.5. The use of transitional forms to bridge the gap between written
symbols and concrete materials. ......................................................... 49
Figure 3.1. Dimensions of research design. .......................................................... 65
Figure 3.2. Original graphic images used on regrouping buttons in software used
during pilot study................................................................................ 72
Figure 3.3. Replacement graphic images used on regrouping buttons in software
used during main study....................................................................... 72
Figure 3.4. Sample Representing numbers task. ................................................... 80
Figure 3.5. Sample Regrouping task..................................................................... 81
Figure 3.6. Sample Use of numeral expander task. .............................................. 81
Figure 3.7. Sample Comparison task. ................................................................... 81
Figure 3.8. Sample Counting task. ........................................................................ 82
Figure 3.9. Sample Addition task, including regrouping. ..................................... 83
Figure 3.10. Diagram showing objects used in interviews for Digit Correspondence
Task with misleading perceptual cues. ............................................... 85
Figure 4.1. Interview scores compared to use of grouping approaches. ............. 115
Figure 4.2. Interview scores compared to use of counting approaches. ............. 122
Figure 4.3. Interview scores compared to use of face-value interpretations of
symbols. ............................................................................................ 132
Figure 4.4. Proportions of feedback from each source for each group. .............. 178
Figure 5.1. Column counters in software representation of 248. ........................ 232
Figure A.1. Screen view of on-screen tutorial question with block representations.
......................................................................................................... 262
Figure A.2. Partial screen image from Rutgers Math Construction Tools, showing
block and symbol representations of a number. ............................... 263
Figure A.3. Screen view of Blocks Microworld showing block representation of a
number, nominating a cube as one. .................................................. 264
Figure A.4. Main screen of Hi-Flyer Maths. ....................................................... 266
Figure A.5. “Show as tens” feature activated. ..................................................... 268
Figure A.6. Number name window and numeral expander displayed................. 269
Figure A.7. A block is “sawn” into 10 pieces...................................................... 271
Figure A.8. “Add blocks” requester..................................................................... 272
Figure L.1. Data entry screen for feedback database. ......................................... 306
Supplementary Material
Hi-Flyer Maths Installation Files [CD-ROM] ......................................................... 385
x
Statement of Original Authorship
The work contained in this thesis has not been previously submitted for a
degree or diploma at any other higher education institution. To the best of my
knowledge and belief, the thesis contains no material previously published or
written by another person except where due reference is made.
Signed:
________________________________
Date:
________________________________
xi
Acknowledgments
The completion of a thesis is a drawn-out, sometimes painful task that cannot
be done without much assistance, both professional and personal, from many others.
I gratefully acknowledge my indebtedness to the following people for their support
over the past six years:
To my principal supervisor, Professor Lyn English, I offer my heartfelt
appreciation for her patience, wisdom and unfailing support since I started this
journey. Your example to me, Lyn, as an academic and colleague has always been of
the highest standard, and I greatly appreciate your patience in leading me to the
completion of the thesis. Thank you for believing in me and for giving me the space
to finish.
To my associate supervisor, Dr Bill Atweh, I thank you also for your patience,
support, and wisdom. Your ability to see past the data to what they reveal has been
invaluable in helping me frame the last few chapters and in structuring what was
quite a mess and turn it into a coherent account.
To my dear wife and partner, Trish, I can only say that a lesser person would
have given up long ago. I deeply appreciate your love and support over what has
ended up as a longer time than we could have imagined when I started. This has truly
been a partnership, in which you have sacrificed your desires and your time to give
me space to study, since 1993. Thank you from the bottom of my heart.
To my lovely, wonderful children, Mary, Andrew and Hannah, I express my
deep love and devotion. You too have had to give up time with me, and to put up
with your Dad’s frequent absences over a substantial part of your lives. I am
immensely proud of each of you, and I look forward to seeing you grow and develop
into the adults God intends.
To my parents, Rev and Mrs Stanley and Eva Price, I express my love and
heartfelt thanks for everything you put into raising me. Though we are separated by
great distance, I am aware of your constant support and prayers that you have
provided all my life. Thanks, Dad and Mum.
To my colleagues and friends at Christian Heritage College, I express my
heartfelt thanks and love for accepting me and supporting me in this endeavour. In
particular, Dr Robert Herschell has been a constant friend, mentor and source of
support over many years. Thanks, Rob, for believing in me, for giving me the chance
to follow God’s call to teach others.
xiii
To many colleagues, mentors and friends at the School of Mathematics,
Science and Technology Education, QUT, thank you. I have had a very rewarding
time at QUT over many years, and appreciate your input into my life and career,
including the writing of this thesis. In particular, a sincere “thank you” to Professor
Tom Cooper, A/Prof Cam McRobbie and Drs Cal Irons, Ian Ginns, Rod Nason and
Jackie Stokes for your wise advice and counsel. And to my fellow PhDers over the
past several years—Drs Neil Taylor, Carmel Diezmann, Kathy Charles, Mary
Hanrahan, David Anderson, Stephen Norton, Anne Williams and Gillian Kidman—
thank you all for your friendship and support.
Finally, but by no means least, I express my love and appreciation to the Lord
Jesus Christ, without whom I could do nothing. My abilities and talents are from
Him alone; my prayer is that I walk worthy of the calling He has placed on my life,
as a faithful witness to His love and power.
xiv
Chapter 1: The Problem
The development of a competent understanding of place-value concepts by
primary students is a prerequisite for the learning of much later content of the school
mathematics curriculum. Children need to learn from the early primary school
grades1 how numbers are written in the base-ten numeration system, and to construct
accurate mental models for numbers, in order to develop a proficiency with
mathematics that will equip them to solve problems in later life. However, several
authors have noted that place-value concepts are difficult both for teachers to teach
and for students to learn (G. A. Jones & Thornton, 1993a; S. H. Ross, 1990). The
study described in this thesis investigated the teaching and learning of place-value
concepts using number representations in two formats: conventional base-ten blocks
and a computer software application.
1.1
Recommendations for Changes in Mathematics Education
Several documents published over the past 20 years have recommended
important changes in the way mathematics is taught in schools. These documents
include Mathematics counts (Cockcroft, 1982), Everybody Counts (National
Research Council [NRC], 1989), Curriculum and Evaluation Standards for School
Mathematics (National Council of Teachers of Mathematics [NCTM], 1989), A
National Statement on Mathematics for Australian Schools (Australian Education
Council, 1990), and Principles and Standards for School Mathematics (NCTM,
2000). Three prominent topics in these documents are relevant to this study: (a) the
development of mathematical understanding, (b) the development of number sense,
and (c) the use of technology in mathematics classes.
The first recommendation for mathematics education identified as relevant to
this study, that more emphasis be given to students’ development of mathematical
1
N.B. Queensland primary schools include Years 1-7; the term primary as used in this thesis refers to
this range of school class levels, which may be considered to be roughly equivalent to primary and
elementary schools in the U.S.
1
understanding, underlies the advice contained in the policy documents listed in the
previous paragraph. The view of the NCTM (2000) is clear: “Learning mathematics
with understanding is essential” (p. 20). The documents embody a view of learning
as a sense-making activity (Mayer, 1996; McIntosh, Reys & Reys, 1992), in which
learners develop their own personal understandings of concepts to which they are
exposed. Thus the act of teaching is seen not as transmitting ready-formed
knowledge from teacher to learner, but rather as encouraging the learner to construct
concepts so that they make sense to him or her (Cobb, Yackel & Wood, 1992; NRC,
1989). The view of learning as a sense-making activity has special relevance for the
teaching of mathematics, because of its focus on abstract entities that need to be
conceptualised by each learner (Davis, 1992). If learners do not form appropriate,
accurate mental models of numbers, they will be hindered in attempting to solve
mathematical problems in meaningful ways. The literature is replete with
observations of students who, though they can do some computation, do so without
understanding the meanings behind the symbols and procedures used (e.g., Kamii &
Lewis, 1991).
Meaningful understanding of numbers is linked to the second
recommendation relevant to this study, that the development of number sense be
made a priority for mathematics teaching (McIntosh et al., 1992; NCTM, 2000;
Sowder & Schappelle, 1994). Number sense is regarded by many as an important
goal of mathematics education, enabling students to answer flexibly non-routine
questions that require a mathematical solution. Traditionally, mathematics was taught
so that students could answer routine arithmetic questions accurately, for future
employment in retail or manufacturing jobs (NRC, 1989). Today there is a greater
need for adults who can think mathematically and who can devise methods of
solving numerical questions in novel ways (NCTM, 1989).
The third recommendation for change in the way that mathematics is taught is
for the use of technological devices—calculators and computers—to be a matter of
course at all school grade levels (Australian Education Council, 1990; NCTM, 2000;
NRC, 1989). The question of how computer technology (referred to in this thesis as
“technology”) can best be incorporated in mathematics education is the subject of
some debate. Research such as that described here is needed to help answer questions
about the effects of technology on students’ learning. In particular, the computational
power and the representational capabilities of computers have the potential to assist
2
students to develop more meaningful concepts for numbers (Clements & McMillen,
1996; NCTM, 2000; Price, 1996, 1997). This potential needs further investigation.
1.2
The Learning of Place-Value Concepts
The development of understanding of the base-ten numeration system is
foundational to all further use of numerical symbols, both in school and outside the
classroom. Thus, understanding how children develop place-value concepts, and the
difficulties they face in doing so, is of great importance to mathematics educators.
1.2.1 Conceptual Structures and Difficulties With Place-Value Concepts
Children’s difficulties in making sense of the meanings represented by
multidigit symbols have been reported widely in the literature (e.g., G. A. Jones &
Thornton, 1993a; Resnick, 1983; S. H. Ross, 1990). In particular, several authors
reported students having difficulty linking the abstract realm of numbers and their
symbolic and physical referents (e.g., Baroody, 1989; Baturo, 1998; Fuson, 1992;
Hart, 1989; Hiebert & Carpenter, 1992). In describing and analysing these
difficulties, several researchers have postulated children’s conceptual structures for
numbers (e.g., Fuson, 1990a, 1990b, 1992; Fuson et al., 1997; Resnick, 1983). A
number of conceptual structures, and several limited conceptions for numbers, have
been reported as being common among primary-age students. Such conceptual
structures feature prominently in much writing about children’s learning of placevalue concepts, and are considered by many, including this author, to be of critical
importance in understanding how children develop place-value concepts.
This thesis includes an analysis of evidence for conceptual structures for
multidigit numbers in the present study, and a comparison between that evidence and
reported findings of other researchers. Finally there is a discussion of possible links
between conceptual structures and participants’ use of two types of representational
material: physical and electronic base-ten blocks.
1.2.2 Use of Number Representations
Physical base-ten blocks.
Physical base-ten blocks, generally known in Queensland schools as
multibase arithmetic blocks [MABs], are regarded by many teachers as particularly
useful for helping students to build meaningful conceptual structures for multidigit
3
numbers (English & Halford, 1995). Developed by Dienes (1960) 40 years ago, they
have become the concrete materials of choice for teaching the base-ten numeration
system in many countries, including the USA, the UK, and Australia. Physical baseten blocks can be thought of as physical analogues of numbers, and mirror the
internal structures and relative magnitudes represented by the digits that make up a
written symbol (English & Halford, 1995). Students must reason analogically to use
the blocks effectively; that is, they must map the relations inherent in the blocks onto
the relations in the target realm (Gentner, 1983), the domain of numbers. In order for
physical base-ten blocks to be effective in representing numbers, it is important that
students’ attention be drawn to the analogical relationships that exist between the
blocks and the numbers they represent (Fuson, 1992).
Electronic base-ten blocks.
In light of the difficulties students have making links between numbers and
their referents, a number of suggestions have been made of teaching methods that
may help students to perceive connections among various forms of number
representation. One such suggestion is the use of computer-generated representations
for numbers (Clements & McMillen, 1996, Hunting & Lamon, 1995; NCTM, 2000).
Several software programs have been designed to model base-ten blocks
electronically on screen (e.g., Champagne & Rogalska-Saz, 1984; Rutgers Math
Construction Tools, 1992; P. W. Thompson, 1992). All use the capabilities of the
computer to enhance the number representations available to the user beyond those
provided by conventional physical blocks. For example, many of these programs
include number representations such as written symbols and representations of
regrouping actions on blocks, and link these representations tightly together so that a
change in one representation is mirrored by an equivalent change in the other
representations (see Appendix A). At the time the study was conducted, apart from
Rutgers Math Construction Tools the author only had access to descriptions of these
programs, and not to the programs themselves. Furthermore, none of the programs
included all the features that were felt to be desirable for teaching place-value
concepts; specifically, the author wanted the software to model multidigit numbers
with pictures of base-ten blocks on a place-value chart, to model regrouping actions
on the blocks, to show various symbolic representations for the numbers represented
by the blocks, and to play audio recordings of the number names. Because of the lack
4
of these features in available software, the author developed a new software program
for teaching place-value concepts, named Hi-Flyer Maths (described in Appendix A;
installation files available on CD-ROM in Error! Reference source not found.).
Central to this study is the effect of base-ten blocks, both physical and
electronic, on Year 3 students’ place-value conceptions of multidigit numbers. The
Hi-Flyer Maths software was used in the exploratory teaching study to assess these
effects.
1.3
The Research Question
Based on the issues outlined in the previous section, the question investigated
in this study is
How do base-ten blocks, both physical and electronic, influence Year 3
students’ conceptual structures for multidigit numbers?
Within the context of Year 3 students’ use of physical and electronic base-ten
blocks, the following specific issues were of concern:
1.
What conceptual structures for multidigit numbers do Year 3 students
display in response to place-value questions after instruction with baseten blocks?
2.
What misconceptions, errors, or limited conceptions of numbers do
Year 3 students display in response to place-value questions after
instruction with base-ten blocks?
3.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in instruction with physical and
electronic base-ten blocks?
4.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in students’ achievement in
numeration?
1.4
Overview of Research Methodology
The research questions were investigated using qualitative case studies
involving Vygotskian teaching experiments and Piagetian clinical interviews
(Hunting, 1983; Hunting & Doig, 1992). The study involved 16 Year 3 students
selected from a single primary school, half of each gender, and half of either high or
low mathematical achievement level (Table 3.2). The students were assigned to 4
5
groups of 4 students, each group comprising 2 boys and 2 girls, all of the same
achievement level. One high-achievement-level and one low-achievement-level
group were assigned to use physical base-ten blocks, and the other 2 groups used
computer software (electronic base-ten blocks). The groups each took part in 10
teaching sessions, involving up to a total of 45 place-value activities designed to
develop two-digit and three-digit place-value concepts. Each student was interviewed
individually both before and after the teaching sessions, to assess their place-value
understanding.
Each teaching session and interview was videotaped and audiotaped for later
transcription and analysis. As well, the researcher took field notes and the students’
workbooks were collected. The raw data from the teaching sessions and interviews
were transcribed for coding, principally using the grounded theory method described
by Strauss and Corbin (1990). Categories for participants’ responses emerged from
the data as they were analysed. These categories were compared with a framework of
conceptual structures identified in the literature.
1.5
Significance of the study
There have been a number of suggestions for teaching strategies to help
students develop good place-value understanding, including the use of some means
of “bridging the gap” between numbers and physical number representations (Hart,
1989). One suggestion for bridging this gap is to use computer-generated
representations of numbers (e.g., Clements & McMillen, 1996; P. W. Thompson,
1992). However, there are few reports of in-depth investigation of the use of such
software, or of research-informed guidelines for future software development. In
particular, there is no evidence of analysis of children’s conceptual structures for
multidigit numbers as they use electronic base-ten blocks to learn place-value
concepts. Considering both the recommendations to use suitable place-value
software and the money invested in its development and purchase, there is a pressing
need for such research.
This study investigates the ideas that students have of numbers, and how
those ideas may be affected by the use of either physical or electronic base-ten
blocks. The study provides important findings in this field with significance for both
the teaching of place-value concepts generally, and the design and use of place-value
software.
6
1.6
Outline of the Thesis
The thesis has 6 chapters. The current chapter provides an overview of the
study. Chapter 2 is a review of literature relevant to the study. Issues addressed are
current issues in mathematics education, place-value understanding, cognitive
science contributions to understanding of learning of place-value, the use of number
representational materials, and the use of computer software for teaching
mathematics. Chapter 3 contains a description of the methodology used in the study,
including assumptions and issues underlying the design, a description of the pilot
study and the main study, and discussion of validity, reliability, and limitations of the
design. Chapter 4 reports results of the study from the teaching sessions and
interviews. Chapter 5 comprises a discussion of the results in the light of other
reported research, and includes a description of a previously-unreported category of
student response to place-value questions, the independent-place construct. Chapter 6
concludes the thesis with a summary of findings, implications for the teaching of
place-value, and suggestions for further research in the area.
7
Chapter 2: Review of Literature
2.1
Chapter Overview
This chapter comprises a review of literature relevant to the study, divided
into 5 main sections. The broad background to the research questions is related to
several current issues in mathematics education. Three issues relevant to this study
are (a) the development of mathematical understanding, (b) the development of
number sense, and (c) the use of technology in mathematics classes. These three
issues are linked in section 2.2 to the teaching of place-value concepts in primary
schools. Section 2.3 defines place value and place-value understanding for the
purposes of this thesis. This section identifies the skills that children need to develop
and introduces the desired mental models of numbers that are an important focus of
the study.
The contribution that cognitive science has made to the study of children’s
understandings of mathematics, and in particular place value, is summarised in
section 2.4. Two areas of cognitive science study in particular are described: mental
models and analogical reasoning. First, based on previous research, a framework of
four conceptual structures considered necessary for children to learn place-value
ideas is proposed, and three common limited conceptions of numbers are listed.
Second, analogical reasoning is an important consideration in the teaching of many
mathematical topics, including place-value concepts. Base-ten blocks are analogues
of the base-ten numeration system, and mirror the relations among digit places. A
focus on understanding of analogical reasoning is therefore important in considering
their use as representations of numbers.
Section 2.5 describes the teaching of place-value understanding, including the
use of physical models of numbers in teaching place-value concepts. It is shown that
there is evidence of a “conceptual gap” in the minds of many children between
written symbols and base-ten blocks, which a number of researchers have attempted
9
to bridge. One solution introduced in this section is the use of computer-generated
manipulatives. Section 2.6 includes a description of capabilities of modern
computers which make them potentially valuable for helping students to make
connections within many domains, including mathematics. Specifically, the
capability to present different representations of a concept shows promise for
representing numbers in several formats, with the aim of helping students to see
connections among them.
2.2
Issues in Mathematics Education
Several issues of current concern in mathematics education are particularly
relevant to this study. This section describes three of these issues: students’ active
involvement in mathematics learning, development of number sense, and the use of
technology.
2.2.1 Students’ Active Involvement in Mathematics Learning
The view that students should actively participate in the process of learning
mathematics is a comparatively new one. As the NCTM (1998) noted, “the notion of
mathematics as something to be deeply understood, so that it can be used effectively,
has not always been a valued outcome of school mathematics” (p. 33). A
“traditional” model of mathematics teaching, typical of the first half of the 20th
Century, has been widely criticised (NCTM, 1989, 1991; NRC, 1989). This model
viewed the teaching-learning process as the transmission of information, and thereby
knowledge, from teacher to student. In this model the teacher was perceived to be the
source of information, “the sole authority for right answers” (NCTM, 1991, p. 3),
and the student was merely a passive recipient of the information. This model owes
much to behaviourist views of learning: namely, that “learning is conceived of as a
process in which students passively absorb information, storing it in easily
retrievable fragments as a result of repeated practice and reinforcement” (NCTM,
1989, p. 10). In contrast, recent recommendations for mathematics teaching and
learning (NCTM, 1989, 1991, 2000; NRC, 1989) portray a very different picture.
First, the learning process is now widely seen as one of individual construction of
understanding, in which new experiences are integrated with prior knowledge to
form understandings that are meaningful to the student (Simon, 1995). Second,
students are seen as “autonomous learners . . . . [who should] take control of their
10
learning” (NCTM, 2000, p. 21), to make sense of it for themselves. Third, the
teacher’s role is to be a “guide for exploring academic tasks” (Mayer, 1996, p. 152;
see also Sowder, 1994, p. 146), or an “[orchestrator of] classroom discourse in ways
that promote the investigation and growth of mathematical ideas” (NCTM, 1991,
p. 1).
A critical component of the view of mathematics learning described here is
the necessity of students making sense of what they learn (Mayer, 1996). If teachers
want their students to develop meaningful understanding of mathematical concepts,
then there is a need to consider many aspects of the learning environment that exists
in the classroom. One aspect of the learning environment of major relevance to this
study is the question of various interactions that take place, described below; this is
an important item of interest in the research described in this thesis. As explained by
McNeal (1995), “[by] studying classroom interactions, the observer could . . . infer a
particular individual’s knowledge . . . from observations of his/her interactions with
the objects or with other individuals” (p. 3). The following subsection addresses
interactions of three kinds that are of relevance to this study.
Student-teacher interactions.
If the view of learning as a constructive meaning-making activity is accepted,
then the interactions between students and teachers are of obvious importance.
“More than any other single factor, teachers influence what mathematics students
learn and how well they learn it” (NCTM, 1998, p. 30). Part of a constructivist model
of learning is a view that students construct mathematical knowledge as a product of
“interaction in social contexts” (Putnam, Lampert & Peterson, 1990, p. 134). As
Cobb and Yackel (1996) stated, “we consider students’ mathematical activity to be
social through and through because it does not develop apart from their participation
in communities of practice” (p. 180). This idea of a community of practice is implicit
in much recent writing about teaching mathematics (NCTM, 1991, 2000; NRC,
1989), and learning in general (Brown, Collins, & Duguid, 1989; Harley, 1993). The
NCTM’s (1991) recommendations for what a teacher can do to encourage the
development of a community of practice included: helping students to work together,
to rely more on themselves, to reason mathematically, to solve problems, and to
connect mathematics and its applications. More recently, the NCTM (2000) stated
the view that “teachers’ actions are what encourage students to think, question, solve
11
problems, and discuss their ideas, strategies, and solutions. The teacher is responsible
for creating an intellectual environment where serious mathematical thinking is the
norm” (p. 18). Similar advice was given by the NRC (1989), who described teachers’
actions as denoting supportive interaction with students: “encourage,” “help students
verbalise,” “build confidence” (pp. 81-82). Clearly, these authors believed that
interactions between a teacher and students are an important aspect of teaching and
learning mathematics.
Student-student interactions.
The second type of interaction, between student and student, is closely linked
to the first and has also received attention in the mathematics education literature.
One aspect of learning theories that typifies the differences between constructivism
and transmission models of learning is the focus on the interactions occurring among
students. As mentioned earlier, under the transmission model the student was
expected to receive knowledge passively without questioning it; modern learning
theories assert that discussion and debate among students is an essential part of the
learning process. Various benefits have been claimed for students learning in a social
community, whether in pairs, a small group, or a whole class (Akpinar & Hartley,
1996; Brown et al., 1989; Fox, 1988). These benefits include opportunity for
collective problem solving, development of skills of collaboration, development of
flexible thinking, and exposure of misconceptions.
Student-materials interactions.
The third type of interaction of interest here is interaction between a student
and learning materials, such as blocks or a computer. Interaction with materials is
linked with the two previous types of interaction, as the materials are an integral part
of the learning environment, and there is assumed to be an “interplay between
students’ cognitive activity and physical and social situations” (Nitko & Lane, 1990,
p. 5). The connection between learning and the learning environment was mentioned
by Kozma (1991), who stated that the learning process involves “extracting
information from the environment and integrating it with information already stored
in memory” (pp. 179-180). Writing specifically about computer learning
environments, Kozma stated that the learning process was “sensitive to
characteristics of the external environment, such as the availability of specific
information at a given moment” (p. 180).
12
The idea of situated cognition (Brown et al., 1989) addresses the question of
learning and its relation to the learning environment. Brown et al.’s idea, that a
learning environment constrains the learning activity of the students in that
environment, is important to this research. A central assumption of the situated
cognition view is that students “reason with what a situation affords them” (Winn,
1993, p. 16). In other words, the particular capabilities, or affordances (Salomon,
1998), provided by the materials available to a student can have an important
influence on the student’s learning. This view is supported by Kozma’s (1994b)
statement that
knowledge and learning are neither solely a property of the individual or of the
environment. Rather, they are the reciprocal interaction between the learner’s
cognitive resources and aspects of the external environment . . . and this interaction
is strongly influenced by the extent to which internal and external resources fit
together. (p. 8)
The research described here involves the investigation of children’s learning
when using one of two types of materials; the author assumed prior to the study that
the materials’ different special characteristics would have different effects on the
students’ learning.
2.2.2 Number Sense
The idea of number sense is related closely to development of mathematical
understanding, as it “typifies the theme of learning mathematics as a sense-making
activity” (McIntosh et al., 1992, p. 3). Number sense refers to a student’s familiarity
with numbers, and the ability to use numbers in sensible ways to answer
mathematical questions. It lacks a precise definition, but has been likened to “road
sense” (familiarity with a particular geographical area; Trafton, 1992) or
“friendliness with numbers” (Howden, 1989). The NCTM (1989) listed five
understandings demonstrated by students with good number sense. They “(1) have
well-understood number meanings, (2) have developed multiple relationships among
numbers, (3) recognize the relative magnitudes of numbers, (4) know the relative
effect of operating on numbers, and (5) develop referents for measures of common
objects and situations in their environments” (p. 38).
The need for students to possess number sense is extensively argued in the
literature (Australian Education Council, 1990; K. Jones, Kershaw, & Sparrow,
1994; McIntosh et al., 1992; NCTM, 1989, 2000; Sowder, 1988, 1992; Sowder &
13
Schappelle, 1994). Good number sense is important for making sense of
mathematical questions and for working out sensible answers. Perhaps the
characteristic that most easily sums up good number sense is flexibility, in finding
solutions to mathematical problems and in being able to see connections among
numbers in different ways. Trafton (1992) described a person with number sense as
having “a well-integrated mental map of a portion of the world of numbers and
operations and [being] able to move flexibly and intuitively throughout the territory”
(p. 79). A similar idea was proposed by Greeno (1991), who likened knowing and
learning “as an activity in an environment” (p. 175). Greeno connected number sense
with situated cognition, stating his view that “knowing the domain [e.g., the
mathematical domain] is knowing your way around in the environment and knowing
how to use its resources” (p. 175). In this view, number sense relates closely to the
ability to use available resources to make sense of the domain. Though Greeno was
writing specifically of mental resources, this idea is assumed here to apply also to
physical resources, as described in an earlier paragraph. In other words, a student
with good number sense could be seen as having not only a good idea of the
cognitive domain, but also of the physical environment, including how to use
available materials to answer mathematical questions.
For teachers to help students to develop good number sense involves helping
the students to develop a range of prerequisite understandings of numbers and
operations. This is borne out by McIntosh et al.’s (1992) description of number
sense:
Number sense refers to a person’s general understanding of number and operations
along with the ability and inclination to use this understanding in flexible ways to
make mathematical judgements and to develop useful strategies for handling
numbers and operations. It reflects an inclination and an ability to use numbers and
quantitative methods as a means of communicating, processing and interpreting
information. (p. 3)
Prerequisite mathematical skills and understanding needed for good number
sense include proficiency with written algorithms, mental computation skills,
problem-solving ability, and place-value understanding (see NCTM, 2000, p. 32).
Teaching techniques suggested for developing number sense include the use of
calculators to investigate number magnitudes (Bobis, 1991; Schielack, 1991),
emphasising and encouraging sense-making (Sowder & Schappelle, 1994), and the
use of estimation activities (K. Jones et al., 1994; Lobato, 1993; Sowder, 1988,
14
1992). The particular focus of this research, the development of place-value
understanding, is directly relevant to the development of number sense. As explained
by the NCTM (2000),
understanding number and operations, developing number sense, and gaining
fluency in arithmetic computation form the core of mathematics education for the
elementary grades. As they progress from prekindergarten through grade 12,
students should attain a rich understanding of numbers—what they are; how they are
represented with objects, numerals, or on number lines; how they are related to one
another; how numbers are embedded in systems that have structures and properties;
and how to use numbers and operations to solve problems. (p. 32)
2.2.3 Use of Technological Devices
The third issue of particular relevance to this study is the use of technology in
mathematics teaching. There is common agreement that technological advances in
the general society outside schools lead to the need for different objectives in
mathematics education (NRC, 1989). These objectives will be seen in (a) the use of
different means of doing mathematics, and (b) having different emphases in the
curriculum.
Much has been written about the need to bring the procedures used in school
mathematics into line with those expected of workers in the 21st Century. It has been
pointed out (NCTM, 1989) that in the industrial age the goal of public schools was to
educate future shop assistants and factory workers, and so schools taught their
students so-called “shop keeper arithmetic” (Cruikshank & Sheffield, 1992; NCTM,
1989; NRC, 1989). There is no longer the same need for adults to be highly
proficient in written computation procedures; in its place is a need for workers and
citizens who possess a broader range of mathematical “concepts and procedures they
must master if they are to be self-fulfilled, productive citizens in the next century”
(NCTM, 1989, p. 3). These skills include developing methods to solve a variety of
problems; working cooperatively in teams; and reading, interpreting, and critically
evaluating quantitative data. The development of these skills is linked to the issues
discussed in the previous two sections—meaningful understanding of mathematics
and number sense—as well as the use of technological devices in mathematics
teaching.
One feature of the mathematics used by adults in homes and workplaces is the
use of calculators and computers to assist with a range of mathematical tasks
(Sparrow, Kershaw, & Jones, 1994). These include computation, storage of data, and
15
presentation of results of mathematical processes such as in spreadsheets and graphs.
It is assumed by writers of mathematics education policy documents that students in
schools will similarly have access to a range of technological devices to assist them
in learning mathematics (Australian Education Council, 1990; Australian Association
of Mathematics Teachers [AAMT], 1996; Cockcroft, 1982; NCTM, 2000; NRC,
1989). As the AAMT (1996) put it, “mathematics education must reflect the
influence of technology upon both mathematics and society” (p. 2). The NCTM
(1998) similarly recommended that schools improve their level of use of technology
also to match what happens in schools with what employers and others expect of
workers in the workforce:
Today’s jobs demand the use of mathematically driven technological tools. If
schools do not have a level of technology equivalent to the level found outside
schools, and if they do not prepare students appropriately with it, then they are
placing their students at a serious disadvantage. (p. 43)
Merely increasing the amount of technology available in classrooms is not
sufficient to bring about the desired improvements in mathematics education,
however, and technological devices should not merely be added to existing programs
of instruction. Changes are also required in the methods of mathematics instruction,
so that appropriate tasks are set to answer with technological devices. As the NCTM
(1989) pointed out, “access to [calculator and computer] technology is no guarantee
that any student will become mathematically literate. Calculators and computers for
users of mathematics . . . are tools that simplify, but do not accomplish, the work at
hand” (p. 8). There is a need for detailed knowledge of the effects that calculators
and computers have on students using them, especially as they represent considerable
investments of finance and time by education departments and teachers. This topic is
reviewed further in section 2.6.
Electronic technology has the potential for several important effects on
mathematics curriculum. Technology makes mathematical skills such as written
computation easier, but it also hides processes that students in earlier times had to
consider, such as regrouping required for operations. Technology can also make
available to students mathematics learning experiences that previously were not
possible. For example, calculators can allow a student to investigate operations on
large numbers that would be too time-consuming with other mechanical computation
procedures. Computers similarly provide students with considerable computing
power, which can be used to represent mathematical and other domains with which
16
students can “interact” in ways not possible with any other technology. This study
investigates one such interactive learning environment in which a computer is used
to represent the domain of numbers for the purpose of developing students’
understanding of place-value concepts.
2.3
Place-value Understanding
One area of the mathematics curriculum where the issues described in the
previous section warrant attention is in place value. The teaching of place-value
concepts is foundational for understanding of the base-ten numeration system, and is
thus central to the primary mathematics curriculum. As the NCTM (2000) stated,
“foundational ideas like place value . . . should have a prominent place in the
mathematics curriculum because they enable students to understand other
mathematical ideas and connect ideas across different areas of mathematics” (p. 15).
Issues such as those described in the previous section all have a potential impact on
the teaching of place value. It is this author’s view that recommendations for
students’ active involvement in learning mathematics and the development of
number sense have direct relevance for how place value is taught. Advances in
technology have a more indirect influence, through the capabilities they offer in the
area of models of numbers for place-value teaching (section 2.6).
The importance of place value in the primary mathematics curriculum and the
difficulty teachers experience in teaching place-value concepts to their students are
well documented (G. A. Jones & Thornton, 1993a; S. H. Ross, 1990). As Resnick
(1983) stated,
the initial introduction of the decimal system and the positional notation system
based on it is, by common agreement of educators, the most difficult and important
instructional task in mathematics in the early school years. (p. 126)
Teachers have difficulty teaching place-value concepts, and their students
have difficulty learning them (S. H. Ross, 1990). One source of difficulty for
teachers is that, as Skemp (1982) pointed out, mathematics’ “[conceptual structures]
are purely mental objects: invisible, inaudible, and not easily accessible even to their
possessor” (p. 281). Thus, teachers are limited in what they can know of what their
students are thinking with regard to mathematical entities. Research investigating
place-value understanding must address students’ conceptual structures for numbers
and how they are developed. This topic is dealt with in more detail in section 2.4.2.
17
2.3.1 Place Value
Place value refers to the feature of the base-ten system of numeration
(sometimes called the “Hindu-Arabic” system) in which each digit in a number
represents a precise amount, dependent on both the face value of the digit and its
position (Baturo, 1998; Miura & Okamoto, 1989). This contrasts with other
numeration systems that do not exhibit place-value, such as that of the ancient
Egyptians, who wrote a different symbol for each power of 10 (Irons & Burnett,
1994). In the base-ten numeration system the value represented by each digit is equal
to the product of the face value of the digit and the value assigned to the digit’s
position, relative to the rightmost whole number digit, or ones place (Fuson, 1990a;
Figure 2.1). Thus, though pairs of numbers such as “25” and “52” look very similar
(and may be confused by young children), the position of each digit determines its
value, giving a unique value to each different written symbol. This idea is at once
both simple and very powerful, and can be extended an indefinite number of places
to the left (for whole numbers) or the right (for decimal fractions).
Figure 2.1. The face value of each individual numerical symbol, together with its
position relative to the ones place, determines the value it represents.
The base-ten numeration system is principally a system of written symbols by
which users record physical quantities and use them in calculations. Its importance
was emphasised by the NCTM (1998), who commented that “mathematical
symbolism and representation is one of the most significant achievements of
humankind” (p. 94). Associated with the written symbols are number words that are
alternative representations of numerical quantities. Whereas the written symbols
18
follow an entirely consistent mathematical system in which a quantity of ten of each
place equals one of the place immediately to the left, number words include
inconsistencies relating to the history of the language used. In English, there are
inconsistent words for multiples of 10, and for numbers from 11 to 19. Once a child
reaches the study of three-digit and four-digit numbers number naming is much more
consistent, but until a child reaches that stage the learning of numbers and their
names is very difficult. Thus, young children have a very challenging task of
developing understanding of a system that in its earliest, numerically simplest,
examples contains numerous inconsistencies.
The base-ten numeration system has been named as an “unnamed-value
positional value system of written marks” (Fuson, 1990a, p. 343) and a “regular
relative positional system” (Fuson, 1992, p. 136). In these two phrases Fuson has
captured three essential features of the base-ten numeration system: (a) Place names
and values are implicit in written numeric symbols; (b) the system is completely
consistent across all symbol positions; and (c) value is assigned according to each
digit’s position, relative to the point of reference, the ones place. The structure and
rules by which the conventional base-ten numeration system operates are not evident
from merely observing the written symbols, even if the meaning of each individual
symbol (“1,” “2,” etc.) is known (Fuson, 1992, p. 138). However, to those who are
familiar with the scheme’s conventions, each numerical symbol uniquely represents a
number. Apart from minor variations of the symbol used for the decimal point
(usually “.” or “,”), and leading or trailing zeros, each rational number is represented
by a unique symbol, and each sequence of digits stands for a unique number.
In summary, children need to understand the features of the base-ten
numeration system (Baturo, 1998; English & Halford, 1995; Fuson, 1990a, 1992;
Hiebert, 1988; Miura & Okamoto, 1989; Sowder & Schappelle, 1994). A number of
key features of the base-ten numeration system, in common with all place-value
numeration systems, are expressed in the following statements:
1.
A discrete set of individual number symbols (base 10: 0 to 9), and a
decimal point marker, used in combination can uniquely represent any
rational number quantity.
2.
Each place represents a power of the base, derived from its position
relative to the rightmost whole-number place.
19
3.
The system is completely consistent across all places, in that the value
of each place is equal to the base number times the value of the adjacent
place on the right, and the value of the adjacent place on the left divided
by the base number.
4.
The value represented by a digit is the product of the face value of the
digit and the value associated with the place of the digit.
5.
The system allows operations on numbers to be represented
symbolically, and these operations work consistently on numbers in all
places.
From the previous discussion, definitions can be given for place value and for
place-value understanding. This study is concerned with understanding of the baseten numeration system only; hereafter “place value” will be used to refer only to
place value in the base-ten numeration system. For the purposes of this thesis, place
value is based on the description given by Miura and Okamoto (1989, p. 109), and
defined thus:
Place value is the property of the base-ten numeration system, by which the
numerical value represented by each digit of a written multidigit symbol is equal to
the product of the digit’s face value and the power of 10 associated with the digit’s
position in the numeral.
2.3.2 Place-value Understanding
For the purposes of this thesis, place-value understanding is described in
terms of both actions and conceptual structures. This approach is supported by Sfard
(1991), who described historical progress made in mathematical understanding as
having to do with both “operational” (actions) and “structural” (objects) conceptions,
dynamically linked together as professional mathematicians have struggled to
advance knowledge in the field. Sfard stated that “the ability of seeing a function or a
number both as a process and as an object is indispensible for a deep understanding
of mathematics” (p. 5). Though school students are not involved in the same level of
mathematical thinking as professional mathematicians, nevertheless students need to
develop both structural and operational conceptions for each new mathematical
concept. These two types of conception of numbers link students’ internal conceptual
structures of numbers and external physical models of numbers. In other words,
students are assumed to possess internal representational structures for numbers and
20
processes that are influenced as the students access and manipulate available external
representations of numbers (Hiebert & Carpenter, 1992; Putnam et al., 1990).
Internal and external representations and manipulations are closely linked, and in this
thesis they are considered together to be involved in students’ place-value
understanding.
The definition of place-value understanding used here takes into account
advice given by several authors, relating to both actions and conceptual structures.
This includes statements that children need to “construct number representations that
reflect the Base 10 numeration system” (Miura & Okamoto, 1989, p. 109),
“coordinate and synthesize a variety of subordinate knowledge about our culture’s
notational system for numbers” (S. H. Ross, 1989, p. 47), “develop flexibility in
representing and understanding multidigit numbers” (G. A. Jones, Thornton, & Putt,
1994, p. 122) and “[develop understanding of] the interpretation of numbers as
compositions of other numbers” (Resnick, 1983, p. 126).
For this thesis place-value understanding is defined thus:
A student possessing place-value understanding is able to use the placevalue features of the base-ten numeration system to form accurate, flexible
conceptual structures for quantities represented by written numerical symbols. The
student is able to manipulate numerical quantities in meaningful ways to answer
mathematical questions.
A student’s place-value understanding must be assessed at a deep level, by
probing the student’s conceptual structures for numbers (Skemp, 1982). Research in
this area generally uses observation of participants’ behaviour to posit “various
cognitive structures and processes believed to produce the behavior” (Putnam et al.,
1990, p. 65). The following section describes research into children’s cognition,
including the investigation of children’s conceptual structures for numbers.
2.4
The Contribution of Cognitive Science to Mathematics
Education
As shown in the previous section, place-value understanding is an internal
phenomenon that a teacher or researcher cannot access directly. Research on
conceptual structures and analogical reasoning has particular applicability for the
study of mathematical understanding, in two respects. Findings about mental
structures and processes can explain how abstract number concepts are represented in
21
the mind, and cognitive science methods of research may be applied to the
investigation of learning of mathematics. This section outlines two aspects of
cognitive science research relevant to this study. First, the study of mental models is
described, including a form of mental model of particular importance to mathematics
understanding, conceptual structures for numbers. Second, analogical reasoning is
defined, and its relevance to the teaching and study of place-value concepts
demonstrated.
2.4.1 Understanding Mathematics
Understanding of mathematics relies on having internal mental
representations of numbers and the ability to manipulate them in meaningful ways,
because of the abstract nature of its content (Hiebert & Carpenter, 1992; Presmeg,
1992; Sfard, 1991). Thus learning mathematics involves representation of abstract
concepts, which may include physical representation of numbers using concrete
materials, but ultimately involves internal, mental, representations held in the mind
of the student (Baggett & Ehrenfeucht, 1992; Resnick, 1988; Sfard, 1991). As Davis
(1992) noted, “after all, mathematics is about thinking; there is a sense in which
mathematics exists only within the human mind” (p. 225). All numbers and
associated processes are abstract ideas rather than physical entities. For example, the
number three may be represented physically, using representations such as the
written symbol “3,” the verbal or written word “three,” a set of three counters, or a
picture of three objects. However, the number itself can never be perceived directly
by the physical senses (Sfard, 1991).
Because numbers are abstract entities, users can perceive and manipulate
them only mentally. For this reason children need proficiency with certain mental
skills to be successful in learning mathematics. For young children, the apparently
simple act of counting a group of objects demands a cluster of skills that must all be
present in order to correctly count, name, and then understand the number of objects
(English & Halford, 1995). These skills include correctly recalling the sequence of
number names, applying exactly one number name to each object, and understanding
that the last number name counted is the number of objects in the group (Fuson,
1992). At a higher level, children in middle primary school must possess more
advanced skills, relating to place-value understanding, numeration, the concept of
subtraction, and the algorithm itself (Fuson, 1990a).
22
The emphasis in mathematics on mental representations and processes makes
it particularly suitable for psychological research. Several authors have pointed out
the special relevance that psychological theories have for the teaching of
mathematics (Beilin, 1984; English & Halford, 1995; Glaser, 1982; Hiebert &
Carpenter, 1992). Cognitive science has supported mathematics education research in
two distinct ways: Firstly, theory derived from research into thinking generally has
been used to explain how students learn mathematical concepts; secondly,
mathematics researchers have used methods from cognitive science in their study of
mathematical understanding. As noted by English and Halford (1995), cognitive
science research findings have great relevance for understanding of mathematical
concepts. Several authors have written about psychological theories and how they
may explain mechanisms underlying mathematical understanding. For example,
Hiebert and Carpenter (1992) reported that they “[drew] quite heavily from insights
provided by work in cognitive science to deal with questions of learning and teaching
mathematics” (p. 66). The assumptions made by Hiebert and Carpenter with regards
to mental representations are adopted also in this thesis (section 3.5.1). The second
connection between psychological theory and mathematics education has been the
application of methods developed for cognitive science research to the investigation
of questions in mathematics learning (Ohlsson, Ernst, & Rees, 1992; Schoenfeld,
1992; Silver, 1994). These methods include think-aloud protocols, computer
simulations of mental processing of information, and generally inferring mental
models and processes from observed actions. Such cognitive science methods have
been applied to research into mathematics learning on topics such as problem solving
(Schoenfeld, 1992), mental computation (Hope, 1987), mathematics as a situated
mental activity (Silver, 1994), the cognitive complexity of subtraction algorithms
(Ohlsson et al., 1992) and geometric problem solving (Chinnappan & English, 1995).
2.4.2 Mental Models
Cognitive scientists are principally concerned with understanding human
thinking and how it is affected by external events (e.g., Greeno, 1991; Halford,
1993a, 1993b; Presmeg, 1992; Shepard, 1978). In seeking to understand the
workings of the human mind, cognitive scientists posit the existence of mental
models. These models are deduced from observations of people, often made under
experimental conditions, and are used to explain the observed behaviour. Mental
23
models have been defined by Halford (1993a) as “representations that are active
while solving a particular problem and that provide the workspace for inference and
mental operations” (p. 23). Greeno (1991) defined a mental model as
a special kind of mental representation, in that the properties and behavior of
symbolic objects in the model simulate the properties and behavior of the objects
rather than stating facts about them. . . . A model is a mental version of a situation,
and the person interacts within that situation by placing mental objects in the
situation and manipulating those symbolic objects in ways that correspond to
interacting with objects or people in a physical or social environment. (p. 177)
Several researchers in the mathematics education field have investigated
mental models used by students as they learn mathematical concepts (e.g.,
Chinnappan & English, 1995; English & Halford, 1995; Fischbein, Deri, Nello, &
Marino, 1985; Hunting & Lamon, 1995).
Conceptual structures for multidigit numbers.
It is common practice for researchers investigating place-value understanding
to make deductions about “children’s inaccessible mathematical realities” (Cobb &
Steffe, 1983, p. 93) based on their performances on mathematical tasks (Davis, 1992;
Putnam et al., 1990; Resnick, 1983, 1987). As seen in Greeno’s (1991) definition in
the previous section, mental model is a broad term encompassing a range of internal
representations of situations. The particular type of mental model of interest in this
study is the mental models that students form to internally represent multidigit
numbers. These have been referred to variously as internal representations (English
& Halford, 1995; Hiebert & Carpenter, 1992), mathematical constructs (Sfard, 1991)
and conceptual structures (Bell, 1990; Fuson, 1990a, 1992; Fuson et al., 1997;
Skemp, 1982). In this thesis the term conceptual structures is used in the same sense
as Bell (1990) and Fuson (1990a, 1990b, 1992), to refer to the mental models
children use “for the formal mathematical words and marks used in the school
mathematics classroom” (Fuson, 1992, p. 56). As Fuson (1992) explained, children’s
conceptual structures vary in quality and usefulness:
Some of the conceptual structures are accurate and some are not; some are efficient
and some are not; some are advanced and some are simple. To help children function
effectively in mathematics, teachers need to reflect on how the classroom
experiences they are providing their children are supporting children’s construction
of accurate, efficient, and advanced conceptual structures for the mathematical
marks, procedures, and concepts addressed in the classroom. (pp. 56-57)
Conceptual structures deduced by researchers and reported in the literature
fall into two broad groups: structures considered by authors to be necessary for the
24
development of place-value understanding, and structures that are limited
conceptions of numbers that hinder children’s mathematical understanding and
performance. Descriptions of conceptual structures of the first group, that are
believed to be necessary for the learning of place-value concepts, are given first in
this section. Common limited conceptions of multidigit numbers are described later
in this section.
The place-value literature includes a number of papers in which authors
provided descriptions of children’s conceptual structures; Table 2.1 shows a
summary of several of these descriptions. Some authors (Cobb, 1995; Cobb &
Wheatley, 1988; Miura & Okamoto, 1989; Miura, Okamoto, Kim, Steere, & Fayol,
1993; Resnick, 1983; S. H. Ross, 1989, 1990; Steffe, Cobb, & von Glasersfeld, 1988)
proposed stages or levels of understanding through which children are purported to
pass as they develop place-value understanding. Aspects of the schemes are
integrated into the proposed framework of conceptual structures described in this
section. Generally, authors devised schemes post hoc, during the analysis of
experimental data (A. Sinclair & Scheuer, 1993, p. 200). Other authors (e.g., Fuson
et al., 1997; Janvier, 1987), however, have disputed the validity of defining stages in
place-value understanding at all.
25
TABLE 2.1.
Aspects of Place-value Understanding Described in the Literature
Researcher(s)
S. H. Ross, 1989,
1990
Aspect of
place-value
understanding
Acquisition of
knowledge
about two-digit
numbers
No. of
stages or
levels
Five
Steffe, Cobb, & von
Glasersfeld, 1988
Concepts of
ten
Five
Cobb, 1995
Cobb & Wheatley,
1988
Concepts of
ten
Five
Fuson, 1990a, 1990b
Fuson & Briars, 1990
Fuson et al., 1997
Number
knowledge /
Development
of decimal
knowledge
Conceptual
structures for
multidigit
numbers
Miura & Okamoto,
1989
Miura et al., 1993
Mental
representations
of multidigit
numbers
Resnick, 1983
Three
Not applicable
Three
Summary of findings
The author identified five
stages in children’s
acquisition of place-value
understanding.
The authors identified five
conceptions of ten
constructed by children.
The authors identified five
conceptions of ten in
children’s counting after
textbook instruction.
The author posited a mental
number line and a part-whole
schema preceding three
stages of place-value
understanding.
The authors identified several
conceptions of numbers,
many of them limited
conceptions.
The authors observed three
types of representation used
by students to represent twodigit numbers with concrete
materials.
A general overview of common features of the various classification schemes
summarised in Table 2.1 can be given, despite the diversity among them. First, the
authors each listed a number of levels at which children may operate in the placevalue domain. Some authors’ levels, the number of which varied from three to five,
were stages through which most children pass (e.g., S. H. Ross, 1990); generally,
however, they represented levels of expertise or maturity of understanding observed
in children (e.g., Miura et al., 1993). In fact, few studies attempted to track individual
students’ understanding over time. Rather, researchers usually described behaviour
common to several students at a particular point in time, as indicative of a particular
level of understanding. Second, the authors’ schemes each presented a sequence,
starting with initial immature understandings, with each successive level representing
better understanding of place value. This is clearly relevant to the teaching of place-
26
value concepts, the implied goal of which is to assist students to move to higher
levels of number understanding. The levels were often used as a means of
comparison of individuals and groups of children, to describe differences in
performance and understanding (e.g., S. H. Ross, 1990).
It is important to note here that the levels and stages proposed by the various
authors do not agree completely. This may be due to factors such as the date of the
research, the aims of the research, the philosophical stance of the author(s), and the
tasks provided to participants. Nevertheless, there does exist substantive agreement
among the various authors on internal structures revealed by observations of
children’s task performance, and so it is useful to draw them together for the purpose
of summarising the current state of knowledge of this field.
A framework of conceptual structures for place value.
This subsection describes a framework of four conceptual structures believed
to be necessary for the development of mature place-value understanding, based on a
synthesis of work in the field of place value research described in the previous
section. This framework was used to inform initial data analysis in this study and
then was subsequently compared with the study’s findings. The following paragraphs
describe the four conceptual structures in the proposed framework, including support
for each structure from the place-value literature.
Conceptual structure 1: Unitary construct. Early in their school years, and
even before the start of formal schooling, children are introduced to the idea of
numerical symbols. They learn to recognise the symbols for numbers 1 to 9 and to
associate each one with a number: the concept that refers to the numerosity of a
group of objects (Fuson, 1990b; Resnick, 1983). Resnick likened this conceptual
structure to a “mental number line” (p. 110), on which cardinal numbers are placed
in sequence from zero or one to the limit of a child’s counting. By having a mental
image of the number sequence, when counting a group of objects a child can
associate each element on the number line with an object, and give the name of the
last-mapped element as the total number in the group. Fuson et al. (1997) explained
the importance of this conceptual structure for later learning about base-ten numbers:
27
Multidigit numbers build on and use the unitary single-digit triads of knowledge for
single-digit numbers. Thus, before children can learn about two-digit numbers, they
must have learned for one to nine how to read and say the number word
corresponding to each number mark, write the numeral corresponding to each
number word, and count or count out quantities for each mark and number word one
to nine. Because the number words for single-digit numbers in most languages and
the corresponding written marks are arbitrary, most children learn most of the
unitary single-digit triads as rote associations. (p. 138)
The unitary construct, though an essential component of early number
teaching and learning, can lead to a limited conception, common in older children,
that multidigit numerals represent only collections of single objects (Fuson, 1990b;
Fuson et al., 1997). This conception is described in more detail later in this section.
Conceptual structure 2: Tens and ones structure. Resnick (1983) proposed
that this structure followed an earlier “part-whole” construct, by which students learn
that quantities may be partitioned in different ways, especially when learning singledigit addition and subtraction operations. Partitioning a multidigit quantity into
whole tens and leftover ones is a “unique partitioning of multidigit numbers”
(Resnick, 1983). At this stage, students learn counting number names for numbers
beyond 9, learn that numbers greater than 9 are separated into “tens” and “ones,” and
learn to write symbols for numbers using two digits. With this level of knowledge a
child may be able to carry out addition and subtraction operations that do not involve
trading, and may also be successful on many typical classroom and textbook
questions such as “How many tens are there in 36?” or “Circle the tens digit in 82.”
However, as several authors have pointed out (e.g., Cobb & Wheatley, 1988), this
type of question does not involve true place-value understanding, as it does not
address the multiplicative idea of 1 ten being composed of 10 ones, and children with
only this level of knowledge are not able to handle demands of operations which
include trading.
Conceptual structure 3: Ten as a unit. The third construct develops from the
second, and focuses on the fact that the tens digit in a multidigit number stands for a
collection of 10 single items. There are many variations of this conceptual structure
(cf. Cobb & Wheatley, 1988; Fuson et al., 1997); the common element of the various
constructs of this type is that ten is a single entity, made up of 10 units. By
understanding this idea, a student thinking at this level can mentally or physically
decompose a ten into 10 ones, or regroup 10 ones into a single ten, as the situation
demands.
28
Several researchers have identified the ten-as-a-unit construct. S. H. Ross
(1989) named this construct the construction zone stage, and explained the
understanding involved in this way: “Students know that the left digit in a two-digit
numeral represents sets of ten objects and that the right digit represents the remaining
single objects” (p. 49). Miura and Okamoto (1989) identified students who chose to
represent two-digit numbers using a canonical base 10 form: that is, so that the
number of tens material and ones material equalled the number of tens and ones,
respectively, in the written numeral. Numbers represented canonically can have no
more than nine in any place, unlike under the following construct, where groupings
of more than nine in a place are allowed. Steffe, Cobb, and von Glasersfeld (1988)
identified a concept of ten that is congruent with the ten-as-a-unit construct in their
study of children’s counting that they named ten as an iterable unit. This concept of
ten was held by students who could count using ten as a composite unit and the
remainder as units of one, and was also identified by Cobb and Wheatley (1988).
Conceptual structure 4: Flexible representations. The flexible representations
conceptual structure develops the understanding in the previous ten-as-a-unit
construct. The base-ten numeration system is written using a strict protocol of having
no more than nine in any one place. When concrete materials represent a number, it
is said to be a canonical representation if it has no more than nine in any single
place, or non-canonical if there are more than nine in any single place. For example,
75 can be represented as 6 tens and 15 ones, or as 4 tens and 35 ones, and so on. The
ability to understand multidigit numbers in non-canonical terms and to represent
them non-canonically is essential for proficiency with mental or written computation
(Greeno, 1991). The flexible representations construct represents a high level of
place-value understanding, and may be exhibited in a variety of ways. These include
the abilities to represent a given number in non-canonical form, to write the
numerical symbol for a number represented non-canonically, and to carry out mental
computation by flexibly partitioning multidigit numbers.
The idea that students need to develop the flexible representations construct is
contradictory to advice contained in a chapter written over 20 years ago, by Merseth
(1978). In her explanation of how to use concrete materials to teach addition and
subtraction algorithms, Merseth advised teachers to institute a trading rule, “that no
player may have more than nine objects in any column at the end of the individual
turn” (p. 64). With further knowledge of necessary conceptual structures for
29
multidigit numbers, many authors today are in favour of encouraging students to
develop more flexible understandings of how a number may be represented (e.g., G.
A. Jones et al., 1994; Resnick, 1983). By providing students with the idea that it is
never permitted to have more than 9 in a place, this “canonical arrangement only”
rule may restrict students’ thinking to the level of this base-ten structure construct.
Several researchers identified the flexible representations construct. Miura
and Okamoto (1989) and Miura et al. (1993) identified it as the highest category of
place-value understanding of their participants. After students represented a two-digit
number using blocks, researchers asked them to “show the number another way
using the blocks” (Miura & Okamoto, 1989, p. 111). The researchers categorised
students who did so using non-canonical arrangements of tens and ones blocks as
using a non-canonical base 10 representation. S. H. Ross (1989) described the ability
to determine the number represented by an arrangement of materials under this
construct as the understanding stage:
Students know that the individual digits in a two-digit numeral represent a
partitioning of the whole quantity into a tens part and a ones part. The quantity of
objects corresponding to each digit can be determined even for collections that have
been partitioned in nonstandard ways. (p. 49)
Children’s limited conceptions of multidigit numbers.
As well as accurate conceptual structures, researchers conducting research in
this field have identified a number of common limited conceptual structures held by
children for multidigit numbers. Though misconceptions of place-value concepts
held by children are “very diverse” (A. Sinclair & Scheuer, 1993, p. 200), the
research literature contains references to a cluster of observed behaviours, each
indicating a basic conceptual misunderstanding. Three limited conceptions for
multidigit numbers commonly observed in children are: a unitary concept of
multidigit numbers, a face value construct, and a counting sequence concept. The
conceptions are outlined in Table 2.2, which also shows task behaviour that
illustrates the presence of each misconception.
30
TABLE 2.2.
Task Performance Illustrating Limited Conceptions in Place-value
Understanding
Limited conception
Task Performance Example
Illustration
Unitary concept of
multidigit numbers:
Multidigit numbers seen
as unitary collections only.
Student represents a
multidigit number as a
collection of ones only. For
example, 21 is represented
as 21 ones only.
21:
Face value construct (a):
Tens digits represent
single units, not multiples
of 10 ones.
Student represents a twodigit number as two sets of
units. For example, 43 is
represented by 4 ones and 3
ones.
43:
Face value construct (b):
Digits representing traded
amounts in written
algorithms represent their
face value only.
Student believes that a
carried “1” in a written
algorithm represents just
one.
Counting sequence
concept: Each number is
one element in the
sequence of counting
numbers.
Student represents
multidigit numbers as
elements in counting
sequence only. For example:
25 is the number after 24
and before 26.
6
7 15
- 4 7
2 8
Limited Conception 1: Unitary concept of multidigit numbers. The unitary
construct is one of the first steps towards understanding the base-ten numeration
system, as described earlier in this section. Single digit numbers are linked both to
single symbols 0 to 9 and to groups of fewer than 10 objects. However, it appears
that many children also retain this conceptual structure for numbers greater than 10
(Fuson, 1990b).
The place-value literature is replete with reports of children who see
multidigit numbers as collections of ones, or single elements, only. In other words,
they see 34, for example, not as 3 tens and 4 ones, but as 34 ones. For example,
Hughes (1995) found that when asked to show $67 with “play money” in $1, $10,
and $100 denominations, some children counted out 67 $1 notes. Miura and
Okamoto (1989) made very similar observations in their study of U.S. and Japanese
students’ cognitive representations of number. The researchers found that certain
participants held unitary concepts for two-digit numbers. These conceptual structures
31
were deduced by researchers from the representations of two-digit numbers which
participants produced using base-ten blocks. If a student showed 28 ones for the
number 28, for example, then the researchers inferred that the student was using a
unitary conception of multidigit numbers. This construct is closely linked to S. H.
Ross’s (1989) whole numeral stage: “[Children’s] cognitive construction of the
whole comes first—the numeral 52 represents the whole amount” (p. 49). Fuson
(1990a, 1990b, 1992) referred to ideas of multidigit numbers held by some children
as collected multiunits. She explained the children’s concepts this way: “The
collected multiunits are collections of single units: A ten-unit item is a collection of
ten single unit items, a hundred-unit item is a collection of one hundred single unit
items, . . . , and so forth” (Fuson, 1992, p. 142). Explaining the considerable
difficulties faced by English-speaking children in linking understanding of multidigit
numbers, their written symbols, and their spoken names, Fuson (1990a) blamed the
common construction by these children of unitary conceptual structures on the
“obfuscation of the underlying tens structure in English number words” (p. 357).
Limited Conception 2: Face value construct. The face value construct is also
a very common conceptual structure among children, according to place-value
researchers. It is defined as the idea that each digit in a multidigit numeral represents
only that number of ones: its face value. S. H. Ross (1989) defined a stage of placevalue understanding at which children exhibited this construct as the face value
stage:
Students interpret each digit as representing the number indicated by its face value. .
. . but these objects do not truly represent groups of ten units to students in [this
stage]; students do not recognize that the number represented by the tens digit is a
multiple of ten. (p. 49)
The presence of the face-value construct points to a critical misconception of
multidigit numbers, but one that may be difficult to detect (S. H. Ross, 1990).
Though it is efficient to compute answers to multidigit questions as if each digit was
a single unit, many children apparently believe that each digit actually represents
only its face value. Researchers who were investigating a variety of mathematical
abilities have reported this construct. These abilities included children’s counting
(Cobb & Wheatley, 1988), representations of two-digit numbers (Miura & Okamoto,
1989; Miura et al., 1993; S. H. Ross, 1989, 1990), comparison of pairs of two-digit
and three-digit numbers (A. Sinclair & Scheuer, 1993), handling two-digit and threedigit numbers in novel problem-solving exercises (Bednarz & Janvier, 1982), and
32
completing written algorithms (Fuson & Briars, 1990; Fuson et al., 1997; Kamii &
Lewis, 1991).
A task where the presence of the face-value construct is particularly important
is that of carrying out written computation. Though they can correctly carry out the
procedure, many children do not have a good idea of the values they are symbolically
manipulating, and regard each digit as representing only its face value. Fuson and
Briars (1990) called this conceptual structure concatenated single digits: “Even many
children who carry out the algorithms correctly do so procedurally and . . . cannot
give the values of the trades they are writing down” (p. 181). Similarly, Cobb and
Wheatley (1988) found that some students had a different conception of addition
questions when written vertically, compared to a horizontal presentation. They
concluded that the students understood two numbers added horizontally as one
number incrementing the other, whereas in vertical format, the operations were seen
either as separate single-digit tasks, or as separate tens and ones addition tasks.
A paradox needs to be clarified here. Place-value understanding requires a
person to understand the value represented by each digit of a number; however,
people proficient with written algorithms treat operations in each place as single-digit
sums, differences, products, and quotients. In other words, it is quicker, and
cognitively less demanding, to operate on digits in each place as if they were all
units, rather than to keep in mind the value actually represented by each digit as each
step is carried out. By doing this, those proficient in written computation thus take
advantage of the efficiency inherent in the base-ten numeration system’s notation,
referred to by other authors as the “unreasonable power of mathematics” (Fuson,
1992, p. 56), and “the beauty and seeming simplicity of the base-ten number system”
(Sowder & Schappelle, 1994, p. 343). Nevertheless, appreciation of this efficiency
does not develop automatically in students (Sowder & Schappelle), and care is
needed to teach this to students without causing lack of understanding. As Resnick
and Omanson (1987) noted, automatic performance in written arithmetic is
incompatible with continuing reflection on principles underlying the written
algorithms. For example, consider the following addition question using a
conventional written algorithm:
274
+ 318
The algorithm is completed by considering a series of three single-digit sums:
33
4
+ 8
1
+ 7
+ 1
2
+ 3
This use of single-digit sums to complete a multidigit addition question is an
example of the trade-off involved in use of conventional written algorithms: In order
to achieve efficiency, the quantities represented by the digits in the various places are
ignored. As pointed out by Carpenter, Franke, Jacobs, Fennema, and Empson (1997),
“standard algorithms have evolved over centuries for efficient, accurate calculation.
For the most part, these algorithms are quite far removed from their conceptual
underpinnings” (p. 5). For children who have not developed accurate conceptual
structures for multidigit numbers, ignoring the meaning behind the symbols in this
way has the potential to hide the connections between symbols and the numbers they
represent.
The above comments illustrate the importance in place-value research of
distinguishing between children who operate according to the face-value construct,
and those who understand multidigit numbers according to the base-ten structure or
flexible groupings constructs. This importance is supported by S. H. Ross’s (1989)
comment that “pupils [holding the face value construct] may appear to understand
more than they actually do” (p. 50). Kamii and Lewis (1991) made the same point,
pointing out the inadequacies of standard achievement tests, which “tap mainly
knowledge of symbols” (p. 50), rather than understanding of numbers.
Limited Conception 3: Counting sequence construct. This limited conception,
similar to the unitary concept, was identified by Fuson (1992), who described a
limited understanding of multidigit numbers that she called “sequence multiunits.”
Students having this understanding of multidigit numbers conceive of them as
“[entities] within the number-word sequence: ‘Five thousand six hundred eighty
nine’ is the word after five thousand six hundred eighty eight and the word before
five thousand six hundred ninety” (p. 143). Fuson based her idea on observations of
two methods that certain students used to carry out multidigit addition or subtraction,
using counting procedures based on the position of each number in the sequence of
number names. Students possessing this construct either counted on (or back) from
one number using the counting number sequence by hundreds, tens, and ones; or
counted on or back within each place separately before combining the partial sums or
34
differences. For example, to compute 596 + 132, a student could count either “596,
696, 706, 716, 726, 727, 728” or “500, 600; 90, 100, 110, 120; 6, 7, 8; 728.”
Sources of children’s limited conceptions of multidigit numbers.
Many factors have been cited as causes of students’ difficulties in reaching a
level of understanding of place value that is robust, flexible, and efficient.
Knowledge of students’ common difficulties, and possible underlying causes for
them, is important in the planning of either teaching for or research into
understanding of the base-ten numeration system. Four sources of difficulty are
discussed in this section: (a) cognitive complexity, (b) over-emphasis on rules and
routines, (c) English language number names, and (d) lack of connections.
Cognitive complexity. As noted in the previous subsection, efficient
arithmetic computation is highly routinised, and effectively unthinking: The person
carrying it out is generally not thinking of the values involved (Hiebert, 1988). This
may be part of the reason that much instruction in mathematics is based on
unthinking use of routines. However, in order to check the accuracy of procedures
and to self-correct errors, it is important that a person carrying out computation has
the ability to reconstruct the meanings of the procedures involved, when needed
(Fuson & Briars, 1990).
The difficulty for primary teachers is that for students to be able to carry out
computational procedures while maintaining a mental representation of the quantities
involved may impose a greater processing load than many primary students can
handle. Boulton-Lewis (1993) and Boulton-Lewis and Halford (1992) referred to
Halford’s (1993a) structure mapping theory to explain the cognitive demands placed
on a student in learning about the base-ten numeration system. Boulton-Lewis and
Halford (1992) pointed out that in order to understand place value a student needs to
be able to make mappings at the system mappings level, possible from about 5 years
of age. They cautioned that processing loads will be increased if unfamiliar or
inappropriate analogues are used, and advised that
in order to reduce the load it is necessary to ensure that the child is able to recall
automatically the relations between quantity, place value, and any symbolic and
concrete representations of the task and uses less rather than more demanding
[computational] strategies. (p. 8)
The question of how different concrete representations may be used in the
teaching of place value is addressed in section 2.5.3.
35
Over-emphasis on rules and routines. Several authors have cited premature or
over-emphasis on procedures as having a detrimental effect on children’s number
learning (e.g., Hiebert, 1988). This style of teaching and learning is often called a
“textbook” approach (Cobb & Wheatley, 1988; Fuson, 1990b, 1992; Kamii & Lewis,
1991). The characteristics of this approach include a strict sequence of instructional
steps according to the perceived difficulty of question types; few pictures, often
poorly linked to symbolic representations; and a rule-based approach to computation
(Fuson, 1992, pp. 149-150). These observations underline the advice, summarised in
section 2.4.1, that mathematics teaching today should be based not on the teaching of
rules and procedures, but on teaching for mathematical understanding and the
development of number sense.
English language number names. The irregular system of number names in
the English language is another source of difficulty in learning place-value concepts,
mentioned by many authors (Bell, 1990; Boulton-Lewis & Halford, 1992; Carpenter,
Fennema, & Franke, 1993; Fuson, 1990a, 1992; Fuson & Briars, 1990; Hughes,
1995; G. A. Jones & Thornton, 1993a; Miura & Okamoto, 1989; Miura et al., 1993).
Whereas the base-ten numeration system is used consistently in most nations today,
the spoken number names naturally vary with the language used. The number names
in the English language are not a consistent system across the range of spoken
numbers, and include a number of irregularities, especially in tens place names and
the teen numbers (Fuson, 1992). Similar irregularities also exist in other European
languages (Miura et al. 1993), such as French, in which are found such irregular
number names as quatre-vingt-quinze (“four-twenty-fifteen”) for the number ninetyfive. Such number name systems obscure the grouped tens structure of the base-ten
number system. There is a growing belief that this is a major cause of the difficulties
which European language-speaking students face in learning place-value concepts,
compared to other language speakers, including speakers of Asian (Bell, 1990;
Fuson, 1992) or Maori (Hughes, 1995) languages. Miura and her colleagues (Miura
& Okamoto, 1989; Miura et al., 1993) have claimed that their research comparing
place-value understanding of students who speak Asian and European languages
demonstrates that the respective structures in these languages influence conceptual
structures held by students who speak them. Hughes (1995) suggested that teachers
of English-speaking students incorporate more “transparent” English number names
in their lessons, to aid the students’ place-value understanding. Whether or not this
36
strategy is adopted, any teaching program that aims to develop efficient conceptual
structures for multidigit numbers in English-speaking students must take into account
the particular difficulties introduced by the number names in the English language.
Lack of connections. It is widely reported in the literature that many children
do not connect number symbols to their real-world referents or to number
representations such as blocks (Baroody, 1989, 1990; Hart, 1989). As a result, there
is considerable support for helping children build strong links between numbers and
number representations (Fuson, 1992; Hiebert, 1988; Hiebert & Wearne, 1992;
Resnick, 1987), including support for the use of computer software for this purpose
(Clements & McMillen, 1996; Fuson, 1992; Hiebert, 1984; Hunting & Lamon,
1995). The topic of building connections in place-value teaching through the use of
concrete materials is discussed in greater detail in section 2.5.3.
2.4.3 Analogical Reasoning
Analogical reasoning is the second branch of cognitive research relevant to
the teaching of place value. Analogical reasoning has been claimed to have a
particular importance in the study of thinking and reasoning (Gentner & Toupin,
1986; Goswami, 1992), and to be “the most important of all our reasoning processes”
(Grandgenett, 1991, p. 30). This section discusses the importance of analogical
reasoning as the cognitive mechanism underlying the use of materials such as baseten blocks to represent numbers.
Definition of analogical reasoning.
Though there is no single generally accepted definition of analogical
reasoning, several authors (English, 1997; Simons, 1984; Vosniadou & Ortony,
1989) have offered definitions for the term that share a number of essential features.
Vosniadou and Ortony’s (1989) definition will be used here: “Analogical reasoning
involves the transfer of relational information from a domain that already exists in
memory (usually referred to as the source or base domain) to that domain to be
explained (referred to as the target domain)” (p. 6).
Research into children’s analogical reasoning.
Analogical reasoning by children has not received much attention until
comparatively recently. As Goswami (1992) explained,
37
the reasons for this neglect were partly historical. According to piagetian [sic]
theory, the ability to reason by analogy was a late-developing skill, emerging at
around 11-12 years of age during the “formal-operational” period of reasoning.
Younger children were thought to be incapable of reasoning by analogy, and
consequently few people investigated their analogical reasoning skills. (p. 3)
However, several more recent studies of children’s analogical reasoning have
put these Piagetian claims in some doubt. Goswami (1992), in particular, argued that
the findings of Piaget and others on this point should be challenged on the
assumptions underlying their research. Goswami argued that when experiments are
designed that ensure that participants fully understand the task and the relations that
exist between terms, even very young children are capable of reasoning analogically.
Successful training in analogical reasoning skills both to children and to adults has
been demonstrated in the work of several researchers (Alexander, White, et al., 1987;
Alexander et al., 1989; Alexander, Wilson, et al., 1987; Bisanz, Bisanz, & LeFevre,
1984; Newby, Ertmer, & Stepich, 1995). Participants included 4- and 5-year-old
children, students aged 9 to 19, college students, and teachers of 4th grade and preschool classes. Results showed that the training was effective in each case. In the
case of the teachers, the training effects were found to transfer to the teachers’
students also (Alexander, Wilson, et al., 1987).
Though the application of analogical reasoning to science education has
received much research attention, there has been little study of the use of analogies in
teaching mathematical concepts. As English (1997) stated, “this appears to be a
serious omission, given the important role of analogy in mathematics learning”
(p. 192). Analogies are used by mathematics teachers, to teach a range of
mathematical ideas, including number, place value, and fractions. Some of the few
studies of analogical reasoning and mathematics have been those by Wilson and
Shield (1993) and English (1993, 1997).
Structural mapping theory.
The structural mapping theory of Gentner (1983, 1988, 1989) provides a
useful explanation of the mechanisms involved in analogical reasoning and the use of
mental models.
Definition of structural mapping. The structural mapping theory (Gentner,
1983) explains how commonalties between target and base domains are perceived
when reasoning analogically: “The central idea is that an analogy is an assertion that
a relational structure that normally applies in one domain can be applied in another
38
domain” (p. 156). In other words, the user perceives parallels between target and
base domain, based upon a common relational structure. The two domains are then
perceived to be correspondent, to the extent that relations among members of the two
domains can be mapped from target to base.
Gentner (1983) used Rutherford’s theory that an atom is like a solar system to
demonstrate the idea of structural mapping. In this example, though some attributes
of the solar system components cannot be mapped to an atom (such as colour and
temperature of the sun), key relations between the sun and planets (such as the
central body being more massive than, and attracting, the orbiting body) are mapped
directly onto relations between nucleus and electrons in the atomic domain. Under
the structural mapping theory, this is a general principle: Analogies have few
attribute matches, but many relation matches. Thus analogies can be distinguished
from literal similarity, abstraction, or anomaly (Gentner, 1983). This distinction also
applies to Halford’s (1993a) definition for cognitive representations:
A cognitive representation is an internal structure that mirrors a segment of the
environment. The representation must be in structural correspondence to the
environment and be consistent. Resemblance between the representation and the
environment is not required, and representations are not ‘pictures in the head.’
(p. 69)
Application of structural mapping theory to place-value instruction. The
following major section (section 2.5) describes the use of manipulative materials for
the teaching of place-value concepts; the remainder of this section describes the
theory behind their use based on the structural mapping theory. Concrete materials,
such as bundling sticks and base-ten blocks, used to represent numbers “are
technically analogues [of numbers], and can be analyzed using analogy theory”
(Boulton-Lewis & Halford, 1992, p. 2). As Boulton-Lewis and Halford pointed out,
concrete representations of numbers mirror the structure of the domain of numbers.
Thus, the structure of the visible sticks or blocks is mapped onto the domain of
invisible numbers. Boulton-Lewis and Halford pointed out that the use of concrete
representations of numbers in mathematics teaching requires attention to two
important points. First, the representation itself should accurately model the structure
of numbers; second, children should be familiar with the representation to reduce the
cognitive load entailed in the use of the representation.
39
Base-ten blocks as analogues of numbers.
Base-ten blocks were developed by Dienes (1960), and are “probably the
most commonly used analogues in the teaching of numeration and computation”
(English & Halford, 1995, p. 105). Base-ten blocks qualify as analogues of numbers,
based on Gentner’s (1983) definition of structural mapping given in section 2.4.3.
First, there are no physical attributes that could be mapped from base-ten blocks to
numbers, since numbers are abstract entities. Second, there are a number of
relational similarities that can be mapped from base-ten blocks to base-ten numbers;
three of these are described in the following paragraphs.
The first feature of base-ten blocks that makes them effective analogues of
numbers is the fact that relative sizes of the four blocks map onto the relative values
of the four places represented (English & Halford, 1995; Figure 2.2). Bednarz and
Janvier (1982) described this feature of materials as representing numbers “so that
the rule of grouping is apparent (visible or explicit)” (p. 36). Individual base-ten
blocks are available in only four standard sizes: the one-block, a 1 cm cube; the tenblock, a rectangular prism 1 cm x 1 cm x 10 cm; the hundred-block, 1 cm x 10 cm x
10 cm; and the thousand block, a 10 cm cube. Each of the three larger blocks has
sawn grooves, at 1-cm intervals, that provide a visual indication of the relation
between each larger block and a number of one-blocks. Thus, the size of each block
in relation to the size of a one-block maps directly onto the value of each place in
relation to the ones place. For example, as 100 is one tenth of 1000, and 10 times 10;
so also a hundred-block is one tenth the size of a thousand-block, and 10 times the
size of a ten-block. Between any pair of the four block sizes, the same mapping can
be made from the relative size of blocks to the relative values of the represented
numbers (see Figure 2.2).
Other materials available for the teaching of mathematics can be
manufactured according to other groupings, or can be so grouped by children using
them. For example, Unifix™ cubes can be grouped arbitrarily in any sized group and
so do not model the base-ten system in particular. Such materials may be termed
unstructured or semistructured analogues of multidigit numbers (English & Halford,
1995), indicating their lack of a built-in structure that directly models the base-ten
numeration system.
40
Figure 2.2. Relationships inherent in base-ten blocks.
Note. Based on figure from Mathematics education: Models and processes (p. 105), by L. D. English
and G. S. Halford, 1995, Mahwah, NJ: Erlbaum.
The second mapping from base-ten blocks to the domain of numbers maps
the numerosity of a group of blocks of one size, onto the number represented in the
associated place. A set of blocks of the same size is used to represent a single digit
from one to nine, with the number of blocks being equal to the face value of the digit
represented. For example, 6 hundred-blocks represent the number 600. A
combination of the first two mappings described in this subsection is available in any
representation of numbers with base-ten blocks. In a base-ten block representation of
a number, such as 752, not only is the block representation of the entire number
proportional to its value (compared to a single one-block), but the representation of
any portion of that number—the tens part of it (50), or the hundreds and tens
expressed as tens (750), for example—is also proportional to that portion. Thus when
base-ten blocks are used to represent the steps in a computational algorithm, they do
so in a manner that preserves at every step a valid mapping from the block
representation for each number to its value.
The third mapping that base-ten blocks exhibit is that of trading relations
(Fuson, 1990a, 1992). In carrying out written or mental computation with multidigit
numbers, it often necessary to regroup a portion of a number in one place to another,
generally adjacent, place. This process of trading one-for-ten is essential for the
41
operations of addition, subtraction, multiplication, and division, which are important
components of the primary mathematics curriculum. Base-ten blocks effectively
model the trading process when one block is swapped for 10 of the next smallest
place, or vice versa. In the process, the size of the representation is preserved and so
can be mapped from the materials to the number. This mapping of size relations does
not occur with materials such as coloured chips or an abacus, as each chip or abacus
bead is the same size, making them less useful as analogues of numbers, particularly
early in children’s learning of place-value concepts (English & Halford, 1995).
The above paragraphs demonstrate that base-ten blocks incorporate the
systematicity principle, a further development of the structural mapping theory
introduced by Gentner and Toupin (1986). They defined the term in this way:
The systematicity principle states that a base [source] predicate that belongs to a
mappable system of mutually interconnecting relations is more likely to be imported
into the target than is an isolated predicate. A system of relations refers to an
interconnected predicate structure in which higher-order predicates enforce
constraints among lower-order predicates. (p. 280)
As demonstrated, base-ten blocks are capable of at least three different
relational mappings: (a) mappings between the sizes of individual blocks and the
values assigned to places, (b) mappings between the numerosity of a group of similar
blocks and the value of an individual digit, and (c) mappings between traded actions
on blocks and the corresponding regrouping carried out on numbers. These three
mappings together form a system of interrelated relations and so satisfy the
conditions for the systematicity principle described above. Thus, according to
Gentner and Toupin (1986) relations among base-ten blocks are “likely to be
imported into the target” (p. 280), adding further support to their use in teaching of
place-value concepts.
Cognitive load theory.
The cognitive load theory is also relevant to the teaching of place-value
concepts with base-ten blocks, being concerned with the demands placed on
students’ thinking processes by various instructional designs. Sweller (1999) pointed
out that current theories suggest that “we can process no more than about two to four
elements at any given time with the actual number probably being at the lower rather
than the higher end of this scale” (p. 5). In order to simultaneously manage larger
numbers of elements in working memory, it is necessary for learners to develop
schemas that “provide the means of storing huge amounts of information in long42
term memory” (Sweller, 1999, p. 11). Sweller defined a schema as “a cognitive
construct that permits us to treat multiple elements of information as a single element
categorised according to the manner in which it will be used” (p. 10). According to
cognitive load theory, base-ten blocks assist children to understand the base-ten
numeration system by helping them form such schemas that relate numerical
quantities, written symbols, and the blocks.
2.5
Teaching Place-value Understanding
This section includes three subsections. Broad approaches to the teaching of
place-value understanding recommended in the literature are described in section
2.5.1. The focus is narrowed in section 2.5.2, to concentrate on the widely-stated goal
of helping students to build connections between numbers and number
representations. Section 2.5.3 describes the reported use of a range of concrete
materials in the teaching of place-value understanding, with particular emphasis on
base-ten blocks.
2.5.1 Teaching Approaches
A number of writers have described different ideas of how to teach placevalue concepts. Each of these teaching methods aims to help students to develop
links among number concepts, their real-world referents, and their representations by
symbols or physical analogues. Four recurrent themes evident in the literature on
place-value teaching are described in this section: (a) use of structured materials to
model numbers, (b) use of real-world problems, (c) teaching place-value concepts in
the context of computation, and (d) adopting a constructivist view of learning. These
four themes are by no means mutually exclusive; several authors included more than
one of these themes in their work.
Some writers have advocated a structured approach to teaching place value,
using concrete materials, and especially base-ten blocks (Fuson, 1990a, 1990b, 1992;
Fuson & Briars, 1990). In this approach the teacher continually reinforces the links
among written symbols, number names, and concrete materials. Bednarz and Janvier
(1982, 1988) recommended another approach that focused children’s attention on the
structure of the base-ten numeration system. Their particular focus was on the
groupings inherent in the base-ten numeration system; G. A. Jones and Thornton
(1993a) called this an explicit grouping approach. In their research, Bednarz and
43
Janvier presented students with various explicit groups of objects, often including
multiple groupings, or groups of groups. The objects used included cereal boxes
grouped by six into cases, and baskets of three cases; peppermints in rolls of 10, and
bags of 10 rolls; and paper flowers, each made of 10 sheets of paper, and grouped
into bouquets of 10 flowers. The research showed that some students did not
understand the groupings inherent in the base-ten numeration system, shown by the
fact that they attempted to answer questions involving multiple groupings without
inquiring about the number of objects in each group.
A second theme in the literature on place-value instruction is the use of realworld problems to help students make links between symbols and real-world
application of mathematics. This approach has been recommended by Bednarz and
Janvier (1982, 1988), Hiebert (1989), and Hiebert and Wearne (1992).
The third theme is shown in a teaching approach recommended by several
writers (e.g., Fuson, 1990a, 1990b, 1992; G. A. Jones & Thornton, 1993b; G. A.
Jones et al., 1994), to teach place value in the context of computation or problemsolving exercises, and particularly multidigit addition and subtraction exercises.
Several researchers, including Carpenter et al. (1993), Fuson (1990b, 1992), Kamii,
Lewis and Livingston (1993), and S. H. Ross (1989, 1990), argued that rather than
attempting to teach place-value concepts first, as a prelude to the teaching of addition
and subtraction, place-value learning should take place within the context of
computation. Fuson (1992) summarised this idea in her comment that
multidigit addition and subtraction are problem situations that permit crucial
attributes of the named-multiunit words and the positional written marks to become
evident and thus are excellent contexts within which children can construct placevalue understandings. (p. 173)
One aspect of the idea that place-value concepts should be taught in the
context of their use in computation is the advice from several authors that children
should be encouraged to invent their own computational procedures, as a means to
gaining proficiency in understanding and using multidigit numbers. This advice has
been given by Carpenter et al. (1993), Duffin (1991), Kamii and Lewis (1991),
Kamii et al. (1993), Resnick and Omanson (1987), S. H. Ross (1989), Sowder and
Schappelle (1994), and P. W. Thompson (1992).
The fourth theme that emerges from the place-value literature is the use of a
constructivist approach to teaching place value. Some authors (e.g., Cobb, 1995;
Kamii & Lewis, 1991; Kamii et al., 1993) specifically mentioned constructivism as
44
the basis of their work; others (e.g., R. Ross & Kurtz, 1993; S. H. Ross, 1990; P. W.
Thompson, 1992) mentioned the idea of children constructing understanding,
indicating their acceptance of constructivist ideas. G. A. Jones et al. (1994) favoured
what they termed an interactive-constructivist approach, recommending that teachers
engage their students in “negotiated learning” as they solve problems that “challenge
and stretch” their abilities.
It is relevant to point out one view of constructivist teaching of place-value
concepts that excludes the use of concrete materials. Constructivism was a part of
Kamii et al.’s (1993) developmental approach that focused on the internal
construction of meanings in children’s minds. However, unlike most authors in the
field, including others who used constructivist teaching methods, Kamii et al. did not
use concrete materials to support learning. In their view, concrete materials are a
hindrance to children’s development of place-value concepts, because these concepts
derive from mental actions and higher-order constructions rather than from objects in
the external world (p. 201). It appears that Kamii et al.’s view is a minority one at
odds with that of the majority of other writers in the field, who recommend the use of
concrete materials for the teaching of place-value concepts.
2.5.2 Building Place-Value Connections
The findings of cognitive scientists are commonly used to inform the teaching
of mathematics, as explained in section 2.4. Mathematics researchers are typically
interested in the conceptual structures for numbers posited by cognitive scientists and
the relations that these structures have to external representations of numbers,
including written symbols and concrete materials. For example, Hiebert and
Carpenter (1992) described two assumptions in mathematics education research: that
“some relationship exists between external and internal representations” and that
“internal representations can be related or connected to one another in useful ways”
(p. 66). There is widespread support in the mathematics education literature for the
view that teaching place-value concepts involves assisting students to build
connections among numbers, verbal names, and written representations of numbers
(Baroody, 1989, 1990; Clements & McMillen, 1996; Fuson, 1990b, 1992; Fuson &
Briars 1990; Gluck 1991; Hart 1989; Hiebert & Wearne 1992; Merseth 1978; Payne
& Rathmell, 1975; Peterson, Mercer, McLeod, & Hudson, 1989; Peterson, Mercer,
Tragash, & O’Shea, 1987). The idea of “drawing connections between a set of
45
understandings and an appropriate symbol system [is] a central feature of all
learning, regardless of content” (Hiebert, 1984, p. 499); in the case of mathematics
learning, connections have to be made between the student’s understanding of
numbers and the written numeration system.
The development of place-value understanding involves the formation of a
number of interrelated connections, or links, in the student’s understanding. Three of
these links are illustrated in Figure 2.3, which portrays one view of the relations
among numbers, written symbols, and physical models. As already mentioned,
numbers are abstract entities that do not exist in the physical world. However, they
are represented in physical form in two principal ways: (a) through the numeration
system of written symbols and associated procedures (Skemp, 1982) and (b) through
various physical models known collectively as concrete materials. Students need to
understand the links among numbers and these two different sets of referents. Figure
2.3 shows that whereas written symbols represent numbers in an abstract, sociallyconstructed manner (Cobb, 1995; S. H. Ross, 1990), concrete materials model
numbers analogically in a physical form that is closer to young children’s experience
(Hiebert, 1988; Hiebert & Carpenter, 1992).
Figure 2.3. Relationships among numbers, written symbols, and concrete materials.
It is clear that, though written symbols and concrete materials can both be
used to represent numbers (Hiebert, 1988), the relationships between numbers and
the two representations are of a different character. As explained in section 2.4.3,
concrete materials such as base-ten blocks model numbers analogically (BoultonLewis & Halford, 1992): There is a direct relationship between the size of a number
and the size, numerosity, or both, of its physical representation. In contrast, written
symbols are “cultural tools” (Cobb, 1995, p. 380), “the shared symbol systems of
46
mathematics” (Putnam et al., 1990, p. 70), and only represent numbers as a function
of their socially-agreed meanings (Kamii & Lewis, 1991). There is no sense, for
example, in which the written symbol “7” or the words “seven,” “sept” (French), or
“qi” (Chinese) represent an actual number seven apart from the convention that
people using each of them agree that it stands for that number. As J. H. Mason
(1987) pointed out, the “symbolizing process” by which symbols stand for numbers
is often forgotten by teachers, and is never understood by many students (p. 76).
Students need to be made aware of the parallel relationship that is meant to
exist between symbols and concrete materials, so that they can take advantage of the
implied connections which exist between the two systems of representations. Hiebert
and Wearne (1992) summarised the idea that students need to see the connections
between written and material representations of numbers with their statement that
from a cognitive science point of view, it can be argued that building connections
between external representations supports more coherent and useful internal
representations. . . . Different forms of representation for quantities, such as physical
materials and written symbols, highlight different aspects of the grouping structure,
and building connections between these yields a more coherent understanding of
place value. (p. 99)
Mathematicians have been concerned with helping children make links
between symbols and the ideas they represent for many years (Hiebert, 1984, p. 500).
For example, over half a century ago Van Engen (1949) wrote of the need for
teachers to consider how to help their students develop meanings for arithmetic.
Hiebert (1984) stated the view that making links is central to mathematics learning
and that “although it is not surprising that students have trouble connecting form and
understanding, the effect of not making these connections may be one of the most
serious problems in mathematics learning” (p. 499). Elsewhere, Hiebert (1989) again
stressed the centrality of this process to mathematics education. He stated that “it is
impossible to overemphasize the importance of helping children establish
connections between quantities and numerals and between actions on quantities and
operation signs” (p. 40).
Several authors have referred to a gap that appears to exist in the minds of
many children between the two systems of number representations—written symbols
and concrete models—and have referred to the need for teachers to work at “bridging
the gap” (e.g., Gluck, 1991; Hart, 1989; Hiebert, 1988). Figure 2.4 portrays what
Gluck referred to as “the very large gap between manipulatives and paper-and-pencil
47
tasks” (p. 10). Hart reported the same gap in preliminary results of a research project
entitled “Children’s Mathematical Frameworks.” She concluded from the poor
performance of the project’s participants that they were not making the needed
connections between manipulatives and written procedures. She summed up her
belief that students see the use of manipulatives and written procedures as
disconnected processes in her suggested subtitle for the project report: “Sums Are
Sums and Bricks Are Bricks” (p. 139). In her final paragraph Hart summed up her
thoughts:
Many of us have believed that in order to teach formal mathematics one should build
up to the formalization by using materials, and that the child will then better
understand the process. I now believe that the gap between the two types of
experience is too large, and that we should investigate ways of bridging that gap by
providing a third transitional form. (p. 142)
Figure 2.4. Conceptual gap between written symbols and concrete materials.
For many years authors have recommended a bridging approach to teaching
place value, that aims to bridge the gap between written symbols and concrete
materials. However, it needs to be realised that solutions to the problem seem rather
elusive. Several authors have warned that bridging the gap between numbers and
symbols, or numbers and materials “will not occur automatically” (Merseth, 1978,
p. 61). Fuson (1992) found that, even in the presence of appropriate concrete
materials, without appropriate guidance from a teacher to link blocks and written
symbols, some children did not make the relevant connections, and errors resulted.
Teaching strategies for improving these links have been suggested by several
authors. One such strategy is to allow students more time to construct links (e.g.,
Carpenter et al., 1993; Gluck, 1991; R. Ross & Kurtz, 1993). As Hiebert (1989)
pointed out, “connecting symbols with understanding is a difficult intellectual task,
48
and does not occur quickly” (p. 40); “students need time and many opportunities to
construct the connections for themselves” (p. 42). A second strategy is to strongly
emphasise connections between symbolic and concrete representations (Fuson, 1992,
p. 165) so helping to draw students’ attention to errors in their written computation.
Fuson explained that “when experimenters forced children to connect … marks
procedures to the blocks, the multiunit quantities always could help the children selfcorrect the incorrect marks procedure” (p. 165). In fact even teachers who were
“forced” to link closely written symbols and blocks improved their understanding of
the represented meanings. Elsewhere, Fuson referred to the need to make “constant
use of the three sets of words” for number names, block names, and digit names
(Fuson & Briars, 1990, p. 182) and stated that links must be made “very tightly and
clearly” (Fuson, 1990b, p. 277). A third strategy for bridging the conceptual gap
between materials and symbols is to use an intermediate representation: This idea is
portrayed in Figure 2.5.
Figure 2.5. The use of transitional forms to bridge the gap between written symbols
and concrete materials.
Various intermediate representations have been suggested to strengthen the
link between written symbols and concrete materials. Three suggestions for items to
bridge the gap between concrete materials and written symbols are (a) using material
that links concrete materials and symbols (Gluck, 1991), (b) pictorial representations
(Baroody, 1990; Peterson et al., 1987, 1989), and (c) computer-generated
49
representations (Champagne & Rogalska-Saz, 1984; Clements & McMillen, 1996; P.
W. Thompson, 1992).
First, Gluck (1991) devised a teaching method that involved the use of a
“place-value board” incorporating flip-over number labels for each digit and base-ten
blocks. Gluck claimed that this material could be used “to take students step by step
from the concrete, through the semi-concrete, and on to the abstract stage of
development” (p. 12). It is not clear what Gluck meant by the “semi-concrete” stage;
it appears that she was referring to activities in which students used base-ten blocks
and flip-books at the same time. The key idea behind Gluck’s place-value board is to
mirror actions on the blocks with changes in the written symbols, as also
recommended by Fuson (1992).
Second, the use of pictorial representations of numbers as intermediaries
between concrete materials and written symbols has a long history, going back at
least to Bruner (1966). Baroody (1990) stated that some writers, including Bruner,
hypothesised that use of pictorial models was “a necessary bridge between concrete
and abstract embodiments” (p. 283). Peterson et al. (1987, 1989) designed two
teaching experiments according to what they claimed was a “generally accepted
hierarchy for presenting a new skill [that] follows a concrete to abstract continuum”
(Peterson et al., 1989, p. i). They found success in teaching place value to students
with learning disabilities using a concrete-semiconcrete-abstract sequence: using
one-inch cubes, pictures of place-value sticks and cubes, and worksheets without
pictures, respectively. However, Fuson (1990a) argued that such a sequence was
based on a faulty understanding of concrete and symbolic representations of number:
The use of Bruner’s concrete-pictorial-abstract continuum in this context ignores the
fact that the blocks and the written marks are not endpoints on a single continuum:
They are structurally different systems that must be connected. Pictures have the
same properties as the blocks (and different properties from the marks). . . . It is not
clear at this time what, if any, advantages are provided by pictures, and there are
definite disadvantages. (pp. 390-391)
A third suggestion for bridging the gap is to use computer-generated
representations of numbers. The software used in this study, like several other placevalue software applications described in the literature (Ball, 1988; Champagne &
Rogalska-Saz, 1984; Clements & McMillen, 1996; P. W. Thompson, 1992),
incorporates pictures of base-ten blocks. The connections made between these
pictures and the on-screen number symbols are provided by the software, so that
50
changes in one representation are reflected quickly in the other representation. Three
software applications that modelled base-ten blocks are described in Appendix A,
and compared to the software application designed for this study.
2.5.3 Use of Concrete Materials
Many types of concrete materials have been mentioned in the literature as
being appropriate for use in teaching place-value concepts (e.g., Baroody, 1990;
Bednarz & Janvier, 1982, 1988; Clements & McMillen, 1996; English & Halford,
1995; Hiebert & Carpenter, 1992; Hiebert & Wearne, 1992; Howard, Perry, &
Conroy, 1995; Nevin, 1992; Peterson et al., 1987; S. H. Ross, 1989; C. Thompson,
1990). These materials include
1.
a wide variety of objects (including pictures of objects) that may be
used singly as counters, including wheels, lollies, flowers, or beans;
2.
materials capable of being grouped into groups of 10, 100, and so on,
including bundling sticks, Unifix™ cubes, Multilink™ material, cereal
boxes, and paper flowers;
3.
materials that include proportionally sized representations for ones,
tens, hundreds, and so on, including bean sticks (wooden sticks each
with 10 beans glued onto it), string lengths, and base-ten blocks;
4.
materials that include representations for various places that are
distinguished from each other by colour or some other arbitrary feature,
including play money and coloured chips;
5.
Cuisenaire rods, that are rods of different lengths and different colours
representing numbers from 1 to 10; and
6.
materials that illustrate the sequence of number symbols, including
hundreds boards.
Some indication of the extent of use of concrete materials in mathematics
teaching is given by the results of studies in which researchers surveyed teachers
about their use of a range of concrete materials. The first, by Gilbert and Bush (1988)
surveyed grade 1, 2, and 3 teachers in 11 states of the U.S.; the second, by Howard et
al. (1995), surveyed 249 primary teachers in a metropolitan education region in New
South Wales. Both studies showed frequent use of concrete materials by the surveyed
teachers. Gilbert and Bush found that 65% of respondents reported using concrete
materials at least once per week; in Howard et al.’s study 62% of respondents
51
reported using concrete materials “often,” with less than 1% of teachers reporting
that they did not use them at all. International data providing indirect evidence of the
use of concrete materials was provided by a report of the primary school phase of the
Third International Mathematics and Science Study [TIMSS]. Over 90% of teachers
of Grade 4 students from every one of 24 nations surveyed agreed that “more than
one representation (picture, concrete material, symbol, etc.) should be used in
teaching a mathematics topic” (Mullis et al., 1997, p. 151). The most common reason
given by teachers in the Howard et al. (1995) study for using concrete materials was
that “they benefit children’s learning,” chosen by 96% of teachers surveyed (p. 6).
The results of these studies make it clear that primary teachers collectively spend a
lot of time using a variety of concrete materials in mathematics lessons, and that
teachers believe that concrete materials have a beneficial effect on the children’s
learning of mathematics. However, results of research into this belief have been
equivocal (Hunting & Lamon, 1995; P. W. Thompson, 1992, 1994). This point is
discussed further in the following subsection.
The material used most often by teachers in the Howard et al. (1995) study
was base 10 material, used by 84% of respondents. The term “base 10 material” was
not defined by Howard et al., but is assumed to refer to base-ten blocks. As noted by
Howard et al., the finding that “number material”—base 10, Multilink, and Unifix—
is used more than any other is “hardly surprising given that the syllabus and many
commercial mathematics programs encourage the use of such material” (p. 6). These
findings underline the importance of research into the use of base-ten blocks, such as
that reported in this thesis.
Research into learning with base-ten blocks.
Despite the very common use of base-ten blocks in primary classrooms
(Gilbert & Bush, 1988; Howard et al., 1995), several authors have pointed out the
lack of consensus in results of research into number learning using base-ten blocks
(e.g., Hunting & Lamon, 1995; P. W. Thompson, 1992, 1994). As Thompson (1994)
pointed out, some research (such as that by Resnick & Omanson, 1987) has shown
little benefit from use of base-ten blocks for students learning place-value concepts,
whereas other studies (e.g., Fuson & Briars, 1990) did show significant gains in
student learning. Hunting and Lamon (1995) noted that results from several studies,
including a meta-analysis by Sowell (1989), “suggest that there is a large host of
52
variables influencing the use of didactic materials, among these, type of material,
length of time used, teacher training, age of the students, whether students or teacher
chose the manipulative” (p. 55). As this statement demonstrates, questions of why
base-ten blocks may be effective in some cases and not in others are not easily
answered. P. W. Thompson (1994) suggested that
these contradictions [among findings of different studies] are probably due to aspects
of instruction and students’ engagement to which studies did not attend. Evidently,
just using concrete materials is not enough to guarantee success. We must look at the
total instructional environment to understand effective use of concrete materials—
especially teachers’ images of what they intend to teach and students’ images of the
activities in which they are asked to engage. (p. 556)
One aspect of the effective use of place-value materials that has been
mentioned by several authors (Baroody, 1989; Hunting & Lamon, 1995; P. W.
Thompson, 1994) is students’ engagement with learning activities, that Hunting and
Lamon referred to as “cognitive engagement in sense making.” As discussed in
section 2.2.1, a prominent issue in mathematics education at present is the
development of number sense. In the context of the teaching of place-value concepts,
it is widely agreed that students must actively and sensibly consider the quantities
represented by place-value materials as they use them (e.g., Resnick & Omanson,
1987). Otherwise there is a risk that “just as with symbols, pupils can learn to use
manipulatives mechanically to obtain answers” (Baroody, 1989, p. 4; see also
Clements & McMillen, 1996).
Comments such as those reported in this subsection demonstrate a need for
further information about how students learn place-value concepts, and in particular
how that learning is influenced by materials such as base-ten blocks. As mentioned
by several authors, the use of concrete materials to teach place-value concepts has
great theoretical and intuitive appeal (e.g., Howard et al., 1995; Hunting & Lamon,
1995; Perry & Howard, 1994; P. W. Thompson, 1992, 1994). In light of this
widespread belief among educators that concrete materials should be effective for
teaching place-value ideas, there is a need for research that endeavours to find
reasons why such effectiveness is not always demonstrated. One aspect of this
research is discussed in the following subsection: There are now available a number
of software titles that model numbers by pictures of base-ten blocks; one hope held
for such software is that it may help overcome some of the drawbacks of physical
blocks.
53
Computer-generated models of numbers.
In recent years a number of software applications have been developed to
model numbers and numerical relations on a computer display (Ball, 1988;
Champagne & Rogalska-Saz, 1984; Clements & McMillen, 1996; Hunting, Davis, &
Pearn, 1996; Hunting & Lamon, 1995; Rutgers Math Construction Tools, 1992; P.
W. Thompson, 1992). Three arguments in favour of the use of computer software to
model numbers are discussed in this subsection. Writers have argued that software
can (a) model numbers just as effectively as physical blocks, (b) provide a “cleaner”
form of manipulative, and (c) provide features not available with physical materials.
The first argument in favour of computer software to model numbers is based
on a view that physical models are not effective purely as a result of their tactile
properties, and that therefore software representations can be just as effective as
numerical models. “Computers might supply representations that are just as
personally meaningful to students as are real objects” (Clements & McMillen, 1996,
p. 271). Clements and McMillen argued for a reappraisal of what constitutes a
concrete manipulative in the context of teaching numeration ideas to children. They
asked the question “What does concrete mean?” (p. 270), and concluded that the
tactile, sensory nature of physical manipulative materials is not what makes them
useful for teaching about number. They argued that physical manipulation of
materials by children does not guarantee that they will generate the mental images
that their teachers expect, and that mathematical meaning is not contained within
physical materials (see also Hiebert et al., 1997; Holt, 1964; Hunting & Lamon,
1995; Perry & Howard, 1994; P. W. Thompson, 1994). Clements and McMillen
argued further that
mathematics cannot be packaged into sensory-concrete materials, no matter how
clever our attempts are, because ideas such as number are not “out there.” As Piaget
has shown, they are constructions—reinventions—of each human mind. “Fourness”
is no more “in” four blocks than it is “in” a picture of four blocks. The child creates
“four” by building a representation of number and connecting it with either real or
pictured blocks. (p. 271)
A second argument advanced for the use of computer representations of
numbers is that they may provide “cleaner” manipulatives (Clements & McMillen,
1996). The difficulties that children have with physical manipulatives are sometimes
due to their “messy” nature. Students may miscount blocks, be distracted by
extraneous features of the blocks, or otherwise use them inaccurately (Champagne &
54
Rogalska-Saz, 1984). Champagne and Rogalska-Saz noted that conventional
physical materials could distract students from the task at hand and cause difficulties
for teachers with managing materials; Hunting and Lamon (1995) and Touger (1986)
noted similar student difficulties produced by features of particular materials used to
represent numbers. In each case, it appears that properties of the materials intruded
on the student’s understanding of the mathematical relations the materials were
meant to model. As Clements and McMillen (1996) noted in relation to base-ten
blocks,
actual base-ten blocks can be so clumsy and the manipulations so disconnected one
from the other that students may see only the trees—manipulations of many
pieces—and miss the forest—place-value ideas. The computer blocks can be more
manageable and “clean.” (p. 272)
The third argument in favour of computer models of numbers is that software
can incorporate features not possible with physical models (NCTM, 1998, p. 112).
Clements and McMillen (1996) compared physical blocks with computer-generated
blocks, and noted several aspects of computer materials that either improved on
features, or added features not available with physical materials. They noted
advantages of computer materials including flexibility of presentations, the ability to
store and retrieve configurations, the provision of aural and visual feedback, and the
capability to record student actions (Clements & McMillen, 1996, pp. 272-273).
Appendix A includes a further discussion of features of computer materials, in
relation to the software designed for this study.
2.6
Computers and Mathematics Education
The purpose of this section of the thesis is to raise a number of issues from
the literature of particular relevance to the study of learning effects of computer
software. Four issues are addressed in this section: claimed educational benefits of
modern computers, features of software design that have the potential to enhance
mathematical learning, design considerations, and research into the use of computers
in mathematics education.
2.6.1 Claimed Benefits of Computers
Recent technological advances in hardware and software design are claimed
to have a number of claimed educational benefits. Three benefits of particular
relevance to this study are discussed in the following paragraphs: (a) improvements
55
in students’ learning, (b) the promotion of interaction among students, and (c)
representation of conceptual relations in a knowledge domain (section 2.6.2).
Learning benefits.
A number of benefits for children’s learning have been claimed for
educational use of computers. Some of these general benefits are summarised here;
benefits that specifically have to do with cognition are described in section 2.6.2.
Fletcher-Flinn and Gravatt (1995) listed several advantages for learners provided by
computers, including
the presentation of realistic problems requiring interactive hypothetical-deductive
reasoning, immediate feedback and self-evaluation, opportunities for collaborative
learning in small groups, and ease of teacher monitoring and control. . . . Welldesigned programs delivered by a computer can provide all of these benefits and, in
addition, seem to be enjoyed by learners as shown by their positive attitudes and
higher expectations about CAI [Computer-Aided Instruction]. (p. 232)
One general benefit claimed for computers is that computers may “provide
learning experiences not available by more ordinary means” (Champagne &
Rogalska-Saz, 1984, p. 44; see also NCTM, 1998, p. 96; NCTM, 2000, p. 25).
Similarly, Clements and McMillen (1996) suggested that in selecting software to
teach mathematics, teachers choose “computer manipulatives that . . . go beyond
what can be done with physical manipulatives” (p. 277). One issue addressed in this
study is the effects that features available only in computer software have on
students’ place-value learning.
Promotion of student interaction.
One benefit claimed for computers in classrooms is the promotion of student
interaction. The first type of interaction that has been claimed as the result of the
educational use of software is interaction between the user and the software (Akpinar
& Hartley, 1996; Helms & Helms, 1992; Kozma, 1994b; Stedman, 1995; Ullmer,
1994). Ullmer believed that the nature of interactive software required users to
change the way they learn:
Users of such [interactive] systems cannot ignore the technology and focus only on
the content; a new level of instrumentality that may affect learning has been imposed
on them. Consequently, the manner in which they perceive their relationship to the
medium is invariably changed. But in this highly responsive environment, they gain
increased control over their own learning activities and enjoy a more constructive
role in learning. The shift in the learner’s role makes interaction, rather than passive
assimilation, the key learning process. (p. 28)
56
Kozma (1994b) described learning with a computer as a “complementary
process” (p. 11), in which both the user and the software construct representations
and perform procedures. This embraces the idea of “distributed cognitions”
(Jonassen, Campbell, & Davidson, 1994), in which the computer and its user form a
partnership, with the computer “assuming a significant part of the intellectual burden
of information processing” (K. E. Sinclair, 1993, p. 21).
2.6.2 Cognitive Aspects of Computer Use
One of the most prominent arguments in the literature in favour of the
educational use of computers is that they directly aid learners’ thinking, in ways not
possible with other teaching methods. This section addresses issues of cognitive
effects of computers on learners.
Computers as cognitive tools.
A number of authors have written that computers should be thought of as
cognitive tools of one sort or another (Edwards, 1995; Jonassen, 1995; McArthur,
1987; Pea, 1985; Salomon, 1988; K. E. Sinclair, 1993). Pea noted that it had been
common for proponents of computers in learning to claim that computers amplified
human thinking. Pea, however, declared that though efficiency may be one result of
learning with computers, other benefits were the result of reorganisation of thinking,
fostered by the software. Clark (1994) similarly maintained that the principal use of
software should be to support cognitive processes.
As mentioned earlier, some computer software has been criticised for its
behaviourist foundations, and for continuing a transmission model of teaching and
learning (Jonassen, 1995; Pea, 1985; K. E. Sinclair, 1993). Several authors have
proposed that educational software be designed so that instead of being used to
transmit knowledge, it constitutes a cognitive tool that can extend and develop a
student’s cognitive abilities (Jonassen, 1995; McArthur, 1987; Salomon, 1988).
Salomon explained this point in this way:
It is often said the computer-based tools can extend not human muscle or sensory
functions, but cognitive, symbolic ones. To be a bit more specific, computer-based
tools extend cognition to accomplish functions the cognitive apparatus could never
accomplish on its own. (p. 129)
Similarly, in place of software that attempts to “control all learner
interactions” (p. 61), Jonassen (1995) suggested that computers be used as
57
intellectual partners or cognitive tools that “support, guide, and extend the thinking
processes of their users” (p. 62). One aspect of support for students’ thinking that
computers may provide is in the provision of an environment that encourages “an
active, experimental style of learning” (Cohen, Chechile, Smith, Tsai, & Burns,
1994, p. 237). McArthur (1987) stated that computers enable students to test “a wide
range of hypotheses . . . [which is] . . . an important way to exercise misconceptions
and learn” (p. 192). McArthur commented that in this respect computers are far
preferable to “the traditional paper-and-pencil medium [that] tacitly encourages the
students to think of such changes as mistakes to be avoided. On the contrary, the
ability to try out hypotheses rapidly, especially incorrect ones, is central to learning”
(p. 194). This feature is incorporated in the software used in this study; with little
effort users can quickly try different number representations to test their ideas.
Representation of conceptual domains.
Many writers have claimed that computers can benefit learning by
representing relations inherent in a conceptual domain (Babbitt & Usnick, 1993;
Becker & Dwyer, 1994; Bottino, Chiappini, & Ferrari, 1994; Cohen et al., 1994; De
Laurentiis, 1993; Edwards, 1995; Marchionini, 1988; Parkes, 1994). Most school
subject areas, including mathematics, involve the understanding of abstract concepts
and relations that exist among elements of the domain. A number of software
designers have used computer software to represent conceptual domains, and to
illustrate important conceptual relations, in ways that are claimed to improve
students’ understanding of the domains. This claimed benefit is widely reported in
the educational software literature, and is an important aspect of this study. These
software applications present what Parkes (1994) described as “a manipulable
problem space representation” (p. 199).
In representing ideas with computer software, designers often include
multiple representations of an idea (NCTM, 1998, p. 112). In so doing, it is hoped
that connections among elements of a domain can become evident to students
(Babbitt & Usnick, 1993; Becker & Dwyer, 1994). As De Laurentiis (1993) asserted,
excellent educational software will make explicit the associations in the body of
knowledge that is being taught. This simplifies the student’s task of integrating this
new body of knowledge into his or her own mesh of concepts. The software should
also make it possible for the student to explore the associations, therefore enhancing
his or her own mesh of concepts, and building an individualized representation of
the world. (p. 7)
58
Software embodying this principle have been developed to teach topics
including theorem-proving problem solving (Parkes, 1994), high school chemistry
(Kozma, 1994a), common fractions (Babbitt & Usnick, 1993), and place value (P.
W. Thompson, 1992). The software designed for use in this study includes pictorial,
symbolic, and verbal representations of numbers that are linked closely together, in
an attempt to help student users develop their own conceptual links in the way
described here by De Laurentiis (1993).
2.7
Chapter Summary; Statement of the Problem
The literature review presented in this chapter gives the background to the
problem investigated, stated below. Specifically, there are five findings from this
literature review that undergird the investigation described in this thesis: (a) Various
authors continue to encourage mathematics educators to develop meaningful
understanding and number sense in their students, (b) teaching of place-value
concepts is an important foundation for later mathematical study, (c) the teaching and
learning of place-value concepts is difficult and incompletely understood, (d)
computer technology appears to offer the promise of more effective teaching of
abstract domains, through its capabilities of presenting information in connected
ways, and (e) there is a need for up-to-date information about learning with computer
technology.
These findings lead to the statement of the problem for this study: How do base-ten
blocks, both physical and electronic, influence Year 3 students’ conceptual structures
for multidigit numbers? The investigation design used to address this problem is
described in chapter 3.
59
Chapter 3: Methods
3.1
Chapter Overview
This study investigated the development of understanding of place-value
concepts as students used either physical or electronic base-ten blocks, each of which
modelled numbers using a structured base-ten representation. In particular, this study
was planned to identify the Year 3 participants’ conceptual understandings of twodigit and three-digit numbers, before, during, and after a program of 10 teaching
sessions. The study had five phases, which were trialled in a small pilot study:
selection of students, first interview, software training session, teaching program, and
second interview.
As shown in chapter 2, problems in teaching and learning place-value
concepts are very common, though they have been the subjects of discussion and
research for several decades. The current study was designed to contribute to this
discussion, by evaluating an innovative method of teaching place value with
appropriate computer software and comparing this with a conventional method using
base-ten blocks. The study design centred on a detailed descriptive analysis of what
happened as students used computer software or base-ten blocks to answer placevalue questions. Analysis of data from videotapes, participants’ written work, and the
researcher’s written records enabled discussion of differences between the effects of
using the software and the effects of using conventional base-ten blocks.
3.2
Aims of the Study
The research question of the study was stated in section 1.3, and is repeated
here:
How do base-ten blocks, both physical and electronic, influence Year 3
students’ conceptual structures for multidigit numbers? Within the context of Year
61
3 students’ use of physical and electronic base-ten blocks, the following specific
issues were of concern:
1.
What conceptual structures for multidigit numbers do Year 3 students
display in response to place-value questions after instruction with baseten blocks?
2.
What misconceptions, errors, or limited conceptions of numbers do
Year 3 students display in response to place-value questions after
instruction with base-ten blocks?
3.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in instruction with physical and
electronic base-ten blocks?
4.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in students’ achievement in
numeration?
3.3
Variables
Two variables were controlled in pursuit of the above aims: mathematical
achievement level and mode of number representation. They were operationally
defined as follows:
3.3.1 Mathematical Achievement Level
This is defined as the level of each student’s mathematical achievement as
determined from results of the previous year’s The Year 2 Diagnostic Net
(Queensland Department of Education, 1996; hereafter referred to as the Year 2 Net).
The Year 2 Net is a diagnostic instrument administered by Year 2 class teachers in
every Queensland state school, and involves the diagnosis of mathematical abilities
in a range of areas including place value. Level of achievement was defined as high,
medium, or low, by dividing the available group of Year 3 students into thirds.
Students from the top third and bottom third were selected for involvement in the
study, to investigate any differences in place-value understanding, including the use
of representational materials, that related to the participants’ level of mathematical
achievement.
62
3.3.2 Number Representation Format
The number representation formats used by participants in teaching sessions
were: (a) physical base-ten blocks, materials in hundreds, tens, and ones sizes that
can be manipulated by hand; or (b) electronic base-ten blocks, computer software
that models numbers using written symbols, verbal names, and pictures of base-ten
blocks that can be manipulated on-screen.
3.4
Data collection and analysis.
A working definition for place-value understanding, given in chapter 2, has
been used to guide data collection and analysis: A student possessing place-value
understanding is able to use the place-value features of the base-ten numeration
system to form accurate, flexible conceptual structures for quantities represented by
written numerical symbols; the student is able to manipulate numerical quantities in
meaningful ways to answer mathematical questions.
The methodology adopted for this study was a combination of clinical
interviews and teaching experiments (Hunting, 1983). Responses of participants were
analysed as they related to the research sub-questions listed previously. Specifically,
evidence was collected as it related to participants’ conceptual structures for
numbers; misconceptions, errors or limited conceptions regarding numbers; and
effects on participants’ thinking about numbers that were apparently influenced by
features of each number representational format. The evidence principally came from
transcripts of videotapes of interviews and teaching sessions, supplemented by
researcher’s notes, participants’ written working, and software audit trails.
3.5
Design Issues
This section comprises descriptions of assumptions underlying the design and
the theoretical stance taken with regard to five dimensions of research design.
3.5.1 Assumptions
Three major assumptions underlie the research described here. These
assumptions relate to how numbers are understood, how mathematics is learned, and
how a person’s thinking may be studied.
63
Understanding of numbers.
The first assumption relates to the question of how people understand
numbers. Although numbers are generally considered to be abstract notions with no
physical objective existence, it is assumed that most people by their actions treat
numbers as entities, indicating that, for them, numbers do exist in some form (Sfard,
1991; Sfard & Thompson, 1994). A person’s conceptions of numbers are believed to
form a system that has a structural form and that incorporates rules by which the
conceptions may be manipulated (Ohlsson et al., 1992; Resnick, 1983). These
conceptions are also influenced by the person’s interactions with other people, and so
form part of the shared understanding of numbers held by the person’s social group
(Cobb & Yackel, 1996; Cobb et al., 1992).
Learning of mathematics.
The second assumption is that a person learning any topic has to construct
personal (and therefore to some extent unique) models of that topic. In learning
mathematics in particular, students are assumed to develop internal models, or
conceptual structures, for numbers (Fuson, 1990a, 1992). It is assumed that these
conceptual structures are influenced by interaction with physical, external
representations for numbers. Furthermore, there is assumed to be some relationship
between internal and external numerical representations, meaning that manipulation
of one has an effect on the other (Hiebert & Carpenter, 1992; Putnam et al., 1990).
Study of thinking.
The third assumption is that the nature of internal representations of numbers
may be deduced from a person’s responses to particular mathematical tasks (Resnick,
1983, 1987). This assumption is basic to cognitive research; it is assumed that a
person’s actions and speech are partially the product of internal mental structures
possessed by that person. Thus is it assumed that by studying a person’s actions, the
internal structures they hold for the domain under consideration may be deduced.
3.5.2 Theoretical and Methodological Stance
There are a number of theoretical and methodological considerations which
underlie this study. The study uses a design that may be described according to its
relation to five aspects of research methodology, illustrated in Figure 3.1 as continua
between pairs of opposing terms. The five aspects of the study’s design are the (a)
64
underlying paradigm, (b) aim with regards to theory, (c) methodology, (d) type of
data collected, and (e) researcher’s role. The following paragraphs relate to Figure
3.1 and describe the study with reference to each dimension.
Figure 3.1. Dimensions of research design.
Paradigm.
The first dimension on which this study may be described is the question of
underlying paradigm. Paradigms that underlie research may be placed on a
continuum that extends from positivism on one side to non-positivism on the other.
Other terms have been used in opposition to positivism by various authors, including
the constructivist paradigm (Guba & Lincoln, 1989), phenomenological inquiry
(Patton, 1990), and the qualitative paradigm (Creswell, 1994); in this discussion the
term non-positivism is used.
The question of underlying paradigm needs to be addressed because of its
bearing on the choice of data collection and analysis methods. Rossman and Wilson
(1985) described “three distinct perspectives about combining methods” that they
labelled “the purist, the situationalist, and the pragmatist” (p. 629). As Rossman and
Wilson explained, the question of paradigm is considered by purists to be of vital
importance. Purists (evidently including Guba & Lincoln, 1989, and Creswell, 1994)
believe that one’s paradigmatic view must necessarily determine the research
methods to be used, because quantitative and qualitative methods derive from
“different, mutually exclusive epistemologic and ontologic assumptions about the
65
nature of research and society” (Rossman & Wilson, 1985, p. 629). However, this
view was contradicted by Miles and Huberman (1984), who stated that the two
positions “constitute an epistemological continuum, not a dichotomy,” and that
“epistemological purity doesn’t get research done” (p. 21). As explained later,
choices of method used for the current study were based on such a pragmatic view of
research design (Patton, 1990; Rossman & Wilson, 1985). Therefore, questions of
underlying paradigm are not given further discussion here.
Aim.
The second dimension of description in Figure 3.1 indicates whether the aim
of the study is to test a theory or theories proposed in advance of the collection of
data, or to generate new theory as a result of data analysis. Some research, especially
when conducted from a positivist perspective, sets out to propose a theory or theories
based on a review of literature and then to test those theories so that they may be
confirmed or refuted. Conversely, a strictly qualitative study generally commences
without any pre-conceived ideas of the likely results of the planned investigation, the
researcher expecting theory to emerge as the study proceeds (Creswell, 1994). This
study uses an adaptation of this approach described by Creswell (1994), in which the
researcher “advances a tentative conceptual framework in a qualitative study early in
the discussion” (p. 97). Theoretical models of children’s conceptual structures for
numbers were identified in the review of literature (section 2.4.2). These conceptual
structures have been used as starting points in the data analysis phase of the study
and have been compared with conceptual structures emerging from the data. These
two sources of data, the literature review and the data collection phase of the study
itself, have been compared and analysed in relation to each other for the purposes of
cross-validation (Wiersma, 1995).
Methodology.
Research studies may be described according to their overall methodology,
and located on a continuum from naturalistic inquiry on one hand, to experimental
research on the other. As already mentioned, some researchers believe that research
methods should be chosen to match the paradigm view that the researcher holds. For
example, Guba and Lincoln’s (1989) work implies two strongly-held assumptions:
(a) Positivism is an inadequate theory of the world and how things happen and
(b) use of quantitative, experimental, research methods is antithetical to the non66
positivist paradigm. Therefore, they argued that qualitative research is the only viable
option for a researcher studying social phenomena. However, this view has been
disputed by others (e.g., Patton, 1990; Yin, 1994). Yin argued against distinguishing
between qualitative and quantitative research on the basis of opposing philosophical
beliefs and contended that “there is a strong and essential common ground between
the two” (p. 15). Similarly, Patton stated that he “preferred pragmatism to one-sided
paradigm allegiance” (p. 38) and maintained that a methodology should be chosen
that is appropriate (a) for meeting the study’s purpose, (b) for answering the
questions being asked, and (c) for the resources available. This study is based on
such pragmatic considerations, although it utilises primarily a naturalistic inquiry
approach.
Type of data.
The fourth descriptive dimension is that of data type, shown in Figure 3.1 as a
continuum from qualitative to quantitative. There is widespread support in the
research design literature for an approach that incorporates both quantitative and
qualitative methods (e.g., S. A. Mason, 1993; Patton, 1990; Rossman & Wilson,
1985; Strauss & Corbin, 1990). Wiersma (1995) described a continuum between
quantitative and qualitative research and stated that “from a practical standpoint of
conducting research, quantitative and qualitative procedures are often mixed” (p. 14).
Likewise Best and Kahn (1993), noting that quantitative research had traditionally
dominated educational research, stated that “some investigations could be
strengthened by supplementing one approach [quantitative or qualitative] with the
other” (p. 212).
The use of mixed methods in a single study was given more detailed support
by Rossman and Wilson (1985, 1991), and Greene, Caracelli and Graham (1989). In
the first of these papers, Rossman and Wilson (1985) described three different
purposes for mixed-methods research. This list was added to by Greene et al. (1989),
and then expanded by Rossman and Wilson (1991) into a typology of four purposes
at the stages of research design or data analysis. The four purposes listed by these
authors are (a) corroboration, (b) elaboration, (c) development, and (d) initiation.
Briefly, corroboration refers to “classical triangulation where different methods are
employed to test the consistency of findings from one method to another” (Rossman
& Wilson, 1991, p. 2). Elaboration, also called “complementarity” by Greene et al.,
67
is used to “illuminate different facets of the phenomenon of interest” (Rossman &
Wilson, 1991, p. 2). Development uses the results gained from one method to inform
subsequent investigation by the other method. Initiation is used only at the analysis
stage of a study to uncover “the unexpected, the paradoxical, or the contradictory”
(Rossman & Wilson, 1991, p. 4); in other words, initiation may be used to lead to
further questions for investigation. Rossman and Wilson (1991) pointed out that the
above four purposes for using mixed methods either may be planned in advance, or
may be decided upon after initial analysis as a study’s findings begin to emerge.
Although the study is predominantly qualitative, it also collects data in a
quantitative form; however, this quantitative data is used descriptively not
inferentially.
Role of researcher.
The researcher’s role is the fifth dimension on which this study is described.
Gold (1969) proposed a continuum of researcher roles, from complete participant, in
which the researcher becomes one of the participants under investigation, to
complete observer, in which the researcher is completely separate from the
participants. Between these two extremes, Gold identified roles of participant-asobserver and observer-as-participant. In this study, the author was a participant-asobserver; by taking the role of teacher for each group of students, he was an integral
part of the interactions that took place in each group. The researcher also observed of
what took place, mostly after the event via videotapes of the sessions.
3.6
Pilot Study
A small-scale pilot study was conducted prior to the main study. The
following sections describe the pilot study’s purposes, the procedures followed, and
the results.
3.6.1 Purposes of the Pilot Study
The pilot study was used to test the feasibility of four aspects of the study
design: software design, teaching program, procedures, and data collection and
analysis. As a result of the pilot study, the design of the main study was modified in a
number of aspects, as explained in section 3.6.5.
The software. The software design (Appendix A) was tested to determine (a)
if the interface was clear to the users, (b) if the program contained any programming
68
bugs that needed correcting, and (c) if any improvements were necessary to make the
software more effective in teaching place-value ideas.
Teaching program. The teaching program (Appendix B) was examined to
check that (a) 10 teaching sessions were sufficient to show some development of
place-value understanding, (b) the instructions to participants were clearly
understood by them, (c) teaching procedures were effective, and (d) there were
sufficient activities for the time available.
Procedures. Procedures including the interviews and teaching sessions
investigated whether (a) duration of teaching sessions and interviews was sufficient
to show development of place-value understanding, but not too long for the
participants’ attention spans; (b) placement of participants, researcher, camera and
microphone was suitable for clear video recording; and (c) arrangements for taking
students to and from their classrooms were suitable.
Data collection and analysis. These methods were examined to check
whether (a) sources of data were sufficient for developing triangulated descriptions
of participants’ actions and speech, (b) interview questions were appropriate to
identify place-value understanding, and (c) analysis methods facilitated the
identification of conceptual understanding of participants. In the end, the longer time
spent on coding and analysing transcript data from the main study led to changes to
data analysis that were not foreseen after the pilot study; see chapters 4 and 5 for
description of results and how they were analysed.
3.6.2 Selection of Pilot Study Participants
The pilot study was conducted at a school similar to that planned for the main
study. Participants in the pilot study were drawn from Year 3 classes at a small
primary school in a rural area north of Brisbane, Australia. Both the pilot and the
main studies were conducted using students at the Year 3 level, as that is the age at
which three-digit numeration is generally introduced in Queensland schools. At the
time of the pilot study (1997), the school had two Year 3 classes with approximately
50 students in total. Participant selection was made based on the previous year’s Year
2 Net (Queensland Department of Education, 1996). Results from this test were used
to divide the population of Year 3 students into three approximately equal groups,
defined as being of high, medium, and low mathematical achievement respectively.
In order to manage the time needed for data collection and analysis, only two pairs of
69
participants were selected for the pilot study. A pair of students was selected at
random from each of the high and low achievement groups. Each pair was of one
gender, on the author’s assumption, based on classroom teaching experience, that
children at this age would commonly prefer to work with a peer of the same gender.
Random selection was used to assign the pair of girls to use the physical base-ten
blocks and the pair of boys to use the computer software (the electronic blocks).
3.6.3 Pilot Study Procedures
Place value of three-digit numbers is generally taught in the second term of
Year 3 in Queensland schools. The pilot study was timed to occur towards the end of
the first term of the school year, leaving time for the main study in the second term.
The researcher interviewed selected students individually in a quiet room, before and
after a teaching program of 10 sessions, described in the following paragraph.
The teaching program for the pilot study comprised 10 sessions for which a
teacher’s script was written in advance. An overview of the pilot study’s teaching
program (Appendix B & Appendix C) and the script used for the first session
(Appendix D) are appended to the thesis. The researcher led the participants through
a series of tasks, progressing from revision of two-digit numeration through to threedigit numeration and two-digit addition and subtraction. If participants were unable
to complete all the tasks planned for a session, as was generally the case with the
low-achievement girls, then tasks were held over for the following session.
3.6.4 Pilot Study Data Collection and Analysis
There were three main sources of data in the pilot study: interviews; teaching
sessions; and software-generated records of user actions with the computer software,
known as an audit trail (Misanchuk & Schwier, 1992; Schwier & Misanchuk, 1990;
Williams & Dodge, 1993). Videotapes of the interviews and teaching sessions were
transcribed, including actions and dialogue by the researcher and the participants.
The transcripts were studied to identify any aspects of the main study which should
be modified in the main study. Preliminary data analysis was also carried out to test
analysis procedures planned for the main study.
3.6.5 Changes Made to Study Design After Pilot Study
As indicated earlier, the purpose of the pilot study was to investigate whether
any changes were indicated for the study design, in the areas of software design,
70
teaching program, procedures, and data collection and analysis. Changes made for
the main study are summarised below in four sections, describing changes in
software design, procedures of participant selection, administration, the teaching
program, and data collection and analysis.
Changes in software design.
Minor changes were made to the software design (see Appendix A) as a result
of the pilot study results. One change was needed due to a bug in the program that
caused a difficulty when children clicked rapidly on buttons to add new blocks onscreen. The Windows operating system recognises a pair of rapid mouse clicks as a
“double click” rather than two single clicks; in response to a double click the
software added only a single block. Thus, for example, if a child rapidly clicked six
times only three blocks were added to those on screen. The software was modified to
produce two blocks if a double click was made. A second modification was made to
the graphic images applied to two of the buttons on-screen. In the pilot study, the
graphics for the buttons by which regrouping actions were accessed were not clear to
the students. The graphics represented symbolically the idea of changing a larger
block for 10 smaller blocks (partitioning), and 10 small blocks for a larger block
(grouping), respectively (Figure 3.2). It was obvious that students did not recognise
these graphics as representing the actions as intended. The metaphors underlying onscreen tools used to achieve these actions are a saw and a net; the button graphics
were therefore changed to pictures to match these tools (Figure 3.3), making the links
between the buttons and the tools clearer.
71
Original Partitioning button
graphic
Original Grouping button
graphic
Figure 3.2. Original graphic images used on regrouping buttons in software used
during pilot study.
Replacement Partitioning
button graphic
Replacement Grouping button
graphic
Figure 3.3. Replacement graphic images used on regrouping buttons in software used
during main study.
Changes to selection procedures.
A difficulty was encountered early in the pilot study, regarding the ability of
one of the low-achievement students to understand the tasks. Of the two girls,
selected at random from the population of low-achievement Year 3 girls at the
school, Jenny (a pseudonym) was much more able than the other, Nina. Early in the
program it was found that whereas Jenny was ready to progress to more difficult
questions, Nina did not understand two-digit numeration concepts needed to make
progress in the teaching program. Consequently, on the one hand Jenny became
frustrated and started to lose interest in the activities, and on the other it was evident
that Nina needed considerable help to understand each question. The decision was
made to continue the teaching program with the girls separately, to continue to trial
the full 10 teaching sessions, and to decide at the end of the pilot study if individual
instruction might be more effective with low-achievement students. The same
difficulties were not experienced with the pair of boys, who for most of the program
worked amicably and cooperatively. There were occasions where one or other of the
boys made mistakes in answering a question, but the other student was able to state
the correct answer without causing any difficulties.
The difficulties described in the previous paragraph underlined the need to
select students for the main study who were able to cooperate in the learning
situation, especially since this study was exploratory, and its aim was not to
72
generalise to all students of Year 3 age. Because of this finding the design of the
main study was modified to exclude students who might find participation difficult,
because of either specific learning difficulties or behavioural problems such as
Attention Deficit Disorder (ADD). This is explained in more detail in section 3.7.1.
Changes to administration procedures.
The pilot study was conducted with pairs of students working with either
base-ten blocks or the computer software; this was changed to groups of 4 students
for the main study. The initial use of small groups was based partly on the need for
each student in the computer group to have ready access to a computer, and partly on
constraints regarding videotaping facilities. To have more than two students using a
single computer would make it difficult for each student to have sufficient access to
the software. However, there was an observed lack of collaborative learning, which
was believed to be due to the small number of students in the pilot study’s teaching
sessions. Students tended to follow the researcher’s directions and answer his
questions, but not to consult with each other. Of course, collaborative learning was
not possible with the girls once they were separated. It was therefore decided to alter
the general design of the administration procedures, to involve groups of 4 students
at a time. To achieve this, it was necessary to use two video cameras for every
session and two computers for the computer groups. The use of two cameras was
needed to capture adequately interactions that occurred among 4 students and the
researcher. The two computers were needed for the computer groups, to give each
student sufficient access to a machine.
Changes to teaching program.
Changes were also made to the teaching program, to take advantage of the
larger groups and to encourage collaborative learning. The tasks planned for the main
study were very similar to those used in the pilot study; however, in place of teacher
directions explaining each step required, tasks were written that required each group
of four participants to discuss and complete the tasks with little direction from the
researcher. In this way, students were expected to exhibit more cooperative learning
and interaction within each group of four than took place with pairs or single
students in the pilot study. Because of these changes to the organisation and content
of teaching sessions, no attempt was made to analyse results of the pilot study in
depth.
73
Changes to data collection and analysis procedures.
Analysis of the pilot study video tapes showed that the position of the video
camera during teaching sessions was of particular importance, and so this was
carefully planned for the main study. For the blocks groups, at times the
manipulations made by participants were not visible to the camera because of
obstructions, including piles of unused blocks. It was thus decided in the main study
to ensure at all times a clear line of sight for the camera. For the computer groups,
there was a different problem. Because the students sat facing the screen, it was not
possible to video both the students’ faces and the screen simultaneously with one
camera. A compromise between videoing the screen and videoing the student was
achieved in the pilot study by placing the camera slightly in front of the computer,
giving a side-on view of the screen and the student that was usually adequate. In the
main study with groups of four, two cameras were used on opposite sides of the
group, to give the best view possible of participants and blocks or computer screens.
This method is unsatisfactory for recording every interaction between participants
and the computer: The author strongly recommends the use of “split screen” methods
of video recording, which record a view of the computer screen and a view of
participants simultaneously, if the requisite technology is available.
The audit trails generated by the software during the pilot study were found to
be of limited usefulness, and so this feature was extended for the main study (see
Appendix E for an example of an audit trail generated during the main study). The
text files generated by the software recorded each time a button was clicked,
including the time on the computer system clock. However, it was found nearly
impossible to match these recorded actions with actions viewed on the videotapes.
The audit trails were modified in two ways for use in the main study. First, the time
recorded for each action included seconds as well as hours and minutes, to provide a
more accurate measure of when each action was taken. Second, more detail of each
action was recorded, to enable more accurate knowledge of what the student(s) did:
Each line of the audit trail identified the button clicked, the time, the blocks present
on the screen, and the number represented by the blocks overall. Audit trail data were
used only where video data were unclear and further information was needed to
determine what a student did with the computer.
Data collection sources were supplemented for the main study. Researcher
field notes and student workbooks were used, in addition to the videotapes and
74
software audit trails. Field notes were not used in the pilot study, but were considered
desirable in the main study to add another source of data to support categories of
responses found through analysis of video transcripts. In the pilot study students’
written work either was collected on loose sheets of paper, or was written in the
students’ regular mathematics exercise books. It was considered desirable to collect
each student’s written work for verification of observations made from the
videotapes. Therefore, each student in the main study was given a workbook in
which all written work was dated and collected for later analysis as required.
Data analysis procedures for the pilot study were limited to transcription of
videotapes and initial coding of students’ responses. Video transcripts of the main
study were subject to analysis that was considerably more detailed, as described in
section 3.7.5.
3.7
Main Study
The main study comprised five phases, summarised in Table 3.1. Four groups
of four Year 3 students were taught by the researcher in a teaching program of 10
lessons. Interviews before and after the teaching sessions were used to identify
differences in participants’ conceptual models of numbers before and after the
teaching phase. Each of phases I, II, and V was identical for both computer and
blocks groups. In phase III the two computer groups had an extra training session
prior to the teaching program, to familiarise them with the software. A parallel
session was not considered to be necessary for blocks groups, as the children were
familiar with the use of base-ten blocks from their class lessons. Phase IV was the
teaching phase, involving different treatments for the two cohorts. The study took
place over a 3-week period immediately prior to the mid-year break, after the
participants had been in Year 3 for almost half a school year.
TABLE 3.1.
Phases of the Research Design
Phase:
Blocks
Groups
(physical)
I
Selection &
assignment
of students
II
First
interview
III
Computer
Groups
(electronic)
Selection &
assignment
of students
First
interview
Software
training
session
75
IV
Teaching
program
using
blocks
Teaching
program
using
computer
V
Second
interview
Second
interview
3.7.1 Selection of Participants
The main study was conducted at a school that, like the school for the pilot
study, was a small state primary school in a semi-rural area north of Brisbane,
Australia. At the time of the study, the chosen school had two classes of
approximately 22 Year 3 students each, and a composite Year 3/4 class, with
approximately 6 Year 3 and 22 Year 4 students. When the teachers of these classes
were approached, one of the two Year 3 classes had already commenced teaching
hundreds place concepts, and so participants were selected from the other Year 3
class and the Year 3/4 class.
The previous year’s Year 2 Net (Queensland Department of Education, 1996)
results were used to rank the population of Year 3 students at the chosen school (see
Appendix F). The class teachers were asked to exclude from the population of Year 3
students any students who had been diagnosed as having either a specific learning
disability or a behavioural disorder, such as ADD, in order to exclude students who
might have difficulty completing the tasks or who might find cooperation in
groupwork difficult. Following this process, the top 4 boys and 4 girls, and the
bottom 4 boys and 4 girls were selected from the ranked list to participate in the
study. The top 8 participants are referred to hereafter as “high achievement level”
participants, and the bottom 8 participants as “low achievement level” participants.
Participants were assigned to 4 groups, as indicated in Table 3.2, each composed of 2
girls and 2 boys.
TABLE 3.2.
Participant Groups for the Main Study
Computer groups
Blocks groups
High Mathematical
Achievement
4 students (2 male, 2 female)
4 students (2 male, 2 female)
Low Mathematical
Achievement
4 students (2 male, 2 female)
4 students (2 male, 2 female)
As in the pilot study, each group was of a single mathematical achievement
level. This approach was supported by Fox (1988), who stated that “learning in small
groups is most effective when gaps in understanding between individuals are neither
too ‘great’ nor too ‘small’” (p. 36). In this thesis the groups are referred to as the
high/computer group, low/computer group, high/blocks group, and low/blocks group.
Appendix G contains a full list of participants, including their dates of birth and the
groups to which they were assigned.
76
3.7.2 Teaching Program
As in the pilot study, the main study included a teaching program of 10
sessions (Appendix H). The following two sections describe the teaching approach
adopted and the lesson content.
Teaching approach.
The teaching program was based on a view of teaching of mathematics
described by G. A. Jones et al. (1994) as having a “constructivist orientation with a
strong emphasis on social interaction” (p. 119). It also agrees with Cobb et al.’s
(1992) view of learning as “an active, constructive process in which students attempt
to resolve problems that arise as they participate in the mathematical practices of the
classroom” (p. 10). This view was operationalised to include cooperative group
work; a sequence of learning activities that build on previously-understood concepts;
and the provision of freedom for students, within reasonable bounds, to choose for
themselves how to answer the questions asked. The groups of participants were
encouraged to cooperate with each other and to negotiate answers to the questions so
that, if possible, each group reached a consensus about the answer to each one. In
each question the students were asked to represent the quantities involved in each
question in symbolic form, with materials (blocks or computer software), or both.
The students were free to use different representations of the numbers to support
their answers, in keeping with the constructivist model of teaching employed.
The researcher took the role of teacher for all teaching sessions and used
appropriate teaching strategies to support and encourage learning by the participants
(see Confrey & Lachance, 2000, for a discussion of having the researcher do the
teaching in a teaching experiment). He encouraged students to discuss and negotiate
meanings of each question, the quantities involved, and possible solutions. The
researcher made suggestions to the participants, such as using the available
representational materials (physical or electronic blocks) to represent the numbers
involved, if the students did not seem to be making progress in answering a question.
He neither confirmed nor denied the validity of any solution proposed by participants
until the group members had discussed it and expressed their individual views of the
problem and possible solution. This was done for the same reasons cited by Fuson,
Fraivillig, and Burghardt (1992), to simulate a situation that is believed to be
common practice in classrooms, in which a teacher does not supervise each group of
77
students continuously but monitors them infrequently as time permits. As Fuson et
al. explained,
an experimenter-intervention strategy was adopted that attempted to let children
follow wrong paths until it did not seem likely that any child would bring the group
back onto a productive path; the experimenter then intervened with hints to help the
group but giving as little direction as necessary. This was done to provide maximal
opportunities for the children to resolve conflicts and solve problems creatively. . . .
This criterion was intended to reflect the reality of a classroom where a teacher
monitoring six or more groups might not get to a given group for a whole class
session but would be able to give support by the end of that time. (p. 47)
The problems were written in a sequence of increasing difficulty. New
problems were presented once the previous question had been answered to the
group’s satisfaction; the researcher inserted supplementary questions similar to any
that caused a group to have difficulties if he felt it was necessary. As is to be
expected, the 2 low-achievement groups did not complete as many questions as the
high-achievement groups by the end of the study.
Lesson content.
The teaching program was written to take into account features of both
physical and electronic base-ten blocks. Where specific mention was made of
features available only in the software, equivalent activities were included for the
blocks groups, using activities that would typically be used in a classroom. For
example, when the number name window feature of the software was used, the
teacher provided written symbols to the blocks participants, either by writing them
on paper or by showing them printed on cards.
There were 45 tasks (Appendix H) in the teaching program. Many of the tasks
were written as non-routine problems, to challenge and motivate the students, and
thereby to promote maximal learning (Sowder & Schappelle, 1994). The tasks all
required understanding of place-value concepts to complete them, and are examples
of five types of task found in the place-value literature. These task types are (a)
number representation, (b) regrouping, (c) comparison and ordering numbers, (d)
counting on and back, and (e) addition and subtraction. These types of task are
described in the following paragraphs, including reference to other researchers who
have used similar tasks. Instructions for tasks are given in full in Appendix H; task
numbers referred to in this section, and elsewhere, refer to the numbers used in the
appendix.
78
For each task type there were two sets of tasks, the first involving two-digit
numbers and the second involving three-digit numbers. The tasks were sequenced
according to their relative difficulty and the need for comparatively basic skills to be
practised before the advanced tasks were attempted. Specifically, number
representation tasks were the first in the program, as these tasks involve basic skills
of demonstrating knowledge of written symbols, concrete representations, and verbal
names for numbers. These tasks were followed by regrouping tasks that also use
block representations and extend the skills needed for the number representation
tasks. Following this were comparison and ordering tasks, in which students
compared and ordered numbers presented as written symbols. The final task type for
each set of two-digit and three-digit number tasks was addition and subtraction.
These computation tasks relied on a number of skills in combination, including
knowledge of symbols, regrouping, and number facts, and thus were the last ones for
each set.
Numbers used for a single task were also sequenced according to the reported
difficulty that children have with different numbers. In particular, teen numbers were
used after other two-digit numbers, in view of the previously-mentioned difficulties
that teen number names introduce. Numbers that include zero digits were introduced
after other two-digit numbers, since zeros also cause well-documented difficulties for
children learning place value.
Description of task types, with examples.
In this sub-section each type of task is described, an example of each type is
given, and decisions made about the sequence of questions in each task type are
described. The full list of tasks is provided in Appendix H.
Number representation (Tasks 1-3, 28-30; see example in Figure 3.4). As
discussed in chapter 2, the ability to make connections among various
representations of numbers is generally considered to be fundamental to place-value
understanding (Fuson, 1992; Janvier, 1987). The tasks followed the “symbol-verbalconcrete” model (Payne & Rathmell, 1975), which has been adopted by many
curriculum writers up to the present. In each task the student was given a number
representation (in written, concrete, or verbal form) and was asked to represent the
same number in one of the other two forms. Tasks of this general type have been
used by several researchers, including Boulton-Lewis (1993), Hughes (1995), Miura
79
and Okamoto (1989), and S. H. Ross (1990). In this study these tasks were written in
sets of three for both two-digit and three-digit numbers; starting with concrete
representations, then verbal, and then symbolic; in each case converting the
representation to the other two forms. For example, in Task 1 (a) students were asked
to look at a block representation for the number 25 and to respond with the verbal
name and the written symbol for 25.
Task 1 - Representing numbers
Look at the blocks put out by Mr Price. Say what number the blocks show. Write the number
in your workbook. [Numbers were not printed on task cards provided to participants.]
25
61
13
40
Figure 3.4. Sample Representing numbers task.
As mentioned earlier, the numbers included in these tasks were sequenced
according the difficulty they provide for children, based on reports in the place-value
literature. For example, in Tasks 1 to 3 four examples were provided, beginning with
a number between 20 and 99 with more than 1 one. Following this was a number
with a number of tens and 1 one, as some children reportedly confuse such numbers
with teen numbers. Thirdly there was a teen number, and finally a number with zero
ones, regarded as the most difficult types of two-digit numbers. Similar sequences
were used in Tasks 28-30, with three-digit numbers.
Regrouping (Tasks 4-7, 31-34; Figure 3.5, Figure 3.6). An important
component of understanding the values represented by symbols is being able to
group or partition quantities represented into different arrangements (G. A. Jones &
Thornton, 1993a; G. A. Jones et al., 1994). For example, to show a sound
understanding of the symbol 35, a student should be able to represent 35 as 3 tens
and 5 ones, as 2 tens and 15 ones, or as 35 ones. This process of regrouping numbers
in different ways is essential for proficiency in written and mental computation,
though many students do not demonstrate this skill (Miura & Okamoto, 1989).
Regrouping tasks in this study required participants to regroup numbers in various
ways, including regrouping a single ten for 10 ones, all available tens for ones, or a
single hundred for 10 tens. Tasks 7 and 34 involved the use of a numeral expander to
investigate regrouping based on the written symbols.
80
Task 4 - Regrouping
Show the number with the blocks. Now swap one of the tens for ones. How many ones do
you need? Record what you have done in your workbook.
77
23
91
58
Figure 3.5. Sample Regrouping task.
Task 7 - Use of numeral expander (Computer groups)
Show the number with the blocks. Turn on the numeral expander. Use the expander to
show the number in different ways. Write the number in two ways in your workbook.
34
96
52
Figure 3.6. Sample Use of numeral expander task.
Numbers included in these tasks were chosen to provide a variety of
examples, including a number with only 1 one. Because regrouping tasks are only
introduced in the Queensland mathematics syllabus in Year 3, more difficult
examples of regrouping with teen numbers and numbers with zero ones were not
included in the two-digit examples of this type of task.
Comparison and ordering numbers (Tasks 8-12, 35-39; Figure 3.7). These
tasks developed the ability of students to use their understanding of quantities
represented by symbols to compare pairs of numbers, or to order three or more
numbers. Tasks of this type have previously been used by G. A. Jones and Thornton
(1993a), and A. Sinclair and Scheuer (1993). To compare or order numbers students
need to have a good understanding of values represented by symbols, in particular
the value represented by each digit. For example, in order to correctly compare 51
and 39, a student must know that the tens only have to be compared, and that 5
represents 5 tens, which is greater than either 3 tens or 9 ones.
Task 8 - Comparing 2 numbers
Tommy and Billy were arguing about who had more marbles. Tommy had 48 marbles, and
Billy had 62 marbles.
Who had more marbles? Show the numbers with the blocks. Explain your answer
in your workbook.
Figure 3.7. Sample Comparison task.
Numbers chosen for comparison and ordering tasks included pairs of numbers
in which one number in each pair had more tens, and the other number had more
81
ones; the number of ones in the latter number was also the largest digit of the digits
involved (e.g., 48 & 62, 51 & 39). Ordering tasks included sets of three numbers in
which two numbers had the same number of tens (or hundreds in three-digit
examples), and in which digits were repeated in different positions in the various
numbers (e.g., 82, 37, & 88; 75, 57, & 54).
Counting on and back (Tasks 13-17, 40-43; Figure 3.8). Another type of task
that requires good understanding of place value is counting forwards or backwards,
also used by G. A. Jones and Thornton (1993a) and Boulton-Lewis (1993). Counting
on or back by ones involves the standard counting sequence that children learn early
in school (Resnick, 1983). This set of tasks also included counting by tens or
hundreds, either forwards or backwards, which requires knowledge of the values
represented by the tens and hundreds digits.
Task 14 - Counting back by 1s
The Sunny Surfboard Company has 75 boogie boards left.
If one is sold, how many are left?
Then how many if another is sold?
Say all the numbers in order from 75 back to 60. Show the numbers with the blocks. Write
them in your workbook.
Figure 3.8. Sample Counting task.
Numbers chosen for these tasks included numbers that allowed the sequence
to proceed for several numbers before either a teen number or a change of decade or
number of hundreds was required. For example, Task 13 involved counting back by
ones from 28, not requiring a change of decade until the tenth number in the
sequence. This was done to allow the participants to recognise the regular pattern in
which only one digit changes before having to deal with two digits changing at once.
Addition and subtraction (Tasks 18-21, 44-45; Figure 3.9). These tasks
involved application of prerequisite skills used in other question types, such as
regrouping and knowledge of digit values. Several researchers have recommended
that students be given tasks that require them to invent strategies to solve problems
of this type (Kamii et al., 1993; S. H. Ross, 1989). For this study these tasks were
presented as word problems, with no particular algorithm mentioned. Each task
involved a single operation, which is appropriate for students at this Year level.
82
Task 19 - Addition
A Space Race video game costs 75 dollars, and a set of batteries costs 19 dollars.
How much will the game and the batteries cost? Show the numbers with the blocks.
Discuss how to work it out with your group. Show how you work it out in your workbook.
Figure 3.9. Sample Addition task, including regrouping.
Numbers chosen for addition and subtraction questions included examples in
which there is no regrouping, followed by harder examples that involved regrouping
in one place (e.g., 28 + 31, 75 + 19, 95 – 23, 83 – 48). The researcher ensured that
students were able to complete addition and subtraction without regrouping before
the more difficult tasks were introduced.
3.7.3 Instruments - First and Second Interviews
Interviews were conducted before and after the teaching program. They were
in the form of “standardised open-ended” interviews, in which “all interviewees are
asked the same basic questions in the same order” (Fraenkel & Wallen, 1993,
p. 387). The following comments about particular items apply to both interviews; the
same questions were asked in each interview, with only the quantities involved
differing. The questions for the first and second interviews are listed in Appendix I
and Appendix J, respectively. The question numbers mentioned in this section apply
to both interviews.
Design criteria. Criteria adopted in designing the interviews were as follows:
1.
Tasks were used by researchers in at least two other published studies;
2.
Each task was to target one or more key components of place-value
understanding, based on the literature review; and
3.
The whole interview was to take no longer than 20 minutes to
administer, considered a suitable length for the age of the students.
Categories of task. There are five task categories included in the interviews:
(a) number representation, (b) counting, (c) number relationships, (d) digit
correspondence, and (e) novel tens grouping. Other researchers have used these types
of task to probe students’ conceptual models of multidigit numbers, as described in
the following paragraphs.
Number representation tasks (Questions 1-3) were previously used by Miura
and Okamoto (1989), and Miura et al. (1993). In each task the participant was asked
83
to represent the number shown by a written symbol using place-value material
(generally base-ten blocks).
Counting tasks (Question 4) were a component in G. A. Jones et al.’s (1994)
“framework for nurturing and assessing multidigit number sense” (p. 121): The
framework included a series of tasks of increasing difficulty, starting with counting
on by ones and progressing through counting on and back by tens to mental addition
and subtraction.
In number relationships tasks (Questions 5-6) participants were asked for a
number a little larger, a little smaller, much larger, and much smaller than a given
two-digit number. This task is an extension of an item used previously (G. A. Jones,
Thornton, & Van Zoest, 1992; G. A. Jones et al., 1994) that required students to
write a number a little more and a lot more than 42. To be successful in such an item,
a student needs to have good number sense; in particular, a clear idea of the relative
magnitude of numbers is required (Sowder & Schappelle, 1994).
Digit correspondence tasks (Question 7) were previously used by S. H. Ross
(1989, 1990) and Miura and colleagues (Miura & Okamoto, 1989; Miura et al.,
1993). Participants were asked to count a number of items between 10 and 40, and to
write the symbol that showed that number. The researcher asked the participant to
explain which of the counted items were represented by each digit in turn. A
variation of the digit correspondence task (Question 8), also used by S. H. Ross
(1989, 1990) and Miura et al. (1993), involves providing misleading perceptual cues
to the child that suggest a face value interpretation of the written symbol. For
example, if 13 objects are shared among three containers with one remaining, some
children will say that the digit 3 represents the three containers and the 1 the
remaining object (see Figure 3.10). This item was included in this study (as in studies
by Ross and Miura et al.) to test the robustness of the child’s understanding of digit
value in the face of misleading evidence.
84
Figure 3.10. Diagram showing objects used in interviews for Digit Correspondence
Task with misleading perceptual cues.
Novel tens grouping tasks (Question 9) were used by Bednarz and Janvier
(1982, 1988), who presented students with problem tasks involving items not
commonly used in place-value lessons, such as peppermints or paper flowers,
grouped in tens and hundreds. Bednarz and Janvier (1988) based the attention they
paid to groupings on their observation that “few children give a true interpretation of
the digit position in terms of groupings” (p. 300). To complete the tasks students had
to identify the groupings involved, deduce the relation between the groupings, and
then operate on the groupings to answer the given questions.
3.7.4 Administration Procedures
As described earlier, the study comprised five phases: selection of
participants, first interview, software training session, teaching program, and second
interview (Table 3.1). Administration procedures for each of these phases are
described in the following paragraphs.
Selection of participants. Sixteen students were selected for participation in
the study, as described in section 3.7.1. The researcher sought the consent of parents
or guardians of selected students for them to take part in the study (see Appendix K).
In all cases the parents or guardians of first 16 students selected for participation
gave their consent, and so selection of alternative participants was not needed.
85
Interviews. Participants were interviewed both before and after the teaching
sessions, as described in section 3.7.3. Details of data collection procedures are
described in section 3.7.5.
Software training session. Participants using physical base-ten blocks were
already familiar with them from regular classroom lessons, but participants using the
software were not familiar with use of a computer for mathematics lessons, and the
software was totally new to them. To make the treatments more similar in this
respect, the 2 computer groups were given an extra session, prior to the first teaching
session, to familiarise them with the software. During these introductory sessions the
students were given the opportunity to experiment with the software and discover its
features. The researcher demonstrated any features that they did not discover for
themselves, or that they did not seem to understand.
Teaching phase. Participants were involved in a teaching program as
described in Appendix H. In the Queensland mathematics curriculum, which was
followed by the school chosen for the study, two-digit numeration is taught in Years
1 and 2, and three-digit numeration is introduced in Year 3. The Year 3 teachers at
the school generally introduced three-digit numeration in the second of four terms in
the school year. The teaching phase of the study was conducted at the end of term 2,
to match the usual timing of the topic. The Year 3 teachers of the study participants
were asked not to teach the topic to their classes until the study had concluded, in
order not to contaminate any learning effects produced during the teaching program;
both teachers involved complied with this request.
Four groups of 4 participants separately took part in the teaching program in a
room separate from the classroom, with the researcher taking the role of teacher.
Two groups used conventional physical base-ten blocks (blocks groups) and two
groups used electronic base-ten blocks (computer groups). Materials used by both
groups included task cards, workbooks, and pencils. Blocks groups used physical
base-ten blocks, and the computer groups had access to two computers with the
software installed (i.e., one computer between each pair of participants). Each
session was recorded using an audio cassette recorder and two video cameras, each
with an external microphone placed near the participants.
The sessions were conducted as follows. The first session commenced with
several activities designed to familiarise the participants with cooperative working, in
case they were not used to that mode of learning mathematics. This approach was
86
recommended by Fox (1988), who commented that “groups must be shown how to
work cooperatively to get best results” (p. 37). When it seemed clear that the students
were comfortable with each other and with the researcher, the first task was
commenced. Each new task was introduced when the researcher was satisfied that
the participants had successfully understood the previous task. If it appeared that
further practice with a given task type was needed, the researcher introduced a
supplementary task or tasks before giving the students a task of the next type. The
researcher planned each session to last 20 minutes; in the main this was followed,
though some sessions exceeded this time by up to 10 minutes. On occasions when
the researcher decided that the students needed further practice with the last task in a
session, the following session commenced with a supplementary task of the same
type. On other occasions, in the interests of time remaining for the study and in view
of the participants’ competence on tasks of one type, the researcher decided that
certain tasks could be omitted. Otherwise, the next task was the next one in the
sequence listed in Appendix H.
3.7.5 Data Collection and Analysis
Guba and Lincoln (1989) summarised the role of a qualitative researcher in
the following statement:
The major task of the constructivist investigator is to tease out the constructions that
various actors in a setting hold and, so far as possible, to bring them into
conjunction—a joining—with one another and with whatever other information can
be brought to bear on the issues involved. (p. 142)
In this study data from five sources were used to progressively triangulate, or
cross-validate, observations and conclusions (Wiersma, 1995, p. 264).
Data collection procedures.
Data came from several sources: a researcher’s journal, comprising field
notes and a field diary; videotapes and audio tapes of interviews and teaching
sessions; software audit trails; and participants’ workbooks. The researcher and an
assistant transcribed data from each source onto a computer. Video recordings were
transcribed, recording both actions and dialogue by the participants and the
researcher. Software audit trails saved as plain text files were copied from the
computers used in the teaching sessions. Hand-written data in the researcher’s field
notes and field diary and participants’ workbooks were transcribed into text files.
87
Data from these various sources were used in combination to triangulate observations
and conclusions. The following paragraphs describe procedures followed for each
data source.
Researcher’s journal. The researcher kept a journal in which he recorded
field notes (notes taken during the teaching sessions) and a field diary (notes written
up at the end of each day of the teaching program; see Fraenkel & Wallen, 1993).
The field notes were used to record the researcher’s impressions of what was
happening as students attempted to complete tasks in the program; in particular, the
researcher commented on the apparent use of conceptual structures for numbers. The
field diary included notes about each day’s teaching, written in more detail. It
included questions about what occurred in the daily sessions, to direct the
researcher’s attention to particular aspects of the following sessions. By recording
comments at a time when they were fresh in the researcher’s mind, it was hoped to
provide insights about the students’ learning that may not have been accessible from
video transcripts alone.
Interviews. All participants were interviewed before the teaching sessions, as
far as possible on the same day. The second interviews were conducted after the
teaching sessions, again mostly on the same day. One participant, Yvonne, had to be
interviewed later than the others, after school resumed from the following vacation
break, as she and her family left in the last two days of the term for a holiday. The
researcher interviewed each participant individually, in a room separate from the
classroom. The researcher explained before each interview began that some items
might be too difficult for the student. This was necessary particularly for the first
interview, as students had not been taught three-digit numeration concepts in class
prior to that time. Each interview consisted of 9 questions (see Appendix I &
Appendix J), and was planned to take approximately 20 minutes per participant. The
two sets of interviews were videotaped and written responses to certain tasks were
collected. The resulting videos were transcribed, as described below.
Teaching sessions. Each lesson was audiotaped and videotaped. Two video
cameras were used simultaneously on opposite sides of the group, to record as many
of the occurring interactions as possible. As described in section 3.6.5, the cameras
were positioned carefully to avoid obstructions hiding the students’ actions, and in
the computer groups to record both participants’ faces and the computer screens as
88
far as possible. All videotapes were transcribed, recording dialogue spoken and
actions taken by the participants and the researcher.
Software audit trails. The computer software used in the study generated an
audit trail as a plain text file for each session, timing and recording each major action
taken by the users, such as dragging a block or clicking the mouse on a button (see
Misanchuk & Schwier, 1992; Schwier & Misanchuk, 1990; Williams & Dodge,
1993). These text files were used to support the video transcripts where necessary;
the audit trails were referred to if actions taken by participants in the computer
groups could not be clearly determined from the videotapes.
Participants’ workbooks. Participants record their work during the teaching
sessions in workbooks provided by the researcher. Each day the participants dated
the page, and the workbooks were collected at the conclusion of each session, and
the contents transcribed onto a computer. Like the audit trails, workbooks were used
to support video transcript data, to clarify any actions of writing responses that were
not visible on the videotapes.
Data analysis.
Analysis of what took place in teaching sessions was centred on several
readings of the transcripts of session videotapes. The transcripts themselves contain
records of actions taken and verbal interactions among participants and the
researcher. At the start of the transcription process virtually all speech and actions
were recorded. However, after about half of the transcripts were completed, it was
clear that little was being revealed in descriptions of speech and actions that did not
relate to the mathematical tasks themselves. Therefore, for the remaining videotapes
only interactions relating to the tasks were transcribed. Transcripts from videotapes
were supported by data from audiotapes, participants’ workbooks, the researcher’s
field notes, and software audit trail records of user actions with the software. Once
the transcripts were completed they were read several times to ascertain categories of
participant action and speech emerging from the data. There were many candidates
for possible categories to consider: Over the 10 sessions the 4 groups attempted
approximately 30 mathematical tasks each, leading to a wide range of responses
relating to numbers, written symbols, and block or software numerical
representations.
89
Once the raw data were transcribed data analysis began that involved
progressively organising and reducing the data until “focused conclusions” could be
made (Wiersma, 1995). The study is primarily exploratory and one of its major aims
is the generation of theory to describe and explain students’ use of place-value
materials in light of inferred conceptual structures for numbers. As Wiersma pointed
out, in qualitative research hypothesis generation and modification proceeds
throughout the study. Analysis of the data was conducted according to the grounded
theory approach of Strauss and Corbin (1990). This approach involves four main
phases: (a) review of literature, (b) open coding of data, (c) axial coding of data, and
(d) final integration of categories to form theory. These phases are summarised in the
following paragraphs.
Review of literature. In the first phase of the grounded theory approach, the
“technical” literature is reviewed, to “stimulate theoretical sensitivity” (Strauss &
Corbin, 1990, p. 50). Strauss and Corbin explained that
though you do not want to enter the field with an entire list of concepts and
relationships, some may turn up over and over again in the literature and thus appear
to be significant. These you may want to bring to the field where you will look for
evidence of whether or not the concepts and relationships apply to the situation that
you are studying, and if so what form they take here. (pp. 50-51)
This is the situation with this study. A number of categories of conceptual
understanding of numbers were found in the place-value literature (section 2.4.2).
These categories were used as starting points for the data analysis, but did not restrict
the search for new categories “that neither we, nor anyone else, had thought about
previously” (Strauss & Corbin, 1990, p. 50). The literature was also used as
“supplementary validation” (p. 52) in the succeeding phases of the study, to check
findings against previous work in the field.
Open coding of data. The second phase of grounded theory research is what
Strauss and Corbin (1990) called open coding of the data, and is linked closely to the
third phase of axial coding of data. Strauss and Corbin described open coding as
“breaking down, examining, comparing, conceptualizing, and categorizing data”
(p. 61). It involves first discovering categories in the raw data and naming them.
Following the naming of categories, they are developed in terms of their properties
and dimensions. This refers to the process of locating properties of each category on
a continuum. Strauss and Corbin describe several further procedures that can be used
in open coding, including questioning, comparing, and “waving the red flag.” Each
90
of these procedures is designed to examine categories in further detail and to “break
through assumptions” (p. 84) regarding what the data show. In this study the
procedure described in this paragraph was carried out initially using Q.S.R.
NUD*IST (1994) software to code the data. Later this method was changed to use a
database designed by the author to analyse one particular category in the data,
feedback (see Appendix L). Categories identified in the review of literature and
categories emerging from the data were compared and used to cross-validate each
other (Strauss & Corbin, 1990; Wiersma, 1995), as mentioned in the previous
paragraph.
Axial coding of data. The third phase involved further refinement of the
categories defined in the previous stage. As Strauss and Corbin (1990) explained,
whereas open coding “fractures the data,” axial coding is used to put the data back
together, “by making connections between a category and its sub-categories” (p. 97).
Sub-categories are specific features of a category that give further detail about the
category, by describing conditions giving rise to it, its context, strategies that apply
to it and the consequences of those strategies. Strauss and Corbin introduced a
paradigm model to guide the process of axial coding. The paradigm model links a
category, or phenomenon, to its sub-categories in a linear fashion, as indicated:
Causal conditions → Phenomenon → Context → Intervening conditions →
Action/Interaction strategies → Consequences
The same procedures used in open coding, comparing and questioning, are
used in axial coding, but in axial coding the procedures are more complex. This
phase in the analysis involves “performing four analytic steps almost
simultaneously” (p. 107): (a) hypothesising the nature of relationships between
categories and sub-categories, (b) verification of hypotheses against the data, (c)
further search for the properties of categories, and (d) initial investigation of
variation in phenomena. Strauss and Corbin explained that in the coding phases
deductive and inductive thinking are used in turn repeatedly as hypotheses are
alternately proposed and checked. The final justification needed for a proposition is
that the relationship has been “supported over and over in the data” (p. 112).
Integration of categories to form theory. Strauss and Corbin (1990) labelled
the fourth phase as selective coding. This involves finally integrating the categories
previously identified and selecting the core category, “the central phenomenon
around which all the other categories are integrated” (p. 116). This phase involves
91
five steps: (a) explicating the core category, (b) relating subsidiary categories to the
core category, (c) relating categories according to dimensions, (d) validating
relationships against the data, and (e) filling in categories that need further
refinement.
3.8
Validity and Reliability
Any report of research should address questions of validity and reliability of
the study being reported on. As Burns (1990) stated,
with all data we must ask: (a) was the assessment instrument/technique reliable and
valid; (b) were the conditions under which the data was obtained such that as far as
possible only the subject’s ability is reflected in the data and that other extraneous
factors had as minimal an effect as possible? (p. 189)
In quantitative research, reliability and validity questions refer to the
consistency and accuracy of test instruments for measuring the variables being
studied. For qualitative research, such as in this study, different reliability and
validity questions are needed. Rather than asking if observations are consistent with
others made at different times, or in different places, the question asked of qualitative
methods is whether observations made faithfully record what actually occurred
(Burns, 1990). These issues are addressed here in relation to three aspects of the
research: accuracy of raw observations, use of triangulation, and rigour of methods
of analysis.
Accuracy of observations. First, the researcher is an experienced primary
teacher, and as such is used to working with students, observing their reactions to
instruction and judging their understanding of subject matter. Videotapes and
audiotapes of each session have enabled actions and dialogue to be examined at a
level of detail that would not be possible in unrecorded situations. Though there is
obvious subjectivity inherent in any qualitative research, it is claimed that this
drawback is compensated for by the depth of insight into participants’ actions and
understandings afforded by the method. Burns (1990) stated that reliability of
qualitative research was enhanced by “delineation of the physical, social and
interpersonal contexts within which data are gathered” and that what is needed is
“careful and systematic recording of phenomena” (p. 246). The present chapter of
this thesis includes detail of reasons for and assumptions behind the research that
thus help to improve its reliability.
92
Use of triangulation. As described in section 3.7.5, several different sources
of data were used for the study, to triangulate observations and findings. This is a
primary method for improving internal validity of observations in qualitative
research (Burns, 1990; Wiersma, 1995).
Rigour of methods of analysis. As has been noted by many qualitative
researchers, qualitative research methods have been criticised for their apparent lack
of rigour. Researchers who favour experimental research have rejected qualitative
research as being unscientific and sloppy. Strauss and Corbin (1990) developed the
grounded theory approach to qualitative research partly to address such concerns.
Conducted according to Strauss and Corbin’s advice, the grounded theory approach
involves a number of internal checks for validity, and requires the researcher to
check and re-check data to confirm conclusions.
3.9
Limitations
Limitations of this study relate to three particular aspects of the design: the
size and representative nature of the sample, possible observer bias, and the use of
qualitative research methods. First, the sample size is just 16 students at one primary
school. This sample is too small and not sufficiently representative to generalise
findings to primary students in general. However, this is not the main intention of
this study. Fraenkel and Wallen (1993) explained that
in qualitative studies . . . it is much more likely that any generalizing to be done will
be by interested practitioners—by individuals who are in situations similar to the
one(s) investigated by the researcher. It is the practitioner, rather than the researcher,
who judges the applicability of the researcher’s findings and conclusions, who
determines whether the researcher’s findings fit his or her situation. (p. 403)
Thus it is argued that conclusions drawn in this study, as in qualitative
research generally, are to be viewed “as ideas to be shared, discussed, and
investigated further” (Fraenkel & Wallen, 1993, p. 403). The study has been used to
generate hypotheses that are likely to be of interest and relevance to primary teachers
and that may potentially lead to further investigation.
The second limitation is that only one researcher carried out all data
collection and analysis, introducing a possible source of bias. This is typical in
studies of this size and nature that do not have external funding. This concern is
addressed using triangulation; well-documented, comprehensive descriptions
(Wiersma, 1995); and an iterative process of hypothesising and checking. As
93
described in section 3.7.5, triangulation of data has been achieved using several
means of collecting data. These data comprise careful, comprehensive notes from
each relevant incident. Data collection and analysis have been through a number of
iterations of hypothesis proposing, checking, and modification. It is claimed that
through these techniques a “logical basis” has been established for the validity of the
study’s findings (Wiersma, 1995, p. 223).
Finally, this study is limited because of its small scale. As is often the case
with qualitative research, the sample size was small, and the data were collected over
a short period of time. The obvious implication of these aspects of the study is that it
is risky to attempt to apply the study’s findings to Year 3 children generally. This
concern is handled by pointing out the different purposes of qualitative research and
its alternative epistemology. Qualitative research such as that described here attempts
to demonstrate a set of findings that applied in one particular situation and then
presents hypotheses about those findings that may be used to foster further study.
The situations investigated are not perceived as obeying certain laws of nature, but
rather as being constructed and understood individually by the participants in those
situations. In this study, the conceptions of numbers held by a small number of
children have been studied in depth, via recordings of their actions and spoken
dialogue. The proposed categories of response are compared with the results of
previous research, which strengthens the conclusions made. Conclusions about these
findings are presented for evaluation by the reader of the research based on the logic
inherent in the report, rather than being presented as a version of “the truth.”
3.10
Chapter Summary
This chapter outlines the methodology employed in the study. The overall
design is exploratory in nature and is used to generate theory regarding Year 3
students’ understanding of two-digit and three-digit numbers. This theory is
investigated with relation to the students’ prior mathematical achievement levels and
to the mode of number representation used, either computer software or conventional
base-ten blocks. Results of these two sets of conditions have been studied in relation
to differences among student interactions and students’ development of place-value
understanding.
The overall design, a teaching experiment, is widely used in research into
mathematical understanding. A pilot study was conducted to trial various aspects of
94
the study and appropriate modifications to procedures were made to the design. The
main study comprised 5 phases: participant selection, first interview, software
training session, teaching program, and second interview. Sixteen Year 3 students
from a single school were selected to take part in the study. Half of the students were
of low mathematical achievement and half of high mathematical achievement, based
results gained in the previous year using the Year 2 Net (Queensland Department of
Education, 1996).
The researcher took the role of teacher in the teaching program, teaching 4
groups of 4 participants for 10 daily sessions of 20 minutes duration. Sessions
involved students being presented with a series of tasks on cards, to be solved
cooperatively by each group. Tasks were written in a sequence of increasing
difficulty and were presented in order. New tasks were given as students appeared to
be ready for them; supplementary tasks were inserted as necessary, for extra practice.
All sessions involved the collection of qualitative data from several sources,
including a researcher’s journal and video transcripts. The method of data collection
and analysis used is the grounded theory approach of Strauss and Corbin (1990). The
method they have described was followed to generate theory regarding how students
learn place-value concepts and how the use of two modes of number representation
influences that learning.
95
Chapter 4: Results
4.1
Chapter Overview
4.1.1 Restatement of the Research Question
The research question for this study, stated in section 1.3, is repeated here:
How do base-ten blocks, both physical and electronic, influence Year 3
students’ conceptual structures for multidigit numbers? Within the context of Year
3 students’ use of base-ten blocks or place-value software, the following specific
issues were stated as being of concern:
1.
What conceptual structures for multidigit numbers do Year 3 students
display in response to place-value questions after instruction with baseten blocks?
2.
What misconceptions, errors, or limited conceptions of numbers do
Year 3 students display in response to place-value questions after
instruction with base-ten blocks?
3.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in instruction with physical and
electronic base-ten blocks?
4.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in students’ achievement in
numeration?
Data from the interviews and the teaching phase of the study are described in
this chapter, as they relate to the above four questions. Section 4.3 comprises an
overview of data from the interviews, summarising the performance of the
participants on place-value tasks before and after the teaching phase of the study.
Section 4.4 includes a discussion of participants’ apparent number conceptions
evident in their responses. Section 4.5 summarises participants’ performance on digit
97
correspondence tasks. Section 4.6 contains descriptions of errors, misconceptions,
and limited conceptions evident in participants’ responses. Section 4.7 includes an
outline of data relating to participants’ use of either base-ten blocks or computer
software to represent numbers.
4.2
Transcript Conventions Used in this Thesis
Table 4.1 comprises a list of notations used in transcript excerpts quoted in
this thesis.
TABLE 4.1.
Transcript Notations
Indication
Notation
Example
Speech
Normal text
Tell me what this means.
Actions
Square brackets []
[Points to blocks.]
Pause or unfinished
Ellipsis (…)
statement
It’s, … um …
Part of transcript
omitted for brevity,
clarity, or both
Em dash (—)
—
Emphasis of point
of analysis
Italic script
It’s still the same number!
Text inserted to aid
clarity
Parentheses ()
Where does this (block) go?
Numbers as written
symbols
Single quotation
marks (‘’)
What does the ‘2’ mean?
Cardinal numbers
referring to
members of a set
Number words or
figures in normal
text
These three blocks go here.
Base-ten blocks
Figure and place
name
Puts out 4 tens and 10 ones.
Identifying information is appended to transcript excerpts to aid the reader.
For interview transcripts, the number of the interview and the question are
abbreviated in parentheses at the end of each excerpt. For example, (I1, Qu. 2c) refers
to Question 2 (c) in Interview 1. In excerpts from teaching session transcripts, the
session number, group, and task number are similarly indicated. High-achievementlevel and low-achievement-level groups are indicated with the letters “h” and “l,”
respectively, and computer and blocks groups by “c” and “b,” respectively. For
example, (S6 h/b, T 23b) refers to Session 6 of the high/blocks group, attempting
Task 23 (b). Similarly, where necessary in the main text, the group to which a
98
participant belonged is indicated by an abbreviation placed after the participant’s
name: for example, “Hayden (l/c).”
The author of the thesis conducted all interviews and all teaching sessions. To
indicate the different roles of the researcher in interviews and in teaching sessions, in
transcripts of interviews he is referred to with the label “Interviewer,” and in
teaching sessions with the label “Teacher.”
4.3
Place-Value Task Performance Revealed in Interview Results
Results from the two sets of interviews are summarised in this section, in
order to provide an overview of performance of the participants at the start and at the
conclusion of the study. The interviews were intended to show any differences in the
learning of participants resulting from their using the two representational formats.
The results reported in this section show that such differences in performance by
participants using the two types of representational material were minor.
4.3.1 Methods used to Analyse Interview Data
Initial analysis of the interview responses was conducted by listing eight
different skills assessed during the interviews, and deciding on a criterion by which
to judge whether each participant had demonstrated each skill. The eight skills,
divided into 21 sub-skills, and the criteria by which the participants’ responses were
judged are listed in Appendix M. The identified skills and sub-skills mirror the
questions and part-questions very closely, because most questions targeted a
particular numeration skill. This varies for interview Questions 5, 7, and 8 only.
Question 5 was asked in four parts, asking the participant to state numbers that were
a little smaller, much smaller, a little greater, and much greater than a particular twodigit number. Participants’ performance on these questions indicated that they were
able to state numbers close to the given number, or far from that number, but not
always both. Therefore the four question parts relate to two sub-skills, numbered 5a
and 5b. In the case of Questions 7 and 8, each question has been collapsed to a single
sub-skill. Questions 7 and 8 both asked participants to count between 20 and 40
objects and then to identify the referents for that number. Question 8 differs in that
the objects were grouped in such a way as to provide misleading perceptual cues
regarding the referents for the digits. The two questions both targeted the same basic
skill, but within two contexts; the sub-skills have been numbered 7a and 7b.
99
The researcher decided on the criterion for each sub-skill by comparing the
intention of each question with the actual responses of participants, and making a
judgement about what was considered acceptable. For example, for sub-skills 1a to
1c (representing numbers with blocks), it was decided to allow at most one error in
counting the blocks for achievement of each criterion. This was found to be
necessary because of several participants who were clearly able to state the number
represented by the blocks, but who made a mistake in their first attempt at counting
the blocks. By stating a criterion for each sub-skill, the possibility of researcher bias
in deciding who had demonstrated each skill was reduced, and the reliability of
reported performance scores is improved. Reliability of these data was further
strengthened by having the coding of responses cross-checked by a second
researcher, also an experienced primary teacher. A score was determined for each
participant at each interview, based on a count of the sub-skill criteria achieved;
these scores are listed in Table 4.3 and referred to elsewhere in this chapter.
4.3.2 Overview of Interview Results
The scores relating to participants’ achievement of place-value criteria at the
interviews are summarised in a series of four tables, based on analysis of the
interview transcripts.
Table 4.2 indicates the numeration skills demonstrated by participants at each
interview. Three symbols are used to indicate the questions where participants’
demonstration of place-value understanding altered between the two interviews. A
vertical line ( | ) indicates that a criterion was achieved in both interviews. An
upward arrow ( ⇑ ) indicates that a participant achieved a criterion at the second
interview, but not at the first; a downward arrow ( ⇓ ) shows that the participant
achieved the criterion at the first interview, but not at the second. Shading is used
with upward arrows to add visual cues to improvements indicated in the table. The
data in Table 4.2 are summarised in Table 4.3, showing the overall improvement or
deterioration in the number of place-value criteria achieved by each of the 16
participants, and the combined score for each group. The data are further
consolidated in Table 4.4 and Table 4.5. Table 4.4 shows the aggregated scores for
participants of high-achievement-level and low-achievement-level, and Table 4.5
shows the aggregated scores for participants using blocks and participants using a
computer.
100
Patterns in the interview data.
A number of comments may be made about the performance of individual
participants and groups at the two interviews, revealed in Table 4.2 and Table 4.3. It
is to be expected that a series of planned teaching sessions would result in
improvement of students’ understanding of place-value concepts; the shaded arrows
in Table 4.2 indicate the specific skills where this appears to have taken place. An
overview of the scores attained by the 16 participants shows that improvement on a
range of criteria occurred between interviews in the case of many participants, and
there were few criteria on which participants did worse at the second interview.
Individual scores ranged from 3 to 19 at the first interview, and 6 to 20 at the second
interview, and changes in individual scores ranged from +7 to -4.
101
TABLE 4.2.
Summary of Participants’ Numeration Skills Identified in two
Interviews
Numeration Skilla
Participantb
High/Blocks
1a 1b 1c 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 6a 6b 7a 7b 8a 8b 8c
Amanda
|
|
|
|
|
|
|
|
|
|
|
|
|
⇑
⇑
⇑
|
⇓
Craig
|
|
|
|
|
⇓
|
|
|
|
|
|
|
⇑
⇑
|
|
|
John
|
⇑
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Simone
⇑
⇑
⇑
|
⇑
⇑
⇑
|
|
⇓
⇑
|
|
|
⇑
Belinda
|
|
|
|
|
|
|
|
⇓
⇑
|
⇑
|
|
|
|
|
|
|
|
⇓
Daniel
|
|
|
|
|
|
|
|
|
⇑
⇑
|
|
|
|
|
|
⇑
⇓
⇑
|
Rory
|
|
⇑
|
|
|
|
|
⇓
|
⇑
|
|
|
|
|
|
|
|
⇓
Yvonne
|
|
⇓
|
|
|
|
|
|
⇓
|
|
|
⇓
|
|
⇓
Clive
⇑
⇑
|
⇓
|
|
|
⇓
Jeremy
|
⇑
|
|
⇓
|
Michelle
|
|
⇑
|
⇑
Nerida
|
⇑
|
|
⇑
Amy
|
|
|
⇓
⇑
Hayden
|
|
|
|
Kelly
|
⇑
⇑
⇑
Terry
|
|
|
|
|
|
⇑
High/Computer
Low/Blocks
|
⇓
⇑
⇓
⇑
⇑
|
|
|
|
⇓
|
|
|
|
⇑
⇑
⇑
⇑
⇑
⇑
⇓
Low/Computer
⇑
⇑
⇓
⇑
⇑
|
|
⇑
⇑
|
|
⇑
|
⇑
|
⇑
⇓
⇓
⇑
⇑
⇑
|
⇓
Note. | - Criterion achieved at both interviews; ⇑ − Criterion achieved at Interview 2 only;
⇓ - Criterion achieved at Interview 1 only; no mark – Criterion not achieved at either interview.
Criteria for numeration skills are described in Appendix M.
a
Numeration skills: 1 – Read block representation; 2 – Show block representation; 3 – Recognising
three-digit block representations; 4 – Skip counting; 5 – Number relationships; 6 – Comparing pairs of
numbers; 7 – Digit correspondence; 8 – Mental computation.
b
Participants’ names are sorted alphabetically within groups in this and later tables. All participants’
names mentioned in this thesis are pseudonyms (Appendix G).
The number of place-value criteria attained by individual participants in the
two interviews are summarised as scores in Table 4.3. The circumstances of
Yvonne’s (h/c) second interview were different from the other 15 participants:
because her family went on a holiday before the end of the term, her second
interview was delayed for over 3 weeks. For this reason, her interview scores have
been discarded when calculating average group scores, and the row in Table 4.3
referring to Yvonne’s scores is greyed out.
102
TABLE 4.3.
Summary of Numeration Skills Demonstrated by Each Participant and
by Each Group
Group
Score at
Interview 1
Score at
Interview 2
Increase
(Decrease)
High/Blocks
15
17
2
Craig
18
20
2
John
17
18
1
Simone
7
14
7
14.3
17.3
3.0
19
19
0
Daniel
17
20
3
Rory
18
18
0
Yvonne
17
13
(-4)
18.0
19.0
1.0
7
6
(-1)
Jeremy
6
6
0
Michelle
3
6
3
Nerida
8
14
6
6.0
8.0
2.0
8
10
2
Hayden
9
12
3
Kelly
4
6
2
Terry
9
14
5
7.5
10.5
3.0
Participant
Amanda
Group Average:
Belinda
High/Computer
Group Averagea:
Clive
Low/Blocks
Group Average:
Amy
Low/Computer
Group Average:
Note. Maximum possible score per participant per interview was 21.
a
In calculating average scores for the high/computer group, Yvonne’s scores have been discarded, as
her second interview was conducted more than 3 weeks after teaching sessions were concluded.
The figures in Table 4.3 show that the aggregate scores for the 4 groups, if
considered on their own, would hide the differences in interview scores within
groups, that in some cases are greater than the differences between groups. For
example, Simone’s achievement of 7 more criteria at the second interview than the
first interview makes up more than half of the improvement (12 points) in the score
of the entire high/blocks group. Similar differences are evident in the scores achieved
by Yvonne (h/c), Nerida (l/b), and Terry (l/c) compared to their respective groups. In
the case of Yvonne, it is likely that her score was influenced by the circumstances of
her second interview, as mentioned earlier.
103
With a small sample such as this, strong claims about the relative benefits of
use of computer software or blocks for learning place-value concepts based on the
interview scores would not be justified. Thus any conclusions that may be drawn
from the data are necessarily tentative; hypotheses that are suggested to explain any
apparent trends in the data will require further large-scale studies for testing. With
these comments in mind, the following observations are made regarding patterns in
the performance of the 4 groups at the interviews shown in Table 4.2:
1.
The most improvement in scores is evident in 2 groups: the high/blocks
group and the low/computer group.
2.
Certain participants appear to have been especially helped by the
teaching program used in the study, particularly Simone (h/b), Nerida
(l/b), and Terry (l/c).
3.
Questions relating to skip counting (Skills 4a to 4d inclusive) showed
greater improvement among participants who had used the computer
than among those who had used the blocks.
One further observation can be made regarding Skill 8c, which involved
subtracting fewer than 10 ones from a number of tens (e.g., I1: 5 tens - 8). Question
9 (c), relating to this skill, was successfully completed by 7 fewer participants at the
second interview than at the first interview, and there was no participant who
improved on that question. The specific numbers used at each interview may help
explain this result. In Interview 1, participants were asked to subtract 8 ones from 5
tens; in Interview 2, the task was to subtract 6 ones from 7 tens. The particular
combination of numbers chosen may have led to a greater chance of error for
participants when considering the second version of the question. The highachievement-level participants who had answered the parallel question correctly at
the first interview but were incorrect at the second interview all gave the answers 63
or 61. This implies that they lost count of the tens and ones parts of the question,
either subtracting 6 from 7 to get the ones part of the answer, or subtracting 7 ones
instead of 6 ones from 70.
Achievement level and interview performance.
Although the differences in group scores between the first and second
interviews can be explained in light of individual performances, there are still
differences worth noting in the results summarised in Table 4.2 and Table 4.3. These
104
two tables show a clear distinction between high-achievement-level participants and
low-achievement-level participants in their ability to meet criteria on interview tasks.
The interview scores are further collapsed in Table 4.4, showing the results from the
interviews arranged according to the mathematical achievement level of the
participants.
TABLE 4.4.
Summary of Place-value Understanding Criteria Achieved by Highachievement-level and Low-Achievement-Level Participants
Achievement
Level
n
Average score at Average score at
Interview 1
Interview 2
Average
increase
High
7
15.9
18.0
2.1
Low
8
6.8
9.3
2.5
Note. Scores of Yvonne (high/computer group) have been discarded. Maximum possible score per
participant per interview was 21.
There was a marked difference in performance between the highachievement-level participants and low-achievement-level participants, with the
high-achievement-level participants achieving an average score that was
approximately twice that of the low-achievement-level participants at each interview.
Both cohorts improved between interviews; the low-achievement-level participants
had more room for improvement, and showed a slightly greater improvement,
increasing their scores by an average of 2.5 points. The difference in scores at the
first interview provides broad justification for the initial identification and selection
of high-achievement-level and low-achievement-level participants to participate in
the study. Though there were some anomalies in the performance of individual
participants, noted in the previous paragraph, in general high-achievement-level
participants showed a much better understanding of place-value than their lowachievement-level counterparts.
Number representation formats and interview score.
The researcher’s intention was to form two equivalent groups of highachievement-level participants and two of low-achievement-level participants, with
one of group of each achievement level to use blocks and one to use computers in the
teaching sessions. However, the interview results show that there were marked
differences when comparing the 2 high-achievement-level groups with each other,
and also when comparing the 2 low-achievement-level groups, that raise a question
of the equivalence or the comparability of the pairs of groups of similar achievement
105
level. Table 4.3 shows that at the first interview the high-achievement-level
participants who were to be in the computer group achieved an average score of 18.0
place-value criteria, compared to 14.3 among the high-achievement-level participants
who were to use the blocks. A similar difference is evident in the scores of the 2 lowachievement-level groups: The computer group achieved an average score of 7.5
points, and the blocks group an average of 6.0 points. These differences are repeated
in Table 4.5, which shows that participants who used the computer started the study
with a score on average 1.9 points higher than participants who used blocks. At first
glance, these figures are cause for some concern, as it appears that the 8 participants
who used the computers started with a higher level of place-value understanding than
those who used the blocks.
TABLE 4.5.
Summary of Place-value Understanding Criteria Achieved by
Participants in Computer and Blocks Groups
Groups
n
Average score at
Interview 1
Average score at
Interview 2
Average
increase
Computer
7
12.0
14.1
2.1
Blocks
8
10.1
12.6
2.5
Note. Scores of Yvonne (high/computer group) have been discarded. Maximum possible score per
participant per interview was 21.
Two factors may help explain the reasons for the apparent inequality in levels
of place-value understanding shown in Table 4.5. First, differences in initial scores
varied among individuals more than expected, considering the results from the
previous Year 2 Net (Queensland Department of Education, 1996). Appendix F
shows that the performance on the Year 2 Net by the high-achievement-level students
selected to participate in the study showed little variation compared to results from
the interviews conducted in the study. In particular, Amanda and Simone, both in the
high/blocks group, were expected to perform better in the first interview compared to
their peers, based on the Year 2 Net results.
The second fact that may help explain the anomalies in the initial scores of
groups of participants is the method used to place participants in groups once the 16
participants had been selected. Initially, the students’ two class teachers assisted the
researcher to select 8 high-achievement-level and 8 low-achievement-level students,
and then the students were placed in pairs of the same gender. Four of the highachievement-level students, 2 girls and 2 boys, came from one class, and the
remaining 12 students were from another class. The researcher made the decision to
106
separate the four children from the first class, so that there was not a group of 4
children from one class and 3 groups from another class. The teacher of the 12
students advised the researcher about which students she felt would work well
together, based on her knowledge of their friendship groups.
The researcher formed groups of 4 participants from these friendship pairs,
and randomly assigned each group to use the computer or the blocks. Thus the fact
that both computer group had somewhat higher levels of place-value understanding
prior to the commencement of the study was the result of a number of decisions
made for various pragmatic and research-oriented reasons, and the random
assignment of groups to each treatment. The variation in place-value understanding
of the 4 groups did not become evident until after the teaching phase had
commenced, as time did not allow the interviews to be transcribed prior to
commencing the teaching sessions.
4.4
Students’ Conceptions of Numbers
Participants’ number conceptions and other information regarding participant
thinking are revealed through detailed analysis of the transcripts themselves, looking
at descriptions of the words spoken and the actions taken by participants as they
answered the questions. This analysis is described in this section, divided into
subsections, describing two broad approaches to interview questions, grouping
approaches (4.4.1) and counting approaches (4.4.2); and a common faulty
conception, the face-value interpretation of symbols (4.4.3). These results are
summarised in section 4.4.4, and comments are made about the changeability of
participants’ conceptions (4.4.5).
4.4.1 Grouping Approaches
A number of participants gave answers to interview questions that referred to
groups of 10 when dealing with numbers in the tens place. This is termed here a
grouping approach, and is considered to imply a concept of multidigit numbers that
recognises the groups of 10 around which the base-ten numeration system is based.
Transcripts of responses to interview Questions 1, 3, 6, 7, 8, and 9 show instances of
participants using a grouping approach. The following paragraphs describe how
individual participants used a grouping approach in answering each of these
107
questions. At the conclusion of this section, Table 4.6 and Figure 4.1 summarise the
use of grouping approaches by each participant.
Question 1 (b) and (c): Interpreting non-canonical block representations
(e.g., asking the participant to say the number represented by 4 tens & 12 ones). In
answering these questions some participants grouped either ones or tens to make a
group of 10 blocks, and then counted the new group as a ten or hundred,
respectively, before finishing the count. For example, to interpret a block
arrangement comprising 4 tens and 12 ones, some participants first grouped 10 of the
ones together, then counted the tens including the new group of 10 ones, and then
added the remaining 2 ones:
Craig (h/b):
I just got all the tens together here and I said to myself there’s 40 there and I
counted these, and there was 10 [ones] there. And so I thought I put them with
the tens so I know that there is 10 here. Then I counted the last two. So it’s 52.
(I1, Qu. 1b)
A comment is needed at this point about the possible use of a grouping
approach when answering Question 2: Using blocks to represent a two-digit or
three-digit number. Base-ten blocks allow students to take advantage of the grouped
structure inherent in the blocks themselves to represent the groups-of-ten structure in
the base-ten numeration system, as described in section 2.5.3. However, base-ten
blocks may also be used to represent numbers using a face-value interpretation of
numbers, as discussed later. Thus if a participant used the blocks to represent a
number canonically it is not possible to tell if the student had in mind the groups of
10 in the number, or a face-value construct for multidigit numbers. Therefore,
whereas it is possible to identify a counting approach (section 4.4.2) in a participant’s
response to Question 2, it is not possible to clearly identify the use of either a
grouping approach or a face-value construct in an answer to this question.
Question 3 (a) and (b): Interpreting non-canonical block representations of
three-digit numbers, and comparing them with written symbols (e.g., comparing
1 hundred, 2 tens, & 16 ones with 136). For these questions, participants were asked
to read a written symbol for a three-digit number. They were then shown three
examples of block arrangements one at a time, and asked if each arrangement
represented the same number as the written symbol. The third example was incorrect,
and targeted the face-value construct for multidigit numbers, discussed in section
4.4.3. The first two arrangements were of non-canonical representations for the
108
number on the card—for example, 1 hundred, 2 tens, and 16 ones for 136—and
could be answered using a grouping approach. This approach is demonstrated in the
following transcript excerpt showing Daniel (h/c) interpreting a collection of 17 tens
and 2 ones and comparing it with the symbol ‘172’:
Daniel:
Mmm … [counts out 10 tens and places them together, then counts remaining
7 tens] — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Yeah that’s 10 there. 1, 2, 3, 4, 5, 6, 7.
Yep (the blocks represent the same number).
(I2, Qu. 3b)
Question 6 – Comparing pairs of two-digit and three-digit numbers (e.g.,
compare 27 & 42, 174 & 147). In Question 6 participants were shown a pair of
written symbols for two-digit numbers, followed by a pair of symbols for three-digit
numbers. In each case they were asked which number was larger, and to explain their
reasoning. It was considered that participants were using a grouping approach if they
referred to the names of the places concerned when justifying their answers. For
example, consider the following transcript showing Rory (h/c) explaining which of
the symbols ‘27’ and ‘42’ represents the bigger number:
Rory:
That one: ‘42.’
Interviewer: And how do you know it’s bigger?
Rory:
Because it has more tens.
Interviewer: Uh-huh. And how many tens does it have?
Rory:
‘4.’
Interviewer: — Can you explain why that one’s bigger? I mean this one [‘27’] has a ‘7’ and
[you say] this one is smaller …
Rory:
Because this one has 4 tens and 2 ones and that one has 2 tens and 7 ones.
(I1, Qu. 6a)
Rory, like several other participants, clearly knew that the position of each
digit determines its place name, and that tens are worth more than ones are. What is
not revealed by this nor other transcript excerpts is whether or not these participants
were aware of the “tenness” of a number in the tens place—the fact that “a ten” is a
collection of 10 ones. As S. H. Ross (1990) commented,
children may sound very knowledgeable as they speak of so many “tens and ones.”
Yet in reality a child may be using a face-value interpretation in which “tens and
ones” are merely names for different objects and have no real connection to
“tenness.” (p. 14)
109
Despite this observation, it appears that responses like Rory’s do show an
awareness that tens and ones are not interchangeable, as students with face-value
interpretations of digits sometimes indicate. Nor did Rory’s response rely on the
counting sequence to justify why one number is larger than the other is: He indicated
that it was sufficient to check individual digits, in particular the tens digit, to
determine the larger number. For this reason, it is decided to include responses to
Question 6 that include reference to place names in the category of using a grouping
approach. Nevertheless, the points raised here should be kept in mind when
considering summaries of grouping and counting approaches given later in this
chapter.
Questions 7 and 8: Explaining referents for the digits in two-digit written
symbols (e.g., count 24 sticks, write symbol, & explain symbol). Participants were
asked to count a number of objects and to write the symbol for the number. They
were then asked to show which objects were represented by each written digit. Many
participants answered with a face-value interpretation of the symbols, but others
correctly showed the objects remaining after the ones had been taken out as the
referents for the tens digit. As in the case of responses to Question 6, again this type
of response may not indicate a complete understanding of the groups represented by
the tens digit. Nevertheless, it does show an awareness that the digit represents more
than its face value, and that the referents for the two digits together make up the
entire collection of objects. To distinguish between (a) the basic understanding that
the sum of the objects corresponding to the digits in a number must equal the entire
collection, and (b) the more advanced concept that a tens digit represents the product
of the digit’s face value and its place value, participants who indicated the correct
number of objects for the tens digit were asked “How can that digit stand for so
many?” If a participant said that the digit was a number of tens, the response was
categorised as showing a grouping approach. For example:
Interviewer: Does this part [‘3’] of your ‘37’ have anything to do with how many sticks
you have? Can you show me?
Rory (h/c):
Yeah. [Picks up remaining 30 sticks] There.
Interviewer: All of them? How does that ‘3’ stand for all of those?
Rory:
Because it’s 3 tens.
Interviewer: All right, so how many have you got in your hands there?
110
Rory:
[Just glances at them] 30.
(I2, Qu. 7c)
Instances of a grouping approach to digit correspondence questions were not
very common; Table 4.6 shows that only 5 participants, all of them highachievement-level participants, showed a grouping approach at either interview in
answering these two questions. Section 4.5 includes a description of four distinct
categories of response to Questions 7 and 8, ranging from the grouping approach
described here to a face-value interpretation of digits. The grouping approach is
classed as a Category IV response, the highest level of response noted in this study.
Question 9 – Mental addition and subtraction (e.g., How many pieces of gum
in 3 packets of 10 sticks + 17 sticks?). In Question 9 participants were asked to work
out the answers to three questions: adding a group of tens and fewer than 10 ones,
adding a group of tens and between 11 and 19 ones, and subtracting fewer than 10
ones from a number of tens. To assist their thinking, at the first interview participants
were provided with packets of 10 pieces of chewing gum, and at the second
interview participants were provided with plastic bags each containing 10 clothes
pegs. In each case participants could handle and count the packets or bags, but they
were not permitted to open the collections to manipulate single items. Bags of pegs
permitted participants to see the pegs, and packets of gum allowed individual pieces
to be felt under the wrapper. Participants adopting the grouping approach used the
groups of 10 in each question to help them answer the question. For example, note
how Belinda (h/c) added 3 groups of ten and 17 single objects:
Belinda:
47. There’s um, three of them and then there’s a one, which would make a 40,
and then you put a ‘7’ on the end and it equals 47.
(I1, Qu. 9b)
The same use of groups of 10 is shown in the following transcript in which
Terry (l/c) calculated 5 tens minus 8 ones. Terry subtracted 8 from 10, and then
added the remaining 4 tens:
Terry:
8, and there’s 10. [Moves packets to the left, counting quietly in tens] 42.
Interviewer: That was quick. How did you work that out?
Terry:
‘Cos I already knew. ‘Cos it’s 10, there only had to be 2 more because 9, 10.
Interviewer: And you know how many are in those packets?
Terry:
Yup. It’s how you tell. ‘Cos there’s only 1 ten, 2 tens, 3 tens, 4 tens. So it must
(I1, Qu. 9c)
be 40.
111
Summary of the use of grouping approaches.
The use of grouping approaches by each participant is indicated in Table 4.6,
and group totals are summarised in Table 4.7. It should be noted that many
participants used a variety of approaches to answer further questions from the
researcher; other approaches are indicated in later tables in this chapter. For each
participant the responses at each of the two interviews are indicated in two adjacent
rows of Table 4.6, and the number of questions for which the participant used a
grouping approach at each interview is indicated in the last column.
112
TABLE 4.6.
Use of Grouping Approaches for Selected Interview Questions
Question
Participant
Interview 1b 1c 3a 3b 6a 6b
High/Blocks
Amanda
Craig
John
Simone
High/Computer
Belinda
Daniel
Rory
Yvonne
Low/Blocks
Clive
Michelle
Nerida
Low/Computer
Amy
Kelly
Terry
x
x
x
x
x
x
1
2
1
2
1
2
1
2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
1
2
1
2
1
2
1
2
Jeremy
Hayden
1
2
1
2
1
2
1
2
x
x
x
x
x
x
7
8
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
9a 9b 9c
Count
x
x
x
x
x
x
x
x
x
x
x
x
x
5
7
7
10
9
8
1
6
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
1
2
1
2
1
2
1
2
x
x
x
x
x
x
x
x
x
x
x
Note. x – indicates use of a grouping approach in responding to the question.
TABLE 4.7.
High
Low
Total
x
x
x
x
x
Use of Grouping Approaches by Each Group
Blocks
53
5
58
Computer
65
13
78
113
Total
118
18
136
10
9
7
8
10
10
5
6
0
1
0
0
0
1
1
2
0
1
3
1
0
0
3
5
It is clear from Table 4.7 that, overall, high-achievement-level participants
used the grouping approach far more often than low-achievement-level participants
did. On average, high-achievement-level participants used grouping approaches to
answer over 7 questions per interview, whereas the low-achievement-level
participants used them for just more than 1 question per interview. The clear
difference in the patterns of response of high-achievement-level and lowachievement-level participants implies a markedly different level of understanding of
place-value. Overall, the computer groups used grouping approaches more often than
did blocks groups; however, this is considered to be due to differences of individual
members of these groups, as described earlier.
Scores achieved by the 16 participants at each interview are compared to the
number of times that a grouping approach was used in achieving those scores in
Figure 4.1. This scatter-plot graph shows a clear pattern of higher numbers of placevalue criteria being achieved by those participants who used grouping approaches the
most. Apart from one participant who achieved 14 criteria while using grouping
approaches only twice, participants who achieved more than 10 criteria at interviews
also used grouping approaches at least 5 times in the same interview. It should be
noted that, in this and later scatter-plot graphs, certain data points overlap others, so
that not all 32 data points are visible. This graph may be compared with Figure 4.2,
which shows a similar comparison between interview scores and counting
approaches.
114
21
20
20
19
18
18
17
15
15
Place-Value Criteria Achieved
18
17
17
19
18
14
14
14
13
12
12
10
9
9
8
6
8
7
7
6
6
4
3
3
0
0
1
2
3
4
5
6
7
8
9
10
Incidence of Grouping Approaches
Figure 4.1. Interview scores compared to use of grouping approaches.
4.4.2 Counting Approaches
A second common approach to interview questions, adopted by several
participants, was based on consideration of individual ones in a number, rather than
groups of 10 ones or 10 tens. Participants’ responses of this type involved either
counting single one-blocks without grouping them first, or reference to the counting
sequence of number names, and so this approach is called a counting approach.
Counting approaches were characterised by participants ignoring the grouped aspect
of base-ten numbers, and treating multidigit numbers as collections of ones.
Representative responses to certain interview questions are summarised in the
following paragraphs.
Question 1 (b) and (c): interpreting two-digit and three-digit non-canonical
block representations. The counting approach was clearly evident among some
participants when attempting to name a number represented by a non-canonical
arrangement of blocks. For example, the following excerpt demonstrates that Jeremy
(l/b) used a counting approach to work out the number represented by 3 tens and 16
ones:
Jeremy:
[Touches tens] 10, 20, 30, [touches ones] 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
(I2, Qu. 1b)
41, 42, 43, 44, 49, 46.
115
Jeremy’s answer in this instance was correct, but the inefficiency of the
method caused him to take longer than he might have using a grouping approach, and
there was a greater chance that a counting error could cause him to arrive at an
incorrect answer. In fact, Jeremy at first gave the answer 36 for this question, perhaps
because of a mistake at the change of decade from 39 to 40. Other participants also
made counting errors when counting 3 tens and 16 ones: Michelle (l/b) and Simone
(h/b) also gave the answer as 36, Hayden (l/c) and Nerida (l/b) as 47, and John (h/b)
as 44. Discussion of the relative efficiency and usefulness of grouping and counting
approaches is continued in section 5.2.2.
Question 2: Using blocks to represent a two-digit or three-digit number.
When asked to represent a two-digit or three-digit number using the blocks, some
participants chose to use what Fuson (1990a) called collected multiunits: They
represented the tens and ones digits of a number using only ones material. For
example, when showing 261, Daniel (h/c) selected 2 hundred-blocks, and then
started to count out 61 ones. He stopped when he had 20 ones, and changed his mind,
putting out 2 hundreds, 6 tens, and 1 one. Amy (l/c) used a similar approach when
asked at her first interview to show the number 134. She started to count out oneblocks, apparently meaning to count 134 ones. She stopped when she reached 59,
and changed her method to putting out 10 tens and 34 ones. This approach, of
choosing multiple ones or tens to represent a multidigit number, is an example of a
counting approach. Rather than making use of the groupings inherent in the base-ten
numeration system, participants using multiunits to represent a multidigit number
count out blocks one at a time in until the end number is reached.
There is evidence that participants who used a counting approach for
Question 2 did so because they had not thought of using the already-grouped baseten material. This is shown in both examples mentioned above. Daniel changed to a
canonical representation for 261 himself, without input from the researcher.
Similarly, after a while Amy decided on her own not to try to count 134 ones, though
nevertheless she still chose to use 10 tens rather than 1 hundred-block and 34 ones
rather than 3 tens and 4 ones. It is quite possible that in these incidents participants
did not use a hundred-block because of a lack of familiarity with both three-digit
numbers and the base-ten blocks used to represent them, as at the time of the first
interview participants’ class teachers had not taught about the hundreds place.
116
Question 6 – Comparing pairs of two-digit and three-digit numbers. Some
participants demonstrated a counting approach when answering Question 6. When
asked to justify their answer stating which of two numbers was larger, some
participants referred to the position of one or both numbers in the counting sequence.
For example, Hayden (l/c) explained in this manner when comparing 138 and 183 at
the first interview:
Interviewer: Which number is larger?
Hayden:
183.
Interviewer: OK, and how do you know it’s bigger?
Hayden:
Because it takes longer than 138.
Interviewer: How do you know it’s going to take longer?
Hayden:
Because you have to count to a 100 and then keep … count to um 83, and [for
the other number] you just have to count to 138.
(I1, Qu. 6b)
A counting approach was also evident in the way that some participants
appeared to be influenced by the verbal names of the numbers in a question. For
example, in the following excerpt Michelle (l/b) appeared to have no reason for
believing 42 to be larger than 27, other than their respective names:
Michelle:
[Points to ‘27’ then changes mind and points to ‘42’] No, that one [‘42’] is
bigger.
Interviewer: — And how do you know it’s bigger?
Michelle:
Because it’s … that’s 27, that’s forty-se … 42.
Interviewer: Uh-huh, so how do you know 42 is bigger?
Michelle:
Because they’re [‘27’] little and they’re [‘42’] bigger.
(I1, Qu. 6a)
It may be that Michelle was thinking of some other reason for believing that
42 is greater than 27 other than the counting sequence. However, other authors (e.g.,
Resnick, 1983) have suggested that many children without an understanding of the
tens and ones nature of two-digit numbers picture numbers only as a sequence of
counting numbers. This would be consistent with Michelle’s statement in the
previous excerpt that (a) one number was 27 and the other was 42, and that (b) 27 is
little and 42 is bigger. Certainly the next example supports this argument, as it shows
Amy starting by referring to the verbal names of two numbers and then referring to
their position in the counting sequence.
117
The name of a number appeared at times to trigger a response in some
participants that focussed on their knowledge of the counting number sequence. For
example, Amy (l/c), in comparing 38 and 61, started to say that 38 was bigger, until
she read the names of the numbers represented by the symbols. She started to say
that ‘61’ was “sixteen,” but corrected herself and immediately said that “That’s
bigger because it’s 61. So and that’s [‘38’] smaller.” She followed this with a clear
example of a counting approach, explaining that 61 was a bigger number than 38
because of their relative positions in the counting sequence: “‘Cos then it goes 40,
50, then 60” (I2, Qu. 6a). Another apparent example of a counting approach was seen
in a transcript in which Clive (l/b) compared the symbols ‘259’ and ‘295.’ Clive had
initially chosen 295 as the larger number, but noted that the written symbols had the
same digits, in different positions. When pressed, Clive said that he knew that 295 is
larger “because um it sounds like it’s the biggest number” (I2, Qu. 6b).
Questions 7 and 8: Explaining referents for the digits in two-digit written
symbols. An interesting phenomenon occurred among some participants when
answering Questions 7 and 8, that again indicates thinking that included the idea of
counting. Certain participants correctly rejected the idea that the two digits each
represented only their face value, but failed to explain the meanings of the digits in
terms of the groups of 10 and single ones. Instead, they explained that the two digits
in the symbol together represented the entire collection of objects, but that each
individual digit did not have a referent. This response ignores the grouped tens aspect
of multidigit symbols, and instead focuses on the entire set as a collection of single
objects: a counting idea. For example, Amanda (h/b) explained the referents for each
digit in the number 13 using a counting approach that incorporated her understanding
that the digits ‘1’ and ‘3’ combined in the symbol ‘13’ somehow represented more
than the sum of their individual values:
Amanda:
If there’s only this three [takes the beads out of one cup and puts three out] by
itself then it won’t be 13.
Interviewer: Yes. All right, not on its own, no. All right, okay. Put them back in the cup
again. Now look at this part [‘1’] of your ‘13’: Does it have anything to do
with how many beads you have?
Amanda:
Because there’s a one and you need another three, but that, it’s not like that,
because it has to be 13.
Interviewer: Uh-huh.
118
Amanda:
One … and three, doesn’t make it.
Interviewer: — So can you tell me why that’s a one? Er … what the one is for?
Amanda:
— You need it because that’s how you count it, that’s how much they are.
Interviewer: Right, but the one … are you saying that the one doesn’t really stand for
anything? It’s just how you write it down, is that right?
Amanda:
It means something, but that’s how … It means it’s part of the number, and
it’s um … you need it because um if you can’t have, if you don’t use it, it will
only be 3, not 13.
Interviewer: Uh-huh … it sounds like it’s to do with how you write it down?
Amanda:
And it needs both of the numbers to make it.
(I1, Qu. 8b)
Several participants explained the referents for individual digits in two-digit
numbers using explanations similar to Amanda’s response. These responses to digit
correspondence tasks are defined as Category II responses, according to the
hierarchy of response categories proposed in section 4.5. Participants responding as
Amanda did in the previous transcript were often forced to deal with contradictions
in their beliefs, due to their not recognising any number of sticks as corresponding to
each individual digit. The place of such contradictions in children’s development of
place-value understanding is discussed further in section 5.4, looking at evidence of
participants’ construction of meaning and how children managed apparent
contradictions as they perceived them in the information available to them.
A further example of counting approaches used when responding to the
question “How can that digit refer to so many objects?” is provided in the following
transcript excerpt. Hayden (l/c) gave a counting explanation for the fact that the face
value of the tens digit did not match the number of objects it represented:
Hayden:
[The ‘7’] is a part of 30 … it’s a part of like in 30 it’s a part like … you count
to 30 and then you count 7 more and it ends up 37.
Interviewer: And what does this ‘3’ here mean?
Hayden:
It’s up … it’s up to 30 … like if you count up to 30.
(I2, Qu. 7b)
Question 9 – Mental addition and subtraction. Several participants opted for
a counting approach to answering mental addition and subtraction questions. In some
cases, participants used their fingers as an aid to counting; in others, they nodded
their heads or pointed at the desk, as if at imaginary objects. Unlike those who used a
grouping approach to consider separately the tens and ones parts of an addition or
subtraction question, participants using a counting approach stepped forward or back
119
in the cardinal number sequence to find the answer. For example, participants using a
counting approach to answer the question 3 tens plus 17 ones generally chose an
inefficient approach of counting on from 30 by 17 steps. The following transcript
excerpt shows Kelly (l/c) using this method, and demonstrates a difficulty that it
introduces for the student. In counting on by 17 from 30, Kelly made an error and
reached the answer 43:
Kelly:
[Touches each packet of gum] 10, 20, 30. [Counts on fingers by touching them
one by one on table] 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43.
43 pieces of gum.
Interviewer: 43. How did you do that?
Kelly:
I counted them in tens and then I counted 17 more on.
Interviewer: Uh-huh. How did you know when you got to 17?
Kelly:
I um in my head I counted out the um numbers and I just um did that [touches
fingers one by one on table] and I knew when I got to 17 ‘cos you have 10 and
you add a couple more on …
Interviewer: How many do you add on to make 17?
Kelly:
Well you add um seven more on.
(I1, Qu. 9b)
The above transcript clearly shows that Kelly was not using a grouping
concept for two-digit numbers. She knew that 17 was made of 10 plus 7, shown by
the fact that she used her fingers to count on 10 and then started again to add another
7. However, she evidently did not perceive of the ten as a group that could be added
straight to the 3 tens to make 4 tens, but rather saw the ten as 10 ones that had to be
added to 30 one at a time. For the same question Nerida (l/b) used an even more
inefficient counting strategy: she started at 17, and then counted on the 3 tens as 30
ones, using her fingers:
Nerida:
[Counts quietly, looking around, then counts on her fingers] 47.
Interviewer: 47, well done. How did you do that?
Nerida:
I counted um these 17 first then I counted 10, counted on by 10. — I went
from 17 and I counted on three times out of tens out of my hands. (I1, Qu. 9b)
In this instance Nerida’s strategy was successful: evidence of the care she
must have taken in carrying out the counting. However, the likelihood of making a
mistake with this method is clearly quite pronounced. Difficulties for students using
a counting approach are discussed further in section 5.2.2.
120
Summary of the use of counting strategies.
Table 4.8 shows use of counting strategies by each participant at each
interview. These data are summarised for each group in Table 4.9. Incidence of
counting strategies by individual participants is compared with their interview scores
in Figure 4.2.
TABLE 4.8.
Participant
Use of a Counting Approach for Selected Interview Questions
Question
Interview 1b 1c 2a 2b 6a 6b 7
8
x
x
9a 9b 9c Count
High/Blocks
Amanda
Craig
John
Simone
High/Computer
Belinda
Daniel
Rory
Yvonne
Low/Blocks
Clive
Jeremy
Michelle
Nerida
Low/Computer
Amy
Hayden
Kelly
Terry
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
x
2
0
0
0
2
1
1
2
x
x
x
x
x
0
0
0
1
0
0
1
1
x
x
x
1
2
1
2
1
2
1
2
x
1
2
1
2
1
2
1
2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Note. x – indicates use of a counting approach in responding to the question.
121
x
x
x
x
x
x
x
x
x
x
x
x
x
x
2
5
3
1
2
2
4
4
5
5
7
4
7
6
1
1
TABLE 4.9.
Use of Counting Approaches by Each Group
Blocks
8
23
31
High
Low
Total
Computer
3
36
39
Total
11
59
70
Table 4.9 shows that, as in the case of grouping approaches (Table 4.6), there
was a clear difference in the frequency with which the high-achievement-level and
low-achievement-level participants used the strategy. However, this trend is reversed
in the case of counting strategies: Whereas high-achievement-level participants were
much more likely to use a grouping approach, the low-achievement-level participants
used counting approaches more than 5 times as often as high-achievement-level
participants. Differences between total use of counting strategies by blocks and
computer groups are minor.
21
20
20
19
18
18
18
17
17
17
Place-Value Criteria Achieved
15
15
14
14
14
13
12
12
10
9
9
9
8
7
7
6
6
6
3
3
8
6
6
6
4
0
0
1
2
3
4
5
6
7
Incidence of Counting Approaches
Figure 4.2. Interview scores compared to use of counting approaches.
Figure 4.2 shows how interview scores related to the use of counting
approaches. Clearly, participants who showed better place-value understanding used
counting approaches infrequently. On the other hand, participants with weak placevalue understanding included participants who used counting approaches frequently
and others who did not do so. Figure 4.2 shows that it is possible to answer questions
like those in the interviews successfully using the less efficient approach of counting.
For example, one particular data point on the graph represents Hayden’s (l/c)
122
performance on the first interview, at which he used a counting approach 7 times,
and achieved a score of 9 out of 21. However, it is highly likely that consistent use of
counting approaches would lead to difficulties in the future if a student did not learn
to switch to using the groups of 10 inherent in the base-ten numeration system: This
point is discussed further in section 5.2.2.
4.4.3 Face-Value Interpretation of Symbols
As discussed in section 2.4.2, a face-value interpretation of multidigit
numerical symbols is very common among children who are learning about the baseten numeration system. Researchers investigating a variety of aspects of place-value
understanding have found children who believe that each digit in a multidigit number
represents only its face value, rather than groups of 10, 100, and so on. Data
collected in this study reveal such ideas among several of the participants. In
particular, Questions 3 (c), 6, 7, and 8 prompted certain participants to use a facevalue construct in answering the question. The ways that face-value ideas were used
in each question are described in the paragraphs following. Note the comments in
section 4.7, regarding the use of base-ten material to represent multidigit numbers:
Responses to other interview questions may have been influenced by face-value
interpretations of digits without this being obvious.
Question 3 (c): Interpreting block representations of three-digit numbers with
misleading perceptual cues, and comparing them with written symbols. The task set
in Question 3 (c) was similar to that in Question 8, in that it offered participants
misleading perceptual cues about how an arrangement of blocks represented a
number. The blocks were arranged so that the numbers of blocks of each size
matched the three digits in the printed numerical symbol in order from left to right,
but so that the values represented by the blocks were incorrect. For example, in the
first interview the participants were shown 1 ten, 3 hundreds, and 6 ones in order
from left to right and asked whether or not they represented the number 136.
Most participants did not initially accept the three-digit block representation
presented to them in Question 3 (c) as correct. Considering the blocks presented, an
in particular the large number of hundred-blocks, it is perhaps not surprising that
even a student who held a face-value interpretation for multidigit numbers would
agree that the blocks represented the number. However, when the researcher offered
the counter-suggestion that each digit could in fact represent the number of blocks
123
presented, some participants did accept the idea, indicating some willingness to
accept a face-value interpretation:
Interviewer: Do these blocks [1 ten, 3 hundreds, 6 ones] show that number [136]?
Terry (l/c):
Well I already know [that they do not], ‘cos there’s a thousand [sic] in this
[top 2 hundreds] and there’s a hundred in this [lowest hundred-block] …
Interviewer: Mmm. So is that [block arrangement] the same as that [symbol on card]?
Terry:
No.
Interviewer: Right, OK. Well let me just ask you another question, then. Could that ‘1’ [on
card] be for that [ten-block] and that ‘3’ be for those three [hundred-blocks]
and that ‘6’ be for those six [one-blocks]?
Terry:
Oh yes! It does add up to that, does it?
Interviewer: Oh, Right.
Terry:
‘Cos it’s a hundred [points to ten-block], thirty [3 hundreds], six [6 ones]. Yes.
(I1, Qu. 3c)
Michelle (l/b) also initially rejected the face-value interpretation of the block
arrangement, but then offered her own, equally incorrect, block arrangement. She
apparently was not content to agree that 3 hundreds could represent the ‘3’ in ‘136,’
and changed the hundred-blocks for 3 tens. However, she left the 1 ten to stand for
the ‘1’ digit. Later she accepted the researcher’s counter-suggestion that the initial
block arrangement did match the written symbol.
Question 6: Comparing pairs of two-digit and three-digit numbers. Questions
6 (a) and 6 (b) required participants to compare two pairs of printed numerical
symbols. The numbers in Question 6 (a) were two-digit numbers, such that the
smaller number had a ones digit that was larger than either digit in the larger number;
the pairs were 27 and 42 in Interview 1 and 38 and 61 in Interview 2. The numbers
for Question 6 (b) were three-digit numbers, that had the same digits, with the tens
and ones swapped; the pairs at Interview 1 were 183 and 138, followed at Interview
2 by 295 and 259. The intention of these questions was to target face-value
interpretations of symbols, as a face-value interpretation should lead a participant to
choose the smaller number in Question 6 (a), and to state that numbers in Question 6
(b) were equal. As in other questions, the researcher offered face-value countersuggestions to participants who gave the correct answer, to test the stability of their
beliefs. Accepted counter-suggestions are indicated in the summary of face-value
interpretations in Table 4.10 by parentheses in the relevant cells of the table.
124
Several participants provided face-value interpretations when answering
Question 6, without any counter-suggestion being offered. For example, Jeremy (l/b)
stated that 38 was bigger than 61 immediately on being asked:
Jeremy:
[Points to ‘38’]
Interviewer: What number is that?
Jeremy:
38.
Interviewer: And why is 38 bigger?
Jeremy:
Because it’s got a … 3 tens and 8 ones.
Interviewer: All right, and what’s the other number?
Jeremy:
61
Interviewer: And which one’s bigger?
Jeremy:
38.
Interviewer: That’s got 3 tens and 8 ones. And what’s this one [‘61’] got?
Jeremy:
6 tens and 1 ones.
(I2, Qu 6a)
It is interesting that although Jeremy could correctly state the name of each
digit’s place, he ignored these labels in favour of a face-value interpretation of each
individual digit. A second example shows Terry (l/c) explaining why he believed that
259 and 295 were equal:
Interviewer: Can you tell me which of these two [‘259’ & ‘295’] is bigger?
Terry:
You’re trying to trick me, aren’t you?
Interviewer: Well, I might be able to Terry.
Terry:
Well, they are both bigger.
Interviewer: They’re both bigger? They’re both the same?
Terry:
Yep.
Interviewer: And why are they both the same?
Terry:
259, 295.
Interviewer: So why are they the same? That doesn’t sound the same.
Terry:
If you just turn around the ‘5’ and put the ‘9’ there, it’d be 259.
(I2, Qu 6b)
Question 7: Explaining referents for the digits in two-digit written symbols.
Questions 7 and 8 were written purposely to target participants’ understanding of
two-digit written symbols, and to identify participants who held either face-value
interpretations of written symbols or correct grouping interpretations. In each
125
question participants were asked to count a set of between 10 and 40 objects, to write
the written symbol for that number, and then to say which objects were represented
by each digit.
The results of Question 7 initially showed a considerable number of
participants who apparently held a face-value interpretation for the two-digit written
symbols involved (see Table 4.10). At the start of the question all participants easily
counted the objects and wrote the correct symbol for the number counted. The
researcher asked each participant if the number the participant had written
represented the entire group of objects, and most participants agreed that it did.
When asked about the referents for each digit, many participants indicated objects
that corresponded with only the face value of each digit. If that was the case, the
researcher asked them about the remaining objects: In Interview 1 there were 18 out
of 24 sticks left over, and in Interview 2 there were 27 out of 37. The following
excerpt is typical of transcripts of participants holding the face-value construct:
Interviewer: Does this part [‘4’] of your 24 have anything to do with how many sticks you
have?
Clive (l/b):
[Frowns, nods]
Interviewer: Can you show me?
Clive:
[Separates four sticks to his left]
Interviewer: Does this part [‘2’] of your ‘24’ have anything to do with how many sticks
you have?
Clive:
[Puts out two sticks]
Interviewer: [Moves two sticks so they are above the ‘2’] So this ‘2’ is for two and then we
have another four [puts four sticks above ‘4’]. What about those [remaining
sticks] there?
Clive:
They are the leftovers.
Interviewer: You said that this number [‘24’] was for all of the sticks. Do you still agree
with that?
Clive:
[Nods]
Interviewer: All right, but you are saying now that the ‘4’ here [points to symbol] is for
those four [points to sticks] and the ‘2’ [symbol] is for those two [sticks] …
Clive:
And they’re left over … by themselves.
126
(I1, Qu. 7b)
Clive’s answer that the remaining 18 sticks were “leftovers” is typical of
responses of many participants who apparently held a face-value interpretation for
two-digit numbers. Considering that it is almost certain that the participants invented
the ideas themselves, the similarity between responses such as the following is quite
remarkable:
Craig:
Um. Oh, they’re extras.
(I1, Qu. 7b)
Michelle:
They’re just extras.
(I1, Qu. 7b)
Terry:
They’ll be left out.
(I1, Qu. 7b)
Nerida:
They’re left over.
(I1, Qu. 8b)
Jeremy:
They stay up because they’re not in there.
(I2, Qu. 7b)
Kelly:
Um, they don’t stand for any of them.
(I2, Qu. 7b)
Simone:
Those don’t count.
(I2, Qu. 7b)
Amanda:
Well they’re nothing then if that’s how that is.
(I1, Qu. 7b)
Amy:
Um, well, they would but they’re not included in that, um, these things.
(I1, Qu. 7b)
This collection of responses is considered important, as it reveals an aspect of
the participants’ beliefs about how symbols represent numbers that is evidently
common, but has not been reported in the literature before. Discussion of these and
other responses are continued in section 4.5, in which four categories of response to
digit correspondence tasks are identified.
Question 8: Explaining referents for the digits in two-digit written symbols
with misleading perceptual cues. Question 8 added another layer of difficulty to the
tasks in Question 7. Participants were asked to share a set of objects evenly into a
certain number of groups, resulting in equal-sized groups and leftover objects that
matched the digits in the written symbol, except that the groups were not groups of
tens and ones. In Interview 1, there were 13 beads to share evenly among three cups,
resulting in three cups of beads and one left over (Figure 3.10). In Interview 2, there
were 26 counters to share evenly onto six circles, resulting in six groups with two
remaining. Reports in the research literature (e.g., S. H. Ross, 1989, 1990) describe
children choosing incorrect interpretations of written symbols in the face of such
misleading perceptual cues.
As with Question 7, there were several variations of face-value interpretation
of written symbols evident in responses to Question 8. Some participants nominated
127
a face-value interpretation without prompting by the researcher, nominating the
groups and leftover objects as referents for the digits in the written symbol. Other
participants initially did not choose these referents by themselves, but accepted them
later when the researcher suggested them. Some participants were unsure about the
researcher’s suggestion, and indicated that the face-value interpretation might be
correct, and still others rejected a face-value interpretation and gave a correct
interpretation of the digits.
The incidence of participants choosing a face-value interpretation for written
digits when faced with misleading perceptual cues was quite low (see Table 4.10).
The research literature, however, indicated that this pattern of response was quite
common. For example, S. H. Ross (1989) found that “nearly half” of the third-grade
participants in her study incorrectly chose a face-value interpretation of 26 objects
grouped in six groups and two single objects. In this study, however, even
participants who associated “remaining” ungrouped objects with the tens digit often
did not also associate the grouped objects with the ones digit: At the first interview, 5
participants chose the remaining bead as the referent for the ‘1’ in ‘13’ without
prompting; at the second interview, 2 participants chose the two single counters for
the ‘2’ in ‘26’ without prompting. On the other hand, no participant chose the three
cups in Interview 1 as referents for the ‘3’ digit for themselves, and at Interview 2
only 1 participant (Simone; h/b) initially said that the six groups were represented by
the ‘6.’
With prompting by the researcher a few participants were willing to accept
the face-value interpretation for the written digits suggested by the grouped objects.
However, even those participants who did accept the incorrect suggestion were
generally still reluctant to agree completely with the idea. In the following excerpt,
Yvonne (h/c) was clearly not totally convinced that the suggested face-value
interpretation was correct:
Interviewer: Let me say something to you: Some people would say that the ‘3’ is the three
cups and the ‘1’ is that one [bead]. Now is that right?
Yvonne:
[Nods slowly]
Interviewer: You look a bit doubtful. Do you think it might be, or you think it is, or you are
sure it is, or … what do you think?
Yvonne:
I think it is.
(I1, Qu. 8b)
128
Interpretation of digits in multidigit numbers is an important component of
understanding the base-ten numeration system. Section 4.5 includes more detailed
analysis of participants’ explanations for the meanings of the digits in two-digit
numbers, and descriptions of four categories of response to digit correspondence
questions.
Summary of the occurrence of face-value interpretations of symbols.
Table 4.10 indicates the incidence of face-value thinking in participants’
responses to Questions 3 (c), 6, 7, and 8, as described in this section. Each “x” in the
table represents a response to a question in which the participant finished answering
the question with a face-value interpretation of numbers. The criterion of noting the
participant’s final answer is adopted here to take into account the fact that in many
cases participants gave several differing answers to a question in the course of the
researcher’s questioning. The table reflects the considered response of each
participant after being questioned, rather than the initial response, or a response that
the participant gave in passing that but later denied or contradicted. Note that
instances in which a participant accepted a face-value counter-suggestion from the
researcher are included in Table 4.10, and again in the summaries of incidences of
face-value interpretations in Table 4.11, as indicated by parentheses. However, these
instances are not counted in the overview of approaches in Table 4.12. Previously
published accounts of digit correspondence tests do not include the effects of
researchers’ counter-suggestions, and so to enable comparison between this and other
studies the same method is applied. By accepting counter-suggestions participants
indicated a certain level of uncertainty in their minds about numbers, however,
supporting conclusions of this study that much of the participants knowledge about
numbers was quite tentative, and still being constructed (section 5.4).
129
TABLE 4.10. Incidence of Face-value Interpretations for Written Symbols after
Selected Interview Questions
Participant
Interview
3c
6a
Question
6b
7
8
High/ Blocks
Amanda
Craig
John
Simone
High/ Computer
Belinda
Daniel
Rory
Yvonne
Low/ Blocks
Clive
Jeremy
Michelle
Nerida
Low/ Computer
Amy
Hayden
1
2
1
2
1
2
1
2
x
(x)
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
x
x
(x)
(x)
(x)
x
(x)
x
x
x
x
(x)
x
x
(x)
x
(x)
x
(x)
(x)
(x)
x
x
x
x
x
x
x
x
(x)
x
x
(x)
(x)
x
x
x
x
(x)
(x)
x
x
(x)
x
x
x
Terry
x
Note. “x” indicates the existence of a face-value interpretation at the conclusion of the participant’s
response. Parentheses () indicate that the participant did not volunteer a face-value interpretation, but
accepted a face-value suggestion made by the researcher.
Kelly
(x)
x
More information about responses indicated in the last two columns of Table
4.10 are included in Table 4.13, which indicates a range of responses to Questions 7
and 8, including face-value interpretations for written symbols. There is evidence
that face-value thinking evident in the answers to Questions 7 and 8 is at one end of a
continuum of responses to digit correspondence questions; this is discussed further in
130
section 4.5. The summary of face-value interpretations used by members of each
group in Table 4.10 shows that low-achievement-level participants initiated facevalue interpretations without the researcher’s suggestion 10 times as often as highachievement-level participants did. There are some differences between blocks and
computer groups, but these appear to be related to differences of individual members
of each group.
TABLE 4.11. Use of Face-Value Interpretations of Symbols by Each Group
High
Low
Total
Blocks
3 (1)
19 (9)
22 (10)
Computer
0 (3)
11 (5)
11 (8)
Total
3 (4)
30 (14)
33 (18)
Note. Values not in parentheses represent incidents of face-value interpretations initiated by
participants. Values in parentheses represent face-value interpretations suggested by the researcher
and accepted by participants.
Figure 4.3 shows that, in general, participants who adopted face-value
interpretations of symbols achieved fewer place-value criteria than participants who
did not do so. This is not surprising, given the fact that face-value interpretations are
incorrect. Nevertheless, there were incidents of participants achieving high scores at
interviews who used face-value interpretations during interviews, supporting reports
in the literature indicating that this particular erroneous idea about numbers is quite
prevalent of among students of this age.
131
21
20
19
18
18
17
17
15
15
Place-Value Criteria Achieved
18
14
14
13
12
12
10
9
9
9
8
6
8
7
7
6
6
4
3
3
0
0
1
2
3
Incidence of Face-Value Interpretations of Symbols
Figure 4.3. Interview scores compared to use of face-value interpretations of
symbols.
4.4.4 Summary of Approaches to Interview Questions
Previous tables in this chapter (Table 4.6, Table 4.8, & Table 4.10)
summarise the incidence of grouping approaches, counting approaches, and facevalue interpretations, respectively. Each of these tables shows thinking about
numbers demonstrated by each participant at each interview. Table 4.12 shows a
summary of each of the three earlier tables, to assist in comparing approaches
revealed by the interview data.
132
TABLE 4.12. Incidence of Approaches Adopted for Selected Interview Questions
Participant
Amanda
High/Blocks
Craig
John
Simone
Belinda
High/Computer
Daniel
Rory
Yvonne
Clive
Low/Blocks
Jeremy
Michelle
Nerida
Amy
Low/Computer
Hayden
Kelly
Terry
Interview
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
I
II
Groupinga Countingb Face-valuec
5
2
0
7
0
0
7
0
1
10
0
0
9
2
0
8
1
0
1
1
1
6
2
1
10
0
0
9
0
0
7
0
0
8
2
0
10
0
0
10
0
0
5
1
0
6
1
0
0
2
3
1
5
1
0
3
4
0
1
4
0
1
2
1
2
3
1
4
2
2
4
0
0
5
1
1
5
2
3
7
1
1
4
0
0
7
0
0
6
3
3
1
3
5
1
1
Scored
15
17
18
20
17
18
7
14
19
19
17
20
18
18
17
13
7
6
6
6
3
6
8
14
8
10
9
12
4
6
9
14
Note. aGrouping approaches were noted in responses to 11 questions (Table 4.6).
b
Counting approaches were noted in responses to 11 questions (Table 4.8).
c
Face-value interpretations were noted in responses to 5 questions. The count of face-value incidents
does not include instances where suggestions by the researcher were accepted (Table 4.10).
d
Score represents the number of criteria achieved at each interview (Table 4.2); maximum possible
score per cell in Score column is 21.
Table 4.12 shows a summary which illustrates remarks made earlier about
differences between high-achievement-level and low-achievement-level participants:
In general, high-achievement-level participants adopted grouping approaches more
often and counting approaches and face-value interpretations less often than lowachievement-level participants. It also appears that high-achievement-level
participants’ understanding of the grouped aspect of multidigit numbers was related
to the fact that they rarely adopted either inefficient counting approaches or incorrect
face-value interpretations of symbols. The relative instability of number conceptions
133
of low-achievement-level participants in particular is addressed in the following
section.
4.4.5 Changeability of Participants’ Number Conceptions
One prominent feature of the interview data is the observation that on many
occasions some participants repeatedly changed their answers to questions as the
researcher continued to probe the reasoning behind their answers. Participants who
were unsure about the meanings of numerical symbols and block representations of
numbers often demonstrated thinking that was characterised by a willingness to
consider a range of ideas, apparently in an attempt to make sense of numbers and
numerical symbols. Often the opinions of these participants appeared not to be
completely formed, and were readily influenced by the researcher’s questions and
suggestions, including successive suggestions that contradicted each other. The
processes used by these participants to make sense of numbers match constructivist
ideas of learning; they used new information presented to them to compare with their
existing ideas about numbers, rejecting ideas that did not fit, and accepting others.
In the following transcript, Jeremy (l/b) compared printed symbols for 27 and
42. His initial response was that 27 was larger, apparently based on a face-value
interpretation of the digits, the ‘7’ being the largest digit present in the two numbers.
Interviewer: Can you tell me which of these numbers [‘27’ & ‘42’] is larger?
Jeremy:
That one. [‘27’]
Interviewer: All right, what is that number?
Jeremy:
27.
Interviewer: Okay, and how do you that number is bigger than the other one?
Jeremy:
Because it’s only got a ‘4’ in front of it.
The researcher twice attempted unsuccessfully to appeal to Jeremy’s
knowledge of the counting sequence, firstly by mentioning the verbal names for 27
and 42, and then secondly by asking which number would be reached first when
counting:
Interviewer: Uh-huh. What’s that number there?
Jeremy:
— 42.
Interviewer: 42. And that’s 27, and 27 is bigger because of the ‘7’?
Jeremy:
[Nods]
134
Interviewer: Uh-huh. If you were counting and you were going to count up to a hundred,
say. Which one of those numbers would you come to first, 27 or 42?
Jeremy:
That one. [‘42’]
Interviewer: 42 because it’s … smaller is it?
Jeremy:
[Nods]
Only when the researcher suggested that 27 is larger because it is in the 20s
and 42 is in the 40s did Jeremy change his answer:
Interviewer: Uh-huh. All right. Someone said to me that this comes first because it’s in the
20s and that one comes later because it’s in the 40s. What do you think?
Jeremy:
That one comes first. [‘27’]
Interviewer: So 27 comes first? So you agree with them that the 20s are first and then the
40s?
Jeremy:
Yes.
Interviewer: All right, so do you want to change your answer? You’re now saying this one
is smaller?
Jeremy:
Yeah, and that one [‘42’] is bigger.
Interviewer: All right 27 is smaller and 42 is bigger. And how do you that 27 is smaller?
It is interesting that when the researcher asked Jeremy to explain how he
knew that 27 is smaller than 42, Jeremy did not merely repeat the researcher’s earlier
suggestion about the counting sequence, but instead referred to the first digit of each
symbol:
Jeremy:
Because it’s got a ‘2’ in front of it.
Interviewer: All right and that one has got?
Jeremy:
A ‘4’ in front.
Interviewer: A ‘4’ in front. All right, well what about this ‘7’? ‘Cos you said the ‘7’ was
bigger before. What do you think?
Jeremy:
‘7’s bigger.
Interviewer: Right. So does that make that one bigger? Or is it still smaller?
Jeremy:
Still smaller.
Interviewer: Right, even though it’s got a ‘7’? Even though the ‘7’ is bigger than the ‘4’?
This is … 27 is still smaller?
Jeremy:
[nods]
(I1, Qu. 6a)
135
Another example of a participant attempting to use different pieces of
information to answer a question is provided in the following excerpt, in which the
researcher had just suggested to Hayden (l/c) that 27 might be larger than 42 because
of the digit ‘7.’ In explaining why 42 was larger, Hayden appealed to evidence from
the respective sums of the digits:
Hayden:
Because um … ‘cos if when that makes 6 [‘42’] and that [‘27’] makes 9.
Interviewer: And 9 is bigger than 6 isn’t it? So does that mean this [‘27’] is bigger?
Hayden:
No.
Interviewer: It’s not bigger? Even though 9 is bigger than 6?
Finding that the sums of the respective face values did not confirm his
answer, Hayden switched to a counting approach, referring to the relative order of 27
and 42 in the counting sequence:
Hayden:
No, because if you count to 40 it takes longer. And if you count to 20 it takes
…
Interviewer: … less time?
Hayden:
(I1, Qu. 6a)
Yeah.
In an extended series of questions Terry (l/c) was questioned about 27 and 42
(see Appendix N for a full transcript). In his response, Terry called on a range of
knowledge he had about numbers and attempted to apply it to the question. In a
series of answers that changed in response to the researcher’s questions, Terry stated
that
1.
42 was larger than 27, because 42 is even;
2.
42 was larger than 57, because 42 is even;
3.
26 was larger than 42, because the ‘6’ was the largest digit;
4.
42 was larger than 26, because 42 is in the 40s and 26 in the 20s;
5.
42 was larger than 57, because 42 is even; and
6.
57 was larger than 42, because it is in the 50s.
Evidence of participants changing their minds when answering interview
questions is discussed further in section 5.4.
4.5
Digit Correspondence Tasks: Four Categories of Response
In questions 7 and 8 the interviewer asked participants specific questions
about values represented by digits in two-digit numbers. Because the quantities
136
represented by the digits of multidigit numbers is at the heart of the place-value
system, this type of question is regarded by other authors as quite critical for
revealing place-value understanding (e.g., S. H. Ross, 1989, 1990). This section
addresses the range of thinking revealed by participants’ responses to these
questions. Interview transcripts show a hierarchy of participant responses, that varied
in accuracy in interpreting two-digit written symbols. Four categories of thinking are
proposed in this section, with examples of each one provided from interview
transcripts.
4.5.1 Category I: Face-Value Interpretation of Digits
The type of response to digit correspondence questions showing the lowest
level of thinking about two-digit numbers is a face-value interpretation of digits,
defined here as Category I. Category I thinking was evidenced by participants’
statements that each digit represented only its face value, and that remaining objects
in the set represented by the two-digit symbol as a whole were not represented by
either digit. Examples of Category I thinking have been provided earlier (section
4.4.3), including a number of statements indicating the belief that not all objects were
represented by the two digits. This idea may set up a paradox for the student to
resolve: The two-digit symbol represents the entire set of objects, but the sum of the
referents for the two digits does not equal the same amount, meaning that some
objects are somehow without representation in the symbol. This problem is
overcome if a participant adopts a Category II response.
4.5.2 Category II: No Referents For Individual Digits
Category II responses indicated that a participant accepted the two-digit
symbol as representing the entire set of objects, but rejected the idea that each digit
had separate referents, on the basis that some objects would be left out. The
following transcript excerpt clearly shows a Category II response from Hayden (l/c):
Interviewer: Does this part [‘7’] of your ‘37’ have anything to do with how many sticks
you have?
Hayden:
No.
Interviewer: I doesn’t? OK, can you tell me what that ‘7’ means?
Hayden:
It’s a part of 30 … it’s a part of like in 30 it’s a part like … you count to 30
and then you count seven more and it ends up 37.
137
Interviewer: And what does this ‘3’ here mean?
Hayden:
It’s up … it’s up to 30 … like if you count up to 30.
Interviewer: Uh-huh. Can I show you something? If we have seven sticks like that [puts out
seven sticks], could we say that ‘7’ is for seven like that?
Hayden:
No, because that … that’s not like [picks up three sticks] … that’s only 11
[sic].
Interviewer: Why have you got those three? That’s for the ‘3’ is it?
Hayden:
No, those aren’t for the ‘3.’
Interviewer: It’s not for ‘3’? So the seven is not for ‘7’ either?
Hayden:
No, because it doesn’t make um 37. It only makes 11 [sic].
Interviewer: But the whole number written down like that is for all of them?
Hayden:
Yep.
Interviewer: But if you take just the seven it’s not … you can’t take part of them and say
that part is for that?
Hayden:
No.
(I2, Qu.7b)
Similar ideas are evident in the following three responses to the question
“What about the remaining objects?” asked after a participant initially gave a facevalue interpretation for the written digits:
Kelly (l/c):
Um, they [individual digit symbols] don’t stand for any of them … If they’re
[the two digits] joined together, both of the numbers are for all of them.
(I2, Qu. 7b)
Jeremy (l/b): Put them together and it makes the number.… You put them all in together,
then you know what number, so you write them down and you get the number
with the sticks.
Amy (l/c):
(I1, Qu. 7b)
Yeah but you can’t make it though, just out of like … three [separates three
sticks] … like out of that [seven sticks]. ‘Cos then it wouldn’t be 37 still
though.
(I2, Qu. 7b).
It appears that responses such as those quoted here represented the
participants’ rejection of face-value interpretations of multidigit symbols. In giving
such a response, the participants apparently recognised that each digit could not
represent only its face value and still be consistent with the meaning given to the
entire two-digit symbol. In trying to come to terms with the apparent contradiction of
their view, participants exhibiting Category II responses sometimes provided quite
creative ideas about how to make sense of the symbols. For example, at his second
138
interview, Jeremy (l/b) indicated that the “extra” objects must be somehow recorded
within the two-digit symbol ‘37,’ though no symbol for them could be seen:
Jeremy:
It’s got all [ruffles all the sticks while talking] … the sevens in here and the
threes.… They’re [the remaining sticks] in there too.
Interviewer: They’re in there too?
Jeremy:
Yeah.
Interviewer: Right. So you know they’re part of this number.
Jeremy:
[Nods]
Interviewer: Where are they written down, though?
Jeremy:
In here [points to space between the three and the seven sticks].
(I2, Qu. 7b)
Terry (l/c) gave a category I response that was similar to category II, in that
he tentatively offered a suggested explanation for no referent being visible for the
remaining sticks. Terry appeared to suggest that, after taking out a set of three and a
set of seven from a set of 37 sticks, all the sticks remaining could somehow be
represented by the two digits ‘3’ and ‘7’:
Terry:
They got gave away to sevens and threes, I suppose.
(I2, Qu. 7b)
Amy (l/c) made another suggestion, indicating that she still believed in a facevalue interpretation of the digits, but that there was another possible reason why the
extra sticks were apparently not recorded in the symbol. Her idea seemed to be that
the entire group was recorded by the two-digit symbol, but that if each digit was
considered in turn, the objects represented by that digit were temporarily isolated
from the rest of the group:
Interviewer: But what about this ‘3’?
Amy:
It means three [picks up three sticks] but it still won’t make 37.
Interviewer: Won’t it?
Amy:
No.
Interviewer: How does that work? Because you said that number is for all the sticks.
Amy:
Yeah and it includes these ones.
Interviewer: Yes.
Amy:
But … when they’re out of a group it means they’re not part of the group.
Interviewer: Sorry, which ones are not part of the group?
Amy:
These ones right now [points to the groups of three and seven sticks].
Interviewer: They’re not part of the group?
139
Amy:
Yeah right now ‘cos they’re out of the group. And ‘3’ means 3 and these three
[sticks] means 3, and ‘7’ means 7 and this seven [sticks] means 7.
Interviewer: Right. But what about these here? [Points to remaining 27 sticks]
Amy:
They’d mean how many there are now. And these [remaining sticks] are still
in a group because they haven’t left the group.
Interviewer: Right. They haven’t left the group.
Amy:
Yeah like … you tooken some away …
Interviewer: Right.
Amy:
… I suppose.
(I2, Qu. 7b)
It appears that Category II responses indicate an intermediate stage of placevalue understanding possessed by some participants, between believing that each
digit represents only its face value (Category I), and understanding that a tens digit
represents a number of collections of 10 units (Category III or IV). Evidence for this
idea comes from the fact that several of the participants who gave a face-value
interpretation for the digits in the first interview changed their responses to Category
II responses at the second interview.
4.5.3 Category III: Correct Total Represented by Each Digit, but Tens not
Explained
In a Category III response the participant knew that the tens digit represented
the remaining objects, once the referents for the ones digit were removed, but could
not explain why that digit represented a number of objects greater than its face value.
In the following excerpt, Yvonne (h/c) indicated that the ‘2’ in ‘24’ represented all
the sticks apart from the four represented by the ‘4,’ and knew that there were 20 of
them, but could not explain the connection between the digit ‘2’ and 20:
Interviewer: Can you explain that for me, ‘cos that’s just a ‘2’ isn’t it? — Does this ‘2’ here
stand for all of those, or just some of them?
Yvonne:
All of them.
Interviewer: How does ‘2’ stand for so many? Can you explain that?
Yvonne:
[Shakes head]
Interviewer: But you’re sure it does stand for that many? Do you know how many there are
here?
Yvonne:
20.
(I1, Qu. 7c)
140
Daniel (h/c) also had difficulty explaining the relationship between the tens
digit and the number of objects to which it referred. He proposed an interesting
explanation based on the efficiency of writing just a ‘2’ instead of the digits ‘20’
before the ones digit, but like Yvonne did not connect the ‘2’ with 2 tens, despite a
series of questions from the researcher, some of which are shown in this excerpt:
Interviewer: Can you tell me why that is a ‘2’ and that’s standing for all those?
Daniel:
— They have to uh be ‘2’ instead of like being ‘20’ then a ‘4’ or otherwise it
would be two hundred and four.
Interviewer: — Uh-huh, but why do you write ‘2’ if it’s 20? — Can you explain it?
Daniel:
Uh, because there’s … I forget … there’s a ‘2’ and there’s a ‘0’ at the end so
they just wanted it, just put it as a um ‘2’ to make it quicker?
(I1, Qu. 7b)
Responses such as those from Yvonne and Daniel indicate knowledge of
numbers that is more advanced than a face-value construct, but still do not meet the
criteria for a conventional understanding of multidigit numbers, Category IV,
described next.
4.5.4 Category IV: Correct Number of Referents, Tens Place Mentioned
Category IV includes responses stating a correct number of objects for each
digit, explaining that the tens digit represents the number of groups of ten. The
following transcript excerpt shows that Rory (h/c) knew what each digit in ‘13’
represented, even in the face of misleading cues of three cups and one remaining
bead (see Figure 3.10):
Interviewer: Does this part [‘3’] of your ‘13’ have anything to do with how many beads
you have? Can you show me?
Rory:
Yes. [Takes three out of one cup]
Interviewer: All right, that’s a good answer. Let’s put them back in there again. Now this
part [‘1’] of your ‘13,’ does that have anything to do with how many you have
here? Can you show me?
Rory:
[Takes out 10 beads]
Interviewer: — OK. Can you explain to me how that [‘1’] stands for these [10 beads] here?
Rory:
‘Cos it’s 1 ten.
(I1, Qu. 8b)
141
4.5.5 Summary of Responses to Digit Correspondence Tasks
A summary of the categories of response demonstrated by participants at both
interviews is provided in Table 4.13. Again, there is clear evidence of the generally
superior place-value understanding of the high-achievement-level participants.
TABLE 4.13. Response Categories for Interview Digit Correspondence Questions
Question 7
Question 8
Participant
Group
Interview 1
Interview 2
Interview 1
Interview 2
Amanda
High/Blocks
III
III
II
III
Craig
III
IV
I
IV
John
IV
IV
IV
III
Simone
II
II
I
I
IV
IV
IV
III
Daniel
III
IV
IV
III
Rory
IV
IV
IV
IV
Yvonne
III
III
I
II
I
I
I
I
Jeremy
I
I
I
I
Michelle
I
I
II
II
Nerida
I
III
I
III
I
II
II
II
Hayden
I
II
II
II
Kelly
II
I
II
II
Terry
I
I
I
III
Belinda
Clive
High/Computer
Low/Blocks
Amy
Low/Computer
Note. Categories: I – face-value interpretation of digits; II – no referents for digits; III – correct total
for each digit; IV – referents for tens digit correctly explained.
Table 4.13 shows that several participants improved in the accuracy of their
response to Questions 7 and 8 from the first to the second interview, though others
achieved scored less in the second interview. It is also interesting to note that for
some participants their responses to Question 7 were quite different from their
responses to Question 8. Response categories of all participants as a group are
summarised in Table 4.14.
142
TABLE 4.14. Summary of Digit Correspondence Response Categories
Category
Interview 1
Interview 2
I
44
25
II
22
25
III
13
28
IV
22
22
Table 4.14 shows that, overall, participants in the study improved in
responses to digit correspondence questions between the two interviews; fewer
participants gave Category I responses and more gave Category IV responses at the
second interview, compared to the first. These data are compared in Table 5.1 with
figures for performance on similar tasks quoted by S. H. Ross (1989).
4.6
Errors, Misconceptions, and Limited Conceptions
One clear pattern in the data from both interviews and teaching sessions was
the large number of errors, misconceptions, and limited conceptions evident in
participants’ responses. In this section these errors are categorised and described
separately: Counting Errors (section 4.6.1), Blocks Handling Errors (section 4.6.2),
Errors in Naming and Writing Symbols for Numbers (section 4.6.3), and Errors in
Applying Values to Blocks (section 4.6.4).
4.6.1 Counting Errors
Counting sequence errors.
The use by participants of counting approaches in responding to interview
questions is described in section 4.4.2. The use of counting approaches to work out
answers to questions involving multidigit numbers requires accurate use of counting
sequences for success. Difficulties in this area for some participants led to problems
in answering interview questions. One common problem was in naming the next
decade in a counting sequence. For example, Kelly (l/c) used the following sequence
when counting one-blocks: “… 40, 51, 52, 53, 54, 55, 56, 57, 58, 59, 30, 31, 32” (I1,
Qu 1b). Another common mistake of this sort is illustrated in this sequence used by
Terry (l/c) when counting tens: “10, 20, 30, 40, 50, 60, 70, 80, 90, 20” (I1, Qu 1c).
Terry evidently knew that this was not correct and restarted the count, only to repeat
the same error. It is likely that children sometimes make this error because of the
143
similarity of the two number name sequences “seventeen, eighteen, nineteen” and
“seventy, eighty, ninety.”
Counting errors were revealed in responses to Question 4 in both interviews,
which required participants to skip count by 1, 10, or by 100 with two-digit or threedigit numbers. These tasks proved to be among the most difficult for the participants
and resulted in a low level of success (see Table 4.2). Four common errors made by
participants answering Question 4 are illustrated in the following transcript excerpts:
(a) Mistakes in the new number at a change of decade or change in the
number of hundreds:
Kelly (l/c):
73, 72, 71, 60, 69, 68, 67, 66, 65, 64, 63, 62, 61, 50, 59, 58.
Yvonne (h/c): 273, 283, 293, 203, 223, 233, 243, 253, …
Terry (l/c):
(I1, Qu. 4a)
(I2, Qu. 4c)
681 … 671, 661, 651, 641, 631, 621, 611, 501, … 591, 581, 571, 561, 551,
541, 531, 521, 511 … 491, 481, 471, 461 …
(I2, Qu. 4d)
(b) Omitting numbers, especially numbers with a “teen” component, or 1 ten:
Yvonne (h/c): 52, 62, 72, 82, 92, 102, 122, 132, 142, 152 …
(I2, Qu. 4b)
Daniel (h/c): 65, 75, 85, 95, 105, uh 125, 135, 145.
(I1, Qu. 4b)
(c) Using an incorrect increment or decrement when asked to count on or
back by 10:
Hayden (l/c): 75, 80, 85, 90, 95, 100, 105, 110 …
(I1, Qu. 4b)
Amanda (h/b): 452, 562, 672, 892, … I don’t know the one after that.
(I1, Qu. 4c)
Michelle (l/b): 204, 205, 206, 207, 208, 209, 210.
(I2, Qu. 4c)
(d) Omitting the ones part of each number:
Simone (h/b): [Asked to count by 10 from 463] 270, 270, 280, 290, …
(I1, Qu. 4c)
Nerida (l/b): [Asked to count by 10 from 681] 670, 660, 650, 640, 630 …
(I2, Qu. 4d)
Lack of knowledge of larger numbers.
The difficulties that some participants had with counting sequences were
compounded by a lack of knowledge about larger numbers, and a lack of familiarity
with hundred-blocks, or both. For example, Terry (l/c) evidently knew the name of
the hundred-block, but did not know how to read 2 hundred-blocks, counting them as
“100, 1000.” Amy (l/c) made similar errors when trying to count 16 tens, clearly
being unsure of how to count beyond 100. She rapidly ran out of number names for
places as she tried to apply a new place name to each new block:
144
Amy:
10, 20, 30, 40, 50, 60, 70, 80, 90, 100 … hmmm … 100, 1000, about 2000.
Um, infinity. [Laughs] Just gets up to infinity and then it gets harder.
(I1, Qu. 1c)
The following transcript excerpt shows an attempt to count a block
arrangement that was hindered by lack of knowledge of larger numbers. Kelly (l/c)
was attempting the task of reading 5 hundreds, 13 tens, and 2 ones, but was unable to
complete the task successfully because of difficulties with both the values
represented by the different blocks and the sequence of three-digit cardinal numbers:
Kelly:
[Counts hundred-blocks] 100, 200, 300, 400, 500 … I’ve worked out a easy
way. There’s a hundred there [counts out 10 tens and puts them together]
there’s a hundred there … so it’s 100, 200, 300, 400, 500, [counts group of 10
tens] 600. [Counts individual “ones” on the next ten-block but miscounts, then
moves it next to the 10 tens] 207, [puts the next ten-block across as she counts
each individual “one” on it] 208, 209, 300, 301, 302, 304 …
The researcher stopped her and asked her to restart at 600:
Kelly:
Oh, go on from 600 … [counts each individual “one-block” on the ten-block]
601, 602, 603, 604, 605, 606, 607, 608, 609, 700. [Gets the next ten-block and
again counts individual “ones”] 701, 702, 703, 704, 705, 706, 707, 708, 709,
800.
Interviewer: Can you count aloud? That’s 800 now is it?
Kelly:
Yes. [Gets the next ten] 900, um 800, 900, 901, 902, 903, 904, 905, 906, 907,
908, 109 … 1000 … [puts 2 ones next to the other blocks] 1002.
(I2, Qu. 1c)
Knowledge of the sequence of cardinal numbers is fundamental to
development of understanding of the base-ten numeration system. The difficulties
illustrated here would clearly cause further difficulties in learning about the base-ten
numeration system unless they were remediated.
4.6.2 Blocks Handling Errors
General handling errors.
Mistakes made when handling blocks were very frequent during interviews
and teaching sessions. Errors reported in this section are closely related to the
counting errors described in the previous section, and to mistakes made in assigning
values to blocks (section 4.6.4). However, the errors in this section are apparently
due to mistakes made in handling the blocks, rather than to either an inability to
145
count or ignorance about the number of tens and ones in a number. Handling errors
made while using blocks to represent numbers included the following:
1.
Simone put out 5 tens and 8 ones for 48.
(h/b S3, T 8)
2.
Clive counted out 6 tens when showing 70.
(l/b S5, T 9)
3.
Michelle counted blocks to show 75, but included 8 tens. (l/b S6, T 12)
4.
When showing 75 with blocks, Nerida miscounted the first 3 tens in her
hand as “20,” finishing with 8 tens and 5 ones.
5.
(l/b S6, T 14)
When Craig and Simone showed 627 and regrouped a hundred into
tens, Craig put the 10 tens on top of the other blocks. Some blocks fell
off, unnoticed by the two children, resulting in the representation being
short by 1 ten.
(h/b S9, T 32b)
Note that the above list includes only handling errors that went unnoticed by
participants for a lengthy period. During the counting of blocks many other handling
errors were temporary, as they were checked and corrected quickly by the participant
concerned.
Trading errors.
The process of trading blocks is an important one for students using base-ten
blocks to model the subtraction and addition algorithms. The participants in the study
had learned about trading with blocks previously. This was confirmed by Amanda
who said “We do it all the time - ‘Swap the Bank’,” to which Craig responded, “I
thought that it was called ‘trade’” (h/b S5, T 14). As Amanda and Craig were from
different classes, this indicates that both teachers of participants in the study had
taught previously about trading with blocks. However, errors made by several
participants indicated that their learning of this process was far from complete.
Participants’ trading errors are described briefly in the following paragraphs, grouped
into three categories: trades to 10, trades of 10 for 1, and trades of 10 for other
numbers.
Trades to 10. One faulty idea relating to block trading that appeared in the
teaching sessions several times was that trading was done up to 10, rather than
trading 1 larger block for 10 smaller blocks. On several occasions participants were
observed to remove a ten and replace it with sufficient ones so that there were 10
ones in all. For example, if there were 5 tens and 8 ones, this error would be revealed
146
by the action of removing a ten and adding just 2 ones, to make 4 tens and 10 ones.
This was the process used in the following examples:
1.
Clive traded a ten in 255 for some ones, then counted the ones,
removing extras so that there were only 10 ones.
2.
(l/b S10, T 31a)
Amanda traded a ten in 255 for 5 ones. Later she wrote that the new
arrangement represented 260 [sic], and did not equal 255.(h/b S8, T 31a)
3.
John traded a hundred in 340 for 6 tens and wrote that there were 2
hundreds, 10 tens, and 0 ones.
4.
Clive traded a hundred in 340 for 6 tens, resulting in 2 hundreds and 10
tens.
5.
(h/b S9, T 32a)
(l/b S10, T 32a)
Daniel, when asked to trade a ten in 77, asked twice if he should add
just 3 ones.
(h/c S1, T 4a)
The last example shows a participant using the computer demonstrating the
idea that trading is done up to 10. This example shows that the trade-to-10 idea was
independent of the representational format provided to participants, at least at first.
Daniel asked about making the ones up to 10 before he had used the saw tool in
completing a task, and while he and his fellow group members were considering how
to effect the trade. However, the researcher reminded the participants that they could
use the saw tool incorporated in the software—which they had used in their initial
training session—to carry out the trade correctly. After this task, the trade-up-to-10
idea did not recur in this group. One purported advantage that the software has over
conventional base-ten blocks is that users can use electronic decomposition and
regrouping tools to produce automatic trades that are always carried out correctly; it
may be in using the electronic tools, computer participants were able to recognise the
fallacies in errors such as trade-to-ten.
Trades of 10 for 1. A number of times participants traded a ten or a hundred
for a single one or ten. For example, in Session 1, carrying out the first trading task
of trading a ten in 77, every participant in the low/blocks group attempted to trade a
ten-block for a single one-block (l/b S1, T 4a). After some discussion, the four
participants agreed that the number represented by the blocks after trading was 68.
The researcher then corrected the participants and showed them that the trade must
always be done so that the blocks swapped were equal to the original blocks. Despite
this, at the low/blocks group’s second session Jeremy again started to trade 10 from
23 for 1, until Clive corrected him:
147
Jeremy:
[Moves a ten away]
Clive:
[To Jeremy] Swap one of the tens for a one. [To teacher] A one?
Jeremy:
Just get a one.
Teacher:
No, it doesn’t say “a one.” It says “for ones.”
Jeremy:
Just get a one. You get a one. Just get a one.
Clive:
[Ignoring Jeremy, counts ones into his hand. Then he checks how many he
has:] 2, 4, 6, 8, 10. There. [Adds ones to the other blocks.]
(l/b S2, T 4b)
Clearly even after having the correct trading procedure explained in the
previous session, Jeremy still believed that a ten could be traded fairly for a one. This
belief recurred among members of the low/blocks group later when trading of a
hundred-block for tens was introduced, when Jeremy and Michelle both stated that a
hundred-block should be traded for a single ten-block (l/b S10, T 32a). The actions of
the participants are consistent with a view that blocks were merely counters, and that
no matter what their size, any block was equivalent to any other. This idea is
discussed further in section 5.3.
Trades of 10 for other numbers. On at least two occasions participants traded
a ten for a number other than 10 ones, and did not trade up to 10. In the first incident,
Simone (h/b) traded a ten in 77 for 7 ones. The other participants in her group all said
that trades must be done for 10 ones:
Craig:
[Quietly] 10. 10 for 10. 10, 10, 10. You swap it for 10.
Amanda:
You have to swap it for 10, ‘cos otherwise it’s not the same.
John:
Well, then [if a ten was traded for 7 ones] it’d just be 17 … no, then it’d just
(h/b S1, T 4a)
be 70 [sic]. You need 77.
When the researcher asked the group if a ten could be traded for numbers
other than 10, there was some uncertainty to start with, with John and Craig saying
that they were not sure, and Simone asserting that “We can swap it for other numbers
too. — Like um, you can swap it for 7s, and 9, and 10, and the other numbers.” In the
ensuing discussion the children all eventually agreed that a ten must always be traded
for 10 ones, “or it wouldn’t be the same.” This question did not recur with this group,
though there was a later incident when trading for a hundred in which John traded up
to 10 tens (see previous discussion).
The second example of a participant trading a ten for other than 10 ones
involved Clive (l/b), who, in attempting to use blocks to calculate 83 - 48, traded a
148
ten for 8 ones. This appears to have been related to the subtraction operation and how
it is modelled using base-ten blocks. Clive started with 8 tens and 3 ones, separated 4
tens, then removed a ten and traded it for 8 ones. He then put the 8 ones with the
removed 4 tens, making a representation for 48, leaving 3 tens and 3 ones which he
believed show the answer to be 33 (l/b, S8, T 21). When the researcher talked Clive
and Jeremy through the block transactions again, Clive said that the ten should be
traded for 10 ones, indicating that he had previously been told that, but had decided
otherwise when attempting to calculate the answer to the question.
4.6.3 Errors in Naming and Writing Symbols for Numbers
Participants made many errors in either naming or writing the symbol for a
number represented by collections of blocks. Some of these errors were due to a lack
of knowledge of names of larger numbers, such as when Clive, attempting to read the
symbol ‘932,’ said “Ninety-th … 9 … 109 … no… Can’t read hundreds; can only
read ones and tens” (l/b S10, T 31b). In other instances participants attempted to name
a number or write a numerical symbol, but applied the knowledge they had of
smaller numbers in incorrect ways. Such errors are described in the subsections
following.
Naming incorrectly concatenated number symbols.
There were two instances in teaching sessions in which participants read twodigit non-canonical block representations as three-digit numbers. In each case the
participant evidently concatenated the symbols for the number of each size of block
and then named the resulting number:
1.
Nerida said that 6 tens and 17 ones showed “six hundred and
seventeen.”
2.
(l/b S1, T 4a)
Daniel said that 1 ten and 13 ones showed “one hundred and thirteen.”
(h/c S2, T 4b)
Note that in example 1 Nerida was looking at the physical blocks, with no
written symbols available. In example 2 there were column labels available, which
may have helped Daniel to visualise the written symbol “113.”
Concatenating tens and ones names.
Similar naming errors were made by several participants who named noncanonical block arrangements using the name of the number represented by the tens,
149
followed by the name for the number of ones. As discussed further in section 5.3,
participants naming block arrangements this way were applying a method that will
work for canonical arrangements of blocks, but which gives non-standard number
names for non-canonical arrangements. This method treats each place as independent
of the other, and is evidence of the “Independent-place construct,” described in
section 5.3. The following examples of this type of error were noted:
1.
Jeremy counted 8 tens and 11 ones, and read them as “eighty-eleven.”
(l/b S2, T 4c)
2.
Yvonne looked at 5 tens and 10 ones, and read them as “fifty-ten.”
(h/c S4, T 14)
3.
Clive said that 2 hundreds, 4 tens, and 10 ones represented “two
hundred and forty-ten.”
4.
(l/b S10, T 31a)
Clive and Michelle both said that 9 hundreds, 2 tens and 12 ones
showed “nine hundred and thirty-twelve.”
5.
(l/b S10, T 31b)
The researcher asked Daniel what the next number would be after 492
in a sequence adding tens, and he answered “four hundred and tentwo.”
(h/c S10, T 41)
Errors in writing three-digit numerical symbols.
Several times participants made mistakes when writing symbols for threedigit numbers. The participants had not been taught about numbers beyond 99 in
their regular mathematics classes, and so it is not surprising that they exhibited
difficulties writing and reading them.
The errors in writing three-digit numbers usually resulted from concatenation
of values for the three individual digits; in other words, participants wrote the
symbols representing the value in each place one after the other. For example,
Jeremy, attempting to write the sequence of numbers counting in tens from 100,
wrote ‘10010, 10020, 10030, 10040’ (l/b, S8, T 24). A similar method was used by
Michelle, who wrote the number 538 as ‘500.30.8’ (l/b, S10, T 29a). It is interesting
to note that Michelle inserted full stops between the symbols for adjacent places; it
appears she believed that there should be something to distinguish each place from
the next.
An incident involving Amanda (h/b) is interesting because it shows that she
was able to tell that her first attempt at writing ‘204’ was incorrect, though she
150
needed assistance to finally write the correct symbol. In the following excerpt,
Amanda had just added 170 and 34 using blocks, and wanted to record her answer:
Amanda:
Two hundred and four. [She writes in her book ‘24,’ stops.] Whoopsies. Two
hundred and four - That’s twenty-four. How do you write that? [She looks at
the teacher, but he does not respond.] Oh, yeah. [She changes what she has
written to ‘240.’]
Teacher:
You’ve written ‘240.’
Amanda:
Oh, yeah, “zero four.” [She corrects her answer to ‘204.’]
(h/b S10, T 36)
In the examples of errors made in writing symbols described here, the
participants appeared to consider each place of the number whose symbol they were
writing separately, rather than combining the places to form a composite number
from the separate places.
Perseveration errors.
The psychological term “perseveration” refers to a response to a stimulus that
continues after the stimulus is removed. Fuson and Smith (1995) used the term to
refer to a particular type of error made by children in which they continue to use a
certain place name or value after a change of place or block value. Examples of this
error included the following:
1.
Michelle counted 3 hundreds, 6 tens, and 9 ones. She continued
counting in hundreds after 300 while counting the tens: “300, 400, 500,
600, 700, 800, 900 …”
2.
(l/b S9, T 28a)
When adding tens together, Kelly stated that the number 10 more than
100 was 200.
3.
(l/c S8, T 24)
Nerida counted 5 tens and 4 ones: “10, 20, 30, 40, 50, 60, 70, 80, 90.”
(l/b S6, T 12)
4.
John counted 5 tens and 1 one: “1, 2, 3, 4, 5, 6.”
(h/b S4, T 10)
5.
When attempting to count 5 hundred-blocks, 13 tens, and 2 ones, Amy
(l/c) counted every block as a hundred: “100, 200, 300, 400, 500,
[continues counting tens] 600, 700, 800, 900, 10 hundred, 11 hundred,
12 hundred, 13 hundred, 14 hundred, … 15 hundred, 16 hundred, 18
hundred [sic], 19 hundred, [continues counting ones] 20 hundred, 21
hundred.”
(I2, Qu. 1c)
151
Though at first glance, it appears that these participants did not know the
correct values represented by blocks of different sizes, this is unlikely to be the case.
These same participants were able to count blocks correctly at other times, and
clearly did know the name assigned to each block. It seems that what happened in
examples like those here is that the participant continued (persevered) with an
auditory counting pattern, without changing the place of the verbal number names
when counting blocks of another size.
4.6.4 Errors in Applying Values to Blocks
The largest and most diverse category of errors made by participants in the
teaching sessions was that of errors made in referring to values represented by the
blocks. These errors included referring to blocks using incorrect values, using blocks
of the wrong size to represent a certain digit, referring to the value of the tens as the
number of tens in a block arrangement, attempting to combine numbers of different
places, and perseveration errors. These types of place error are described in the
following subsections.
Size and position misunderstandings.
Some participants were confused about two aspects of blocks used to
represent different places in a multidigit number: their size and their position. The
task in Interview Question 3 (c) (see Appendix I & Appendix J) was deliberately
framed to target the incorrect idea that the value assigned to each block is determined
by its position relative to other blocks, rather than by its size. Participants were asked
to say whether a given arrangement of blocks matched a printed three-digit
numerical symbol. The blocks were arranged so that the number of blocks of each
size matched the digits of the printed symbol, in spatial order. However, the sizes of
the blocks did not correspond to the values of the places. The following transcript
shows that Jeremy (l/b) had some uncertainty in his mind about the two aspects of
the blocks, size and position:
Interviewer: Do these blocks [1 ten, 7 hundreds, 2 ones] show that number [172]?
Jeremy:
… Yes it does.
Interviewer: … So that ‘1’ is for the one there and ‘7’ is for seven and the ‘2’ for two?
Jeremy:
[Nods]
152
Interviewer: OK, what if I turn it round like that? [Arranges blocks in order of 7 hundreds,
1 ten, 2 ones.] What number is shown there? Is that still the same as that?
Jeremy:
Yeah, because it’s changed around.
Interviewer: Right, but it’s still the same number?
Jeremy:
[Nods]
Interviewer: So these blocks here show 172? Is that right?
Jeremy:
[Puts hand on top of the hundreds pile] But that’s not a ten. It’s a hundred.
Interviewer: Right. So does it show the same as that or not?
Jeremy:
No.
Interviewer: It doesn’t. All right, do you know what number this is here with the blocks?
Jeremy:
71 … seventy-hundred … 1, 2. [The correct answer was 712.]
Interviewer: Uh-huh. What if I turn it round that way [rearranges blocks as 1 ten, 7
hundreds, 2 ones from left to right].
Jeremy:
172.
Interviewer: OK, so when it’s like that it’s the same as that, but if I turn it round it’s
different?
Jeremy:
Mmmm. [Nods]
(I2, Qu. 3c)
Initially Jeremy agreed that the “face-value” representation was correct. The
researcher swapped the hundreds and tens blocks, to which Jeremy said that the
number represented was still the same. However, when the researcher mentioned the
name of the number in question he changed his mind, arguing that the hundredblocks were not tens. When the blocks were returned to their first position Jeremy
agreed that the number represented had changed back, though it seems that he was
not entirely convinced, as he merely nodded in response to the researcher’s question,
rather than verbalising an affirmative response.
The following excerpt shows a similar confusion between block sizes and
their relative positions when deciding the value they represented. Michelle (l/b)
accepted a face-value interpretation of blocks almost straight away. However, when
explaining her answer, she used block names and values that did not agree: She
referred to the 7 hundred-blocks in the middle position as hundreds, but counted
them as tens when attempting to confirm the value they represented:
Interviewer: Do these blocks [1 ten, 7 hundreds, 2 ones] show that number [172]?
Michelle:
— [Counts the pile of tens] Yes.
153
Interviewer: OK. How can you tell?
Michelle:
Because there’s one block, 1 ten [puts hand on the 1 ten]. There’s supposed to
be … 7 hundreds, [counts the 7 hundreds one by one] 10, 20, 30, 40, 50, 60,
70, [touches the 2 ones] 2.
Interviewer: OK, so this number shown here [blocks] is 172?
Michelle:
Yeah.
At this point in the interview the researcher explored Michelle’s
understanding of the block values, starting with a hundred-block:
Interviewer: What’s that block [holds up one of the hundreds blocks]?
Michelle:
Hundred.
Interviewer: All right, what’s that one [continues to hold up hundreds one at a time]?
Michelle:
200, 300, 400, 500, 600, 700.
Interviewer: Keep going.
Michelle:
700, [touches the 1 ten, now in middle position] eighty-hue … 800, [touches
the ones 1 by 1] 802, 803.
Interviewer: Right, OK, that’s fine. Thank you.
Michelle:
You changed it around.
Michelle’s comment about the order of placement of the blocks, which had by
this time been altered, prompted the researcher to probe her beliefs further about
values assigned to blocks:
Interviewer: I did … well actually, I’ll ask you about that. ‘Cos I did turn them around
[arranges blocks as they were initially]. You said before that that [1 ten, 7
hundreds, 2 ones] is the same as that [‘172’]?
Michelle:
Yeah.
Interviewer: So that’s 172. But if I change them around [7 hundreds, 1 ten, 2 ones], it’s
different?
Michelle:
Yeah.
Interviewer: OK. So it matters which way round you put the blocks.
Michelle:
Yeah.
(I2, Qu. 3c)
As with the instances of confusion about the values assigned to blocks,
confusion about block size and position also appears to have links with face-value
interpretations of digits. However, this cannot be the complete explanation, since
154
confusions described in this section are more to do with values represented by
blocks, than with values represented by digits.
Counting ten-blocks as fives or twos.
On a few occasions, participants assigned incorrect values to base-ten blocks
in answering interview questions. For example, both Kelly (l/c) and Hayden (l/c)
counted ten-blocks in 5s, implying a value of 5 for each block. Kelly counted 4 tens
and 12 ones in this way, reaching the answer 32 (4 x 5 + 12). Later in a teaching
session Kelly counted 9 tens as 45 (l/c S4, T 7c). In a similar incident, at the second
interview Hayden counted 6 tens and 7 ones and arrived at the answer 37 (I2, Qu. 1a).
When the researcher queried this answer, he changed his answer to 67, admitting that
he had been counting the ten-blocks “in fives.” Jeremy made a similar error,
apparently because of a misunderstanding about how Clive used counting by twos to
speed up the counting of blocks. Jeremy, hearing Clive count two blocks at a time,
counted 9 tens as if each represented a value of 2: “2, 4, 6, 8, 10, 12, 14, 16, 18” (l/b
S2, T 4c).
Reversing values of tens and ones.
A different sort of error in assigning values to blocks was made by Kelly (l/c)
when she was asked to show 28, and then 134, with the blocks:
Interviewer: Can you show me 28 with the blocks, please?
Kelly:
[Puts out 2 ones, then 8 tens]
Interviewer: Okay do you know another way of showing 28? Can you show me a different
way?
Kelly:
[Puts out another 2 ones & 8 tens, arranged differently] There.
Interviewer:
Kelly:
(I1, Qu. 2a)
OK, now can you show me 134 using the blocks?
[Puts out 1 hundred, 3 ones & 4 tens]
Interviewer: Uh-huh. Is there another way of showing 134, do you think, can you show
me?
Kelly:
[Puts out another hundred, on right, then 4 tens on left, & 3 ones in the
middle]
(I1, Qu. 2b)
Apparently, for some reason, Kelly had some confusion in her mind at the
time of the first interview regarding the values represented by each size of block, and
she decided to use one-blocks to represent tens, and ten-blocks to represent ones.
155
Nevertheless, whatever the difficulty she had with tens and ones it did not seem to
affect her use of the hundred-blocks, as she used the correct block to show the
hundreds digit. This may be due to the fact that the hundreds place was new to her
and also she had never used those blocks before in mathematics lessons, and so she
guessed correctly that the new block represented the new place. At the second
interview Kelly used the blocks to show the numbers in Question 2 using a correct,
canonical, representation in each case, implying that whatever misunderstandings she
had in Interview 1 were corrected during the teaching phase. This error made by
Kelly is another apparent example of a participant considering places independently,
without regard for values of places relative to each other. Further discussion of the
independent-place construct is in section 5.3.
Applying incorrect values to blocks.
Participants were observed on many occasions to refer to blocks verbally or
in writing using incorrect values. For example:
1.
Michelle, Daniel, Terry, Amy, and Kelly all read 4 tens as “4.”
(l/b, h/c, & l/c S1, T 1d)
2.
Simone stated that 8 ones followed by 2 tens showed “82.” (h/b S1, T 2a)
3.
Michelle was asked to write the blocks she had before and after trading
both tens in 21 for ones, and wrote “2 ones, 1.”
(l/b S3, T 5a)
4.
Simone wrote “36 = 3 ones 6 ones.”
(h/b S2, T 5b)
5.
Clive attempted to convince Jeremy that 51 is greater than 39. In his
explanation, he asked Jeremy to compare the digits ‘5’ and ‘3.’ Several
times Clive referred to the ‘3’ as “3 ones.”
(l/b S5, T 10)
6.
John read 9 tens and 6 ones as “960.”
(h/b S6, T 16)
7.
When attempting the task asking participants to add 31 and 28, John
held some ten-blocks in his hand, and twice referred to them as
“hundreds.”
(h/b S6, T 18).
8.
Amy read 12 tens as “112.”
(l/c S8, T 24)
9.
Jeremy counted 1 hundred and 1 ten as “101,” then 1 hundred and 2
tens as 102. Clive counted 1 more ten than 110 as “111.” (l/b S8, T 24)
10.
11.
Michelle stated that 8 hundreds and 2 tens showed “eight hundred and
two.”
(l/b S10, T 29d)
Terry stated that 14 tens showed “104.”
(l/c S10, T 32a)
156
12.
Kelly stated that 16 tens and 7 ones showed “607.”
(l/c S10, T 33)
Similar errors were evident when participants chose incorrect blocks to
represent a particular digit. For example:
1.
Belinda put out 4 tens to represent the number 4.
(h/c S4, T 13)
2.
Amy started to put out only tens when trying to represent the number
99. She stopped when Kelly pointed out that she had 39 tens.(l/c, S8, T 25)
3.
Jeremy chose 2 tens to represent 200.
4.
Clive added 8 tens to 2 hundreds and 4 tens when attempting to
represent 248.
(l/b S9, T 27)
(l/b S9, T 27)
As in the case of the previous type of error, this error indicates that the
participant had assigned an incorrect value to certain blocks. Note that in examples
(a) and (b), Belinda (h/c) and Amy (l/c) demonstrated this error, although they were
using the software, which would have indicated in the column labels and the number
window, if visible, that incorrect blocks were being used. Since this error was
observed even when contrary evidence was available to the participants, it appears
that the error is quite a common one among children of this age.
A number of times participants referred to the value represented by the
blocks when asked the number of blocks in a block arrangement. For example:
1.
Asked how many tens there would be if a ten in 62 was traded for ones,
Belinda responded several times “fifty tens.”(h/c S2, Supplementary task)
2.
Comparing representations for 73 and 29, Kelly said that there were “70
tens” and “20 tens.”
(l/c S4, T 9)
3.
Nerida said that 6 tens was “60 tens.”
4.
Asked how many tens would be needed if a hundred-block in 340 was
traded for tens, Jeremy said “a hundred of the tens.”
(l/b S9, T 28a)
(l/b, S10, T 32a)
The phrase used by Jeremy in the last example is interesting, and seems to
indicate the confusion he was feeling between the number of blocks and the value
that that number of blocks represented. By saying “a hundred of the tens,” Jeremy
may have been indicating that he meant “a hundred-worth of the tens,” rather than
“100 tens.”
In another example of applying incorrect values to blocks, several participants
were observed to combine values from different places without converting or trading
them. By combining quantities represented by blocks or digits in different places, the
157
participants were combining numbers from different places as if they were alike.
Examples included the following:
1.
Amy attempted to work out the number represented by 16 tens and 7
ones, and said it was “23 hundred.”
(l/c S10, T 33)
2.
Terry said that 15 tens and 17 ones represented 32.
(l/c S10, T 33)
3.
Daniel wrote that 14 tens and 11 ones represented 25.
(h/c, S9, T 33)
4.
Daniel and Rory both said that 41 tens and 9 ones represented 50.
(h/c, S10, T 34b)
4.7
Use of Materials to Represent Numbers
Materials used to represent numbers may be used in a variety of ways, some
of which were not intended by either their developers or teachers using them with
their students. One sub-question of this study, given in section 1.3, is “Which of
these conceptual structures for multidigit numbers can be identified as relating to
differences in instruction with physical and electronic base-ten blocks?” This section
contains descriptions of two aspects of the participants’ use of physical or electronic
materials: counting and use of trial-and-error methods.
4.7.1 Counting of Representational Materials
In order to use materials, whether physical or electronic, to represent
numerical quantities, a count must be made of the number of the various materials
present. In the case of the software used in the study, counting by participants was
not necessary, as the software keeps a continuous check of the number of blocks in
each place displayed in a text box at the head of the column; the column counters are
always visible to users. On the other hand, users of physical base-ten blocks must
count the blocks themselves when using them to represent a number. This subsection
includes descriptions of participants counting both electronic and physical blocks,
and discussions of links between participants’ favoured approach to numeration
questions and their counting of materials.
Counting justifications for answers.
As in the individual interviews, several participants referred to counting when
justifying answers to questions during teaching sessions. For example, in the
following transcript the researcher asked the high/blocks group how they could be
sure that 62 is greater than 48:
158
Teacher:
Mmm-mm. How do you know that he has more? That’s my question. I mean,
we know 62 is more than 48.
John:
Yeah.
Amanda:
You count.
Teacher:
Why is it more than 48? — Counting is OK, but supposing you didn’t have the
blocks, and we were just talking, and someone said “Well I’ve got 62, and
you’ve got 48,” How do you know 62 is bigger? How can you prove that it’s
bigger?
All:
[Many children talking together]
Amanda:
Because when you were … we knew how to count, and we’ve got bigger.
John:
Because there’s only this many, and they think it’s bigger because this,
because the ones are more [in 48], but the tens are less, and he reckons his is
bigger because the tens are more [in 62].
(h/b S3, T 8)
It is interesting to notice that John recognised for himself that the idea could
be given that the digit ‘8’ indicates that 48 is larger, but that the tens digit is of
greater value. On the other hand, Amanda referred twice to a counting approach to
decide which number is larger.
In the previous example, participants used their knowledge of the counting
number sequence to justify an answer. Counting was also used to justify answers on
occasions when participants counted physical or electronic blocks. For example, the
researcher asked members of the low/blocks group to justify their belief that 2 tens
and 8 ones could be placed in spatially different arrangements and still represent 28.
Clive responded by counting the blocks:
Clive:
10, 20, 21, 22, 23, 24, 25, 26, 27, 28.
(l/b S1, T 2a)
One participant’s preference for counting.
One particular participant, Hayden (l/c), showed a clear preference for a
counting approach at both interviews, and during the group sessions. This case is of
interest because it demonstrates how a student’s preference for a counting approach
can apparently influence that student’s use of representational materials. Hayden’s
preference for counting materials came to the author’s notice because Hayden was in
a computer group, and yet he frequently counted on-screen blocks by pointing to or
touching the computer screen.
159
Table 4.8 shows that Hayden used a counting approach more often than any
other participant did, apart from Kelly (l/c). This subsection includes a brief
discussion of Hayden’s use of a counting approach in the interviews, and evidence
that his preference for this approach had a bearing on incidents in the teaching phase
of the study. First, Hayden’s responses to a question at the second interview show his
reliance on counting to answer questions about numbers. The question required the
participant to compare 38 and 61; excerpts from the relevant transcript follow (see
Appendix O for the full transcript). To start with Hayden used a counting approach to
explain why he believed that 61 was larger:
Hayden:
Because it’s six … 38 takes shorter and 61 takes longer.
Interviewer: If you’re counting, you mean?
Hayden:
Yeah.
The researcher continued to ask Hayden about the numbers 38 and 61, asking
him specifically about the digits in the two numbers. It is interesting to note that
Hayden was unable to use the information contained in the respective digits of the
two numbers to understand which is larger:
Interviewer: OK, what about the numbers that are in [points to ‘38’ & ‘61’]? Does that tell
you anything?
Hayden:
No, it doesn’t … like it still doesn’t mean that it’s got an ‘8’ on the end and
it’s got a ‘1’ on the end [points to respective digits] … because that’s um …
like that … like if you get 1, 2, 3, 4 … like 10, 20, 30, 40 … no 10, 20, 30 and
you just count to 8 … in the 30s, like it’s only the 38.
Interviewer: Uh-huh.
Hayden:
And if you count the 61 it’s a “60 one.”
(I2, Qu. 6a)
It is interesting that when the researcher asked further questions about the
digits in the two numbers Hayden rejected the suggestion that the digits indicate
anything on their own about the size of a number, particularly because the ‘8’ would
give an incorrect result if a face-value interpretation was used. There is a clear sense
that Hayden’s concept of numbers was based around the sequence of cardinal
numbers, based on his frequent use of counting-based justifications for his answers.
From the previous transcript excerpt it is evident that he was able to say quickly
which number was larger just by looking at the digits, implying some knowledge of
the place-value system. However, when the reasons behind his answer were probed
160
further he responded in terms of where the two numbers were in the sequence of
cardinal numbers.
Excerpts from the teaching phase of the study show that Hayden’s ideas about
numbers influenced how he approached questions when using the computer.
Although the software displays counters that indicate the number of blocks in each
column, Hayden still chose to count blocks himself. For example:
1.
He counted 4 tens put out to represent 40 to check that the
representation was correct.
2.
(l/c S1, T 1d)
He started to count the blocks after trading a ten from 77, before Terry
told him the computer could do the counting.
3.
(l/c S2, T 4a)
He started to count 6 tens and 11 ones to check that they represented 71.
(l/c S6, T 15)
The preference that several participants had for either a counting approach or
a grouping approach is one of the clearest findings of this study, discussed in section
5.2.1. Because a grouping approach is more efficient and more useful with larger
numbers than a counting approach, a relevant question is whether or not use of either
representational format might help a participant with a preference for counting to
switch to a grouping approach. In Hayden’s case, his use of counting was quite
successful, and it may have been some time before he considered changing his habit
of counting to answer numerical questions. The last example above supports this
idea. The researcher had asked Hayden’s (low/computer) group how many tens
would be present after a hundred in 340 was traded for tens. No other participant
could work out the correct answer: Amy and Terry said the answers “3” and “4,” and
Kelly did not reply. Hayden counted the blocks on screen, using the marks on the
picture of a hundred-block to count the tens that would result from a trading
procedure, then counted on the 4 tens in the tens column. After counting Hayden was
sure that the answer was 14 tens, and tried to convince the others in the group that he
was correct. The fact that Hayden’s answer in this instance was correct, and that
other participants were unable to work out the answer, would presumably have acted
as a reward to Hayden for using the counting approach, making it more likely that he
would do so again when the opportunity arose.
Apart from Hayden, very few computer participants actually counted the onscreen blocks for themselves. It appears that Hayden’s strong preference for using a
counting approach led him to count on-screen blocks, although the software could
161
have done it for him. Details of participants’ use of counting to gain feedback about
their answers are found in Table 4.16; it shows that participants in blocks groups
received much more feedback about their answers from counting blocks than did
participants in the computer groups. Feedback received by participants is discussed
in section 4.7.7.
4.7.2 Use of Trial-and-Error Methods
One aspect of use of materials to represent numbers that emerged in the data
was the incidence of trial-and-error methods by some participants when putting out
blocks to represent a number. Though this appeared a few times in blocks groups
(see transcript excerpt later in this section), it was most noticeable in transcripts of
computer group sessions. Computer participants were frequently observed to use
backtracking when clicking the relevant buttons to place blocks in the three places.
Analysis of the transcripts did not initially reveal this behaviour, as the videotapes
merely showed students using the computer to place blocks on-screen, without
indicating clearly how many blocks of each size were put out, and in which order.
However, the software audit trails revealed that on several occasions participants
overshot the number of blocks needed in a place and backtracked (see Appendix E
for a sample audit trail, showing the trail generated by Hayden and Kelly’s computer
at session 9). For example:
1.
When representing 538, Terry put out 5 hundreds and 5 tens [550], took
away 3 tens [520], added 2 tens again [540], removed a ten [530], and
added 8 ones [538].
2.
(l/c S9, T 29a)
When representing 712, Kelly put out 7 hundreds and 2 tens [720],
removed the 2 tens [700] and added 12 ones [712].
3.
(l/c S9, T 29c)
When putting out blocks to show 147, Amy put out 1 hundred [100],
removed it [0], put out 1 ten [10] and 7 ones [17]. She then removed the
ten [7], added 1 hundred again [107], and added 4 tens [147].
(l/c S10, T 30a)
4.
When representing 516, Yvonne put out 5 hundreds and 6 tens [560],
removed 6 tens [500], added a ten [510], and 6 ones [516].
(h/c S8, T 30c)
As mentioned earlier, this category of response was noticed first in software
audit trails, which logged participants’ mouse clicks when using the software. When
162
consideration was given to whether participants using blocks exhibited the same
behaviour, it became obvious that blocks participants frequently used checks of the
blocks they put out, by checking the number required or by recounting the blocks.
The example below shows Jeremy (l/b) using both these checking methods as he
used blocks to show 95:
Jeremy:
When representing 95 with physical base-ten blocks, uses the blocks left-over
from the previous question [94], removes the ones [90], adds a ten [100],
recounts the tens and removed the ten again [90], rechecks the card, counts out
5 ones and adds them to the representation [95].
(l/b S7, T 20)
This example is typical of participants using blocks, particularly with more
difficult numbers including three-digit numbers. It seems clear from transcripts that
participants had some difficulty remembering three-digit numbers after reading the
task instructions, and there were frequent examples of participants checking the
number to be displayed after starting to put out blocks, especially after completing a
place or two. This observation is not very surprising; clearly three-digit numbers are
cognitively more demanding on students than two-digit numbers, and students may
require more support when asked to carry out procedures using these larger numbers.
This point is discussed later in section 4.7.3.
Despite the apparent similarity between examples of trial-and-error behaviour
by participants using physical or electronic base-ten blocks, it appears likely that the
two representational formats had different effects in this regard. In the case of
participants in blocks groups making intermediate checks of the numbers of blocks
put out, it is likely that the participants were having to refresh their memory of the
number required, and to check that the blocks put out matched the number(s) in the
instructions. One does not get a sense that these participants were actually trying out
arrangements to see if they fit the requirements of the task at hand; rather, they were
using a “feedback loop” to check their progress as they chose blocks for each place.
However, in the case of computer participants’ frequent backtracking when showing
some numbers, it appears that the method they adopted was genuinely one of trialand-error: They appeared to be testing their ideas by putting out blocks and looking
at the available on-screen numerical symbols to see if they were correct. Because the
electronic blocks may be put out very quickly by clicking with the mouse, it does not
take a user long to put out some blocks to see if they are correct, looking at the
column counters and number window that are updated continuously as each new
163
block is put out. Then it is a simple matter to remove blocks quickly by clicking on
various buttons. As the feedback from the software about the number of blocks put
out is virtually immediate, it enables, or affords (Salomon, 1998) such trial-and-error
methods, whereas to do this with physical blocks would be much more timeconsuming and cumbersome.
It may be noted in the examples given in this subsection that trial-and-error
methods were used most often by the low/computer group; examples were found of
every participant in that group using trial-and-error methods at some time. However,
in the high/computer group few examples were evident, and all by Yvonne, the
participant with the lowest interview scores in that group (see Table 4.3).
Furthermore, the examples of this behaviour all involved three-digit numbers: No
examples were found of participants using trial-and-error methods when showing
two-digit numbers. It appears that this behaviour was related to uncertainty in the
minds of participants who had weaker understanding of the base-ten numeration
system about which blocks they needed for the more complex numbers, and that the
participants used the screen blocks, counters, or both, to revise their decisions as
each block was put out. It is assumed that high-achievement-level participants using
the computer did not need to use trial-and-error methods because they had a
knowledge of base-ten numbers that was sufficiently sound that they did not to have
to make several attempts to show the numbers using on-screen blocks.
4.7.3 Handling Larger Numbers
The transcript examples cited in the previous subsection, showing participants
using trial-and-error methods to handle larger numbers, were all from the last few
teaching sessions, involving three-digit numbers. As mentioned earlier, larger
numbers involve more complex mental mapping between symbols, number names
and representational materials (section 2.4.3; see also Boulton-Lewis & Halford,
1992). Thus, it is to be expected that participants would have more difficulty with
these numbers than the cognitively simpler and more familiar two-digit numbers.
One aspect of participants’ attempts at tasks involving three-digit numbers that is
quite noticeable in the transcript data is the apparent difficulty some participants had
with holding the necessary information in their minds all at once. This was observed
in both blocks and computer groups.
164
Difficulties with three-digit naming tasks.
One apparent example of the cognitive demands imposed by a task being too
great for participants to manage at once is found in the transcript of the low/blocks
group attempting to complete Task 28 (a), which required them to say the number
represented by 3 hundreds, 6 tens and 9 ones put out by the researcher (see Appendix
P for the full transcript). Every participant in the group appeared to have
considerable difficulty managing the task. The participants’ behaviour was consistent
with the conjecture that they were attempting to reduce the amount of information
they had to remember all at once. The following behaviour was observed:
1.
saying part of the number name aloud,
2.
counting blocks aloud,
3.
touching the blocks,
4.
separating blocks of different sizes from each other,
5.
writing an answer down immediately after stating it,
6.
looking at the answer of another participant, and
7.
using the digit in each place and the place name to state the number
represented (i.e., “3 hundreds, 6 tens, 9 ones”), rather than saying the
complete number name (“three hundred and sixty-nine”).
(l/b, S9, T 28a)
The behaviour described here was quite common among participants in the
blocks groups when the numbers involved were larger. Whereas they could manage
to count blocks and write the number they represented in one step when the blocks
were in a canonical arrangement for a two-digit number, when three-digit numbers or
non-canonical arrangements were involved, they generally broke the task of checking
the number and writing its symbol into several steps. Typically, this involved
counting the blocks of one place, recording the appropriate digit, then moving to the
next place, and so on. Participants in the computer groups exhibited similar
behaviour, except that they did not need to count the blocks. On occasions they were
observed instead to check the place counters one by one, writing each digit before
checking the next counter. It is suggested that participants used these strategies
because otherwise they were unable to manage the cognitive demands imposed by
the tasks.
165
Handling a non-canonical arrangement task.
One task in particular caused participants more difficulty than any other,
apparently due to the cognitive demands it placed on participants. Task 33 required
participants to put out more than 9 tens and more than 9 ones, then say what number
the blocks represented. The researcher asked participants to first work out the
number represented without regrouping, then to regroup and check their answer. This
task caused all 3 groups that attempted it some difficulty, in some cases to a
considerable amount:
1.
John and Amanda had difficulty counting the number of tens and
reaching a consensus about the number. They eventually agreed that
there were 24 tens and 15 ones [255]. John said that the blocks
represented 219, which Amanda agreed with until the blocks were
regrouped. They counted the blocks, and John wrote 245, but Amanda
wrote 255.
2.
(h/b, S9, T 33)
Craig and Simone put out 19 tens and 52 ones [242]. Craig calculated
mentally that the number represented by the blocks was 242. When the
pair regrouped and counted the blocks, they reached an answer of 232,
which both participants accepted.
3.
Daniel and Rory put out 14 tens and 11 ones [151]. Rory wrote that this
showed 151, but Daniel wrote that it represented 25.
4.
(h/b, S9, T 33)
(h/c, S9, T 33)
Hayden and Terry put out 14 tens and 22 ones [162]. Terry said that this
showed 32, then revised it to 162.
(l/c, S10, T 33)
Responses shown here of participants using electronic blocks were quite
similar to responses of participants using physical blocks, as they apparently
attempted to manage what to them was clearly a difficult thinking task. However,
this similarity in responses of participants using both representational formats is not
generally seen in the transcripts; it appears that the features of this task somehow
made the experiences of participants using either material very similar. Because of
the numbers of blocks involved, participants using either physical or electronic
blocks had to manage two trades at once to determine the number represented by the
blocks. The researcher did not permit the participants using the software to regroup
blocks or to use the number window, which would have told them the number
represented by the blocks. Thus, they were forced instead to use other methods that
they evidently found difficult. It appears that the level of support provided by the
166
software under these conditions did not give participants in the computer groups
much of an advantage over participants using the physical blocks. In comparison,
when completing other tasks involving only a single trade participants in the
computer groups were often successful in working out the number involved from the
block pictures and the column counters.
Handling skip counting tasks.
Table 4.2 reveals that the tasks that caused participants the most difficulties in
both interviews were the skip counting tasks in Question 4. The four subtasks
required participants to count (a) back by 1 from a two-digit number, (b) on by 10
from a two-digit number, (c) on by 10 from a three-digit number, and (d) back by 10
from a three-digit number (see Appendix I & Appendix J). In each question, the
participant was asked to continue the sequence past the first necessary regrouping.
For example, in Question 4 (d) at Interview 1 the participants were asked to count
back from 496 by 10, and encouraged to continue until they were past the number
400. Results in Table 4.2 indicate that of all the interview tasks, performance on the
skip counting tasks showed the greatest difference between participants who used
physical blocks and participants who used electronic blocks. Based on the criteria
adopted for assessing skip counting performance (see Appendix M), participants of
the computer groups improved their combined score on skip counting skills by eight
criteria at the second interview, whereas the combined score of participants of the
blocks groups was unchanged. Reasons for these differences may be related to the
way that numerical symbols can be shown by the software when it is used to carry
out skip counting; this idea is discussed in section 5.5.5.
4.7.4 Interpreting Non-Canonical Block Arrangements
When blocks are used to represent “trading” actions, in which a block in one
place is exchanged for 10 of the next place to the right, the result is a non-canonical
arrangement of blocks that cannot be mapped onto the corresponding numerical
symbol by merely counting the number of blocks in each place. At the interviews the
participants were asked to interpret several non-canonical block arrangements and to
answer a question involving collections of 10 and more than nine single objects (see
Questions 1 (b), 1 (c), 3 (a), 3 (b), & 9 (b) in Appendix I & Appendix J). Though
most participants were able to work out the numbers represented by non-canonical
167
arrangements of blocks, either by arranging blocks into groups of ten or by counting
all the blocks, several participants were unable to interpret correctly non-canonical
arrangements. Difficulties with non-canonical block arrangements were also
observed during teaching sessions. The following paragraphs describe the actions of
two participants which indicated that they found non-canonical block arrangements
difficult to understand.
The high/blocks group worked on Task 5 (a), representing 21 with the blocks
and then trading both tens for ones. Simone did the trade, then counted the ones
before recording her answer, which she wrote as ‘24.’ In the subsequent discussion
between the researcher and the participants, Simone found that her answer was
incorrect. This may have prompted her to change the way she handled such
questions, as described in the following paragraph.
On several subsequent occasions Simone (h/b) correctly traded blocks, but
kept the blocks from the original arrangement separate from the 10 “new” traded
blocks. When she had recorded her answer, which was generally correct, she
immediately swapped the blocks back to the original canonical arrangement. An
answer Simone gave to a question from the researcher supports the idea that she had
difficulty understanding non-canonical block arrangements: When the researcher
questioned her group about why, after a ten-block in the representation for 255 was
traded, the new arrangement still showed 255, Simone responded “You don’t count
the ones.” When the researcher queried her about this statement, she responded “You
swap 10 [ones] for a ten” (h/b S8, T 31a). It appears that Simone was saying that the
new arrangement of 2 hundreds, 4 tens, and 15 ones represented 255 only because it
could be swapped back for the original canonical arrangement of blocks, and not
because the non-canonical arrangement also represented 255.
Another participant, Michelle (l/b), was also observed to keep traded blocks
separate from other blocks of the same place. When answering Task 5 (b) Michelle
traded all 3 tens in 36 for ones, but kept the 30 traded one-blocks separate from the
original 6 ones. She then counted only the 30 traded ones and recorded her answer as
‘30 3 tens 0 ones.’ By this answer Michelle may have meant “30 = 3 tens 0 ones.”
Thus she did not record the number of ones after the trade, but gave a standard,
canonical, answer based on the values of the digits in ‘30’ rather than on the number
of blocks before her. During another task soon after this Michelle made specific
168
mention of her method of separating traded blocks from the blocks originally present
when trading all the tens in 64 for ones.
4.7.5 Face-value Interpretations of Symbols
One conception of numbers that has been widely reported in the literature is
the “face-value construct” (see section 2.4.2). Students who hold this conception
regard each digit in a multidigit number as representing only its face value, and not
that number of tens, hundreds, thousands, and so on. There is evidence in the
teaching session transcripts that participants using both representational formats
demonstrated face-value interpretations of digits at various times, shown in the
following paragraphs.
Face-value interpretations among users of physical blocks.
There is evidence that when calculating answers to questions some
participants used blocks as counters, and in so doing gave each digit a face-value
interpretation. For example, see the following excerpt, in which Jeremy (l/b)
suggested that 39 was greater than 51 because its block representation comprised
more blocks:
Clive:
[Puts out blocks to show 39 and 51.] That’s … 39, and that’s 51.
Michelle:
Which one is most?
Clive:
This one [51], because it’s 51 … 5 tens is most, and 3 ones [sic] is less.
Jeremy:
But what about this one [39]? [Implied: it has a greater number of blocks.]
Clive:
I talked about that one.
Jeremy:
[Moves the blocks in 39 a little, so that all blocks are visible. Perhaps he
thought that Clive was mistaken because he couldn’t see all the blocks]
Teacher:
What do you think, Jeremy? It has got a lot of ones there, hasn’t it?
Jeremy:
[Nods.]
Clive:
[Re-counts ones. Perhaps he thought the teacher believed that there were too
many ones]
Teacher:
So, do you think this one [39] might be bigger than that one [51]?
Jeremy:
That one [39] would be bigger, because it’s got heaps of ones.
(l/b, S5, T 10)
Table 4.10 shows that Jeremy consistently used a face-value interpretation of
digits in both interviews: It was clearly a persistent idea that he had about numbers.
This transcript is interesting in that to Jeremy, his face-value interpretation of the
169
digits in 39 and 51 appeared to be supported by the blocks themselves. In the
transcript, Jeremy referred to the blocks when arguing for a face-value interpretation
of the two symbols. It is clear that Jeremy was not looking at the size of the blocks at
all, but merely at the number of blocks. This implies that he did not see, or that he
ignored, the markings on ten-blocks indicating the shape and size of ten ones joined
together, and instead used each block as a counter. By counting the number of
blocks, no matter their size, he apparently believed he could make judgements about
the size of the number they represented.
Face-value interpretations among users of electronic blocks.
There is evidence from a teaching session with the low/computer group that
the software may have had the effect of supporting face-value ideas. This is revealed
by written responses to Task 27 (b), which required participants to state the meaning
of each digit in the number 248 after representing it with blocks. Transcriptions of
the participants’ written answers are shown in Table 4.15. Three of the four
low/computer participants gave a face-value interpretation of the digits in ‘248,’
which no other participant did. Two participants in the low/blocks group gave
partially incorrect responses—Nerida and Michelle wrote ‘200 hundreds’ or ‘2
tens’—but even so they still interpreted the ‘4’ and the ‘8’ correctly.
170
TABLE 4.15. Participants’ Written Responses to Task 27 (b)
Group
Participant Workbook Responsea
High/computer
Belinda
[no written response]
Daniel
it mens are 2 Hundreds and 4 Tens and 8 ones
Rory
every number is even / 2 Hundreds 4 tens 8 ones
Yvonne
The 2 is hundred 4 is tens 8 is ones.
Amy
2 means 2 4 means 4 8 means 8
Hayden
2 means 2 4 means 4 8 means 8
Kelly
2 = 2 H 4 = 4 Tens 8 = 8 ones
Terry
2 mns 2 cow / 4 m 4 bes / 8 m 8 horse
[2 means 2 cows, 4 means 4 bees, 8 means 8
horses]
Amanda
two hundred 4 tens 8 ones
Craig
The 2 stands for 200 and the 4 stands for 40.
Low/computer
High/blocks
The 8 stands for 8
John
Hto
248
Low/blocks
Simone
2 hander and 4 tens and 8 ones
Clive
2 h 4 tens 8 one
Jeremy
2 h 4 Ten 8 one
Nerida
200 tens 4 tens / 200 Hundreds and 4 tens 8 ones
Michelle
2 tens 4 tens 8 ones / 200 Hundreds
Note. Task 27 (b): Explain the Meaning of the ‘2,’ ‘4,’ and the ‘8’ in ‘248.’
a
“/” indicates new line started by participant.
It is possible that participants using the electronic blocks were influenced by
the presence of single-digit counters in the software, resulting in different responses
to Task 27 (b). It may be that participants in the high/computer group were able to
interpret the digits correctly without referring to the counters, but that the participants
in the low/computer group used the counters to guide their responses to the task.
Further evidence of face-value interpretations of symbols being held by
participants in the low/computer group is found in the following descriptions:
1.
When answering Task 29 (c), low/computer participants were asked by
the researcher about the three digits in ‘712,’ after both pairs had put
out 7 hundreds, 0 tens and 12 ones. The researcher referred to the ‘7’
representing 7 hundreds, then pointed out that there was no ten-block.
171
Terry responded “Oh no, 1 one! Means 1 one, and ‘2’ means 2 two, and
‘7’ means 7 seven.”
(l/c S9, T 29c)
2.
Hayden wrote about 80, ‘9 [sic] means 9 and 0 means 0.’ (l/c S8, T 26)
3.
Amy wrote about 80: ‘8 means 8 and 0 means 0.’
4.
Terry wrote about 126: ‘1 means 1 cow, 2 means 2 beds, 6 means 6
5.
(l/c S8, T 26)
horses.’
(l/c S8, T 26)
Hayden wrote about 126: ‘1 means 1 and 26 means 26.’
(l/c S8, T 26)
There is insufficient evidence in the data from this study to decide with any
confidence the extent to which the software may have influenced participants to hold
a face-value interpretation of digits in written symbols. Results in Table 4.10 show
that low-achievement-level participants generally used face-value interpretations of
written symbols more often than did high-achievement-level participants, but it is not
possible to point to a definite effect by either representational material on
participants’ face-value interpretations. Whereas at the second interview there was a
reduction in the incidence of face-value interpretations given by Nerida (l/b), and
Terry and Hayden (l/c), other participants in the low/blocks group showed no
improvement, and Amy and Kelly (l/c) showed some deterioration on the relevant
questions at Interview 2. Any statements about the influence of each representational
format on participants’ face-value ideas must therefore be tentative.
4.7.6 Predictions About Trading
When blocks are used to trade a block for 10 blocks of the next smaller size,
an important concept for students to grasp is that the quantity represented does not
change. This concept is particularly important for learning how to handle each of the
four operations with multidigit numbers, each of which can include the need to
regroup from one place to an adjacent place.
Data from the first task involving trading.
Transcript data from all 4 groups showed that in every case at least one
participant in each group noticed when completing the first task involving trading,
Task 4 (a), that the number represented by blocks after regrouping a ten in 77 was the
same as the number represented before the trade. Transcripts show that there was a
clear difference in how participants using the two representational formats responded
to this discovery. Specifically, participants using electronic blocks made frequent
172
mention in later sessions of the equivalence of block arrangements after trading, but
participants using physical blocks did not do so. Excerpts from these transcripts are
shown next (see Appendix Q for full transcripts of Task 4a for each group):
High/computer:
Belinda:
[Looking at screen] 77! There's still 77! Cool.
Yvonne:
No, there’s sixty …
Rory:
It’s 77.
Yvonne:
Oh, yeah it is [laughs].
Computer:
77.
(h/c S1, T 4a)
Low/computer:
Hayden:
[Uses mouse to have computer read representation.]
Computer:
77.
Hayden:
[To Terry, with surprised expression] 77!
Terry:
Oh! We’ve still got … Oh, cool, that’s easy! [Writes in workbook] Seventy …
77! [To teacher] How does it do that? It’s still got 77. [Teacher looks at him,
but does not respond] Oh yeah! [Look of recognition. Bangs himself on his
head with his hand]
Hayden:
[Points to screen] It’s still … You cut it up, and it’s still 77! [Looks at Terry]
Terry:
Mmm. [Pencil in mouth, apparently thinking.]
(l/c S2, T 4a)
High/blocks:
Teacher:
The last question I have to ask you is: Are the two amounts [indicates the two
block representations for 77: 7 tens 7 ones, 6 tens 17 ones] the same?
Simone:
No.
Amanda:
No. Y … [Stops, seems unsure]
Craig:
No.
Simone:
No.
Teacher:
And I want you to discuss that with each other. I mean, you know what
number that is [7 tens and 7 ones]. Is that [6 tens and 17 ones] the same
number?
Amanda:
— Yeah, ‘cos those ones, just for ten. Still the same. Make ‘em for ten.
Craig:
— [Counts ones under breath] … 10. 1, 2, 3, 4, 5, 6, 7. Oh yeah, they’re the
same.
173
Teacher:
[Separates 7 ones from the other 10 ones and the 6 tens] So what number is
shown by these blocks [6 tens and 17 ones]?
Amanda:
77.
Craig:
Er … 77.
(h/b S1, T 4a)
Low/blocks:
Michelle:
[Counts just the regrouped ones] 17.
Clive:
Wrong!
Teacher:
Well, you better count them, Clive. Michelle’s saying it’s 17.
Clive:
2, 4, 6, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, … What
was that one? 76, 77! Huh. — So we got double 77s. Mmm? That was tricky.
(l/b S1, T 4a)
The transcripts for this task show that not all participants made mention of the
equivalence of the two representations for 77, and that even after spending extended
periods of time looking at the block representations some participants still did not see
that the numbers represented by each were the same. For example, Simone (h/b) was
convinced that the traded blocks did not show the same number as the untraded
blocks had, and did not change her mind until the researcher demonstrated the
equivalence of the two arrangements for her.
Place-value software and understanding of trading.
The responses of several participants to the discovery that blocks still
represented the same number after a trading process indicated that they were
surprised that this should be the case. This was particularly the case for Belinda in
the high/computer group, and Terry and Hayden in the low/computer group. Both
Belinda and Terry used the colloquialism “cool” at the information provided by the
software that after trading a ten the blocks represented the same number. The voices
and faces of participants in both computer groups indicated a degree of both surprise
and pleasure, apparently showing that (a) they did not already expect blocks to
represent the same quantity after trading, and (b) they enjoyed having the computer
reveal this discovery to them. These reactions are in contrast to reactions of
participants in the two blocks groups that indicated that they were less convinced
about the equivalence of values.
The difference in the reaction of participants using physical blocks or
electronic blocks to the discovery that traded blocks represent the same quantity
174
apparently extended beyond the incidents quoted earlier. Participants in both
computer groups mentioned the equivalence of traded blocks repeatedly during later
teaching sessions, without prompting by the researcher. On the other hand, the same
was not true of blocks participants: Not one participant in a blocks group mentioned
the equivalence of traded blocks after the initial discussion recorded in the excerpts
cited in this subsection. Appendix R contains many statements, made by at least 6 of
the 8 participants in the two computer groups, regarding the equivalence of traded
blocks. The transcripts show that participants using the electronic blocks commented
often about the fact that trading blocks produces representations that show the same
value. It is of particular interest that participants using the software were also able to
predict accurately the numbers of blocks that would be in each place after the trade,
and when questioned about this frequently mentioned that the new blocks would
represent the same number. The transcripts indicate that the participants had started
to develop considerable confidence in the fact that the equivalence of traded blocks
was always true, no matter what the number being represented.
Physical base-ten blocks and understanding of trading.
Whereas participants using electronic blocks demonstrated confidence in
traded blocks representing the same amount, such confidence was not evident among
participants using physical blocks. It appears that these latter participants were still
developing their understanding of trading, and that whereas participants using
electronic blocks were able to develop a generalisation that traded blocks are always
equivalent in represented value, physical blocks provided much less support for this
construction, and so did not help participants using them to develop the same level of
understanding. Members of the high/blocks group at times showed an awareness that
traded blocks represented the same quantity, but at other times made mistakes when
trading that indicated that their knowledge of trading was still not completely secure
on this point. For example, when trading a ten in 255, Amanda (h/b) initially traded a
ten for 5 ones, making the number of ones up to 10, and wrote in her workbook that
the new representation did not show the same number.
In the case of participants in the low/blocks group, they were clearly quite
confused in many instances about what to expect when manipulating blocks or
quantities. One characteristic of their use of the physical blocks that became quite
evident was that they had few expectations about the results of numerical processes.
175
The frequency of errors in counting and handling blocks made by low/blocks
participants was so high, and their knowledge of the base-ten numeration system was
so weak, that their manipulations of blocks frequently produced incorrect answers.
At the same time, these participants often expressed confidence in the answers they
derived from manipulating the blocks, a confidence that was often misplaced; this
reaction to numbers revealed by one’s own counting is discussed further in the
following section. In this situation, it is not very surprising that they did not appear to
develop the understanding that blocks represented the same value before and after a
trade. There were many instances in which participants in the low/blocks group
argued about an answer, due to one or more errors made by different participants,
that ultimately had to be resolved by the researcher because the participants were
unaware of the errors made in the course of working out their answers.
4.7.7 Feedback
As mentioned earlier, an analysis was made of the data from teaching
sessions looking for essential differences between the responses of participants using
physical blocks and of those using electronic blocks. One super-category that
emerged was that of feedback: the receipt of information about an answer, indicating
whether or not it was correct. It became clear from the transcripts of the teaching
sessions that many of the interactions among participants, the researcher, and the
materials could be interpreted as feedback provided to participants regarding their
answers. Appendix L contains a description of the method used to identify and
analyse incidents of feedback in teaching sessions.
It should be noted that, as defined here, feedback includes information
derived from blocks, in the sense that by counting or otherwise manipulating blocks
a participant could have an answer confirmed, disconfirmed, or provided by the
blocks. Thus certain incidents of feedback were the result of participants’ actions that
led to their receipt of information about how to proceed. For the discussion in this
section, in such incidents the source of the feedback is considered to be the materials,
as the result of actions by the participant.
Sources of feedback.
Five sources of feedback became evident in the videotape data as the analysis
was conducted: the researcher, other participant(s), self-checking, counting of
176
physical or electronic blocks, or electronic feedback from the software. These
sources of feedback are referred to in this section as Teacher, Peer, Self, Materials,
and Electronic, respectively. A list of descriptors used in analysis of feedback is
provided in Table L.1 in Appendix L. A summary of feedback data, indicating the
number of feedback incidents from each source for each group is presented in Table
4.16.
TABLE 4.16. Incidents of Feedback of Each Source per Group
Group
Feedback Source
Teacher
Peer
Self
Materials
Electronic
Total incidents
High/
blocks
96
89
9
40
Low/
blocks
130
115
1
73
234
319
High/
computer
70
63
5
17
119
274
Low/
computer
103
59
8
33
104
307
The data in Table 4.16 are presented in Figure 4.4 in the form of a stacked
column graph. This graph makes it clear that, although the total numbers of incidents
of feedback received by the 4 groups are different, the proportions of feedback
received by each pair of groups using the same material were very similar.
177
100%
Electronic
Self
Materials
80%
Peer
Teacher
60%
40%
20%
0%
High/Blocks
Low/Blocks
High/Computer
Low/Computer
Group
Figure 4.4. Proportions of feedback from each source for each group.
Possibly the most striking aspects of the data in Figure 4.4 are (a) the
similarity of each pair of adjacent columns representing groups using the same
representational format, and (b) the difference between the two pairs of columns. It is
clear that though there were differences in the feedback received by highachievement-level and low-achievement-level participants using the same
representational materials, there were many similarities between participants using
the same materials. On the other hand, comparing participants using physical blocks
and participants using electronic blocks, there are quite marked differences in the
patterns of feedback received. Participants using the software received proportionally
less feedback from the teacher, from each other, and from counting electronic blocks
themselves; however, they received a large number of instances of electronic
feedback from the computers.
One obvious question about the data in Figure 4.4 is: Did participants using
the software receive electronic feedback in addition to feedback received from other
sources, or instead of that feedback? In other words, were the actual numbers of
incidents of feedback from non-electronic sources similar to those of participants
using physical blocks? This question is answered by data presented in Table 4.16,
which contains numbers of incidents of feedback rather than percentages of the
totals. These figures show that the actual incidence of feedback from each non-
178
electronic source was lower for computer groups, compared with the equivalent
blocks groups. If electronic feedback is ignored, the high/computer group received
79 fewer incidents of feedback than the high/blocks group. Similarly, the
low/computer group received 106 fewer incidents of non-electronic feedback than
the low/blocks group. It is interesting to note that this difference is made up almost
exactly by the number of incidents of electronic feedback received by participants in
these groups. As electronic feedback very nearly makes up the “shortfall” of
feedback incidents among computer participants, it appears that electronic feedback
provided by the software was not simply added to the interactions that would
otherwise have existed among participants and the teacher. Rather, the availability of
electronic feedback appears to have reduced the incidence of feedback from both
humans and materials.
Effects of feedback.
Feedback provided by physical base-ten blocks is limited, as mentioned
previously. Participants may count blocks for themselves, but other than that, there is
no direct feedback possible from the blocks. This point is borne out by the data in
Table 4.16. Participants were able to count or recount blocks, but other sources of
feedback had to be human: another child, the teacher, or themselves. On the other
hand, participants using the software were able to access electronic feedback that
gave information about the number of blocks in each place, a numeral expander
representation of the written symbol, and the written symbol and number name for
the number represented by the entire block arrangement.
Feedback and answer status.
Incidents of feedback were coded to show the status of the answer of the
participant receiving the feedback and the effect(s) that the feedback had for the
participant receiving it. On some occasions, participants received feedback that was
in response to their correct or incorrect answer. On other occasions, the participant
had either no answer or an incomplete answer, and feedback was accessed to provide
an answer. In a few incidents, no judgement was possible about a participant’s
answer before receiving feedback; in these cases the answer status was coded as
“unknown.” Table 4.17 summarises feedback received by participants in each group,
summarised according to the status of the answer held by each participant before
receiving the feedback.
179
TABLE 4.17. Percentage of Feedback Compared With Answer Status
Answer Status
Correct
Incorrect
Incomplete
Nil
Unknown
High/blocks
33
44
7
15
2
Low/blocks
22
51
6
21
0
High/computer Low/computer
54
48
27
35
7
2
11
13
0
2
Note. Values represent percentages of feedback incidents for each group.
The greatest differences between groups using the two representational
formats regarding status of answers prior to feedback being received are evident in
data for correct and incorrect answers. Overall, low-achievement-level participants
received proportionally more feedback for incorrect answers, and less feedback for
correct answers, than high-achievement-level participants did, whether using
physical or electronic blocks. When results for computer participants are compared
with blocks participants, an interesting pattern emerges. Whereas users of physical
blocks received on average more feedback for incorrect answers, participants using
electronic blocks received more feedback when their answers were correct.
Another aspect of feedback that is important when considering the assistance
that it provides to students is its quality: This is defined as the likely effect that
feedback would have on the participant receiving it, with regards to the recipient’s
confidence in the answer. In other words, if the feedback is likely to have encouraged
a participant to retain the answer, whether correct or not, then the feedback is said to
be positive. If, on the other hand, the feedback is considered likely to have
encouraged its recipient to reject the answer, then it is said to be negative. Table L.2
provides a list of feedback effects and the associated quality descriptions. The quality
of feedback is compared to the answer status of its recipients in Table 4.18, for
blocks and computer groups.
180
TABLE 4.18. Quality of Feedback Provided for Correct or Incorrect Answers
Answer Statusb
Correct
Incorrect
Computer Correct
Incorrect
Groups
Blocks
Feedback Qualitya
Positive
Negative
Neutral
43
36
22
2
80
19
87
6
7
4
77
19
Total
146
263
294
181
Note. aValues in each row represent percentages of the total in the right-most column.
Lists of feedback categories coded for each quality category are described in Appendix L.
b
Feedback coded with the following categories of answer status are not included, as feedback quality
is not considered relevant in these cases: Incomplete (63 incidents), Nil (174), Unknown (13).
Table 4.18 reveals several interesting aspects to the feedback experienced by
participants using physical or electronic blocks. Participants using physical blocks
received many more incidents of feedback for incorrect answers than for correct
answers. On the other hand, computer users received more incidents of feedback for
correct answers than for incorrect answers. Cells of the table that reveal the most
dramatic differences between blocks and computer participants are those recording
feedback for correct answers. It appears that computer participants received a greater
proportion of positive instances of feedback for correct answers than did blocks
participants: Almost 90% of feedback for correct answers received by computer
participants was positive. For blocks participants, less than half of their feedback for
correct answers was positive, with 36% of feedback negative, and 22% neutral. In
other words, participants using physical blocks were less likely to receive
confirmation for correct answers than were participants using electronic blocks.
Interestingly, feedback for incorrect answers by both blocks and computer
participants had a very similar profile, with about 80% of instances of feedback for
incorrect answers being negative. To summarise these figures: Participants using
electronic blocks seem to have received much more feedback when they had a
correct answer than participants using physical blocks, and proportionally much
more feedback for correct answers given to computer participants was positive than
was the case for blocks participants.
These figures do not reflect the number of mistakes made by participants
using either type of materials. Mistakes were not specifically counted in analysis of
the data, but interview data lead to the conclusion that computer users were no better,
overall, than blocks groups at understanding numbers (see Table 4.3). Thus it is
assumed that when answering questions during teaching sessions the computer users
181
had just as much difficulty understanding the concepts as their peers using physical
blocks. However, the feedback received by each set of participants is substantially
different. Blocks groups received more feedback for their mistakes than for their
successes, and more negative feedback overall. Moreover, when their answers were
correct they received almost as much negative feedback as positive. On the other
hand, computer groups received feedback that was mostly positive, and very few
instances of negative feedback given for correct answers.
Given these differences in feedback provided to participants using physical or
electronic blocks for correct or incorrect answers, it is relevant to inquire of the
source of the feedback in each case. The sources of feedback for correct answers and
incorrect answers are shown in Table 4.19 and Table 4.20, respectively.
TABLE 4.19. Percent of Feedback for Correct Answers from Each Source
Groups
Blocks
Totals
Computer
Totals
Source
Teacher
Peer
Self
Materials
Teacher
Peer
Self
Materials
Electronic
Quality
Negative
1
32
0
3
36
0
6
0
0
0
6
Positive
20
10
0
12
42
17
5
1
3
61
87
Neutral
16
2
2
2
22
4
1
1
0
1
7
Totals
37
44
2
17
100
21
12
2
3
62
100
Note. Values represent percentages of the total feedback for either blocks or computer groups,
rounded to the nearest percent.
182
TABLE 4.20. Percent of Feedback for Incorrect Answers from Each Source
Groups
Blocks
Totals
Computer
Source
Teacher
Peer
Self
Materials
Teacher
Peer
Self
Materials
Electronic
Totals
Quality
Negative
43
32
0
5
79
43
24
0
0
10
77
Positive
0
0
0
1
2
0
2
0
0
3
4
Neutral
15
2
0
2
19
12
6
0
0
2
19
Totals
58
34
0
8
100
54
31
0
0
15
100
Note. Values represent percentages of the total feedback for either blocks or computer groups,
rounded to the nearest percent.
Data in Table 4.19 and Table 4.20 show that incorrect answers received
similar patterns of positive and negative feedback for both computer and blocks
groups, from the teacher, from peers and from materials. Furthermore, most feedback
for incorrect answers received by participants using either representational material
was accurate; very little positive or neutral feedback was provided for incorrect
answers. The exception to this is feedback from the teacher, which was often neutral
for incorrect answers. The reason for this is pedagogical: The researcher deliberately
gave neutral responses to incorrect answers on occasions to encourage participants to
reconsider the question before the researcher provided the correct answer. The
feedback category that makes up the bulk of the difference between the two pairs of
groups is electronic feedback for correct answers: Over 55% of all feedback given to
computer participants for correct answers came from the software. Feedback from
the teacher and from materials was in similar proportions for the groups using
physical or electronic blocks, but feedback from peers was less common for users of
the software. However, the experience of users of electronic blocks appears to have
been dominated by positive feedback from the software for correct answers.
Responses to feedback.
One further aspect of feedback of interest in the data analysis was the
category of response labelled “expressing satisfaction.” This category was introduced
to categorise a large number of instances of feedback in which the participant
183
responded by expressing by either body language or verbal utterances that the
participant was pleased with the feedback. In many instances, this satisfaction was
expressed by the participant saying “Yes!” sometimes accompanied by a gesture with
the arms reinforcing the impression of pleasure and satisfaction. Overall, 95
instances of participants expressing satisfaction in response to feedback were
recorded. Of these 95 incidents, 10 were in response to feedback from the teacher, 4
to feedback from a child, 14 to feedback from materials, and 67 to feedback from the
computer. It was the frequency and character of participants’ responses to electronic
feedback that led to the creation of this category in the data analysis, and it is
considered interesting enough to describe in more detail at this point.
Selected examples of the expression of satisfaction made by computer
participants are given following:
1.
After having computer trade a ten in 58 then read the name of the
resulting representation, Belinda said “Yep, that’s true.” (h/c S2, T 4d)
2.
Belinda predicted that after trading a ten in 62 there would be 12 ones.
When the computer showed this was true, Belinda said “12! Yep, I was
right.”
3.
(h/c S2, T 4d)
Hayden checked the number represented by the computer blocks with
the verbal number name, smiled, and said “Yeah! We got it.”
(l/c S1, T 2d)
4.
Kelly checked the symbol for the number 90, and said “Yep. Yeah, it's
right. That's how you write it.”
5.
(l/c S1, T 2d)
Amy put out blocks to show 15, had the computer read the number
name, and said “Yes,” and made a gesture of “success” with two fists
and bent arms.
6.
(l/c S1, T 3c)
Hayden put out blocks for 77, used the computer to check the verbal
name. Terry commented, “Yep. I believe it’s 77.”
7.
After Amy put out blocks for 23, Kelly commented “Yep. We got it”
when the computer confirmed the verbal name.
8.
(l/c S2, T 4a)
(l/c S2, T 4b)
Terry checked the block representation after a hundred had been traded
from 340. As the computer read the number name, he held his hand
behind his ear, and commented “Good! Just to make sure!”
(l/c S10, T 32a)
184
Participants expressed satisfaction at feedback they received from the
software on many occasions. Often the participants were pleased merely to have the
computer “read” the block representation. At other times they were pleased to see
their answers confirmed by the number of blocks after a trading procedure, the
symbol for a number, or a number of ones equivalent to a multidigit number. Early in
the teaching phase the researcher encouraged participants to use the facilities of the
software to check the block representations they formed. The researcher made
frequent reference to the fact that the computer had the capacity to confirm the
number represented by the blocks on screen. The participants were quick to accept
this idea, and after a while used these facilities without any prompting by the
researcher. It was assumed by the researcher that once the participants had the idea
that the blocks represented numbers as they expected, they would stop using the
verbal name and number symbol features except when beginning a new type of task,
or a task with larger numbers. However, it was evident that the participants enjoyed
hearing and seeing the computer confirm their block representations repeatedly, even
when to an adult the confirmation was no longer needed. Participants frequently
accompanied their response to the computer feedback with comments like “Yep,
that’s right,” “Yep, we’ve got that number,” “I believe that,” or “Yep, that’s true.”
It is relevant to ask if the same category of expressing satisfaction to feedback
was observed among users of the physical blocks. There were just a few instances:
1.
The researcher told John that his answer regarding the blocks left after
trading of a ten in 77 was “a good way of doing it.” John was clearly
pleased, and showed the other participants his book.
2.
Amanda confirmed that her answer was the same as Craig’s, saying
“Yep, that’s what I got.”
3.
(h/b S2, T 4b)
The researcher told John that his answer was correct. John told the
others that he was correct.
4.
(h/b S1, T 4a)
(h/b S5, T 14)
The researcher said that Amanda, Craig, and Simone were correct in
saying that 75 + 19 = 94. They expressed satisfaction at the researcher’s
comments.
5.
(h/b S6, T 19)
Simone recounted the blocks that she and Craig put out to show 394,
and said “Yep.”
(h/b S8, T 30b)
185
6.
The researcher told John that his answer of 12 tens resulting from
trading a hundred in 627 was correct; John expressed his satisfaction at
being correct.
7.
(h/b S9, T 32b)
Craig used a ten-block to check the height of a stack of 10 hundreds,
and was clearly pleased to find that he was correct.
(h/b S10, T 45)
The examples given here show similar reactions of participants using physical
blocks to positive feedback for correct answers. Interestingly, this feedback was
usually from the researcher, who told participants that their answers were correct. In
the case of computer groups, when participants expressed satisfaction in response to
feedback, the feedback always came from the software. The author considers it likely
that the role of the researcher in the examples given in the previous list was in some
way the same as that of the software when it confirmed participants’ answers.
4.7.8 Using Blocks To Discover Number Relationships
For someone who possesses enough familiarity with numbers, blocks may be
used to confirm or illustrate a numerical relationship. However, if this familiarity is
missing, and in the absence of other sources of information, use of physical blocks is
likely to prompt the user to count the blocks in order to discover the result of a
numerical process. One use of base-ten blocks reported in the literature (e.g., Fuson,
1992), which in this context may be extended to the use of suitable software, is to use
them merely as calculating devices for finding the answers to computational
questions. Rather than using their knowledge of number facts and the base-ten
numeration system to work out what an answer should be, some children use
materials in an attempt to discover the answer. Note, in relation to the approaches
identified in analysis of the interview data, this behaviour often involved a counting
approach (section 4.4.2): Blocks were manipulated to replicate the numbers and
processes involved in the question, then counted to discover the answer.
The expectation by participants that the blocks would reveal numerical
relationships was evident in several transcripts. In each of the following examples, it
would have been possible for a student with sufficient number fact or place-value
knowledge to answer the question without using blocks at all. In the examples here,
however, participants using physical blocks gave no indication of knowing what the
answer was until they had manipulated and then counted the blocks:
186
1.
Amanda and Simone each separately used blocks to calculate 31 + 28.
(h/b S6, T 18)
2.
After trading a hundred from 627 for tens Simone counted the tenblocks before writing her answer.
3.
4.
(h/b S9, T 32b)
After trading a ten from 23 for ones Clive counted the one-blocks,
finding that the number represented was still 23.
(l/b S2, T 4b)
Michelle used blocks to calculate 95 - 23.
(l/b S7, T 20)
Counting blocks to discover answers is an example of feedback received by
participants during teaching sessions. During teaching sessions there were five
different sources of feedback available to participants, one of which was the physical
or electronic blocks. The relative frequency of counting blocks to discover answers
by participants in each group is shown in Table 4.21. It can be seen that in both
blocks groups participants’ favoured source of answers was the blocks themselves
(“Materials”), whereas in both computer groups the favourite source of answers was
“Electronic,” via on-screen number counters.
TABLE 4.21. Feedback Providing Answers from Each Source for Each Group
Source
Teacher
Peer
Self
Materialsa
Electronic
High/Blocks
9
27
18
46
Group
High/Computer Low/Blocks
0
1
32
39
11
1
14
59
43
Low/Computer
3
18
18
18
44
Note. Values in each column represent percentages of feedback incidents used to discover answers for
each group.
a
Incidents were included if representational materials were counted to discover an answer; incidents in
which materials were counted only to make a block arrangement have been excluded.
Table 4.21 shows selected data from analysis of feedback incidents observed
on the videotapes, showing only incidents in which participants accessed a source of
information to find an answer that they did not already have; this represents
approximately 18% of all feedback incidents recorded. Clearly, the participants using
physical blocks counted the available representational materials to discover answers
much more often than did the participants using electronic blocks. However, the
proportions of feedback of this category are much closer if electronic feedback is
included in the figures for the computer groups. This seems to indicate that there was
a similar reliance on the available materials to provide answers when the participants
187
could not work out answers themselves, by participants using both representational
formats, though the actual uses of the materials were very different. As mentioned
previously, without other inherent mechanisms for feedback in the blocks,
participants using physical blocks sometimes counted the blocks to reveal answers.
On the other hand, computer participants occasionally counted the on-screen blocks,
but more often used other forms of electronic feedback to discover answers. Note
that when the purpose of the question was to find out the number represented by an
entire arrangement of blocks, the researcher did not permit computer participants to
have the number window, displaying the number represented by the entire block
collection, visible.
Confidence expressed in the results of counting.
One feature of data from teaching sessions involving blocks groups was the
confidence expressed by several participants in the accuracy of physical block
representations. On several occasions participants were faced with two conflicting
answers to a question, one resulting from their count of the blocks and one resulting
from another source such as another participant’s count, or their own mental
processing of the numbers involved. Often a participant resolved the conflict by
expressing trust in his or her own counting. In light of the frequency of counting
errors made by participants, discussed in section 4.6.1, it would appear that such
confidence in their own counting accuracy was unwise. Indeed, on several occasions
participants were forced to retract their answers when they found that their count was
mistaken. For example, in the following excerpt Clive (l/b) miscounted the tens and
ones after trading a ten in 58. He decided that there were 17 ones, insisting that he
was correct until the researcher and other participants convinced him that the ones
were made up of the original 8 plus the extra traded 10, and therefore the correct
answer was 18. Following is a shortened excerpt; see Appendix S for the full
transcript.
Clive:
[Writes in book] 58 equals 4 tens and … [counts ones] 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17! 17 ones.
Teacher:
The boys and girls have two different answers again. Clive? You have
different answers again.
Clive:
— Youse are wrong.
Michelle:
— We traded it for …
188
Nerida:
… for ten ones and we kept our 8 ones already there.
Teacher:
And would that make 18, or would that make 17?
Nerida:
18.
Clive:
[With arms folded; in the previous dialogue of the girls, he has not been
showing agreement with what they said, or any apparent willingness to listen]
17.
Teacher:
— If you had 8 to start with, and then you swapped and had another ten, what
number would that make, without counting?
Clive:
17.
Teacher:
Ten and 8?
Nerida:
[Shakes head] 18.
Clive:
18, I think. Think.
(l/b S2, T 4d)
Jeremy (l/b) demonstrated an extreme example of confidence in the results of
his own counting, shown in the following example. The task was to regroup all the
tens in 21 and record the resulting number of ones. Jeremy was clearly unsure about
what to do, and watched the other two participants to see what they did (Clive was
absent that day). Jeremy took away 2 tens from his representation for 21, then put out
several ones. He watched the other participants carefully for a time, apparently trying
to produce the same arrangement as them, but without counting the blocks. He failed
to remove 1 of the ones, so did not actually carry out the trade. He counted the ones
he had put out, and wrote ‘21 = 17 one’ (l/b S3, T 5a). In trying to copy the other
participants, Jeremy evidently did not know how many ones they had, so he
estimated the number. When he counted his one-blocks he found that he had 17, and
so he wrote that for his answer.
In a similar incident, Michelle and Jeremy (l/b) miscounted the blocks in a
task asking them to add 10 to 26. The children chose to add 10 ones rather than a
single ten. In the process, one of the traded ones became mixed up with the initial 6,
and was removed by Jeremy. When Michelle counted, she reached the answer 35,
which she accepted. In response to a statement by Clive that the answer was 36, she
commented “36? It can’t be 36” (l/b S7, T 16).
Participants used physical blocks to find answers on many occasions in
addition to those incidents already mentioned. Though mistakes in general were
more common among low-achievement-level participants than high-achievementlevel participants, two incidents in which participants in the high/blocks group made
189
errors are particularly revealing, in that they demonstrate that the more able
participants also placed considerable trust in the results of manipulating blocks. On
each of these two occasions a participant in the high/blocks group correctly worked
out an answer to a question using mental computation, but then counted the blocks
and found the two answers to differ. In both cases, the participant rejected the earlier
mental answer in favour of the incorrect counted answer. The following example
shows an incident in which John counted the ten-blocks after trading a hundred in
627, after Amanda accidentally removed a ten-block:
Amanda:
[Puts out 6 hundred-blocks and 2 tens, and starts to remove a hundred.]
John:
[Counts 7 one-blocks into his hand, puts them on the table.]
Amanda:
[Removes a hundred-block, and adds 10 ten-blocks to the representation. She
starts to count the ten-blocks. She starts to write in her book, absent-mindedly
picking up a ten-block and putting it on her book. Then she pauses to count
the ten-blocks. She finds that there are 11 ten-blocks, which she writes into
her answer.]
John:
[Picks up the ten-blocks to count them] I don’t think there are 12. I mustn’t
have counted them properly. [He briefly looks at the floor as he replaces the
tens with the rest of the representation on the table. He writes his answer as 11
tens.]
Teacher:
[To John & Amanda] Do you both say the answer is 11 tens? [They both nod.]
Well, I’m sorry, but you are both wrong.
John:
[Immediately] I had ‘12,’ but I wrote ‘11.’
Teacher:
Did you expect the answer to be ‘12’?
John:
Yeah, but there were only 11 blocks.
Teacher:
Well, you should expect 12, because that is the correct answer. [John is clearly
pleased that he was correct.]
(h/b S9, T 32b)
It is evident that John was not completely happy with the answer found by
counting the blocks, but he still accepted it in preference to the answer he had
calculated mentally. A similar incident occurred when Craig and Simone (h/b) were
working out the number represented by a handful each of tens and ones blocks. The
children put out 19 tens and 52 ones, which Craig correctly calculated to total 242.
The researcher asked the pair to justify their answer, at which they proceeded to trade
groups of 10 ones or tens until they had a canonical arrangement. However, in the
190
process of trading a handling error was made, resulting in the answer shown by the
blocks being 232:
Craig:
[With Simone checks the total representation, now that all the trades have
been done; they find it is 232. Craig is obviously surprised that this is the
answer.]
Teacher:
What do you think?
Craig:
I added an extra ten onto it. [He starts to rub out his previous answer of ‘242.’]
The answer is 232. [He writes ‘232.’]
Teacher:
Where was the mistake made?
Craig:
I put in an extra ten. I thought there was 142, because, 50, no, um, 190 plus
50, I um, it was um 40, 2 hundred and um 42, but instead, um, I forgot I have
to count an extra 10 ones.
Teacher:
OK, so the right answer is 232? How do you know that’s right, and not 242?
Craig:
[Looks very unsure.]
Simone:
Because we did it with the blocks.
Craig:
‘Cos we traded …
Teacher:
And that proves that it’s right? Could you have made a mistake, do you think?
Simone:
[Shakes her head slightly.]
Teacher:
Could you have got it right in your head, and wrong with the blocks?.
Simone:
[Shakes her head.] No.
Teacher:
Which do you trust — your heads or the blocks?
Simone:
[Points] Trust the blocks.
Craig:
Blocks.
Teacher:
Well, with the numbers you started with, the correct answer was 242. You
may have made a mistake with the blocks, and missed a ten. [Craig & Simone
look quite surprised.]
(h/b S9, T 33)
It seems clear that participants’ levels of confidence in the results of mental
computation were related to their computation abilities. In the previous example,
Simone was evidently unable to calculate an answer mentally, whereas Craig did so
correctly. It is not surprising, therefore, that Simone had greater confidence than
Craig in the result reached by counting the blocks.
191
4.8
Chapter Summary
This chapter includes description of a wide range of topics relevant to this
study, regarding the use of the two representational formats in the group teaching
sessions. The size of the study and the type of data collected constrain the
conclusions that may be drawn from the data. Without a strict experimental design,
and with a small number of participants, conclusions from the data have to be made
tentatively. Nevertheless, there are a number of trends in the data collected in the
study that are worthy of serious consideration in discussions of how different
representational materials are used by children learning about the base-ten
numeration system. The following chapter contains discussions of the findings
reported in chapter 4.
192
Chapter 5: Discussion
5.1
Chapter Overview
This chapter comprises discussion of results from the interviews and from the
teaching sessions, divided into four major sections, corresponding with four major
findings of the study:
1.
Many participants demonstrated a preference for either grouping or
counting approaches to place-value questions (section 5.2);
2.
A new category of conceptual structure for multidigit numbers, the
independent-place construct, is needed to explain evidence of
participants’ ideas about numbers (section 5.3); and
3.
Several participants were evidently constructing their ideas about
numbers in light of new information and their prior knowledge about
numbers (section 5.4).
4.
The different features of physical and electronic base-ten blocks
apparently caused the two types of material to have differing effects on
participants’ actions and conceptual structures for numbers (section
5.5);
5.2
Participants’ Ideas About Multidigit Numbers
The literature search conducted prior to this study indicated several
conceptual structures for multidigit numbers identified by other researchers (section
2.4.2). These conceptual structures were used in initial analysis of data in this study.
However, the conceptual structures described by other authors were found to be
rather unhelpful in considering the responses of participants in this study. The
interview data in this study, rather than revealing neat, clear-cut categories of number
conceptions held by participants, show somewhat untidy patterns in participants’
ideas. Although some participants clearly had well-developed conceptions of two-
193
digit and three-digit numbers, the responses of many other participants indicated
mixtures of correct ideas, incorrect ideas, and incompletely formed opinions about
numbers. In this section the conceptual structures described in chapter 2 are
compared with the analysis of this study’s data. Sections 5.2.1 and 5.2.2 contain
discussion of participants’ preferences for grouping or counting approaches in light
of their responses at the interviews and in the teaching sessions. Section 5.2.3
contains a critique of previously published schemes for categorising children’s placevalue understanding in light of this study’s data. Lastly, the “face-value construct”
described in the research literature is compared with this study’s findings in section
5.2.4.
A note about conceptual structures.
At this point, it is appropriate to explain the use in this thesis of the term
approaches rather than the more common “conceptual structures” or “concepts”
when describing children’s thinking about numbers. The term “approaches” has been
adopted here in light of the data collected in interviews and teaching sessions. As
explained later, the data collected in this study does not support the idea that the
participants had stable, coherent ideas about numbers, as is implied by the term
“conceptual structures.” On the contrary, the overwhelming impression given by the
data is that many participants adopted one of two clearly-distinguishable approaches
to number questions, counting or grouping, which individual participants used with
varying levels of consistency when answering various questions. Furthermore, the
approaches used by many participants appeared to be guided by often creative and
flexible use of a variety of numerical knowledge possessed by the participants. This
knowledge was often misapplied or misunderstood, but the fact that participants
attempted to apply several different pieces of information to a numerical question
indicated that their ideas about numbers were not fixed, and so could not be
described simply as belonging to a particular category.
Analysis of previously-described conceptual structures.
The conceptual structures described in section 2.4.2 can be compared with the
findings in this study. To reiterate, there were two sets of conceptual structures found
in the literature. Firstly, four conceptual structures were identified as being necessary
for the development of place-value understanding: (a) the unitary construct, (b) the
tens and ones construct, (c) the ten as a unit construct, and (d) the flexible
194
representations construct. Secondly, three conceptual structures that constituted
limited understanding of base-ten numbers were listed: (a) a unitary concept of
multidigit numbers, (b) a face value construct, and (c) a counting sequence concept.
Descriptions in section 4.4 of participants’ approaches to interview questions agree
with descriptions of conceptual structures summarised in section 2.4.2 from the
literature. Grouping approaches (section 4.4.1) clearly show evidence of both the
“ten as a unit construct” and the “flexible representations construct”; counting
approaches (section 4.4.2) include both the “unitary concept of multidigit numbers”
and the “counting sequence concept”; and face-value interpretations of symbols
observed in this study (section 4.4.3) agree with descriptions in the literature.
However, as already discussed in this section, the idea that any participant
exhibited one of these conceptual structures as his or her principal model for
multidigit numbers was not demonstrated. For example, there was no participant who
was found to use a “unitary construct” generally when answering interview
questions. There were examples of participants using single one-blocks to represent
numbers (e.g., see section 4.4.2 for a description of Daniel’s [h/c] and Amy’s [l/c]
use of multiple one-blocks), but these participants did not show a unitary construct
model for multidigit numbers for other questions, or even as their preferred model of
numbers. Similarly, some more able participants used the “flexible representations”
construct, as shown by many examples of the grouping approach. However, that
construct could not be applied to the thinking of any particular participant, as that
participant’s primary conceptual structure for numbers. The following section
describes the preferences exhibited by participants for grouping or counting
approaches.
5.2.1 Participants’ Preferences for Grouping or Counting Approaches
This study has revealed a preference held by some participants for one
approach or another to answering number questions; the consistency with which
these approaches were adopted varied among the participants. The incidence of the
three main categories of responses to interview questions—grouping approaches,
counting approaches, and face-value interpretations—is summarised in Table 4.12.
This table shows that though a few participants were observed to adopt just one type
of response to the interview questions, the majority of participants used two or three
of the types of response during the course of a single interview. A comparison is
195
made later in this section of the effects of grouping or counting approaches among
the participants. Face-value interpretations are discussed separately from the
grouping and counting approaches (section 5.2.4), because its use did not fit the idea
of a preferred approach in the way that counting and grouping approaches did.
Preference for grouping approaches.
The type of approach used most consistently by participants was grouping
approaches. Six of the eight high-achievement-level participants—Amanda, Craig,
John, Belinda, Daniel, and Rory—each used a grouping approach far more often than
either counting or a face-value interpretation of symbols. Based on observations
reported in Table 4.12, these 6 participants between them used grouping approaches
100 times at the two interviews, and used either counting approaches or face-value
interpretations of symbols only 8 times. The place-value criteria scores of these 6
participants ranged from 15 to 19 at Interview 1, and 17 to 20 at Interview 2; Figure
4.1 shows a clear tendency of participants with better place-value understanding to
use grouping approaches.
The prevalence of use of grouping approaches by high-achievement-level
participants (Table 4.7), and the apparent correlation between the use of grouping
approaches and interview scores (Figure 4.1), are quite striking. It seems likely that
there was a relationship between participants’ place-value understanding and their
use of grouping approaches. In order for a student to use a grouping approach in a
meaningful way, the student must already possess a certain understanding of the
base-ten numeration system that takes into account the groups of 10 at its foundation.
It appears that the more able students had previously developed meaningful, accurate
conceptual structures for multidigit numbers, which enabled them to develop and use
efficient methods when answering mathematical questions. A corollary of this
conjecture is that less able participants did not possess the knowledge of the base-ten
numeration system to enable them to use a grouping approach.
Preference for counting approaches.
The data for use of the other majority approach, the counting approach, are
far less clear-cut. Counting approaches were used most by 4 low-achievement-level
participants; Clive (l/b), Amy (l/c), Hayden (l/c), and Kelly (l/c); each of these
participants used a counting approach at least five times during at least one interview
(Table 4.8). However, these 4 participants together used counting on a total of only
196
41 occasions, and with the exception of Kelly each of these participants also used
both grouping approaches and face-value interpretations of symbols, on a total of 17
occasions. Clearly, participants who favoured a counting approach did not use it to
the exclusion of other approaches. Furthermore, Figure 4.2 shows that counting
approaches were used both by participants with relatively poor place-value
understanding and participants with moderate or good place-value understanding,
indicating at best only a weak correlation between the use of counting approaches
and performance on the interview tasks. The discussion in section 5.2.2 includes
comments about the use of counting by children when first learning about singledigit numbers, and its implications for learning place-value ideas. In light of that
discussion, it is quite possible that participants using counting approaches had not
changed the approach they learned when using single-digit numbers. Some of these
participants used counting approaches quite successfully on interview tasks, but
others clearly had many confusions about place value. It appears that the use of
counting approaches by participants was affected by several factors, and that simple
conclusions about levels of place-value understanding and the use of counting are not
justified.
Inconsistency of preference for approaches to problems.
Apart from the 6 participants listed previously who favoured grouping
approaches, few other participants showed much consistency in their approach to
answering interview questions, as shown in Table 4.12. Of the remaining 10
participants, 2 used only counting approaches or face-value interpretations, 1 used
only grouping or counting approaches, and the remaining 7 used all three response
types—grouping approaches, counting approaches, and face-value interpretations of
symbols—at least once during the two interviews. These figures show that
categorising the place-value understanding of these participants is not a simple task.
Each participant’s approach to answering each question could be described, and the
apparent lacks in knowledge of the base-ten numeration system noted; chapter 4
contains many such descriptions. However, participants’ use of multiple approaches
in answering different questions implies that they did not have a single idea of
numbers, that could be labelled for example as a “unitary construct” or a “face-value
construct,” that they used consistently in their thinking about numbers.
197
Conclusions about preference.
Conclusions that may be drawn from this discussion of participants’
preferences for approaches to place-value questions include the following:
1.
The most successful participants had developed an understanding of
multidigit numbers in terms of groups of ten, and used this
understanding to assist them in answering a range of mathematical
questions. The understanding of numbers held by these participants was
characterised by the ability to use the grouped nature of the base-ten
numeration system to answer a variety of questions involving
representational materials, written symbols and oral questions about
numbers. These participants rarely used counting approaches to answer
questions, and were not often convinced by incorrect countersuggestions, including face-value interpretations, offered during
interviews.
2.
A few participants had developed a clear habit of using counting
approaches. These participants were often correct in their answers,
though their favoured approach is less efficient and more difficult to use
with larger numbers.
3.
The least successful participants often held a variety of ideas about
numbers, including the incorrect face-value interpretation of symbols,
and used a variety of approaches to answering questions.
4.
Few participants, except for the most mathematically able, could be
classified as having a single, particular, concept of numbers. Most
participants answered several questions incorrectly, and drew on a
variety of information they had learned about numbers in attempting to
answer mathematical questions. Even the most able students were
observed at times to use inefficient or incorrect approaches.
5.2.2 Comparison of Grouping and Counting Approaches
Whereas participants who used a grouping approach were usually successful
in answering interview questions, those using a counting approach often had
difficulties. Reasons for this appear to be related to two main factors: the generally
better knowledge of the base-ten numeration system of participants using grouping
approaches, and the fact that grouping approaches are easier to use successfully. This
198
subsection contains a comparison of these common approaches of participants, and a
discussion of features of each approach that are relevant for teachers.
Counting is the first approach used when learning about numbers.
It has been pointed out by several authors (e.g., Booker, Briggs, Davey &
Nisbet 1997; Fuson et al., 1997; Resnick, 1983) that counting approaches are the first
methods used by children when learning to link single-digit numbers, number names,
and their referents. Thus, it is not very surprising to find that many children of the
age of the participants in this study use counting for managing two-digit numbers.
The principal way to discover the number of objects in a small collection is to count
them, and young children learn to associate symbols, number names and collections
of objects using counting-based methods. If a child persists with counting to make
sense of larger two-digit numbers, then that child is using a “unitary concept” of
numbers that merely continues the earlier approach. There are variations of this
approach, such as a “decade and ones” conceptual structure (Fuson et al., 1997), but
for the present discussion they may be considered together as counting approaches
that regard numbers as collections of single items that are apprehended by counting
them one by one.
Cognitive demands of counting and grouping.
To understand single-digit quantities, children need merely to associate each
of nine individual symbols with one of nine small words, and learn to apply correct
counting procedures to groups of less than 10 objects to determine the cardinality of
the group. Each process involves only a single mapping, between the objects and the
symbol representing them, the objects and the number name, or the symbol and the
number name (Boulton-Lewis & Halford, 1992).
If children extend their use of the unitary concept to numbers greater than 10
then the cognitive demands imposed by counting become greater. Two-digit symbols
and their associated number names, except for multiples of 10, all involve two parts,
representing the tens and ones parts of the multidigit whole. This is true even in the
case of numbers 13 to 19, in which the tens and ones parts of the number names are
obscured, and in 11 and 12 in which they are missing entirely. By retaining a unitary
concept for multidigit numbers, in which the number name and symbol are
considered to apply to an undifferentiated collection of single items, a child is
effectively attempting to retain a single mapping between the collection and its name
199
or the collection and the related symbol. However, as explained by Boulton-Lewis
and Halford (1992), “place value because it rests on a binary operation, is at the
system-mapping level” (p. 5; see also section 2.4.3). Thus, even if a collection of
more than 9 items is regarded as a single group, the symbol and the number name
applying to the collection necessarily each involve at least two parts. Added to this
difficulty is the extra cognitive demands imposed on those using counting
approaches by the rules of the base-ten numeration system and the English language.
As described in section 4.6.1, counting across changes of decade, changes in the
number of hundreds, or from tens to hundreds involves simultaneously keeping track
of the numbers in several places. A grouping approach is often simpler to execute
because it allows a student to (a) count blocks in each place separately before
combining them, and (b) count only a single-digit number of blocks per place. In the
case of non-canonical representations, the extra blocks can first be regrouped,
leaving only single-digit counting to be done. This is precisely the approach adopted
in the interviews by several mathematically-able participants. Claims made here of
higher cognitive demand placed on those who use counting approaches are supported
by transcript excerpts showing participants’ counting errors, including counting
sequence errors (section 4.6.1) and perseveration errors (section 4.6.3).
Use of counting by children with poor knowledge of the base-ten numeration system.
Many participants who had difficulties with skip counting (section 4.6.1)
were the same participants who generally chose counting approaches when
answering interview questions. Students with difficulties remembering the counting
sequence correctly who choose a counting approach rather than a grouping approach
are then in a double bind: Firstly, their approach is less efficient and more likely to
lead to errors, and secondly, limitations in their knowledge of counting sequences
mean that they are also more likely to make counting errors before they reach the
answer.
Counting of larger numbers is inefficient.
A counting approach is clearly less efficient than a grouping approach, taking
the student more time and introducing the potential for more errors. The relative
lengths of previous transcript excerpts from interviews with Kelly (l/c) and Belinda
(h/c) answering the same question (3 tens + 17 ones), repeated here, reflect the
200
difference in time that it takes a student to use either counting or grouping
approaches:
Kelly:
[Touches each packet of gum] 10, 20, 30. [Counts on fingers by touching them
one by one on table] 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43.
43 pieces of gum.
Belinda:
(I1, Qu. 9b)
47. There’s um, three of them and then there’s a one, which would make a 40,
and then you put a ‘7’ on the end and it equals 47.
(I1, Qu. 9b)
As already indicated, there are at least two difficulties facing students who
regularly use counting strategies. Such strategies are vulnerable both to counting
errors and to higher cognitive demands. Students may either lose count and arrive at
the wrong answer, or they may not be able to hold all the parts of a question in their
heads, and so be unable to complete the task. Such problems will surely become
more pressing as students progress in school and the numbers involved become
larger. It is highly unlikely that children could continue using such strategies, once
the questions facing them involved three-digit numbers.
Limitations of using counting to understand numbers.
Though there are other reasons for using grouping approaches rather than
counting approaches, the most serious limitation of counting approaches to number
questions from a teaching perspective is that counting is much less helpful to
students to help them see the wider picture of the base-ten numeration system. The
system of base-ten numbers is made up of a number of repeating patterns, the most
fundamental being the repeated pattern of groups of 10 from place to place. The
sequence of counting numbers also contains patterns, but unlike the totally consistent
patterns of the symbol system, counting patterns include a number of inconsistencies
making them harder to follow and remember. In particular, the “grouped tens” aspect
of two-digit numbers is obscured “because the spoken numerals lack reference to the
tens and ones that are contained in them (e.g. eleven, twelve, thirteen, etc. and
twenty, thirty … one hundred)” (Boulton-Lewis & Halford, 1992, p. 9).
Every primary-age student is expected to learn the sequence of counting
number words in their spoken language, no matter how complex or difficult that may
be. Several writers have pointed out that, because of the inherent inconsistencies in
all European languages, learning to count and to use multidigit numbers to solve
mathematical problems is much more difficult for European-language-speaking
201
students than for their Asian-language-speaking counterparts. Thus, the route to
understanding the base-ten numeration system for speakers of European languages
who habitually use a counting approach is likely to be circuitous and difficult.
Changing a child’s preferred approach from counting to grouping.
The reasons underlying a child’s use of one particular concept of numbers to
answer a variety of mathematical questions are not directly discernible from the data
in this study. However, it seems likely that the use of a certain approach to thinking
about numbers is the result of habit, of having successfully used that approach in the
past to answer mathematical questions. This certainly seems to have been the case
for certain participants who preferred the counting approach. As mentioned
previously, counting is often the first method used by young children when dealing
with single-digit numbers. Unless a child is assisted to see other, more efficient,
approaches, if that child experiences success with counting approaches, it will not be
surprising if the child continues to use counting when asked a range of mathematical
questions.
There is evidence from transcripts of interviews with one participant, Hayden
(l/c), that habitual use of a counting approach may actually have prevented him from
using other approaches (section 4.7.1). Hayden was one of the more successful
participants at the interviews, and often he was the only one in his group to have a
correct solution to a question in the teaching sessions. This is notable as he was also
one of the participants to use counting approaches the most at the interviews. It is
relevant to consider how easy it would be for a student like Hayden to develop an
understanding of the grouped aspect of the base-ten numeration system, and to alter
his favoured approach from counting to grouping. Considering the two types of
approach, there is little common ground between them, implying that there may need
to be a fundamental shift in thinking about multidigit numbers before a child could
change from counting to grouping. There is evidence in this study and from Fuson et
al. (1997) that students sometimes use more than one concept of numbers at different
times. However, it appears that participants using counting will need support from
teachers to understand grouping concepts and to adopt grouping approaches to
number questions.
202
Summary of comparison between counting and grouping.
There are various interrelated difficulties likely to be faced by students who
favour a counting approach:
1.
By focussing on a unitary conceptual structure for multidigit numbers,
either a single continuous number line or a sequence of cardinal
numbers containing all numbers in order, participants using counting
approaches are very unlikely to see the repeated grouped-by-10 pattern
inherent in the base-ten numeration system.
2.
The seemingly random rules of the sequence of counting numbers
obscure the regularities in the sequence of numerical symbols. For
example, thirty, forty, and fifty do not clearly relate to three tens, four
tens, and five tens; furthermore, thirteen, fourteen, and fifteen are
similar in sound, but very different in meaning.
3.
Sequences of number names become much more complex and more
difficult to manage mentally as the numbers involved become larger.
4.
Counting of blocks representing multidigit numbers involves skip
counting with changes of increment at each new place. For example,
counting 5 hundreds, 8 tens and 2 ones: 100, 200, 300, 400, 500,
[switch to adding tens] 510, 520, 530, 540, 550, 560, 570, 580, [switch
to adding ones] 581, 582. By contrast, a grouping approach to the same
task involves counting three separate sequences of single-digit numbers
before combining them. For example: 1, 2, 3, 4, 5 hundreds - 500; 1, 2,
3, 4, 5, 6, 7, 8 tens - 80; 1, 2 ones – 2; 582.
5.
Errors made when counting can prevent a student from reaching a
correct result. Frustration and an inability to continue with a task are the
likely immediate results; in the long term, learning of place-value
concepts is likely to be slower because of a lower incidence of success
on number tasks.
6.
Finally, the habit of using counting approaches may blind a student to
the possibility of using other, more efficient methods. Switching to a
grouping approach is going to be beneficial in the long term, but it
seems likely that a student who has used counting approaches for a long
time, and who has not recognised the grouped aspect of multidigit
numbers, might find the change quite difficult to manage.
203
5.2.3 Difficulties With Existing Conceptual Structure Schemes
It appears from a perusal of the literature on research into children’s
understanding of place-value that children’s thinking can be categorised according to
their thinking about numbers (e.g., Cobb & Wheatley, 1988; Miura et al., 1993; S. H.
Ross, 1990). The implication of the various proposed schemes is that individual
children possess stable ideas about numbers that influence their performance on
number-related tasks. Thus, several authors have devised models that comprise
stages or levels of understanding according to which an individual child’s
understanding of number concepts may be categorised. There is a good deal of
overlap among these schemes, with several conceptual structures being identified in
more than one study (see section 2.4.2). For these reasons, the data collected in this
study were expected to replicate many of the earlier findings, which could then be
analysed in relation to the two representational formats used in the study.
The intention to use the list of conceptual structures synthesised from the
literature search was found difficult to carry out, however. Results from this study
disagreed with those cited by other authors, on at least three grounds: (a) the great
variety in the responses of individual participants to different interview questions,
(b) a lack of confirmation in this study of both the frequency and the character of
certain conceptual structures previously reported, and (c) the generally limiting effect
that placing a student into a certain category was felt to impose on a researcher’s
understanding of students’ place-value concepts. The principal research on placevalue reported in the literature that was reviewed for this thesis was conducted by
Ross (S. H. Ross, 1989, 1990; S. H. Ross & Sunflower, 1996), Miura and colleagues
(Miura & Okamoto, 1989; Miura et al., 1993), Cobb and Wheatley (1988), Resnick
(Resnick, 1983, Resnick & Omanson, 1987), and Fuson and colleagues (Fuson &
Briars, 1990; Fuson et al., 1992; Fuson, et al., 1997). The research reported by each
of these groups is discussed in the following paragraphs and compared with the
findings of this study.
Ross’s five-stage model of digit correspondence performance.
Ross’s research on place-value understanding (S. H. Ross, 1989, 1990) has
received much publicity; in particular, her digit correspondence tasks have been
replicated in several other studies (e.g., Carpenter et al., 1997; Fuson & Briars, 1990;
Miura et al., 1993) or described by other authors (e.g., G. A. Jones & Thornton,
204
1993a; C. Thompson, 1990). As described in section 5.2.4, there is broad agreement
between Ross’s five-stage model and this study’s proposed four-category model in
several descriptions of participant responses to digit correspondence tasks. Where
results of this study differ from Ross’s, however, is in the categorisation of students
according to their purported stage of development of place-value understanding.
Ross categorised the 60 students participating in her study according to the stage in
her five-stage model to which their thinking apparently belonged, stating that
each of the sixty students in the reported study was assigned to one of the five stages
according to performances on six digit-correspondence tasks and a positionalknowledge task in which students were asked to identify, in a two-digit numeral,
which digit was in the “tens place” and which was in the “ones place.” (pp. 49-50)
Thereafter in the paper, Ross referred to students as being “at” a particular
stage, or as “using a stage-n understanding.” Results of this study, on the other hand,
revealed students whose responses varied as they attempted different tasks. The
students themselves were not “at” a certain stage, in the narrow sense described by
Ross; the researcher was unable to neatly categorise their thinking based on
responses on one particular type of task. This applies particularly to what Ross
labelled the “face value” stage. As discussed in section 5.2.4, face-value
interpretations of symbols were demonstrated in this study, but were not
demonstrated on all relevant tasks by even one participant. Based on results of this
study it is suggested that face-value interpretation of symbols is but one symptom of
a general confusion about numbers possessed by many students of this age, rather
than an identifying characteristic of their mathematical thinking generally. The
results of asking children a variety of place-value questions in this study were messy,
often inconclusive, and not easily summarised. The contribution Ross has made to
the place-value literature is significant, and her descriptions of interpretations for
digits in multidigit numbers given by students are very valuable. However, results
from this study indicate that describing students’ mathematical thinking may be more
untidy and difficult than Ross suggested.
Miura’s three categories of place-value conceptions.
Miura and her colleagues (Miura & Okamoto, 1989; Miura et al., 1993), who
used some of Ross’s ideas, developed another scheme by which students’
understanding of place-value could be categorised. In both studies Miura et al.
investigated children’s perceptions of base-ten numbers via tasks that focussed on
205
how the children represented multidigit numbers using base-ten materials. Results in
the two studies were used to categorise participants’ “cognitive representation of
number” as belonging to one of three categories. Like Ross’s research, Miura et al.’s
research has been widely reported by other authors, and there is apparently wide
support for her ideas about the effects of number names in European and Asian
languages on children’s conceptions of numbers (though see Saxton & Towse, 1998,
for a critique of Miura et al.’s findings). However, Miura et al.’s findings are open to
some of the same criticisms directed at Ross’s findings in the previous paragraph,
and again results in this study did not reproduce Miura et al.’s discrete categories of
place-value conceptions. Via the use of a narrow range of tasks and a limited set of
categories with which to describe student understanding of base-ten numbers, Miura
et al. claimed to have identified significant differences that existed between the
thinking of U.S. and other (principally Japanese) children with regards to numbers.
There is no doubt that differences in the thinking of children of different nationalities
and backgrounds do exist, and questions of effects of culture and language on
children’s learning of mathematics are worthy of investigation. However, it is
entirely possible, and based on this study seems quite likely, that the thinking
investigated in such studies is far more complex and less tidy than Miura et al.’s
three categories of student thinking would suggest.
Cobb & Wheatley’s three levels of children’s ideas about ten.
Cobb and Wheatley wrote an influential paper (1988) describing in some
detail the conceptions of ten held by young children. Their research is particularly
useful in pointing out the difference between children thinking of ten as a collection
of 10 single items, ten as a single unit, and ten as a collection of 10 that can be
counted as an item. Similarly to the research by Ross and Miura reviewed above,
Cobb and Wheatley also claimed to have categorised participants in their study
according to their performance on certain number tasks: “On the basis of their
performance on the counting-by ones, thinking strategy, and subtraction tasks, the
fourteen children were placed at three levels with respect to their addition and
subtraction concepts” (p. 10). Space does not permit a lengthy discussion of Cobb
and Wheatley’s research. In brief, however, it appears that the small sample size in
their study gives cause for questioning the descriptions of identifying characteristics
of categories, which seem unnecessarily rigid and even somewhat arbitrary. In fact,
206
in several places the authors “hedged” over descriptions of a participant’s responses,
claiming that variations in participant responses did not invalidate the authors’
classification of the participant’s place-value understanding. Cobb and Wheatley’s
research, like that of S. H. Ross (1990) and Miura (Miura & Okamoto, 1989; Miura
et al., 1993) discussed earlier, has made a valuable contribution to the place-value
literature. However, like Ross and Miura et al., Cobb and Wheatley claimed to have
developed a scheme by which the place-value thinking by children in general can be
categorised. Results of this present study do not support such claims, implying
instead that children’s place-value thinking often defies researchers’ efforts to place
it in a stage- or level-based scheme.
Fuson et al.’s six conceptual structures.
The researcher whose work on place-value understanding most closely agrees
with the findings of this study is Fuson. Her work includes detailed analysis of the
base-ten numeration system and the necessary skills needed for children to learn to
use it proficiently (Fuson, 1990a, 1990b, 1992), as well as research into children’s
thinking in a variety of number tasks (Fuson & Briars, 1990, Fuson et al., 1992).
Fuson and her colleagues (Fuson et al., 1997) proposed a model of six conceptual
structures used by children, including five more or less accurate conceptions and the
incorrect “concatenated single-digit conception,” or face-value construct. However,
unlike other authors Fuson et al. (1997) did not try to fit the six conceptual structures
into a scheme by which a child may be categorised, or a stage model purporting to
show how each child’s thinking develops over time. Instead, the authors noted that
children may hold more than one conception at one time and that the conceptions are
used in ways that depend on the child’s background and the particular situation in
which they are accessed:
Children who have more than one multidigit conception may use different
conceptions in different situations. . . . Furthermore, not all children construct all
conceptions; these constructions depend on the conceptual supports experienced by
individual children in their classroom and outside of school. Therefore, children’s
multiunit conceptions definitely do not conform to a stage model [italics added].
(p. 143)
The findings of this study agree with the above statement by Fuson et al., in
(a) finding that individual students use a variety of number ideas rather than one
main idea, and (b) rejecting the idea that students move through various clearlydefined stages or levels.
207
5.2.4 Face-value Interpretations of Symbols
As discussed in the previous section, results in this study give only qualified
support for many of the conceptual structures identified in the literature search
carried out before this study, described in section 2.4.2. One conceptual structure that
is especially common in the literature is the “face-value construct” (e.g., S. H. Ross,
1989), or “concatenated single digits” conceptual structure (Fuson & Briars, 1990;
Fuson et al., 1997). There is evidence in data collected in this study for this
conceptual structure; however, this author believes that there are pertinent aspects of
this construct that have not been identified in previous literature. Firstly, descriptions
in the research literature of apparent evidence for the face-value construct do not
agree entirely with findings in this study, especially in light of participants’ responses
to the digit correspondence tasks in the interviews. S. H. Ross (1989, 1990) described
a five-stage model of children’s interpretations of two-digit numerals. In this thesis, a
four-category model is proposed to describe participants’ understanding of similar
two-digit symbols (section 4.5). Data from the two studies are compared in the
following subsection.
Comparison with Ross’s digit correspondence test data.
S. H. Ross (1989) reported the data from one particular task given to Grade 3
participants, upon which this researcher based Question 7 in each interview. In each
study, the researcher asked participants to count some sticks (25 in the case of Ross’s
study; in the present study, 24 in Interview 1 and 37 in Interview 2), and then to write
the symbol for the number. The researcher then asked the participants to say which
sticks corresponded with each of the two digits. Since participants in the two studies
were of similar age and school experience, it is valid to compare the reported results
of the two studies (Table 5.1). Such a comparison leads to three conclusions. First, it
appears from descriptions given by each researcher that the behaviours observed
were broadly similar; second, each researcher identified a category of behaviour not
mentioned by the other; and third, researchers applied different interpretations to the
results from the two studies. These conclusions are discussed in the following
paragraphs.
208
TABLE 5.1.
Comparison of Results of Digit Correspondence Tasks Between
This Study and Ross (1989)
This study (Question 7)a
Cate- Interpretation of digits in
gory “24”
I
“2” meant two sticks, “4”
meant four sticks.
Ross’s (1989) studyb
%
Interpretation of digits
(N = 16) in “25”
38
%
(N = 60)
“2” meant two sticks, “5”
meant five sticks.
13
Invented numerical
meanings: e.g., that 5 meant
half of ten.
23
II
Individual digits had nothing
to do with how many sticks
were in the collection.
16
Individual digits had nothing
to do with how many sticks
were in the collection.
20
III
“4” represented four sticks,
“2” represented 20 sticks.
22
“5” represented five sticks,
“2” represented 20 sticks.
43
IV
“4” represented four sticks,
“2” represented 20 sticks,
“2” meant “2 tens.”
25
Note. Results on the same row represent similar descriptions of students’ interpretations of digits from
the two studies. Blank cells indicate that no equivalent category was described matching the category
opposite in the other study.
a
Results from interview Question 7 are quoted, as this task matches the one used by Ross. See Table
4.13 for a summary of results from digit correspondence tasks for each participant.
b
From S. H. Ross, 1989, Parts, wholes and place value: A developmental view. Arithmetic Teacher,
36, p. 48.
S. H. Ross (1989) identified a number of behaviours that broadly match
observations made in this study, as shown in Table 5.1. At the lowest levels of
performance, both Ross and this author identified participants who gave face-value
interpretations of digits. At higher levels of performance, both researchers observed
participants explaining the values represented by each digit in terms of tens and ones
language. In between the two extremes, there were participants who did not accept
face-value interpretations of symbols, but who also did not explain digit
correspondence in terms of groups of ten.
S. H. Ross (1989) and this author each included a category of response not
identified by the other. Firstly, Ross distinguished between participants who gave a
straight-forward face-value interpretation of the digits and participants who
responded with “invented numerical meanings, such as that the 5 meant ‘half of ten,’
that the 5 meant that groups contained 5 sticks, or that the 2 meant ‘count by twos’”
(p. 48). It is apparent that the “invented numerical meanings” category does not
209
exclude a face-value interpretation of the digits. If Ross’s lowest two categories
(face-value and invented meanings) are added, their combined incidence (36%) is
close to each incidence of face-value interpretations in this study (44% and 31% at
Interviews 1 and 2, respectively). Secondly, this author distinguished between
responses explaining the value of the tens digit in terms of the name of the multiple
of ten (20 or 30), and responses referring explicitly to the number of tens (two tens or
three tens). It appears that Ross’s highest category of participant response might
include both of this author’s two highest categories, if Ross did not make the
distinction described in the previous sentence. If the assumptions described in this
paragraph are accepted, this then adds support to the claim that the two studies have
identified similar patterns of response. However, Ross’s and this author’s
interpretations of these responses differ markedly, as explained in the following
subsection.
Differing interpretations of face-value responses.
Interpretations of children’s face-value behaviour made by this author differ
from S. H. Ross’s (1989) interpretations of similar behaviour (see Appendix T), for
two reasons. First, there is evidence of an inconsistency in Ross’s interpretations of
children’s responses indicating that a two-digit symbol represents the entire referent
set, without referents for each digit. Second, this study does not support that idea that
individual children possess stable mental models for numbers that can be used to
describe their number understanding generally, as explained in section 5.2.3.
S. H. Ross (1989) and this author give different interpretations for the facevalue responses made by participants in their respective studies. The first column of
the table in Appendix T includes descriptions of the four categories of digit
correspondence task response from section 4.5. Adjacent to most descriptions is an
excerpt from Ross’s paper that apparently describes similar behaviour. When the two
columns are compared, a striking difference emerges between the categories defined
by the two researchers. In particular, Ross and this author disagree regarding
participants’ responses indicating a belief that the entire collection of objects was
represented by the entire symbol, but that each digit did not have its own referents.
Ross defined this type of response as being at the lowest level of understanding of
digits, and in particular, below the face value stage. In analysis of participants’
responses in this thesis, this type of response was categorised as Category II, above
210
Category I, face value. The author considers a Category II response to demonstrate
superior understanding of the digits to a straight-forward face-value interpretation of
digits, for three reasons: (a) it rejects the incorrect face-value interpretation, (b) it
explains correctly that the entire symbol represents the entire collection of objects,
leaving none out, and (c) participants exhibiting this category of response generally
demonstrated superior performance on other interview tasks than participants who
answered with face-value interpretations (see Table 4.13). If indeed the two
categories (Stage 1 / Category II) essentially describe the same behaviour, then it
appears that this study has revealed an aspect to children’s thinking about digit
correspondence that has not been widely reported before. This aspect is that, as
described in section 4.5, some participants were not comfortable with a face-value
interpretation of the digits, and appeared to operate at a higher level of thinking about
digit correspondence in rejecting face-value interpretations. This was despite the fact
that they did not fully understand the grouped aspect of base-ten numbers and were
not able to explain numbers in terms of place value.
Despite the similarities in data collected in S. H. Ross’s (1989) study and the
present study, one particular aspect of the data, already alluded to, points to a
difference in interpretation of children’s place-value understanding. Whereas Ross
categorised the children themselves, in this study it is the children’s responses that
are classed as belonging to a particular category. Furthermore, there is compelling
evidence in this study that such categories were not fixed, but altered with the
particular context in which the response arose. In short, results of this thesis did not
demonstrate even one participant who held a consistent belief that each digit
represents only its face value. The participant who was the most likely candidate for
possessing a face-value construct for multidigit numbers is Jeremy (l/b). He had one
of the lowest scores at both interviews, and he was observed to use a face-value
interpretation of symbols at least eight times during the two interviews, in every
instance unprompted by the researcher. However, despite this pattern of responses,
he clearly rejected a face-value interpretation of symbols on several occasions. For
example, the following excerpt shows Jeremy’s response to the question asking
which is bigger, 183 or 138:
Jeremy:
That one [‘183’].
Interviewer: Okay, what is that number?
211
Jeremy:
one eighty ... 183.
Interviewer: Uh-huh, and why is that one bigger?
Jeremy:
Because it’s got a ‘1’ and it’s a ‘8.’
Interviewer: What about the other number?
Jeremy:
It’s got a ‘1’ and a ‘3.’
Interviewer: Okay why does this one ... see this has got an ‘8’ as well as that one and it’s
got a ‘3’ like that one. Why is that one bigger than that one if it has the same
numbers in it? — Is there a chance these two are the same, do you think?
Because they’ve got the same numbers … or is this [‘183’] going to be
bigger?
Jeremy:
That one [‘183’] will be bigger.
Interviewer: — If you were counting would … do you know which one of these numbers
you’d come to first?
Jeremy:
That one [‘138’].
Interviewer: Uh-huh. Do you know why you’d come to that one first?
Jeremy:
Because it’s down lower.
(I1, Qu. 6b)
Considering Jeremy’s comments about the face values of single digits (e.g.,
“It’s got a ‘1’ and a ‘3’”), rather than the values represented by the digits, it might be
inferred that he was using a face-value interpretation of the two symbols. However, if
Jeremy believed that digits only represented their face value, he would not have
rejected the researcher’s counter-suggestion that 183 and 138 are equal because they
have the same digits. Even though Jeremy did not know what the two numbers were,
and could not read them, he still believed that the values they represented were
different, and that the order of the digits indicated which one was bigger. Clearly
Jeremy’s thinking about these two symbols cannot be summed up with the label
“face-value construct,” even though at other times he clearly demonstrated a facevalue interpretation of digits; this observation is repeated many times in the interview
transcripts.
This author contends that a new category is needed to describe children’s
numerical thinking that may help to interpret response patterns such as those
described in this section. The following section contains a description of such a
category, the independent-place construct.
212
5.3
Independent-Place Construct
Results of this study indicate the presence of a previously unreported
conceptual structure for numbers in the minds of some participants, here named the
independent-place construct. Discussion of the independent-place construct in this
section is arranged in the following subsections: a description and definition (5.3.1),
comparison between the independent-place construct and the face-value construct
(5.3.2), evidence for the independent-place construct in this study (5.3.3) and in the
research literature (5.3.4), and the effects of the independent-place construct on
written computation (5.3.5) and on place-value tasks (5.3.6). Implications of the
independent-place construct for teaching are discussed later in the final chapter, in
section 6.3.3.
5.3.1 Description & Definition of the Independent-Place Construct
As explained later in this section, the independent-place construct includes
aspects of face-value interpretations of symbols and the use of materials as “column
counters.” Use of this construct is indicated by participants’ actions indicating that
they regarded individual places in multidigit numbers, block representations, or both,
to be separate and unrelated. In other words, they did not see any link between
“hundreds,” “tens,” and “ones” places, but regarded them as independent categories
of quantity with separate names, separate digits, and separate block representations.
In doing so, though participants were able to complete certain simple tasks, it appears
that they were not able to appreciate the value represented by an entire multidigit
symbolic or block representation in any meaningful way. For the reasons discussed
in following sections, the author contends that the independent-place construct is
essentially different to face-value interpretation of symbols, and provides a better
explanation of patterns of computation behaviour previously labelled by other
authors as examples of the face-value construct.
The following definition for the independent-place construct is used for the
subsequent discussion:
The independent-place construct occurs when a student treats symbols or
concrete materials representing values in one place in the base-ten numeration
system as separate from other places, and does not attempt to relate one place to
another.
213
5.3.2 Comparison of the Independent-Place Construct and the Face-Value
Construct
The independent-place construct is proposed here as a means of explaining
apparent anomalies in this study’s data when considered in the light of previously
published research in the field. As discussed earlier, although certain responses by
participants indicated that they believed that individual digits in multidigit numbers
represented only their face values, data in this study do not support the idea that
participants held these beliefs consistently as they answered place-value questions.
The author asserts that much behaviour previously identified as revealing a facevalue construct is better understood as demonstrating a perception that digits are
independent of each other.
The independent-place construct proposed here and the face-value construct
widely reported in the literature are similar and yet distinct from each other.
Although both constructs have the effect of leading a student to ignore the values
represented by individual symbols, the essential natures of the two constructs are
quite different. Whereas a person possessing a face-value construct denies that each
digit in a multidigit number represents anything other than its face value, a person
with the independent-place construct considers each place separately from other
places, while taking no account of what each digit actually represents. Furthermore,
whereas the presence of a face-value construct indicates a serious misunderstanding
of the base-ten numeration system, and rightly attracts attention from teachers and
researchers, the independent-place construct is consistent with computation practices
that ignore actual values represented by digits for the sake of efficiency.
5.3.3 Evidence for the Independent-Place Construct in This Study
Evidence of an independent-place construct is found in several patterns of
participants’ responses reported in chapter 4, including (a) trading 1-for-1, (b)
choosing incorrect blocks, (c) number naming errors, (d) use of place names merely
as labels, (e) errors made in writing numerical symbols, and (f) a reluctance by some
participants to consider non-canonical arrangements of blocks.
(a) Trading blocks one-for-one.
Firstly, as described in section 4.6.2, several participants proposed to trade a
block of one size for one block of another; that is, to trade a ten-block for a single
one or a hundred-block for a single ten. This idea may be an example of believing
214
that blocks are merely counters, and that each has a “value” of one, no matter what
size it is; it is difficult to understand how children could think that a ten and a one
could be swapped unless they perceived each to be merely a single block.
(b) Choosing incorrect blocks to represent numbers.
The error of choosing the “wrong” blocks to represent places, was shown by
Kelly (l/c), as described in section 4.6.4. Kelly used 2 ones blocks and 8 tens to show
28, then used 1 hundred, 3 ones and 4 tens to show 134. It could be argued that
Kelly’s response to this question indicated a face-value interpretation of digits, since
she did not realise that the quantity represented by the tens digit was in groups of 10
ones, and was happy to use blocks that were ten times the size of her “tens blocks” to
represent ones. However, two aspects of this incident make it appear that this was not
the case: (a) When asked to show each number in another way she retained the same
block-value assignments, merely changing their relative positions; and (b) the
numbers were given to her verbally, so there were no written symbols for her to
interpret.
In light of the current discussion, Kelly’s response can be interpreted as
demonstrating an independent-place construct: In Kelly’s thinking there was
apparently no relationship between the ones and tens places, with regards to the size
of the blocks representing digits in each place. Furthermore, Kelly’s consistency in
using small cubes (“ones”) to represent tens digits and long blocks (“tens”) to
represent ones seems to indicate that she did not believe that the blocks could be
applied arbitrarily to any place, which presumably would have been the case with a
true face-value construct. On the contrary, when asked by the interviewer if there
was any other way to represent each number, Kelly consistently used the same
blocks to represent digits in both the ones and tens places four times over the two
questions, changing only the spatial arrangement of blocks of exactly the same sizes.
(c) Errors naming non-canonical block arrangements.
The third evidence for the independent-place construct is provided by certain
examples of participants mis-naming numbers represented by non-canonical block
arrangements. For example, some participants incorrectly stated the number
represented by a non-canonical collection of different blocks by applying a name to
each size of blocks in turn. In such incidents participants “read” the block
arrangement using an “x-ty y” form, where x is the number of tens blocks, and y the
215
number of ones. For example, as reported in section 4.7.1, Jeremy (l/b) read 8 tens
and 11 ones as “eighty-eleven,” and Yvonne (h/c) read 5 tens and 10 ones as “fiftyten.” Fuson et al. (1992) commented on such non-standard number names that “one
can easily say more than nine of a given multiunit and such constructions have a
quantitative meaning even though they are not in standard form” (p. 42). Despite this
point, since such constructions are not standard English number names, it seems
likely that, if asked, the children themselves would regard these number names as
incorrect. If this is so, then it appears that the participants were merely applying a
linguistic procedure that is successful with canonical block arrangements, of naming
the tens and then the ones, without taking account of the meaning of the resulting
number name.
(d) Use of place names merely as labels.
The fourth type of evidence for the independent-place construct in the study
is participants’ use of place names “hundred,” “ten,” and “one” with no apparent
notice paid to the numerical basis for the names. For example, section 4.6.4 includes
mention of an incident in which Clive attempted to explain to Jeremy (both l/b) why
51 was greater than 39. It is interesting that Clive, who clearly knew that 51 was
greater than 39 because of their respective tens digits, could not explain why the tens
should be regarded as having greater value than the ones. When asked by the
researcher he replied that “the tens are first on the tens mat, ten sheet, so . . .” but he
was unable to say why the digits’ positions on a place-value chart made a difference
to the values represented. Clive was correct in noting the relative positions of tens
and ones, both on place-value charts and in written symbols. However, he apparently
did not make a connection between the name “ten” and the idea of 10 ones (see
Fuson et al., 1992; NCTM, 2000).
(e) Errors in writing numerical symbols.
The fifth type of evidence in the study for the independent-place construct
relates to certain attempts by participants to write numerical symbols for multidigit
numbers, either represented by blocks or spoken verbally as a number name. For
example, section 4.6.3 includes a description of Amanda (h/b) having some difficulty
in writing the symbol for 204; writing ‘24,’ then ‘240’ before writing the correct
symbol. It is possible that Amanda had become used to writing a single digit for each
part of a two-digit number’s name, and tried to use the same process with three-digit
216
numbers. Such a method would work with all two-digit numbers except those ending
with zero. Thus, fifty-six can be written by recording a digit for the fifty [5] and
another for the six [6]. It would also work with many three-digit numbers, such as
two [2] hundred and ninety [9]-eight [8]. However, the method fails if there is a zero
in the tens or ones place: two [2] hundred and four [4] has only two place number
words, resulting in just two digits if the “each-number-word-is-a-digit” method is
used. Nerida (l/b) used a variation of this method, writing ‘617’ when attempting to
write the symbol for the number represented by 6 tens and 17 ones. She then read the
symbol and stated that the blocks represented “six hundred and seventeen.” This
incorporates the idea of independent places, since writing ‘617’ for 6 tens and 17
ones involves concatenating the symbols for the two subsets of like-sized blocks
without regard for their respective values.
(f) Reluctance to consider non-canonical block arrangements.
As already mentioned in this section, the independent-place construct appears
to be linked with various behaviours associated with non-canonical block
arrangements. Further support for this is provided by observations of participants
who were apparently reluctant to consider non-canonical arrangements of blocks
(section 4.7.4). When asked to trade a block for 10 of the next place, at least two
participants, Simone (h/b) and Michelle (l/b), attempted to keep traded blocks
separate from non-traded blocks so that a canonical arrangement could be made as
soon as an answer was recorded. When considered alongside other difficulties
participants had with non-canonical arrangements, the desire of these participants to
revert to canonical block arrangements is consistent with an understanding of
multidigit numbers that relies on counting blocks and naming and recording numbers
in each column separately.
5.3.4 Evidence of the Independent-Place Construct in the Literature
Observations made by several other researchers lend support to the proposed
independent-place construct. Evidence is given in this section of reports of students
constructing tens as abstract singletons, calculating answers column by column,
choosing incorrect blocks to represent places, and choosing misleading independentplace materials to represent two-digit numbers.
217
(a) Tens as “abstract singletons”
Cobb and Wheatley (1988) made an important discovery in their study of
children’s abilities to manage a variety of tasks involving tens and ones. The authors
found that several children in their study perceived of tens and ones as “abstract
singletons” and “abstract units,” respectively. These children were evidently unable
to perceive of a ten as comprising a collection of ten ones, but instead saw it only as
an abstract, indivisible unit that could be counted separately from ones units. This
finding gives clear support to the idea that some children operate on numbers using
an independent-place construct. The independent-place construct encompasses this
and other behaviours, as described in this section, and thus is considered to include
the abstract singleton and abstract unit constructs described by Cobb and Wheatley.
(b) Column-by-column computation.
A number of authors (Cobb & Wheatley, 1988; Fuson & Briars, 1990; Fuson
et al., 1997; Nagel & Swingen, 1998) have noted students adding or subtracting
numbers by considering numbers in each column separately. Cobb and Wheatley
(1988) asked second-grade children to add pairs of numbers such as 16 and 9,
presented either horizontally or vertically. The authors commented that “the children
seemed to operate in two separate contexts: (a) pragmatic, relational problem solving
and (b) academic, codified school arithmetic” (p. 1). When researchers presented
numbers horizontally, the children typically used a counting-on procedure that
necessarily incorporated some notion of the sizes of the two numbers. When the
same numbers were presented in a conventional vertical algorithm format, several of
the same children made concatenation errors of the type described in an earlier
paragraph, with several students writing that 16 + 9 equalled 115. Fuson and Briars
(1990) also noted arithmetic performance of this type, and referred to it as addition
“column by column: . . . The sum of each column was written below that column
even when the sum was a two-digit number (e.g., 28 + 36 = 514)” (p. 189). Fuson et
al. (1997) commented on this phenomenon:
The vertical presentation elicits an orientation of vertical slots on the multidigit
numbers that partitions these numbers into single digits. The physical appearance of
the written multidigit marks as single digits and the nonintuitive use of relative leftright position as a signifier may combine to seduce children to use a concatenated
single-digit conceptual structure even if they have a more meaningful conception
available. (p. 142)
218
Further evidence of independent-place thinking is provided by Fuson et al.
(1992), who investigated the effects of using base-ten blocks with groups of secondgrade children to investigate written symbols, number names, and base-ten blocks.
Researchers presented groups of participants with four-digit addition problems, and
asked them to solve the problems using base-ten blocks. The authors noted that
every group immediately added the like multiunit blocks. After making each addend
with blocks, they . . . pushed the addend blocks of each kind together and counted all
the blocks of a given kind. . . . Evidently the visually salient collectible multiunits in
the blocks supported the correct definition of multiunit addition as adding like
multiunits. . . . All groups also added two four-digit written marks addends by
adding together the marks written in the same relative positions. (p. 76)
Fuson et al. (1992) commented that it was difficult to tell if children linked
the idea of adding like digits with the place names, or if their actions were only
“based on a procedural rule and did not imply understanding of adding like
multiunits” (p. 76). This author suggests that at least some of the children may have
had an independent-place construct that enabled them to correctly add column
amounts separately even though they had not been formally taught procedures for
adding four-digit numbers prior to the study. Support for this suggestion is found in
Fuson et al.’s (1992) comments that some children found difficulty when attempting
to add numbers when trading was needed.
(c) Students choosing the “wrong” blocks.
Another example of apparent independent-place thinking in Fuson et al.’s
(1992) paper is that when asked to add pairs of three-digit numbers, many
participants chose to use incorrect blocks, starting with the largest available block,
the thousands block, to represent the first digit, the hundreds. By starting from the
left-most digit and the largest block size, the children were able to make a
representation for numbers that was mathematically sound, providing that the “tenblock” was given a value of 1. This also is consistent with independent-place
thinking, as each place is mapped to a block size, without regard for the “absolute,”
“correct” value represented by each block.
(d) Students choosing independent-place materials.
An unintended illustration of the effects of certain representational materials
on students’ actions is provided by a recent report of place-value research. Saxton
and Towse (1998) designed their study to test the central claim by Miura et al. (1993)
219
that a child’s spoken language affects the way the child represents numbers using
base-ten materials. Saxton and Towse introduced an important change to the method
used in the earlier research by Miura et al., in making a critical alteration to the
representational materials provided to participants. Whereas Miura et al. provided
participants with standard base-ten material to represent numbers, Saxton and Towse
asked 6- and 7-year-old children to represent two-digit numbers using orange and
green cubes, arbitrarily assigned to represent tens and ones digits. Saxton and Towse
justified this change to the test procedure used by Miura et al. by arguing that baseten blocks “concretised” the abstract relationship between tens and ones material:
In principle, a child could represent a multi-digit number with blocks of ten units
without any clear understanding of place value, simply by counting the component
units in each block. The use of single cubes to represent tens avoids this possibility,
and moreover, ensures that block counting is not prompted by the increased
perceptual salience of large blocks over single cubes. (p. 69)
In view of the large volume of literature on children’s faulty face-value
conceptions of number, and the present discussion of the independent-place
construct, Saxton and Towse’s (1998) argument for using cubes of the same size but
different colours to represent tens and ones seems particularly problematic. Rather
than forcing their participants to focus on the abstract relationship between tens and
ones, as they intended, the researchers may instead have prompted the participants to
use face-value or independent-place interpretations of the digits to represent the
numbers asked of them. Instead of “avoiding [the] possibility” of children counting
units in ten-blocks, this method is likely to promote face-value or independent-place
ideas about what each digit represents. Tellingly, when the researchers did not model
the use of “tens” and “ones” cubes, most participants (over 90% of some cohorts)
used only ones cubes to represent two-digit numbers. However, when the researchers
demonstrated how cubes of two colours could be used to represent numbers, the use
of both “tens” and “ones” increased dramatically. It seems quite possible that
children to whom the researcher modelled the use of two arbitrary colours to
represent tens and ones were thereby encouraged to use an entirely false and
misleading face-value construct or independent-place construct. Furthermore, using
such materials, the responses of children who possessed good understanding of the
base-ten numeration system would be completely indistinguishable from responses
of children who thought either that each digit represented its face value, or that the
220
tens and ones places were independent and could be represented separately by
“unitary” material.
5.3.5 Written Computation and the Independent-Place Construct
There is an apparent contradiction between the teaching of place-value
concepts and the practice of competent users of written or mental algorithms. On the
one hand, students are taught to recognise the different values that digits assume
according to where they are found in a number; on the other hand, efficient use of
computational algorithms requires the user to ignore actual values represented by
individual digits and to focus instead on their face values. These differing
conceptualisations make recognising the conceptual structure possessed by a student
who is carrying out written computation very difficult, especially if the computation
involves no regrouping. For example, a student adding 47 and 31 may arrive at the
correct answer merely by adding pairs of digits in each place, without any regard for
the values represented by the tens digits. A student doing so may have a good
understanding of place-value concepts, or may be operating from an independentplace construct; an observer would be unable to distinguish one from the other
without further questioning. On the other hand, evidence of faulty or limited
conceptions of number may emerge in examples requiring regrouping, as illustrated
by earlier-mentioned examples reported by Cobb and Wheatley (1988), Fuson and
Briars (1990), and Nagel and Swingen (1998).
It is relevant to point out that regarding symbols in written algorithms as only
composed of single digits, and adopting a face-value interpretation of them when
carrying out the computation, is not inherently incorrect, and may have merit when
compared with counting approaches. Clearly it is more accurate to use counting on
to arrive at an answer which, even with minor counting errors, is close to the correct
sum, than to add columns separately and arrive at an answer that could be several
times too large (such as “16 + 9 = 115”). However, there are other factors to consider
before judging an independent-place method too harshly. Counting approaches may
give results that are more or less accurate, but they are prone to errors, inefficient,
and cumbersome for large numbers. Independent-place methods, on the other hand,
take advantage of the power of the base-ten numeration system to represent
quantities with a small number of written digits by considering each place in turn. A
child who says that the sum of 16 and 9 is 115 needs help to see why that is not a
221
reasonable answer, and to interpret, add, and record the partial sums 1 and 15
correctly. However, the method of separating places is quite sound, and is the basis
of conventional computation procedures, providing that the separated partial sums
are correctly interpreted.
5.3.6 Place-Value Tasks and the Independent-Place Construct
Like the face-value construct, the independent-place construct can be difficult
to recognise in responses to many place-value tasks. Firstly, the independent-place
construct does not preclude the use of terms such as “tens” and “ones,” if they are
used only to name places and not to refer to values represented by objects or symbols
in places. As mentioned by other authors (e.g., S. H. Ross, 1990), such terms can be
perceived merely as labels, with no particular meaning with respect to value. As C.
Thompson (1990) pointed out,
having students mechanically put numerals in columns is of no value if the complex
and difficult grouping concepts have not already been constructed by the students.
There is little doubt that young children can count the number of sets of ten sticks
and write that number in a box labeled TENS and similarly count single sticks and
write that number in a box labeled ONES. But such activity does not help students
construct the relationships between tens and ones or the concept of representing
larger quantities by using groups of ten and singles. (p. 90)
Secondly, the presence of the independent-place construct does not
necessarily cause students to arrive at incorrect answers, depending on the nature of
the questions asked (see also Reys & Yang, 1998). If tasks given to students rely
only on the student being able to link each place with a block size or with a set of
number names, or both, then students can consider places to be independent of each
other with no detrimental effect on task performance. A student may (a) consider
separately each digit in a symbol, (b) consider separately each block size in a block
arrangement, (c) use the intuitive mapping that exists between digits and subsets of
like-sized blocks, or (d) state a number name by considering each place separately,
and will often receive correct answers as a result. The following comment by S. H.
Ross (1989) regarding the face-value construct applies equally to the independentplace construct:
222
Students who use a … face-value interpretation of digits succeed on a … [wide]
variety of tasks, including many that use manipulative materials. In many
instructional tasks students are asked to make correspondences between digits and
materials. If a collection is already grouped into a standard place-value partitioning
of tens and ones, a student who is asked to make correspondences for the digits in
52, for example, need only look for “five of something and two of something else.”
(p. 50)
To this observation, we may add that if asked to represent a number using
base-ten blocks, a student need only choose the “right” block size to represent each
digit to be considered correct. If there are only two sizes of blocks to choose from,
and two places to represent, the only remaining problem is to know which block
represents which place. Simple training to associate two block sizes with labels “ten”
and “one,” and teaching of the number names for multiples of ten, would be
sufficient to ensure that many students could correctly show blocks to represent a
number while having no real idea of the values represented by the digits or the
blocks. An example from section 4.6.4 illustrates the importance of the point that
correct task behaviour does not necessarily indicate correct numerical understanding.
In her first interview Kelly (l/c) was asked to show a two-digit and a three-digit
number with blocks. In response to both questions she consistently used one-blocks
to represent the tens, and ten-blocks to represent the ones. By making the mistakes
that she did Kelly drew attention to her faulty ideas. However, the fact that other
participants generally chose conventional block sizes for each digit to represent
multidigit numbers does not rule out the possibility that they may have had similar
misconceptions about digit referents to those apparently demonstrated by Kelly. The
task given by Saxton and Towse (1998), described earlier, illustrates this point:
Students were asked to represent two-digit numbers using green and orange cubes to
stand for tens and ones digits, according to the researchers’ arbitrary assignment of
each colour to represent a place.
5.4
Participants’ Construction of Meaning
One prominent feature of the data in this study, mentioned several times
previously, is the noticeable changeability of participants’ responses to questions,
both in interviews and in teaching sessions. This general observation has led the
author to the view that, in the majority of cases, the participants’ number conceptions
were not characterised by fixed conceptual structures. On the contrary, the
participants appeared often to be weighing up the evidence before them and making
223
the best sense of it they could, altering their answers as and when inconsistencies
appeared between their responses and other information. This sense-making
character of participants’ responses is believed to show that for many participants
their conceptions of numbers were still in a “construction zone,” subject to influence
by outside information such as visual cues provided by representational materials or
the interviewer’s questions.
5.4.1 ‘Organic’ Understanding
Results of this study show a picture of children whose ideas about numbers,
symbols, and representational materials fluctuated with the introduction of further
data to challenge those ideas. The understanding of many participants could perhaps
best be described as organic, rather than as belonging to a particular fixed category:
Participants’ understanding appeared often to be in a developing state, subject to
various influences in the surrounding “environment.”
One particular type of question that elicited frequent changes of opinion was
the digit correspondence questions. As demonstrated in section 4.5.2 and elsewhere,
several participants who believed that each digit represents only its face value
nevertheless understood that there was a contradiction between their explanation of
digit referents and the evidence of the objects before them, and apparently tried to
resolve the contradiction by generating other explanations. This phenomenon is
captured in response Category II in digit correspondence tasks; participants giving
these responses rejected face-value interpretations of symbols in favour of a different
explanation, that individual digits had no meaning in the multidigit symbol, but
together represented the whole collection of objects. S. H. Ross (1989) believed that
this type of response represented the lowest level of understanding of digits, showing
that the child had little idea at all of what the symbol meant. However, as explained
earlier in this chapter, it is believed by this author that, to the contrary, such
responses show a willingness on the part of the child to forgo the immediate
suggestion that each digit represents just what it would represent if it were on its
own, and instead to find some other explanation for two symbols each of small value
representing a comparatively large collection of objects.
Even in the case of participants holding apparent face-value interpretations of
the symbols, there was evidence of construction of meaning about the symbols.
Section 4.4.3 includes a series of statements made in the interviews by 9 participants
224
who gave a face-value interpretation of digits, explaining the reason why the
remaining sticks “left out” of their face-value interpretation of digits apparently had
no written representation. It is clear immediately on watching videotapes or reading
transcripts of these responses that the participants did not appear to be troubled by
the question. In most cases participants stated the answers without hesitation,
apparently indicating that they had already decided on an interpretation of the digits
before being asked by the researcher. In fact, there was not one participant holding a
face-value interpretation of the digits in Question 7 who did not offer an explanation
for the remaining sticks. The second thing that is surprising to an adult observer is
that the illogicality of their view either did not occur to the children, or at least did
not trouble them. They accepted a situation in which the symbol ‘24’ represented 24
sticks, and simultaneously the ‘2’ represented two sticks and the ‘4’ represented four
sticks, with 18 sticks not represented by any symbol at all.
The acceptance by participants of two mutually exclusive propositions is a
characteristic of several responses made by participants that, again, supports the idea
that the participants were actively trying to make sense of a situation about which
they did not have fully-formed opinions. Section 4.4.5 includes a transcript excerpt
showing Jeremy (l/b) attempting to explain which is larger, 27 or 42. This account
typifies several transcripts showing participants weighing up various pieces of
information in justifying their responses. Jeremy did not merely accept every
suggestion made or implied by the researcher, but considered each one in turn. When
the researcher finally convinced him that 42 was later in the counting sequence,
Jeremy mentioned again the sizes of the digits and supported his answer by referring
to the relative order of the digits in the two symbols. He was also able to defend his
revised belief about the two numbers against another incorrect face-value suggestion
from the researcher about the larger ones digit in ‘27.’ This pattern of responses is
indicative of on-going development of number conceptions, not simply of a fixed,
incorrect face-value construct.
5.4.2 Participants’ “Invented” Answers
One feature of the construction of meaning evident in transcripts is that
participants often referred to ideas that they had evidently invented in order to answer
questions about numbers. This is shown in the discussion of Terry’s (l/c) explanation
about 27 and 42 in section 4.4.5 (the full transcript of which is in Appendix N).
225
Terry’s idea that even numbers are larger than all odd numbers is a good example of
an invented response. No teacher would teach this idea, but it is conceivable that if a
teacher presented diagrams showing odd and even numbers in a certain way, a child
might arrive at Terry’s rule in the absence of further information to challenge it. It
appears that some of the participants had been taught about even and odd numbers
shortly before the study, as several participants referred to odd and even numbers in
the interviews, though there was no mention of these numbers in any question.
Some participants found they could not interpret three-digit numerical
symbols using their existing knowledge of two-digit symbols, leading to some
interesting ideas. Clive (l/b) and Daniel (h/c) both suggested that ‘138’ had 1 ten and
38 ones, and ‘183’ had 1 ten and 83 ones. Daniel clearly attempted to interpret these
symbols using his knowledge of two-digit symbols. He said that the ones column
was “round here somewhere,” indicating rather vaguely the tens and ones digits, and
seemed amused to find that it had “two numbers in it,” the ‘3’ and the ‘8’
(I1, Qu. 6b).
It is evident that the changing of opinions when responding to interview
questions was exhibited most often by low-achievement-level participants. It is to be
expected that low-achievement-level participants would have ideas about numbers
that are less fully developed than high-achievement-level participants. Thus, it may
be deduced that low-achievement-level changed their ideas about numbers more
often than high-achievement-level participants did. However, such an observation
may be misleading. As discussed earlier, apparently correct responses to
mathematics questions can obscure faulty understandings if the questioner does not
probe the reasoning behind responses. In cases in which participants responded
quickly with correct answers, the researcher often did not probe their thinking to any
great extent, assuming that their correct answer represented a sound understanding of
the topic. However, such quick, correct responses may hide very similar processes of
testing tentative theories that are taking place mentally, and therefore out of sight.
Thus, it is quite possible that high-achievement-level participants may also have been
engaged in the construction of understanding of numbers, and considering multiple
interpretations before giving their answers.
226
5.4.3 Teaching, Learning, and Constructing Meaning
The idea that participants in this study were engaged in meaning construction
closely matches the literature on constructivism. Constructivist ideas focus on the
individual nature of understanding, and on the notion that each student constructs an
understanding of each concept that is unique to that student. There is no place in a
constructivist pedagogy for a teacher to try to give information to a student, because
it is not possible to transmit ideas directly from one person to another, or from
another source of information to a person. Instead, teachers are exhorted to
encourage each student to develop ideas personally, to allow students space to
develop unique understandings of each topic in the curriculum.
Evidence in this study’s data of participants apparently thinking actively
about numbers and what they mean, implies that ideas that teachers present to
students may not be received as the teachers intend. In fact, depending on how an
idea fits with a student’s already-existing concepts about numbers, the student may
interpret it in ways that the teacher could hardly imagine. This thesis includes many
examples of such unusual ideas held by children that an adult is unlikely to have
predicted. In some cases, these involved the simultaneous acceptance by participants
of contradictory or inconsistent beliefs as participants attempted to make sense of the
information available to them. There are clear implications in these data for how
teachers present mathematical information to students and how teachers ascertain
their students’ understandings of numbers. These points are mentioned again in
section 6.3.4.
5.5
Effects of Physical or Electronic Base-Ten Blocks on Place-Value
Understanding
This study has explored a wide range of issues relating to children’s
understanding of numbers when using materials. In this section the effects of
physical or electronic base-ten blocks on how children represent numbers are
analysed in light of the study’s findings. Four aspects of the relevant results are
discussed in this section: (a) minor differences that were evident between the
learning that occurred among participants who used physical blocks and learning
among those who used electronic blocks; (b) the sensory impact of both types of
material; (c) the facility that each material offers to aid the development of links
227
among blocks, symbols, and numbers; and (d) the support that each material provides
for the development of number concepts.
5.5.1 Differences in Learning of Participants Using Physical or Electronic Blocks
As reported in section 4.3, though there were clear differences between the
learning of high-achievement-level and low-achievement-level participants during
the course of the study, differences in learning that occurred among participants
using blocks and learning that occurred among participants using software were
minor. Based on performance on place-value tasks at the interviews before and after
the teaching sessions (Table 4.3), individual participants such as Simone and Nerida
did show improvement in their understanding of place-value concepts. However,
aggregate scores of the 4 groups show no differences in learning about place-value
that could be attributed to use of one material or the other. It must be pointed out that
the small scale of this study does not support strong claims of such differences,
which on the basis of the data here appear to be rather subtle. Similar studies
conducted with larger numbers of participants and over longer periods of time would
be needed to confirm initial ideas of differences between the use of physical and
electronic number models mentioned here.
5.5.2 Sensory Impact of Physical or Electronic Blocks
One apparently common view of teachers when it is suggested that software
could be used to take the place of base-ten blocks to teach place-value concepts is
that children using software would be somehow missing out because of a lack of
tactile contact with the medium. As noted by Clements and McMillen (1996)
“manipulatives are supposed to be good for students because they are concrete”
(p. 270). However, as pointed out by other authors (Hunting & Lamon, 1995; Perry
& Howard, 1994; P. W. Thompson, 1994) the mathematics is not contained in the
material, and so benefits from conventional blocks’ physical attributes may be more
imaginary than real.
Though students have no direct physical contact with computer-generated
blocks, they do have access to other sensory input that differs from that offered by
conventional blocks. First, there is the auditory input of the audio recordings of
number names used to read the numbers represented by blocks, the number
represented by the blocks of one size, and the numeral expander. As shown in section
228
4.7.6, participants who used the software enjoyed using the audio capabilities of the
software to gain confirmation of their block representations. Furthermore, section
4.7.1 demonstrates that the audio recordings in the software acted as one source of
feedback available to users of the software. No such features are available to users of
blocks; other sources of auditory feedback such as a teacher, if available, must be
accessed instead.
The second source of different sensory input provided by the software is the
visual arrangements of blocks, coupled with counters, labels, number window, and so
on. Though the on-screen blocks appear to be quite similar to their physical
counterparts, there are several differences that the study data showed to be important.
One difference is the juxtaposition of several representations of a number
simultaneously. It has been mentioned several times that the blocks and the on-screen
numbers changed at virtually the same time, to provide a continually updated set of
parallel representations for numbers, in close proximity to each other. Users of the
software were able to take in visually the various representations for numbers with
little effort, and watch changes occur in all representations at the same time. The
other major visual difference between blocks and software was mentioned by
Clements and McMillen (1996), writing about computer manipulatives in general:
“[computer] representations may also be more manageable, ‘clean,’ flexible, and
extensible” (p. 272). There is no question that the computer representations of
numbers were much neater than representations made with physical blocks. As
mentioned in several places in the thesis, counting and handling errors with the
blocks were quite common. These errors mostly resulted from difficulties with
managing the material so that numbers and processes could be correctly represented.
Some participants were generally careful when handling blocks to check counted
arrangements to ensure that the correct quantities were put out. However, other
participants were less careful and made frequent handling errors that led to incorrect
answers. Errors made with computer blocks were less frequent, apparently because
of the feedback provided by column counters and the number window.
Related to the sensory impact of the computer blocks is the ease with which
they may be used to demonstrate numbers and numerical processes. Not only does
the software provide on-going feedback about the number of blocks displayed, it also
enables very rapid placement of blocks via clicks with the computer mouse. Each
click of the mouse on the appropriate button results in the placement of a block in a
229
particular place. As the participants using the software became familiar with it, they
became adept at placing blocks very quickly. On occasions participants using the
software overshot the number they required, but they were able to assess this and
correct it without much delay. The ease of use of the software is particularly
noticeable when it comes to trading processes. The regrouping (10 for 1) and
decomposing (1 for 10) tools ensure that each trading action is done accurately and
quickly. In the case of trades performed with conventional blocks, mistakes were
quite frequent, particularly with low-achievement-level participants, and the
researcher had to step in to correct errors before they caused the participants further
difficulties.
5.5.3 How Numbers Are Represented by Physical or Electronic Blocks
As described in section 2.3, the base-ten numeration system has a number of
features with which primary school students need to become proficient. These
features include the place-value system underlying the written symbols, the system
of naming numbers, and the trading processes necessary for multidigit computation.
Students learning place-value concepts in their second or third year of schooling face
several difficulties, for a number of reasons: (a) the “collected multiunit” idea
(Fuson, 1992) is far more complex than single digit representation of numbers up to
9, (b) the system of English number-naming words contains many inconsistencies,
and (c) trading processes produce non-canonical arrangements of tens and ones that
temporarily break the normal rules of the base-ten numeration system.
Base-ten blocks and place-value software incorporate features that may
support or hinder students as they face these obstacles to understanding base-ten
numbers. For example, it is important to consider how each representational format
helps students to (a) represent numerical quantities, (b) name quantities and written
symbols, (c) carry out trades, and (d) recognise different numerical representations,
such as non-canonical arrangements of blocks. These considerations are addressed in
the following subsections.
Physical base-ten blocks and the independent-place construct.
As explained in section 2.4.3, base-ten blocks are a type of analogue of
numbers. The relative sizes of the blocks map directly onto the relative values
230
represented by the first four places of the base-ten numeration system (English &
Halford, 1995).
However, despite the apparent transparency of the mapping between blocks
and places, data in the study show quite convincingly that some children do not
regard the sizes of base-ten blocks when they use them, but merely use them as
“place counters.” As described in more detail in section 5.3, children possessing the
independent-place construct match block sizes, written digits, and number names
without regard for the values involved. Such children are able to use blocks quite
successfully on routine tasks; however, it is probable that they could be just as
successful using materials that did not act as a proportional analogue for the base-ten
numeration system, such as coloured chips or other materials arbitrarily assigned to
represent each place. On more difficult tasks involving trading the independent-place
construct leads to errors, such as writing concatenated place symbols, like “215” for
the sum of 17 and 18. However, if early work with two-digit numbers does not
involve trading or non-canonical representations, children with an independent-place
construct can use materials such as physical base-ten blocks without revealing any
errors in their thinking.
Electronic base-ten blocks and the independent-place construct.
Place-value software such as Hi-Flyer Maths can be used in similar ways to
physical base-ten blocks, and may also fail to challenge students who possess the
independent-place construct. Though the various counters incorporated in the
software were designed to assist students to make connections among numbers,
written symbols and block representations, there is some evidence that, like physical
base-ten blocks in the previous discussion, they may have helped support face-value
interpretations of symbols among low-achievement-level participants. The software
used in the study incorporates a counter at the top of each column of hundreds, tens,
and ones blocks that displays a continuous tally of the number of blocks in that
column. Because these are only counters of the number of electronic blocks, in
themselves they do not indicate anything of the represented values. A label is
included below each counter to indicate the place name “hundreds,” “tens,” or
“ones”; but a student could possibly see these as words only, rather than as numerical
values. Thus, for example, if there are 2 hundreds, 4 tens and 8 ones on the screen, it
is possible that a student may notice only the digits ‘2,’ ‘4,’ and ‘8’ (see Figure 5.1),
231
and interpret the symbol ‘248’ as being composed merely of a concatenation of these
face values.
Figure 5.1. Column counters in software representation of 248.
It is clear that students who possess an independent-place construct could use
the software to represent numbers, noting the column counters and the number
window, and listening to the number name read to them by the software without their
independent-place ideas being challenged. On the other hand, certain features of the
software are likely to cause some conflict with an independent-place construct. The
on-screen numeral expander will show different ways of grouping digits to form
symbol-based non-canonical representations for numbers, having the effect of
transferring a digit into an adjacent place. For example, the number 267 can be
shown on the expander as “26 hundreds 7 ones,” “2 hundreds 67 ones,” or “267
ones.” Each of these representations for 267 breaks the central idea behind the
independent-place construct by showing different ways of interpreting the values of
the digits. Similarly, the software will quickly and accurately demonstrate either noncanonical arrangements or trading processes that could be used to challenge an
independent-place construct held by a student. Though physical blocks and numeral
expanders could be used with the same effect, the speed and accuracy of the software
provides extra convenience. Using electronic blocks, a student could witness many
more examples of number representations to challenge the independent-place
construct than could be shown by physical base-ten blocks in the same time.
5.5.4 Development of Links Among Blocks, Symbols, and Numbers
Section 2.5.2 includes a discussion of difficulties students have in using
materials to model numbers that have been identified by several researchers, and
focuses particularly on the idea that there is a conceptual gap in children’s thinking
232
between symbols and number material such as base-ten blocks. Data gathered in this
study provide information that adds to the available knowledge of how children use
representational materials, and may help inform discussions of why some material is
not always successful in teaching students about the base-ten numeration system.
This study also provides the opportunity to compare physical base-ten blocks with
electronic blocks, to see if the different features of the two representational formats
make a difference in helping children make connections between symbols, numbers,
and the material.
An important issue that has a bearing on how well children develop
conceptual links among numbers, symbols, and blocks when using physical base-ten
blocks is the accuracy of the block representations formed by the children. As
mentioned previously, one major difference between physical and electronic blocks
is the facility of the software for providing accurate counters for the number of
blocks in each column and an accurate symbol for the entire number represented by
the blocks on screen. To generate equivalent symbols when using physical blocks it
is necessary to count the blocks; if sufficient care is not taken with counting, errors
can be introduced that require remediation before correct ideas can be gained from
the blocks. In light of the large number and variety of errors made by participants,
described in section 4.6, clearly users of physical blocks need to take great care when
counting blocks to ensure that counting or handling errors do not give an incorrect
impression. Later in this section the tendency of some participants to trust their own
count of the blocks, even in the face of other contradictory information, is discussed.
It is clear that one solution to difficulties that students have in using physical blocks
to understand numbers is for the teacher to stress the importance of care in handling
the blocks, and the frequent use of checking procedures to attempt to trap errors.
However, such procedures will only assist students when the errors made have been
counting or handling errors, and if the students already have enough understanding of
numbers not to introduce incorrect ideas, such as trade-up-to-10 (section 4.6.2).
Conceptual errors, as opposed to counting or handling errors, cannot easily be
checked by the person possessing them; they require another person or agent to point
them out before their effects can be countered.
Thus, support provided by a representational format for the development of
accurate conceptions of numbers depends to a large degree on the accuracy and
correctness of the manipulations of the material carried out by the user. No material
233
is going to demonstrate correct ideas about numbers if the user makes fundamental
errors in using the material that remain uncorrected. In the teaching phase of this
study the researcher was always present to “pick up the pieces” if participants made
errors they were unable to correct themselves. The researcher also made sure that any
errors were corrected and faulty ideas challenged before participants started new
tasks. In a busy classroom with 30 or so students, a teacher does not often have time
for this sort of management of the learning environment, and so errors and
misunderstandings can easily go unchallenged.
5.5.5 Support for the Development of Number Concepts
The one component of the two interviews in which there was a notable
difference between the performance of participants who used physical blocks and
those who used electronic blocks was skip counting. It is noted in section 4.3.2 that
interview results appear to indicate a higher performance on skip counting tasks by
participants who used electronic blocks than by those who used physical blocks (see
Table 4.2). The following transcript excerpt illustrates some skills required to skip
count successfully:
Daniel:
681, 671, 661, … 661, 651, 641, 631, 621, 6 hundred and … uh … 11, 601, 6
hundred … no, so that must be … 5 hundred and … 91, 581, 571, 561 …
(I2, Qu. 4d)
In order to complete this task correctly, Daniel (h/c) had to keep track of (a)
the number of hundreds; (b) the number of tens; (c) the names of each decade,
including the “teen” number 11; and (d) the rules of the base-ten numeration system
that define how to count 10 less than 601. In completing this task successfully,
Daniel was able to use a regular “six hundred and n-ty-one” pattern in naming the
numbers to 621. However, this pattern is not used for the number 611 or the numbers
less than 600, causing Daniel to pause in his counting while he thought about those
numbers.
The features of the software available to its users may help explain why
participants from the two computer groups were better able to skip count after the
teaching phase than were participants who had used physical blocks. During the
teaching phase the researcher encouraged participants to use the number window
when doing tasks that involved skip counting (such as Tasks 13-17 and 40-43; see
Appendix H). The effect of the number window during these tasks was to show a
234
counter that changed instantaneously when a block was added or subtracted, showing
clearly the changing digit and how the numbers changed at the change of a decade or
a hundred. For example, using the number window while completing Task 41 would
show the symbols 462, 472, 482, 492, 502, 512, 522, and so on. Experiences with
this “odometer effect” may have helped participants from the computer groups to
improve their skip counting abilities.
5.6
Place-Value Understanding Demonstrated by High- and LowAchievement-Level Participants
5.6.1 Similarities in Place-Value Understanding of High- and Low-AchievementLevel Participants
Many of the observations made in chapter 4 and earlier in this chapter apply
to both high-achievement-level and low-achievement-level participants. First, though
high-achievement-level participants in general performed much better on the placevalue tasks they were set, at times they also demonstrated similar misconceptions and
errors to the low-achievement-level participants. Specifically, at various times a
small number of high-achievement-level participants used inefficient counting
approaches (Table 4.8) or face-value interpretations of digits (Table 4.10), and gave
the lowest categories of response (Category I or II) to digit-correspondence tasks
(Table 4.13). High-achievement-level participants also made similar errors to lowachievement-level participants, including each of the types of counting, blockhandling and naming errors described in section 4.6.
Second, a few low-achievement-level participants at various times showed
similar abilities to high-achievement-level participants. For example, Table 4.2 and
Table 4.3 show that certain low-achievement-level participants demonstrated similar
numbers of numeration skills as certain high-achievement-level participants. Also, as
shown in Table 4.6, Table 4.8, and Table 4.10, there was some overlap regarding the
frequency of use of counting and grouping approaches and face-value interpretations
of digits by low-achievement-level and high-achievement-level participants.
Similarities in responses of low-achievement-level and high-achievementlevel participants may partly be due to the nature of the Year 2 Net (Queensland
Department of Education, 1996) and how students’ scores are determined, as
intimated in section 4.3.2. Another factor is the small number of available children at
the school. It can be seen in Appendix F that differences in mathematical
235
achievement between low-achievement-level participants and high-achievementlevel participants were not very great; in a school with a larger pool of Year 3
students from which to select participants, it may have been possible to have
participants who demonstrated more widely separated mathematical achievement
levels.
More importantly, similarities in response patterns of high- and lowachievement-level participants may point to important factors regarding the learning
of place-value concepts by Year 3 students generally. As discussed in section 5.4,
one notable feature of the data in this study has been the changeability of
participants’ ideas about numbers. As already discussed, though changeability of
ideas was more commonly exhibited by low-achievement-level participants, it is
quite possible that high-achievement-level participants also used their existing
knowledge of numbers to test hypotheses regarding questions put to them, before
responding to the researcher’s questions. This idea is supported by incidents in which
high-achievement-level participants changed their answers or accepted incorrect
counter-suggestions offered by the researcher. These observations are important
because they show that even the high-achievement-level participants at times
demonstrated ideas about numbers that were being developed and subject to change,
rather than being fixed and immutable.
5.6.2 Differences in Place-Value Understanding of High- and Low-AchievementLevel Participants
In spite of the similarities in responses of high-achievement-level and lowachievement-level participants reported in the previous section, a number of clear
distinctions were observed between them. Several tables in chapter 4, including
Table 4.13, show dramatic differences between responses of participants in highachievement-level and low-achievement-level groups. Table 4.2 and Table 4.4 show
that in the interviews high-achievement-level participants outperformed lowachievement-level participants by an average of about 8 or 9 place-value criteria.
Table 4.6, Table 4.8, Table 4.10, and Table 4.12 show clear differences between
high-achievement-level and low-achievement-level participants’ approaches to
place-value questions, with high-achievement-level participants adopting grouping
approaches much more often, and using counting approaches or face-value
interpretations of digits much less often. Table 4.13 shows that high-achievement-
236
level participants generally gave more advanced interpretations of digits in digit
correspondence tasks. These tables together appear to illustrate important distinctions
between the two groups of children. High-achievement-level participants
demonstrated more effective, accurate approaches to place-value questions, and
demonstrated better grasp of place-value concepts, than low-achievement-level
participants. It is not difficult to believe that these two observations are related. The
participants who exhibited best knowledge of the base-ten numeration system, and
the best understanding of place-value concepts, also demonstrated more efficient and
accurate strategies for answering place-value questions. As mentioned earlier in the
discussion of the use of counting, it appears likely that use of more accurate and
efficient strategies enable students to perform better on place-value tasks, and
understand place-value concept better, than their peers who use less accurate or
inefficient strategies, or both. In effect, by having better knowledge of numbers and
the base-ten numeration system, more able students have access to better strategies,
that in turn give quicker, more accurate results, leading to further improved
knowledge and skills. This apparent “Matthew effect” (Burstall, 1978), leading to
improved understanding and performance by those students who start in front, has
clear implications for teachers who attempt to provide equal opportunities for
successful learning to all their students; this point is taken up later in section 6.2.4.
237
Chapter 6: Conclusions
6.1
Chapter Overview
This chapter is divided into three major sections. Section 6.2 includes
discussion of emerging answers to the study’s research questions, section 0 addresses
implications of the study findings for the teaching of place-value concepts, and
section 6.4 outlines recommendations for future research into place-value
understanding.
6.2
Conclusions About Answers to Research Questions
This study was conducted to address the following research question: How do
base-ten blocks, both physical and electronic, influence Year 3 students’
conceptual structures for multidigit numbers? To answer this question, four subquestions have been addressed within the context of Year 3 students’ use of physical
or electronic base-ten blocks. Each of the four following subsections addresses one
of these four questions.
6.2.1 Conceptual Structures for Multidigit Numbers Evident in Participants’
Responses
1.
What conceptual structures for multidigit numbers do Year 3 students
display in response to place-value questions after instruction with baseten blocks?
As a result of the literature search conducted early in this study, certain
conceptual structures were identified as part of a sequence of essential conceptual
structures adopted by children as they developed their place-value understanding. It
was hoped to be able to analyse the interactions observed in the interviews and
teaching sessions and compare participants’ conceptual structures for the two
representational formats, blocks and software.
239
However, as discussed in section 5.2.1, what emerged from this study’s data
was a pattern of preferences held by participants as they answered a range of
questions, rather than stable conceptual structures. In some cases, participants’
preferences for a particular approach were quite well defined, especially for those
participants who favoured grouping approaches; there is good evidence that these
participants had well-developed conceptual structures for multidigit numbers that
included the grouped, multiplicative aspect of the base-ten numeration system.
However, many participants did not use a single approach to the questions, and
appeared not to have developed stable ideas about multidigit numbers. Their
responses were characterised by the adoption of a variety of approaches and a
marked changeability of opinion about the questions asked.
To summarise this section, conclusions about Year 3 students’ conceptual
structures drawn from the data in the study are as follows:
1.
Conceptual structures described by other authors (e.g., Miura &
Okamoto, 1989; S. H. Ross, 1990) were evident in participants’
responses.
2.
However, in many cases conceptual structures were not held firmly, but
were altered in response to further information or further questioning.
3.
Thus, in light of the changeability of students’ opinions, it is more
accurate to categorise a student’s response, than to categorise the
student per se.
6.2.2 Misconceptions, Errors, or Limited Conceptions Evident In Participants’
Responses
2.
What misconceptions, errors, or limited conceptions of numbers do
Year 3 students display in response to place-value questions after
instruction with base-ten blocks?
As mentioned in chapter 4, the large number and variety of misconceptions,
errors, and limited conceptions of numbers evident in the study data have important
implications for how place-value topics are taught in primary school. These
implications include, but are not limited to, a lower likelihood of success that errors
impose on those making them. Other implications are the greater difficulty added to
the learning of topics and the greater cognitive load certain errors cause. Such errors
are related to broader topics in this study, including the preference of some
240
participants for counting and the independent-place construct. In fact, many errors
observed in the study would be less serious if one could be sure that the participants
making them were doing so accidentally. However, in many cases participants
making errors appeared to have such deep-seated confusions about numbers that they
were unable to carry out successfully all but the most basic of place-value tasks.
Such errors evident in this study as trading a ten-block for a one-block, or counting a
collection of ten- and one-blocks as if each represented only one, lead the author to
the conviction that children making them had little real understanding of the base-ten
numeration system. If a teacher ignores such mistakes in the belief that children
demonstrating such errors are merely being careless, the children will be denied help
they need to develop accurate understanding of base-ten numbers.
The diverse errors made by participants in the study are summarised in
chapter 4 as being errors of counting, handling errors, errors in trading, errors in
naming and writing symbols for numbers, and errors in assigning values to blocks.
The root causes of such errors can be summarised as being of one of three
fundamental problems: (a) lack of knowledge of base-ten number naming
conventions (such as the pattern “x hundred and y-ty z,” etc.); (b) lack of familiarity
with base-ten blocks; or (c) or the independent-place construct, characterised by a
lack of understanding of the relationship between each place and the places either
side of it. This latter misconception is particularly difficult for the teacher to
recognise, as it is often disguised by the tasks typically given in textbooks and some
classrooms, that ask children merely to state which digits, number names, places, or
base-ten blocks to associate with each other. If a question about hundreds, tens, and
ones can be answered by making single-dimensional associations between a digit, a
place-specific number name, and base-ten blocks of a particular size, then the
response of a child possessing an independent-place construct will be entirely
indistinguishable from the response of a child who understands the groups of 10
behind multidigit base-ten numbers.
One other aspect of the data in this study has relevance for helping children
overcome erroneous ideas they have about base-ten numbers. The evidence of many
incidents of participants inventing or mis-applying ideas to explain features of baseten numbers leads to a certain confidence that children will be able to understand the
base-ten numeration system for themselves, provided that their teachers give them a
logical basis for understanding the relations that exist between numbers and their
241
symbolic and concrete referents. Since participants were evidently comfortable in
applying knowledge about numbers to novel questions, this gives good reason to
believe that with accurate information in a form that is accessible to them, children
will be able to develop correct, coherent, sensible conceptual structures for base-ten
numbers.
Evidence in the study for children’s construction of knowledge of numbers
has relevance for the view taken of children’s errors in understanding numbers. Since
it is evident that children are prepared to use a wide variety of information to help
them make sense of numbers, consistent with current advice that children should be
encouraged to develop their own understandings of the world, it is likely that in the
process errors will eventuate. Furthermore, it is clear that if children are taught
merely to follow procedures with blocks or written symbols, their attempts to make
sense of numbers are likely to be frustrated, and may result in the development of
faulty ideas.
Teachers must recognise the great leaps in conceptualisation that have to take
place at various points in the teaching of mathematics topics. In particular, the step
from recognising one-digit symbols as standing for collections of so many single
items, to seeing that two-digit symbols stand for collections of 10 items and left-over
single items, can be accurately labelled a “conceptual leap” (labelled by Baturo,
1998, a “cognitive leap”), rather than a mere progression based on previous ideas.
This study set out to investigate the teaching of the hundreds place; what has
emerged is a clear problem in the learning of the tens place for many children of this
age group. The majority of the low-achievement-level participants had such a limited
understanding of two-digit numbers that questions involving hundreds were really
beyond their abilities. Even many of the high-achievement-level participants had
limited understanding of the base-ten numeration system, and though they could
often answer a question successfully, their understanding of what is represented by
tens digits was based often on independent-place ideas.
Based on the context in which limited and faulty ideas about numbers
emerged in this study, it is clear that the mathematics tasks presented to students
have a strong bearing on the mental models for numbers that they will accommodate.
As mentioned several times, tasks based on matching numerical symbols, number
names and base-ten blocks can often be answered without addressing relationships
between places, and with very limited understanding of the base-ten numeration
242
system. Thus the type of task given to students has an important bearing on the
place-value understanding that is revealed by student responses. Teachers and
curriculum writers need to be aware of these points, and to limit tasks that rely only
on knowledge of place names, block names and number names. Rather than
questions such as “Show me the tens part of this number,” questions which challenge
children, such as “What is another way to use base-ten blocks to represent this
number?” will help to distinguish between children who understand the grouped-ten
aspect of base-ten numbers and those who do not.
6.2.3 Effects of the Two Materials on Students’ Learning of Place-Value Concepts
3.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in instruction with physical and
electronic base-ten blocks?
The central question of this study is how each of the representational formats,
physical or electronic blocks, affect Year 3 students’ learning of place-value
concepts. This question is addressed principally via the data from the two interviews,
discussed in chapter 4. Results from the two interviews show that many participants
did improve their understanding of the base-ten numeration system over the course
of the study. However, there was no marked trend that could be identified to compare
differential effects of the two representational materials on student learning. Gains of
conceptual understanding of the base-ten numeration system were quite conservative,
and neither cohort using physical or electronic blocks appears to have done
significantly better than the other. Certain individual participants showed pleasing
improvement on interview questions over the course of the study, but others showed
no improvement or even deterioration in place-value understanding.
Use of physical base-ten blocks to learn place-value concepts.
The clearest finding about the use of blocks in the data collected in this study
is that students using physical blocks need support to represent numbers and
numerical processes. In this study, this fact has come into sharp focus, as
comparisons with the use of electronic blocks to represent numbers show that
physical blocks lack certain features that appear to have made electronic blocks
easier for participants to use. Specifically, physical blocks lack any sort of counting
device to inform the user of the number of blocks present, or the number represented
243
by a blocks arrangement. Furthermore, unlike electronic blocks, physical blocks do
not have any mechanism for carrying out trading actions that will ensure that such
actions are done correctly. These facts, coupled with the high incidence of errors in
counting and handling blocks, meant that there was the potential for the participants
using physical blocks to face many difficulties in learning about the base-ten
numeration system. In this study’s teaching sessions the researcher was able to give
physical blocks users feedback about their use of the blocks, and thereby to correct
mistakes and misconceptions before they could become entrenched in the
participants’ thinking. This might not be the case in a typical classroom, as it is
unlikely that a teacher with an entire class to supervise would be able to monitor the
use by individual students of base-ten blocks very closely.
Transcripts of blocks groups indicate cause for some concern about how
useful physical base-ten blocks are for teaching number concepts. The approaches
taken by both high-achievement-level and low-achievement-level participants using
physical blocks often were not conducive to the generation of understanding of baseten numbers. Firstly, high-achievement-level participants showed reluctance to use
physical blocks to illustrate numerical processes that they evidently understood; it
appeared that at times these participants regarded the block representations as
redundant because they felt they already understood the concepts illustrated by the
blocks. When the researcher required high-achievement-level participants to use the
blocks, however, on a number of occasions participants expressed greater confidence
in the blocks than in their own thinking, and accepted incorrect answers produced in
mishandling the blocks in preference to correct answers they had worked out
mentally. Secondly, low-achievement-level participants typically used the base-ten
blocks as calculating devices, and evidently had no idea of the answers to many
questions until they counted the blocks. The large number of errors made, both in
counting and handling blocks and in thinking about numbers, meant that in many
instances low-achievement-level participants received misleading information from
the blocks, and the researcher needed to correct their mistakes.
The difficulties described in the previous two paragraphs resulted in many
instances of feedback being provided to participants using physical blocks by the
teacher and by their peers. Rather than thinking about the numbers involved and
attempting to work out answers to questions mentally, on many occasions
participants in the blocks groups used other sources of information to tell them
244
answers. Often the source of information was the blocks themselves, counted by a
participant. On other occasions participants received help from their peers, or relied
on the researcher to tell them if they were correct or not. The researcher attempted to
reduce the amount of feedback he gave to blocks participants, to encourage them to
use other resources, including their own thinking, to come up with answers.
However, in many instances, this was not successful, and the only source of accurate
information available to the participants was the researcher.
Use of electronic base-ten blocks to learn place-value concepts.
As noted in the previous section, the software incorporates features to provide
users with feedback about the blocks on the screen, and the numbers they represent.
Descriptions in chapter 5 of interactions among the researcher, participants, and the
software indicate that these feedback-providing features influenced the ways
participants used the materials, and the frequency with which they accessed feedback
from non-electronic sources. Specifically, these features include an uncluttered view
of blocks, electronic counters of three types that keep a track of the numbers of
electronic blocks present, audio number name recordings, and accurate trading
transactions. The software was designed to incorporate these features in the hope that
they would assist students in learning about the base-ten numeration system; though
results are far from conclusive, there are positive indications of the effects of the
software on student thinking about numbers. Specifically, compared to users of
physical blocks, participants using electronic blocks received considerably less
feedback either from the researcher or from each other, instead using the software to
inform them about the numbers they were representing. In the process they received
far more positive, more accurate feedback overall, which implies a likely positive
effect on students who use similar electronic blocks for learning about numbers.
On one aspect of number processes in particular, trading operations,
participants who used electronic blocks demonstrated great confidence in the
equivalence of traded blocks, after observing accurate trades many times, supported
by electronic symbols (section 4.7.6). Though the same information was available to
participants using physical blocks, and though the researcher pointed out the
equivalence of traded blocks, blocks participants did not exhibit any marked
awareness that traded blocks always represent the same quantity. This positive
learning effect evidently resulting from the use of electronic blocks implies that such
245
software could be very useful for teaching such concepts to students, provided the
software was designed to incorporate carefully-planned support for the concepts.
Comments about learning effects observed in this study.
The comments in the previous paragraphs seem at odds with other aspects of
the results reported in this chapter. In particular, it seems reasonable to expect that
participants using electronic blocks would improve their understanding of certain
aspects of numbers more easily than would participants using physical blocks, based
on descriptions of the apparent effects of using the software.
A number of comments may put this in perspective. First, at both interviews
the students were provided only with physical base-ten blocks with which to answer
questions regarding use of blocks to represent numbers. It is possible that participants
who had used the software were at some disadvantage at the second interview,
having just spent 2 weeks using only electronic blocks to represent numbers; on the
other hand, participants from blocks groups had just had practice in using physical
blocks for the same 2 week period. Some evidence for this is found in an interesting
excerpt from Hayden’s (l/c) second interview, mentioned in section 4.7.1. In the very
first question of Interview 2, Hayden counted 6 tens and 7 ones, counting the tenblocks as five each, reaching the answer 37. He quickly corrected himself when the
interviewer asked him if he was sure, but it is possible that Hayden had momentarily
forgotten how to use physical blocks, after having used only electronic blocks for a
fortnight.
Secondly, the time for this study was quite short. If lasting effects were to be
produced by the use of either representational format, it is likely that it would require
a longer period for these effects to become evident. Since the participants had all
used physical base-ten blocks in class for a considerable time prior to the study, it
may be less likely that electronic blocks would produce a marked effect without a
longer period of exposure, given that the electronic blocks constituted a novel
representational format to these students.
Thirdly, the teaching phase comprised group sessions in which there was one
teacher for four students. This favourable teacher:student ratio provided participants
with accurate, timely feedback from a teacher regarding their deliberations about
numbers that is unlikely to be available in a normal classroom. This would have
particularly helpful to participants who used the physical blocks, in light of the
246
frequency of feedback they received from the researcher (section 4.7.7), apparently
because of a lack of other sources of accurate information. Users of the electronic
blocks, however, could receive similarly accurate feedback from the software in
place of that from the researcher, and so were able to attempt the tasks set for them
with less need for adult intervention. Thus in a classroom with group activities being
conducted, electronic blocks may prove to be more useful than physical blocks for
helping students understand numbers, because of the software’s capacity to provide
feedback without the need for constant adult supervision. Further classroom-based
research would be needed to test this idea.
Effects of feedback provided by physical or electronic blocks.
The differences in the effects of physical or electronic blocks appear to be
rather subtle, except with regard to the provision of feedback. As discussed in
chapter 5, participants who used physical blocks received their most accurate
feedback from the researcher, with less accurate feedback coming from their peers or
from the blocks. On the other hand, participants who used electronic blocks received
less feedback from the researcher or their peers, compared to feedback from the
representational materials available to them. In the sense that the software presented
information to the children, it provided users with feedback for their ideas that was
much more accurate than similar information available to users of physical blocks.
The computational facility of the computer running the software provided nearly
instantaneous feedback about the number of blocks on screen, the number
represented by the blocks, the symbol for the number, and the verbal name of the
number. Clearly, physical blocks offer none of these facilities, meaning that such
information must come from some other source.
It could be argued that much of the feedback provided by software could
easily be provided to users of conventional blocks by well-designed worksheets or by
a careful teacher or adult helper. However, the reactions of participants to feedback
that they received indicates that there were important differences in the participants’
confidence in the feedback, with important implications for use of materials to
represent numbers.
To summarise this section, conclusions about the learning effects produced by
physical or electronic base-ten blocks are as follows:
247
1.
With a teacher available to provide assistance and correction, Year 3
children are able to learn place-value concepts using either physical or
electronic base-ten blocks.
2.
Children using electronic blocks are able to rely on the material for
accurate feedback regarding their ideas about numbers, whereas users
of physical blocks need other sources of accurate information.
3.
During the short time of this study little difference was evident in the
place-value learning by participants using either material. Over a longer
period of time, and with more limited teacher assistance in a regular
classroom, students using electronic blocks may have an advantage in
learning place-value concepts over students using physical blocks.
6.2.4 Differences Between Place-Value Understanding of High- and LowAchievement-Level Participants
4.
Which of these conceptual structures for multidigit numbers can be
identified as relating to differences in students’ achievement in
numeration?
As discussed in section 5.6, there were both similarities and differences
between the performance of high-achievement-level and low-achievement-level
participants. Similarities in responses related especially to the idea of knowledge
about numbers being constructed by participants; high-achievement-level
participants were observed to change their responses in light of further information
or challenges to their initial answer, much as low-achievement-level participants did.
This supports one of the main contentions of this author in this chapter, that the
participants’ knowledge of the base-ten numeration system did not fit into any neat
set of categories, but was marked by flexible, changeable ideas that the children
altered in light of further information.
Differences between responses of high-achievement-level participants and
those of low-achievement-level participants were quite pronounced, as demonstrated
by several tables in chapter 4 (see section 5.6 for discussion). These tables reveal
clear and significant distinctions between both the conceptual structures and the
place-value task performance of the two groups of participants. Not only did highachievement-level participants demonstrate clearly better understanding of placevalue concepts, as a group, than their low-achievement-level counterparts; they also
248
adopted more accurate, efficient and useful strategies for answering place-value
questions.
In summary, these points are noted in comparing the overall performance of
high-achievement-level and low-achievement-level participants:
1.
There was some overlap of performance levels achieved by highachievement-level and low-achievement-level participants, so that some
low-achievement-level participants achieved higher results than some
high-achievement-level participants did.
2.
In general, high-achievement-level participants achieved more placevalue understanding criteria in interviews, on average achieving more
than 8 more criteria on each interview.
3.
High-achievement-level participants demonstrated the use of more
efficient and more accurate approaches to place-value questions, and
adopted the incorrect face-value construct for multidigit numbers much
less often, than low-achievement-level participants did.
4.
High-achievement-level participants on average demonstrated much
better performance on digit correspondence tasks than did lowachievement-level participants.
5.
Despite their better performance on place-value questions generally,
high-achievement-level participants still exhibited similar changeability
of answers and ideas about numbers.
6.
It appears that a “Matthew effect” (Burstall, 1978) existed, by which
those participants who had better understanding of the base-ten
numeration system used more accurate and efficient strategies when
answering place-value questions, leading to further improvements over
participants with more limited place-value understanding to start with.
6.3
Implications for Teaching
6.3.1 Implications of Using Physical Base-Ten Blocks to Teach Place-Value
Concepts
One of the biggest hurdles to overcome in teaching with physical base-ten
blocks may be the knowledge that teachers themselves have about the base-ten
numeration system, and the apparently transparent way in which blocks represent
249
that system. Cobb and Wheatley (1988) and Clements and McMillen (1996) have
pointed out that teachers should not assume that children see numbers and block
representations of numbers the way that adults do; evidence of a number of unusual
ideas held by participants in this study has supported these statements. A related
point is that children’s ideas about blocks and about numbers are often not made
visible by typical classroom mathematics tasks. Both the face-value construct,
previously identified and extensively discussed in the literature, and the independentplace construct, proposed in this study, are ideas apparently held by children that are
not revealed if mathematics questions are kept simple. Routine questions such as
“show me the number in the tens place” can be answered with very limited
knowledge of the way symbols represent numbers, and can be answered quite
successfully while holding any of a number of faulty or limited conceptions for
numbers. Thus, it is important for teachers to attempt to find as much as possible
about how children perceive the “mathematical objects”—including written
numerals, base-ten blocks, and electronic blocks—used in the classroom. One way to
foster this is to assign tasks that are likely to reveal incorrect thinking, including digit
correspondence tasks, trading tasks, and tasks requiring the production and
interpretation of non-canonical block representations. The other aspect of this
recommendation is for teachers to monitor children’s use of the materials quite
closely. This study revealed a large number of errors which, except for the presence
of the researcher, would most likely have gone unnoticed by the participants. Left
alone, children are going to make errors in manipulating materials and answering
mathematical questions. There needs to be some procedure in place in a classroom to
identify and remediate these errors in a timely way.
Provision of tidy, structured working spaces.
Another clear aspect of block use revealed in this study is the “messy” nature
of physical block representations (Clements & McMillen, 1996). When using blocks
to represent two-digit numbers this problem is not likely to be very serious, but with
three-digit numbers and beyond the sheer number of blocks can cause significant
difficulties for children if they do not adopt orderly practices. There were many
instances in this study in which participants in blocks groups made counting or
handling errors that were likely to have been at least partly due to this problem. One
particular error noted at times during the teaching sessions was a difficulty
250
participants had with keeping the current block representation under consideration
separate from the rest of the available blocks. With a large collection of blocks on a
desk, participants in the study sometimes found it difficult to remember whether a
certain block, “found” near others that were being counted, was part of that counted
set, or if it was an “extra” from the other uncounted blocks. Such situations appeared
to cause several errors by participants.
One idea to assist children in keeping track of blocks is to provide containers
for extra blocks, and to use some sort of structured “mat” on which to place block
representations. This idea appeared to have been used by the classroom teacher of
some of the participants, as a “tens mat” was mentioned by participants during the
teaching session. This mat could be as simple as a place-value chart on a piece of
paper marked “Tens” and “Ones,” with a vertical dividing line between the places.
More complicated structured material on which to place base-ten blocks could be
devised that assists students in counting the blocks. A similar idea is commercially
available for use with Unifix™ cubes in the form of a shallow plastic tray that is the
right size to contain a certain number of cubes. If such a device was available for use
with base-ten blocks, having counters to judge how many blocks of each place were
present, it could overcome a major disadvantage for users of blocks over users of the
software, the fact that blocks have to be counted frequently to determine the number
present. This idea may be useful, but is likely to add to the cost of the material, and
may introduce other unforeseen difficulties of interpretation. Whatever method a
teacher adopts for use of base-ten blocks, it is recommended that students be
encouraged to work neatly, to count blocks carefully, and to recheck answers if they
seem unusual. Had participants in the blocks groups routinely used such an approach
they may have made considerably fewer errors.
Base-ten blocks are no substitute for number sense.
The use of base-ten blocks by participants in this study revealed a number of
difficulties if teachers believe that the base-ten blocks show children accurate models
of numbers and associated processes. Participants often did not appear to regard a
block representation holistically, but place-by-place. Rather than engaging with twodigit or three-digit numbers as complete entities, participants often seemed to use an
independent-place construct that enabled them to manage task demands with less
cognitive effort necessary. As revealed in the transcripts, participants often seemed to
251
use base-ten blocks as “place counters,” mapping each number of like-sized blocks
onto a place digit or onto a number name. Kamii et al. (1993) did not use base-ten
blocks or any other representational material in their study, arguing that base-ten
blocks promote the idea that mathematical knowledge is somehow contained in the
blocks, rather than in a person’s “mental action” (p. 201). This author agrees with the
basic thrust of this argument, but would urge better use of base-ten material rather
than a complete abandonment of it. Nevertheless, Kamii’s argument is supported to
some extent by the results of this study, in that participants on many occasions
seemed to be “missing the point” that the blocks were supposed to illustrate,
manipulating blocks in procedural, unthinking ways that did not appear to assist
participants in developing better concepts about numbers. The solution to this
problem might be either to follow Kamii et al.’s advice and stop using the blocks, or
to interrupt children’s manipulations to ask pertinent questions about the quantities
they are modelling. The point made several times in this thesis and elsewhere is that
blocks themselves are only a means to understanding numbers, not the end purpose
for their use. They are no substitute for having an internal understanding of numbers
that includes knowledge of number facts, computation skills, and number sense.
One implication of the independent-place construct and its potential to render
invisible many errors in interpreting values represented by base-ten blocks is that
students’ use of base-ten blocks must be closely monitored. For the sake of
children’s development of number ideas, teachers cannot afford to allow students to
use materials such as base-ten blocks without checking the children’s interpretations
of the representations produced. This may involve greater use of questioning of
students to probe what they believe the blocks demonstrate about numbers, and the
earlier introduction of questions involving trading and other non-canonical
representations of numbers (see Fuson, 1990b, for similar recommendations).
Summary of teaching recommendations for use of base-ten blocks.
The following recommendations to teachers are made for the use of base-ten
blocks, and particularly physical blocks, in primary classrooms:
1.
Challenge students’ ideas about numbers by asking them a variety of
non-routine place-value questions that include non-canonical
representations of numbers and trading.
252
2.
Closely monitor students’ use of base-ten materials to identify various
counting, handling, or conceptual errors that can be made.
3.
Provide help for students to keep their block arrangements neat and
orderly. Use place-value charts or other materials to help add perceptual
structure to the block arrangements.
4.
Be prepared to interrupt children’s use of base-ten blocks to challenge
possible faulty concepts about numbers. If necessary, stop children
using blocks for a time and challenge them to think about numbers in
different ways.
5.
Do not use base-ten blocks to teach numeration concepts to young
children. Use material that includes grouped single material instead,
such as bundling sticks, at least until children understand the grouped
aspect of the base-ten numeration system.
6.
Be aware of and alert for signs of common misconceptions held by
children about multidigit numbers, in particular the face-value construct
and the independent-place construct.
6.3.2 Implications of Using Electronic Base-Ten Blocks to Teach Place-Value
Concepts
Though many software titles to teach mathematics are currently available, it
is not clear how many of them are designed specifically to teach place-value
concepts, nor how many include representations for numbers similar to the software
used in this study. Furthermore, there are no data available to the author of the
proportion of primary teachers who use such software in their teaching of
mathematics. There is clear anecdotal evidence, however, that the use of computers
generally in Queensland primary schools has increased rapidly in recent years, and it
seems likely that the trend is similar in other school regions. The recommendations
in this section are directed towards designers of mathematics software for teaching
place-value concepts, and towards teachers in the position of choosing software for
use with their class. As the designer of software (Price & Price, 1998) that is used in
primary schools in Australia, the author is aware of the wide range of skills needed
by software designers and programmers, and the need for up-to-date information
about children’s mathematics learning; it is hoped that results of this study will lead
to further software for use in primary schools.
253
Generally, the results of participants’ use of the software in this study are
encouraging. Results of the two interviews (Table 4.3) indicate that learning occurred
both for participants who used physical base-ten blocks and for those who used
electronic blocks. It appeared that participants using the software were content to
regard the pictures of blocks on screen as actual entities, and to manipulate them
using on-screen tools to represent numbers and number processes. In particular, the
representation of trading processes seems to have been very successful; participants
were very confident in the idea that traded blocks are always equivalent in value to
the blocks before the trade. There were few aspects of the software that appeared to
introduce misconceptions in participants’ thinking, except perhaps for column
counters. It is believed that counters above the three columns on screen may have
promoted or supported either face-value constructs or independent-place constructs.
On the positive side, feedback mechanisms incorporated in the software were used
often by participants to confirm their ideas. It appears that participants enjoyed
having their answers confirmed by the various electronic means of feedback, and that
the feedback received was more accurate and more encouraging than the feedback
received by users of physical blocks.
Certain tasks were more difficult to manage with electronic blocks than others
were, prompting recommendations for further features to be incorporated in placevalue software. In particular, the software does not easily represent two quantities
simultaneously, as there is just one set of column counters and one number window.
Tasks involving the comparison of two numbers were handled in the study by having
participants at each computer represent one of the numbers, which would not be
possible if there was only one computer available. Similar difficulties were evident in
the representation of arithmetic operations. Participants were able to carry out
addition by adding extra blocks to a representation, and subtraction by taking blocks
away from a representation. However, the software has no facility for keeping a
record of the blocks added or subtracted, making the modelling of these operations
difficult for children to visualise. It would be useful to have a feature that enables
two quantities to be represented at the same time, and to keep a record of
manipulations made in the course of carrying out an operation.
If classroom teachers are in the position of choosing software to use with their
students to teach place-value concepts, it is recommended that teachers choose
254
software that includes features that appear to have been successful with the software
used in this study. Among these features are
1.
linked block and number symbol representations of numbers, so that
both blocks and symbols always show the same number;
2.
dynamic representation of trading processes that demonstrate the
decomposition of a ten into 10 ones and the recomposition of 10 ones
into a ten; and
3.
audio recordings of number names that can be accessed to compare
with block and symbol representations.
One other feature if incorporated in the software could enhance the software’s
versatility and enable its use by children unattended; if children’s tasks were
presented by the software itself on the screen, children could use the software with
less adult supervision.
6.3.3 Implications of the Independent-Place Construct for Teaching Mathematics
As discussed previously, students’ responses to place-value questions may
not reveal faulty conceptual structures, including the independent-place construct, if
students can answer by considering only one place at a time. If, however, students
have to deal with relationships between places, the independent-place construct will
not help students get correct answers. Questions that do foster thinking about places
in relation to each other include questions involving the interpretation of noncanonical arrangements of blocks.
Another type of question that may help students to overcome an independentplace construct is computation questions involving regrouping. Each of the four
arithmetic operations involves thinking about adjacent places whenever regrouping is
carried out. For example, when answering “43 – 28,” one method is to regroup one
of the 4 tens into ones, making 3 tens and 13 ones, and then subtracting each place in
turn. Similar examples could be given for addition, multiplication, and division. If
students are faced with problems that require regrouping when they first learn about
an operation, they will find that independent-place thinking will not allow them to
find a solution. Thus, it is recommended that teachers present to their students
operations that require regrouping from the very first examples, to avoid reinforcing
any independent-place thinking that the students may have. An associated
255
recommendation for the teaching of operations is to assist students to visualise the
quantities being used, through the use of blocks or place-value software.
The alternative, of teaching students to follow a rote written procedure, does
not equip students to handle further examples of that operation in other, perhaps
more efficient, ways. For example, in the case of the addition operation 49 + 35, one
efficient method would be to adjust the addends to 50 and 34, making the task a
simple mental arithmetic question. A child with an independent-place construct,
however, may only be able to answer the question 49 + 35 strictly as written, by
calculating 9 + 5 ones, and 1+ 4 + 3 tens.
In summary, recommendations to teachers that may reveal and remediate
examples of the independent-place construct among their students are to:
1.
Give students place-value questions that involve non-canonical
arrangements of materials.
2.
Challenge students to think of a digit in terms of the adjacent places, to
regroup quantities in different ways.
3.
Give computation examples that require regrouping from the start.
4.
Encourage students to develop creative methods of answering placevalue and computation questions, based on flexible regrouping of
numbers.
6.3.4 Implications of Construction of Meaning for Teaching Mathematics
Results of this study show that participants’ thinking was sometimes difficult
to interpret from an adult perspective. The results also show that one possible reason
for the difficulty in interpreting the actions or statements of the participants was that
they were attempting to make sense of questions posed to them using a wide range of
knowledge that they had about numbers. During that process, participants were
observed to make statements that appear to be illogical or absurd. One temptation for
teachers hearing such statements might be just to tell the student what the answer
should be, or to move on to new material. However, results in this study agree with
statements appearing in the writing of other authors, to the effect that many
apparently nonsensical statements made by children are actually the product of
rational thinking processes, using whatever knowledge the children possess at the
time. In this regard, Cobb and Wheatley’s (1988) advice to researchers is equally
relevant for teachers when responding to children’s sometimes unusual ideas:
256
A fundamental assumption of conceptual analyses is that children’s actions are
always rational given their understandings. We have all seen children who, from our
adult perspective, do some strange things as they attempt to solve mathematical
tasks. One reaction is to wonder how the children could be so stupid or to ask what
is wrong with them. . . . An alternative approach is to readily admit the inadequacy
of adult mathematics for understanding children and for planning instruction. From
this perspective, children’s apparently strange actions are viewed as problems for
the observer to solve. The trick is to develop an understanding of children’s
mathematics so that their actions can be seen as rational and sensible. (p. 2)
It is recommended that teachers take this attitude in attempting to make sense
of what children are thinking. If they are successful, teachers will take on the role of
a researcher, making sense of children’s thinking in order to tailor instruction to help
them understand the realm of mathematics. In conclusion, the following
recommendations are offered for teachers to manage the demands of teaching
students who are making sense of what they experience in the classroom:
1.
Teachers must recognise the changeability of students’ ideas, and the
fact that children will think quite rationally about number concepts
based on their perceptions of the realm of numbers.
2.
A teacher may have a useful role to play in introducing new information
that conflicts with a student’s incorrect stated belief. Without a person
with expert knowledge pointing out the inconsistencies in a student’s
conception of numbers, that conception may remain unchallenged for
some time; if a faulty belief is accepted for a long time it is likely to be
more difficult to correct than if its inconsistencies were pointed out
earlier.
3.
Without asking the right sort of probing questions a teacher is unlikely
to discover what students actually believe about the base-ten
numeration system and about how numbers are represented by it. As
already mentioned, some types of school mathematics questions are
easily answered by a student who has incorrect conceptions about
numbers, such as face-value constructs or independent-place constructs
(section 5.3.6; see also S. H. Ross, 1989).
4.
A teacher is unlikely to have a meaningful impact on a student’s
number conceptions by merely repeating the procedure to use in
answering a certain type of number question. Some students definitely
appear to accept procedures that they do not understand, and of which
they cannot make sense. Such unthinking acceptance of taught
257
procedures does nothing to help a student tackle novel problems, and
will allow the student to answer only questions of the type to which the
learned procedures applies.
5.
In view of the “head start” that more able students seem to have in the
area of understanding place-value concepts and answering place-value
questions, it is important to give extra support to less able students to
enable them to understand place-value foundations and to adopt
efficient strategies that utilise the grouped aspect of the base-ten
numeration system. Without this extra support it appears that those
behind in understanding place-value concepts will fall further and
further behind as they continue to use inefficient, labour-intensive
methods of dealing with multidigit numbers.
6.4
Recommendations for Further Research
This is an exploratory study, designed to explore a wide range of factors in
two versions of a particular learning setting. The results of this study have led to
several proposals for explaining what appeared to be happening in the situations
investigated. Each of these proposals is a possible topic for further research to
increase knowledge of children’s learning of place-value concepts.
As described in chapter 2 of the thesis, the children’s number conceptions
have been heavily researched in the past 20 years or more. Nevertheless, results in
this study suggest that some schemes for classifying children’s number concepts may
have other interpretations that need investigating. In particular, the trend towards
classifying children’s number concepts based on certain limited number tasks seems
particularly problematic. It is suggested that further research should be directed
towards finding out more about children’s knowledge of numbers. In particular, in
light of this study it would be appropriate to test the range and character of number
conceptions held simultaneously by individual children.
Associated with research into children’s number conceptions, research into
children’s understanding of base-ten blocks and other representational materials can
be pursued further. There appears to be a range of opinions about the use of base-ten
blocks to teach about the base-ten numeration system. Some writers do not advocate
their use, perhaps because of the observed mishandling of blocks by children. Other
writers caution teachers about allowing children to believe that mathematical
258
knowledge is contained within the material. There appears still to be considerable
faith in the mathematics education community in the use of representational
materials, and clearly many teachers use base-ten blocks. However, it is also clear
that the use of base-ten blocks by many children is error-prone and based on faulty
ideas about the base-ten numeration system. These varying opinions about
representational material point to a dilemma regarding advice to give to teachers,
which in the context of published research papers is contradictory. There is
promising work being done by researchers, including Fuson and her colleagues
(Fuson et al., 1997), who are involved in various research projects investigating this
important topic and collaborating in reporting the results. This present study points to
a need for such research to continue.
This thesis contains a proposal for the existence of a concept apparently held
by some children, named the independent-place construct. Difficulties applying the
face-value construct to certain responses in the data led to the proposal of the
independent-place construct; the author felt that the differences between the two
concepts could not be overlooked, and so the new label was proposed. The
independent-place construct may be misleadingly similar to the face-value construct,
which has received considerable attention in the literature in recent years. However,
evidence presented in section 5.3.2 suggests that the independent-place construct is
different to the face-value construct, and equally difficult to identify in responses to
certain routine number tasks. In the final analysis, it may be found that the
independent-place construct is so similar to the face-value construct that it can be
considered as a variant of it. However, the status of the independent-place construct
cannot easily be judged without further research.
Finally, this study points to a theme in the data that appears to have great
relevance for the use of educational software, the place of feedback. Results of this
study show that participants using the software were able to access information about
represented numbers from more sources than participants using the physical blocks
could. In particular, electronic forms of feedback were used by the participants to
assess whether their block representations were accurate, and apparently also to
check their ideas about the numbers. Electronic feedback appears to have taken the
place of feedback from human sources, in particular the researcher, that were
accessed more frequently by participants using physical blocks. Implications are that
software that incorporates feedback mechanisms can give students valuable
259
information about their use of electronic blocks that helps students adjust their ideas
and their manipulation of software artefacts being used to present information or
answers. There is a place for continued research to test the relative effectiveness of
various forms of electronic feedback; in light of these results research should be done
to assess the usefulness of feedback specifically for the purpose of representing
mathematical knowledge.
Summary of research recommendations.
The following recommendations are made for research topics that may
continue addressing certain issues discussed in this thesis:
1.
Research into children’s learning about the base-ten numeration system,
and in particular the use of multiple conceptions by individual children.
2.
Research into the effective use of base-ten representational materials,
including physical and electronic base-ten blocks.
3.
Research into the independent-place construct, to investigate if it is a
separate category of children’s responses to number tasks, and to find
out how it is influenced by the use of various representational materials
and teaching practices.
4.
Research into the effects of feedback mechanisms contained in
representational software for teaching mathematics, and how positive
effects from the feedback could be maximised.
260
Appendix A – Design of Software used in the Study
This appendix comprises a description of the software designed for use in this
study and a comparison between it and other software applications that share similar
design features.
Computer Software Incorporating Base-Ten Blocks
Several researchers have described, or developed themselves, software that
generated pictorial versions of base-ten blocks for students to manipulate; three such
programs are discussed here (Champagne & Rogalska-Saz, 1984; Clements &
McMillen, 1996; P. W. Thompson, 1992). The various computer programs have at
least four features in common: Each one (a) represented numbers primarily as
pictures of base-ten blocks; (b) could present written symbols for numbers; (c)
allowed manipulation of the blocks, especially to regroup blocks in 10-for-1 trades,
and (d) modelled basic actions taken with physical base-ten blocks. In this section
comparisons are made between the software used in the study, called Hi-Flyer
Maths, and three similar computer programs: untitled software described by
Champagne and Rogalska-Saz (1984), Rutgers Math Construction Tools (1992), and
Blocks Microworld (P. W. Thompson, 1992).
Champagne and Rogalska-Saz (1984) described software that consisted of 15
lessons presented on-screen, each comprising an instructional and a practice
component. The computer presented users with questions, such as “How many cubes
are there?” (Figure A.1). Pictures of blocks could be “regrouped” to show
equivalence of different groupings, to answer the questions. The authors referred to
three representations used by the software: pictorial, verbal, and numerical. It needs
to be noted that the “verbal” representation was number name displayed on screen as
text; this is generally considered by mathematicians to be another form of numeral.
The software used in this study uses audio facilities of computers that have become
readily available only since the time when Champagne and Rogalska-Saz conducted
their study.
261
Figure A.1. Screen view of on-screen tutorial question with block representations.
Note. From A. B. Champagne and J. Rogalska-Saz, 1984, Computer-based numeration instruction. In
V. P. Hansen & M. J. Zweng (Eds.), Computers in mathematics education: 1984 yearbook, p. 48.
Reston, VA: NCTM.
Rutgers Math Construction Tools (1992) presented pictures of base-ten
blocks on-screen that could be dragged to any position in the main working area,
“broken” into 10 of the next smaller block, or “glued” together to form a next-larger
size block. Symbolic representations available were the standard numerical symbol
(e.g., 3428), an expanded numerical symbol (e.g., 3000 + 400 + 20 + 8), or a number
and place name symbol (e.g., 3 thousands + 4 hundreds + 2 tens + 8 ones; see Figure
A.2). The screen could be divided into two sections, each with a symbolic
representation, for addition of two numbers or partition of one representation into
two subsets.
262
Figure A.2. Partial screen image from Rutgers Math Construction Tools, showing
block and symbol representations of a number.
Note. From Rutgers Math Construction Tools [computer software], 1992. NJ: Rutgers University.
P. W. Thompson (1992) used a computer program called Blocks Microworld
to investigate “students’ construction of meaning for decimal numeration and their
construction of notational methods for determining the results of operations
involving decimal numbers” (p. 125). Thompson’s software included pictures of
base-ten blocks and a symbolic representation in the form “1 cube 1 flat 11 longs 1
single = 1211” (Figure A.3). By using a “unit menu,” a user could nominate any one
of the four block sizes as the unit, so making possible the representation of decimal
fractions. The previous example then became “1 cube 1 flat 11 longs 1 single =
0.1211” (p. 129).
263
Figure A.3. Screen view of Blocks Microworld showing block representation of a
number, nominating a cube as one.
Note. From P. W. Thompson, 1992, Notations, conventions, and constraints: Contributions to
effective uses of concrete materials in elementary mathematics. Journal for Research in Mathematics
Education, 23, p. 127.
One particular feature of Blocks Microworld sets it apart from the other
software reviewed here. Other programs allowed users to manipulate the blocks on
the screen, by using the computer mouse; changes were then reflected in the
numerical symbol, where available. In contrast, P. W. Thompson’s (1992) software
allowed users to manipulate only the numerical symbol displayed on the screen,
which was then mirrored in the blocks; blocks themselves could not be directly
manipulated. Thompson justified this as an example of “constraints on students’
concrete actions in places that are likely to draw their attention to relationships
among meaning, notation, and expression” (p. 127). In light of many findings that
students are prone to manipulate symbols without reference to the numbers they
represent (e.g., Hart, 1989), Thompson’s idea may be brought into question.
However, Thompson’s results with students using the software were generally
encouraging, in that they seemed to make better sense of decimal fractions than the
group using only physical blocks.
Rationale Behind Hi-Flyer Maths
As explained in chapter 2, there is widespread support for the idea that the
key to the development of higher level conceptual structures for numbers is making
connections between numbers and their referents (Fuson & Briars, 1990; Hiebert &
264
Carpenter, 1992). Despite the popularity of base-ten blocks among teachers for 40
years, research shows that learning effects from their use are equivocal (Hunting &
Lamon, 1995; P. W. Thompson, 1992); this may be due to difficulties that students
have in making links between symbols and the blocks.
Clements and McMillen (1996) listed several advantages that computer
manipulatives can have over their physical counterparts. These included avoidance of
distractions, flexibility, dynamic linking of representations, encouraging problem
solving, and facilitating explanations. Clements and McMillen advised teachers to
“choose meaningful representations then guide students to make connections
between these representations” (p. 278). They recommended a broader view of
manipulatives than just physical materials and stressed the potential advantages that
computer software could offer to counter some of the problems of conventional
materials.
It is not surprising, given the dates of the papers reported in this section, that
the assumptions underlying the design of Hi-Flyer Maths and the methods of this
study are closer to those of Hunting and Lamon (1995) and Clements and McMillen
(1996) than those of Champagne and Rogalska-Saz (1984). This study is based on
the belief that students have to actively engage in learning activities in order to
benefit from them (Baroody, 1989). Students need to construct their own conceptions
of numbers through interacting with learning resources. As several writers have
pointed out (Clements & McMillen, 1996; Hunting & Lamon, 1995; P. W.
Thompson, 1994), mathematical meaning is not inherent in materials themselves.
Rather, the source of meaning is located “in students’ purposeful, socially and
culturally situated mathematical activity” (Cobb et al., 1992, p. 6).
The other major assumption underlying this software’s design is that in order
to build up accurate conceptual structures of numbers students need to make
meaningful connections between numbers and symbols, and between symbols and
referents. This topic has been adequately covered in chapter 2. This assumption is
operationalised in the inclusion of at least five different representations of a number
possible with the software, described in detail later in this appendix.
Features
Though Hi-Flyer Maths shares a number of similarities with the other titles
described in the previous section, it incorporates several innovations that were not
265
seen in any other software evaluated for use in the study. As these features are an
integral part of the rationale for the central investigation of this study, they are
described in some detail in the following section.
General Description
The main screen (Figure A.4) presents a workspace with “source blocks”
from where blocks may be taken to form representations of numbers. Next to the
source blocks are buttons that enable access to various features. Most of the screen is
presented as a “place-value chart” with three columns, labelled “hundreds,” “tens,”
and “ones.”
Figure A.4. Main screen of Hi-Flyer Maths.
As blocks are placed on the place-value chart, the labels above the three
columns simultaneously show the number currently in each column. Similarly, if the
number symbol box or numeral expander are visible, changes to blocks are reflected
immediately by changes in the relevant text boxes.
Numerical Representations
Any number from 1 to 1599 can be presented by the software in five different
ways: (a) as a canonical arrangement of base-ten blocks, (b) as a variety of noncanonical arrangements of base-ten blocks, (c) as a written symbol, (d) as a numeral
266
expander, or (e) as an audio recording of its verbal name. An important characteristic
of the different representations is that changes in the block representation are
mirrored by changes in the written symbol representations (if shown) virtually
simultaneously. In other words, as an extra ten-block is added to a representation, for
example, the written symbol(s) for the represented number change to reflect the
change in the blocks. The rationale behind designing the software in this way is that
in order to enable students to make necessary links between numbers and their
various representations, it is desirable to present changes (such as trading a ten for 10
ones) in all representations at the same time. This idea embodies the approach using
base-ten blocks recommended by Fuson (1992), that every change in written symbols
be mirrored in the concrete materials as close in time to the change as possible.
Block arrangements.
Base-ten blocks can be added to the display either by dragging a copy from
one of the source blocks, or by clicking on the relevant “plus” button next to the
source blocks. If a plus button is clicked, a new copy of the associated block is added
in the correct column, in an ordered arrangement: ones and tens are placed in rows of
10 blocks and hundreds are placed 3 across. In that way, every block placed is visible
and none are overlapped. If blocks are dragged, then the user can overlap them.
However, the block arrangement is “cleaned up” if blocks are regrouped.
Non-canonical representations are achieved by adding blocks to a column
until there are more than nine in a place. A temporary non-canonical arrangement is
also achieved by clicking on either the “show as tens” or the “show as ones” button.
When the show as tens button is clicked, any hundred-blocks present on the placevalue chart are changed into a representation of 10 ten-blocks (Figure A.5). When the
show as ones button is pressed, hundreds and tens blocks are changed into
representations of 100 or 10 ones, respectively. Simultaneously, the column labels
are altered to indicate the new represented number of blocks. When the mouse is
clicked again, the pictures of blocks and column labels are reverted to their previous
state.
267
Figure A.5. “Show as tens” feature activated.
Written symbols.
The “show number” button may be clicked to reveal a “number name
window” showing the written symbol for the number represented by the blocks. The
symbol always refers to the total number shown by the blocks, whether they are
arranged canonically or non-canonically. For instance, if 3 hundreds, 14 tens, and 17
ones were placed on the place-value chart, the number name window would show
“457.” This would not change if the blocks were regrouped, or if the show as ones or
tens buttons were pressed.
A variation of the numerical symbol that may be displayed is the numeral
expander (Figure A.6). This is an on-screen version of a device made from light card
used in many primary schools to show equivalence of various representations of a
number. In the non-electronic version a number is written in blank spaces on the
card, with the names of the places hidden by folding a section of the card behind the
number spaces. Then the expander may be pulled open to reveal one or more of the
place names. For example, the number 518 could be shown as “5 hundreds 1 ten 8
ones,” or “51 tens 8 ones,” or as “518 ones.” The software reproduces this with a
picture of an expander that may have the place names hidden or revealed one place at
a time, by clicking on a place with the cursor. As with the number name window, the
numbers on the numeral expander do not change if the blocks are arranged noncanonically; the numbers shown represent the value of the entire display of blocks.
268
Figure A.6. Number name window and numeral expander displayed.
Verbal number names.
The software incorporates 36 audio files that are accessed by the computer to
enable any number from 0 to 1599 to be “read” aloud. For example, to read the
number name of 324, the computer plays the audio files for the words “three,”
“hundred,” “and,” “twenty,” and “four” in succession. Though the speed at which
each successive file can be accessed depends on the computer used, the gap between
words has been found acceptably short on most machines.
Verbal names can also be accessed for the numeral expander, so that the
numbers and place names shown are “read” as they are shown. Similarly the column
labels can be read individually if desired, as “four hundreds,” for example. Lastly,
clicking the “speech bubble” cursor onto one of the blocks can access the number
represented by blocks in each column alone. For example, if there were 5 ten-blocks
and 7 one-blocks, the computer would read the tens blocks alone as “fifty.”
Regrouping Blocks
A primary use of base-ten blocks by teachers is to model each step in
computational algorithms, especially those for addition and subtraction. The
269
processes involved as numerical quantities are altered in the steps of a computation
algorithm can be modelled using base-ten blocks. In common with other place-value
software, Hi-Flyer Maths will also demonstrate combining and separating of
quantities with the on-screen blocks, though the column labels will show only the
total number of blocks in each column, rather than two separate quantities.
When using physical base-ten blocks, regrouping in both directions is carried
out as a “trade” or a “swap.” In other words, to change 10 of a block for one of the
next larger block, or to change a block for 10 of the next smaller block, the blocks
must be traded or swapped for other blocks from a supply container. It is appropriate
to note at this point that some teachers and authors (e.g., Resnick & Omanson, 1987;
P. W. Thompson, 1992) refer to this process as “borrowing”; however, this is not an
accurate description of what is represented. As there is no “paying back” (as there is
when the equal addition algorithm is used), the use of the terms “trading” or
“swapping,” with their connotations of an equal transaction seem much more
appropriate. Consequently in the teaching phase of this study these terms were used
when referring to regrouping.
This process of 10-for-1 trading of blocks has been pointed out as precisely
the point where students may misunderstand what is happening in the numerical
realm. In the world of numbers 10 of one place is equivalent to one of the next larger
place: Thus one may imagine, for example, 38 being regrouped into 2 tens and 18
ones without pause. However, when using physical blocks this same transaction
would require the physical act of removing a ten-block to another place and replacing
it with 10 one-blocks. This process does not accurately mirror what happens with
numbers, and in children’s minds confusion may exist about what the trading means
in the numerical realm. The same problem does not exist with some materials where
ten or hundred material are not pregrouped (Baroody, 1990), such as Unifix™ cubes
or sticks, that may be combined or separated by the child, without having to do any
trading.
Hi-Flyer Maths, in common with each of the other computer programs
reviewed at the start of this chapter, allows combining and separating of blocks to be
dynamically displayed on the screen. Each program handles this process differently.
The software described by Champagne and Rogalska-Saz (1984) required users to
type instructions to initiate regrouping actions on screen. For example, typing “TH”
caused the display to regroup 10 ten-blocks to form a hundred-block. Rutgers Math
270
Construction Tools (1992) includes a “hammer” tool that can be clicked on a block
to cause it to change into 10 of the next smaller piece and a “glue” tool that has to be
clicked onto 10 blocks of the same size to cause them to join together. P. W.
Thompson’s (1992) software required users to click on a digit (for example, the 6 in
“2 hundreds 6 tens 4 ones”) and then click on a “borrow” button, causing the display
to “explode” the relevant block into 10 blocks of the next smaller size.
The software in this study was designed to show dynamically on screen the
processes that are understood mathematically when quantities are regrouped. To
regroup a hundred-block or ten-block into 10 blocks of the next smaller place, a
“saw” tool is used that causes the computer to display the block being progressively
sawn into 10 pieces. The 10 new blocks are then moved into the correct place and the
column labels are changed to show the new number in each place. For example, in
Figure A.7, after the ten-block is sawn up, the 10 ones move to the ones place and
the labels change to show “5 hundreds, 2 tens, 14 ones.”
Figure A.7. A block is “sawn” into 10 pieces.
It is hoped that by showing an on-screen block being progressively sawn, the
software will provide a useful analogue for what happens with abstract numbers. It is
crucial that children understand that a ten is both a separate entity and a composite of
10 ones. This point is not necessarily clear to children when using base-ten blocks, as
they have to be traded. By showing blocks broken up and recombined, the software
shows an analogue of the regrouping process on numbers.
Regrouping in the other direction, from 10 smaller blocks to one larger block,
is achieved with a “net” tool that is placed over either the ones or the tens place and
clicked. If there are at least 10 in that place, the first 10 blocks are highlighted with a
271
surrounding red line, and then progressively moved together to form a new block that
is then moved to the next place to the left. If there are fewer than 10 in the place, a
message is given that there are insufficient blocks to regroup.
Addition and Subtraction
In order to model addition and subtraction, the software allows up to nine
blocks to be added or subtracted consecutively in any column. The user presses
either the “add blocks” or the “subtract blocks” button, and the software responds by
displaying a box asking how many to add or subtract, and which place (Figure A.8).
If, when subtracting blocks, there are insufficient blocks to remove the chosen
number in that place, a message is displayed to that effect. Otherwise, the software
adds or subtracts the requested number of blocks. No further action is taken by the
software, so if a non-canonical arrangement results, then the user has the option of
regrouping blocks if desired.
Figure A.8. “Add blocks” requester.
The addition and subtraction features are designed to enable accurate
modelling of the written algorithms for these operations. Rather than simply adding
or subtracting the entire amount of the second addend or the subtrahend, the software
allows for working on one column only, as is done using the written algorithm. For
272
example, to model the algorithm for 72 - 34 using the software, the student has to
first regroup a ten-block for 10 one-blocks, before using the “subtract blocks” button
twice to subtract separately the 4 ones and 3 tens.
The previous paragraph raises the question of whether or not students should
invent their own algorithms, as recommended by several authors (e.g., Kamii et al.,
1993). The process described above mirrors the standard written algorithm, in which
the right-most place is subtracted first. However, research has shown that students
inventing their own addition or subtraction methods invariably choose to start on the
left (Kamii et al.). The software will support either method (and others also), as with
the blocks representation there is no need to record intermediate calculation steps
(the “carry marks”) of the written algorithms.
The question of how to teach students computation is beyond the scope of this
thesis. However, it is important to point out that a question is raised in presenting
students with a (physical or pictorial) block representation of the amounts to be
added or subtracted of why students should always start on the right, when the blocks
indicate that starting on the left would work as well.
Other Features
Requesting a number representation.
The usual method for putting out blocks to represent a number with the
software is by dragging a source block, or by clicking on the “add” buttons once for
each block. Once a user is familiar with this procedure, another quicker method is
available. There is a menu item named “Choose number” that brings up a text box
requesting a number up to 999. If the user types a number and clicks “OK,” the
software will put out a canonical display of blocks representing the typed number,
block by block. This feature is convenient for representing a number quickly for
further investigation.
A similar feature accessed via the menu bar is a random number requestor.
The user can choose a range from which the software will choose a random number
and then display a block representation of that number. The ranges available are 119, 11-99, 101-499, 101-999, and tens from 10-990. This feature was not used in the
study.
If the number name window, the numeral expander, or both are visible when
either of the number-requesting methods are used, the symbols displayed change as
273
each block is added. For example, if the number to be displayed is 126, the number
name window will display the numbers 100, 110, 120, 121, 122, 123, 124, 125, and
126 as each block is placed on the screen, starting with the left-most column.
Sounds.
Sounds are used in the software in three different ways: (a) as motivational
devices, (b) as reinforcement of metaphors, and (c) as information sources.
A few sounds are provided to add to the appeal of the software for children.
The opening screen shows a balloon picture, which when clicked causes an aeroplane
to move across the screen accompanied by the sound of a plane. There is a “bomb”
tool on the main screen, which when clicked on a block causes it to return to the
source area as a whistle sound is played. To remove all blocks from the screen at
once, a button is clicked; a short “reveille” is played as the software removes the
blocks. These sounds are assumed to add to the novelty effect of the software, but do
not have any other educational purpose.
The second category of sounds has a much more important role, in
reinforcing metaphors shown pictorially. The regrouping features of the software are
designed as an essential part of the modelling process to indicate block
transformations. As explained above, the software demonstrates block regrouping in
a dynamic way not possible with physical base-ten blocks. The idea of sawing a
(wooden) block is reinforced by the button icon (see Figure A.4), by the animation
shown as the block is changed (Figure A.7), and by the sound effect played. As each
smaller block is “sawn off” the larger block, the computer plays a short sawing
sound. Thus as a block is regrouped into 10 smaller blocks, the sound is played nine
times as the transformation takes place. In a similar way, as 10 blocks are placed next
to each other to form one larger blocks, a “gluing” or “zipping” sound is played as
each one is placed. This use of sounds is in accord with advice by Hereford and
Winn (1994), that sounds may be used to “refer metaphorically to qualities of
objects” (p. 217). One aspect of this study is to investigate whether the use of these
sounds assists students to develop accurate understandings of the numerical
processes and relationships.
The third group of sounds used in Hi-Flyer Maths is audio recordings of
numbers’ verbal names. It is common practice for teachers to ask students to link
verbal names, written symbols, and concrete materials representations as place-value
274
concepts are taught (Fuson, 1992). Students need to be able to move among these
three forms of representation to develop accurate understandings of the numbers. The
software will play audio recordings of the number names in a variety of forms, as
described above. Again, it was hypothesised that this feature adds richness to the
information presented to students to assist them in constructing understandings of
multidigit numbers.
The other software reviewed in this chapter did not include this feature;
Champagne and Rogalska-Saz (1984) mentioned “verbal representations,” but
referred to text displayed on screen only. As mentioned above, the lack of this
feature in other programs may be due to technical restrictions. The current program
requires a computer sound card for the sounds to work, which was not widely
available until comparatively recently.
Summary
Several computer programs have been developed that display pictorial
representations of numbers in the form of pictures of base-ten blocks. Each of the
programs includes written symbol representations and permits manipulation of onscreen blocks to model regrouping of numbers. The programs differ in style of
presentation and the means by which users manipulate the blocks and symbols.
An original computer program has been written specifically for this study,
incorporating the same basic features of the other programs reviewed, as well as
several features not previously seen in such software. These innovations include
presentation of several novel representations of numbers and the use of animation
and sound to reinforce analogues of number processes. It is hypothesised that the
incorporation of these features is beneficial to students in developing accurate
conceptual structures for multidigit numbers. Specifically, the inclusion of features
not available in physical base-ten blocks is expected to produce different and better
conceptions of numbers and associated processes.
275
Appendix B - Overview of Teaching Session Content for
Interviews and Teaching Phase of Pilot Study
Session:
P 1
r
e
Diagnosis of place value
understanding
Introduction to program
Review of use of base-ten blocks
Two-digit numbers
Verbal to Concrete
representations
Verbal to Symbolic
Symbolic to Verbal
Symbolic to Concrete
Concrete to Verbal
Concrete to Symbolic
Regrouping
Use of numeral expander
Comparing 2 numbers
Ordering 3 or more numbers
Counting on and back by 1
Counting on and back by 10
Addition
Subtraction
Three-digit numbers
Introduce hundreds place
Introduce notation
Verbal to Concrete
Verbal to Symbolic
Symbolic to Verbal
Symbolic to Concrete
Concrete to Verbal
Concrete to Symbolic
Regrouping
Use of numeral expander
Comparing 2 numbers
Ordering 3 or more numbers
Counting on and back by 1
Counting on and back by 10, 100
Addition
277
2
3
4
5
6
7
8
9
1
0
P
o
s
t
Appendix C – Summary of Pilot Study Teaching Program
Session
Session 1
Session 2
Session 3
Session 4
Session 5
Session 6
Activities
[] Register students’ details on screen.
["] Record students’ details on paper.
[] Introduce basic software features to students, allow students to
experiment with them. Compare software with base-ten blocks.
["] Show base-ten blocks to students. Revise the use of base-ten blocks to
represent numbers.
[] Ask students to show two-digit numbers with base-ten blocks, and the
same numbers with the software.
["] Ask students to show two-digit numbers with base-ten blocks.
Show students numbers represented in one of three forms (Verbal name–
Concrete representation–Written symbol), ask them to give the other two
equivalent representations.
Revise questions about two-digit numbers using the blocksa, asking
students to make translations among three representations (Verbal name–
Concrete representation–Written symbol).
Add sequence of ones to a two-digit number. Regroup 10 ones for 1 ten.
Discuss with students the idea of regrouping of two-digit numbers, in both
directions.
subtract sequence of ones from a two-digit number. Trade a ten for 10
ones.
Revise questions about two-digit numbers using the blocks, asking
students to make translations among three representations (Verbal name–
Concrete representation–Written symbol).
Regroup two-digit numbers in various ways.
Introduction of numeral expander (on paper and on screen). Show twodigit numbers as tens and ones, or just as ones.
Discuss with students pairs of numbers and how they are represented in
symbolic form or with blocks. Compare and discuss which is larger, and
why.
Ask students to order three or more two-digit numbers in each
representational form.
Revise use of numeral expander.
Compare pairs of two-digit number sense, discuss which is larger, and
why.
Order three or more two-digit numbers.
Show students representations of two-digit numbers, ask students to count
on or back by ones.
Ask students to add pairs of two-digit numbers with and without
regrouping, using blocks and written symbols.
Revise regrouping of two-digit numbers.
Revise use of numeral expander.
Ask students to count on and back from chosen two-digit numbers, by
ones and tens.
Add two or more two-digit numbers.
Subtract two-digit numbers with and without regrouping using blocks and
written symbols.
Add two or more two-digit numbers.
Subtract two-digit numbers.
Have students make blocks up to 100, by starting from a number of tens
and adding tens one by one to reach 100. Prompt the students to see that
279
Session
Session 7
Session 8
Session 9
Session 10
Activities
regrouping of 10 tens is needed when 100 is reached.
Discuss with students the size of 100 and situations where it is used.
Introduce written notation for hundreds place.
Ask students to show three-digit numbers between 100 and 200 using the
blocks. Ask how each is represented using both blocks and written
symbols.
Revise addition and subtraction of two-digit numbers with and without
regrouping, using the blocks.
Revise notation of three-digit numbers to include hundreds beyond 200.
Ask students to make translations among Verbal name–Concrete
representation–Written symbol with three-digit numbers.
Revise translations among the three representations of three-digit
numbers.
Ask students to regroup three-digit numbers, regrouping tens or hundreds,
in various ways.
Revise translations among the three representations of three-digit
numbers.
Ask students to regroup three-digit numbers, regrouping tens or hundreds,
in various ways.
Re-introduce numeral expander for three-digit numbers.
Compare pairs of three-digit numbers, using blocks and written symbols.
Ask students to regroup three-digit numbers, regrouping tens or hundreds,
in various ways.
Compare pairs of three-digit numbers, using blocks and written symbols.
Order three or more three-digit numbers.
Count on and back from three-digit numbers, in ones, tens and hundreds.
Ask students to add pairs of three-digit numbers, using written symbols
and blocks.
Note. [] – Computer Group. ["] – Blocks Group.
a
“Blocks” refers to on-screen blocks, or base-ten blocks, for Computer and blocks groups,
respectively.
280
Appendix D - Excerpt of Teaching Script of Pilot Study:
Session 1
1-1.
Hello, N__ and N__. You and I are going to spend some time together in the
next few weeks, doing some interesting activities with MAB blocks. I will ask
you some questions, and I would like you to answer them as well as you can.
If you don’t understand anything, please ask me to explain it again. I think
you will enjoy the activities I have for you. I will be asking you to reach
answers together as a team. I am interested in how you reach your answers,
and what you understand and don’t understand, so please ask me questions if
you have any. Do you have any questions before we start?
1-2.
Introduce students to software on computer. Help students to log on to
software, entering name and date of birth.
" Ask students for name and date of birth, write on record sheet.
1-3.
Show base-ten blocks to students. Ask them what they know of them, and
how they are used. If necessary, revise the value of each size block, and how
to use them to represent a number.
1-4.
Can you tell me what these blocks are called? Do you use them in class?
What do you use them for? Can you show me how to use MABs to show the
number 25? (Correct if necessary.)
1-5.
Introduce the students to the basic features of the software: How to drag a
block with the mouse, how to clear all blocks from the desktop.
1-6.
This computer program shows pictures of MABs, and will help you to learn
about numbers. You will find that the computer can do different things from
the base-ten blocks, as we go through the lessons. Do you have any questions
about the program?
1-7.
(Both blocks and computer groups) Ask students to show a series of two-digit
numbers with base-ten blocks: 16, 38, 60, 82.
1-8.
e.g., Can you show me the number 16 using the base-ten blocks? . . . Explain
what you have shown.
1-9.
Ask students to show the same numbers with the software. e.g., Can you
show me the same number using the computer? Help students if necessary, by
showing how to drag blocks from the “source blocks” on the left of the
screen.
1-10. Ask students to write symbols for various two-digit numbers: 73, 91, 45, 27,
13. Have students compare each other’s answers. Correct if necessary.
e.g., Write the number 73 for me.
281
1-11. Show students numbers written on cards, ask them to represent them using
blocks: 57, 39, 84, 22, 17. e.g., Look at this number written here. I want you
to show this number using the blocks. Are you sure that you are correct?
Explain it to me.
282
Appendix E – Audit Trail Example
Note that each line in the audit trail records the time, the mouse action, the number
represented by the software and the number of blocks in each column.
Date Today: 17 June 1997 9:36:37 AM
Session 9
Heron Group - Kelly & Hayden
Session Start.
9:40:40
9:41:14
9:41:15
9:41:21
9:41:21
9:41:22
9:41:22
9:41:25
9:41:26
9:41:26
9:41:27
9:41:27
9:41:27
9:41:28
9:41:29
9:41:46
9:46:11
9:46:20
9:46:29
9:46:37
9:46:41
9:46:43
9:46:48
9:46:52
9:46:53
9:46:58
9:47:05
9:48:14
9:48:20
9:48:21
9:48:24
9:48:30
9:48:33
9:49:52
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
Click: 1 on Sounds
Number:
Click: 1 on PickHun
Number:
Click: 1 on PickHun
Number:
Click: 1 on PickTen
Number:
Click: 1 on PickTen
Number:
Click: 1 on PickTen
Number:
Click: 1 on PickTen
Number:
Click: 1 on PickOne
Number:
Click: 1 on PickOne
Number:
Click: 1 on PickOne
Number:
Click: 1 on PickOne
Number:
Click: 1 on PickOne
Number:
Click: 1 on PickOne
Number:
Click: 1 on PickOne
Number:
Click: 1 on PickOne
Number:
Click: 1 on Show Number
Number:
Click: 1 on Subtract
Number:
Click: 1 on Expander
Number:
Click: 1 on hunEx
Number:
Click: 1 on tenEx
Number:
Click: 1 on
Number:
Click: 1 on tenEx
Number:
Click: 1 on oneEx
Number:
Click: cursor "speak" on Speech
Click: cursor "speak" on hunDrop
Click: cursor "speak" on Speech
Click: 1 on hunEx
Number:
Click: 1 on Restart
Number:
Click: 1 on hunEx
Number:
Click: 1 on tenEx
Number:
Click: 1 on oneEx
Number:
Click: 1 on Expander
Number:
Click: 1 on Show Number
Number:
Menu Item Selected: ChooseNumber,
Blocks:
Number requested: 369
9:50:21 AM Click: 1 on Show Number
Number:
9:50:26 AM Click: cursor "speak" on Speech
9:50:31 AM Click: cursor "speak" on hunDrop
9:50:50 AM Click: 1 on Expander
Number:
9:50:51 AM Click: 1 on Expander
Number:
9:51:15 AM Click: 1 on Restart
Number:
9:51:18 AM Click: 1 on Show Number
Number:
9:51:21 AM Menu Item Selected: ChooseNumber,
Blocks:
Number requested: 541
9:52:12 AM Click: cursor "speak" on Speech
9:52:13 AM Click: cursor "speak" on hunDrop
9:52:25 AM Click: 1 on Show Number
Number:
9:52:40 AM Click: 1 on Show Number
Number:
9:52:41 AM Click: 1 on Restart
Number:
9:52:53 AM Menu Item Selected: ChooseNumber,
Blocks:
Number requested: 215
283
0 Blocks: 0 0 0
100
Blocks: 1 0 0
200
Blocks: 2 0 0
210
Blocks: 2 1 0
220
Blocks: 2 2 0
230
Blocks: 2 3 0
240
Blocks: 2 4 0
241
Blocks: 2 4 1
242
Blocks: 2 4 2
243
Blocks: 2 4 3
244
Blocks: 2 4 4
245
Blocks: 2 4 5
246
Blocks: 2 4 6
247
Blocks: 2 4 7
248
Blocks: 2 4 8
248
Blocks: 2 4 8
248
Blocks: 2 4 8
248
Blocks: 2 4 8
248
Blocks: 2 4 8
248
Blocks: 2 4 8
248
Blocks: 2 4 8
248
Blocks: 2 4 8
248
Blocks: 2 4 8
Number: 248 Blocks: 2 4
Number: 248 Blocks: 2 4
Number: 248 Blocks: 2 4
248
Blocks: 2 4 8
0 Blocks: 0 0 0
0 Blocks: 0 0 0
0 Blocks: 0 0 0
0 Blocks: 0 0 0
0 Blocks: 0 0 0
0 Blocks: 0 0 0
alias MakeNumMAB
Number:
0 0 0
8
8
8
0
369
Blocks: 3 6 9
Number: 369 Blocks: 3 6 9
Number: 369 Blocks: 3 6 9
369
Blocks: 3 6 9
369
Blocks: 3 6 9
0 Blocks: 0 0 0
0 Blocks: 0 0 0
alias MakeNumMAB
Number: 0
0 0 0
Number: 541 Blocks: 5 4 1
Number: 541 Blocks: 5 4 1
541
Blocks: 5 4 1
541
Blocks: 5 4 1
0 Blocks: 0 0 0
alias MakeNumMAB
Number: 0
0 0 0
9:53:23
9:53:25
9:54:12
9:54:15
AM
AM
AM
AM
Click: cursor "speak" on Speech
Click: cursor "speak" on hunDrop
Click: 1 on Restart
Number:
Menu Item Selected: ChooseNumber,
Blocks:
Number requested: 670
9:54:53 AM Click: cursor "speak" on Speech
9:54:54 AM Click: cursor "speak" on hunDrop
9:55:02 AM Click: 1 on Show Number
Number:
9:55:08 AM Click: 1 on Show Number
Number:
9:55:33 AM Click: 1 on Restart
Number:
9:56:55 AM Click: 1 on PickHun
Number:
9:56:56 AM Click: 1 on PickHun
Number:
9:56:56 AM Click: 1 on PickHun
Number:
9:56:56 AM Click: 1 on PickHun
Number:
9:56:57 AM Click: 1 on PickHun
Number:
9:57:01 AM Click: 1 on PickTen
Number:
9:57:01 AM Click: 1 on PickTen
Number:
9:57:02 AM Click: 1 on PickTen
Number:
9:57:05 AM Click: 1 on PickOne
Number:
9:57:05 AM Click: 1 on PickOne
Number:
9:57:06 AM Click: 1 on PickOne
Number:
9:57:06 AM Click: 1 on PickOne
Number:
9:57:07 AM Click: 1 on PickOne
Number:
9:57:07 AM Click: 1 on PickOne
Number:
9:57:07 AM Click: 1 on PickOne
Number:
9:57:08 AM Click: 1 on PickOne
Number:
9:57:18 AM Click: cursor "speak" on Speech
9:57:19 AM Click: cursor "speak" on Number:
9:58:29 AM Click: 1 on Restart
Number:
9:58:40 AM Click: 1 on PickHun
Number:
9:58:42 AM Click: 1 on PickTen
Number:
9:58:42 AM Click: 1 on PickTen
Number:
9:58:43 AM Click: 1 on PickTen
Number:
9:58:43 AM Click: 1 on PickTen
Number:
9:58:43 AM Click: 1 on PickTen
Number:
9:58:43 AM Click: 1 on PickTen
Number:
9:58:46 AM Click: 1 on TakeTen
Number:
9:58:47 AM Click: 1 on TakeTen
Number:
9:58:49 AM Click: 1 on PickTen
Number:
9:58:51 AM Click: 1 on PickOne
Number:
9:58:52 AM Click: 1 on PickOne
Number:
9:59:39 AM Click: 1 on Toolbox
Number:
9:59:41 AM Click: 1 on Restart
Number:
9:59:52 AM Click: 1 on PickHun
Number:
9:59:52 AM Click: 1 on PickHun
Number:
9:59:53 AM Click: 1 on PickHun
Number:
9:59:53 AM Click: 1 on PickHun
Number:
9:59:53 AM Click: 1 on PickHun
Number:
9:59:54 AM Click: 1 on PickHun
Number:
9:59:55 AM Click: 1 on PickHun
Number:
9:59:58 AM Click: 1 on PickTen
Number:
9:59:58 AM Click: 1 on PickTen
Number:
10:00:12 AM Click: 1 on supply
Number:
10:00:13 AM Click: 1 on supply
Number:
10:00:14 AM Click: 1 on TakeTen
Number:
10:00:14 AM Click: 1 on TakeTen
Number:
10:00:32 AM Click: 1 on supply
Number:
10:00:32 AM Click: 1 on supply
Number:
10:00:33 AM Click: 1 on PickOne
Number:
10:00:34 AM Click: 1 on PickOne
Number:
10:00:35 AM Click: 1 on PickOne
Number:
10:00:35 AM Click: 1 on PickOne
Number:
10:00:36 AM Click: 1 on PickOne
Number:
10:00:37 AM Click: 1 on PickOne
Number:
10:00:37 AM Click: 1 on PickOne
Number:
10:00:37 AM Click: 1 on PickOne
Number:
10:00:38 AM Click: 1 on PickOne
Number:
284
Number: 215 Blocks: 2 1 5
Number: 215 Blocks: 2 1 5
0 Blocks: 0 0 0
alias MakeNumMAB
Number: 0
0 0 0
Number: 670 Blocks: 6 7 0
Number: 670 Blocks: 6 7 0
670
Blocks: 6 7 0
670
Blocks: 6 7 0
0 Blocks: 0 0 0
100
Blocks: 1 0 0
200
Blocks: 2 0 0
300
Blocks: 3 0 0
400
Blocks: 4 0 0
500
Blocks: 5 0 0
510
Blocks: 5 1 0
520
Blocks: 5 2 0
530
Blocks: 5 3 0
531
Blocks: 5 3 1
532
Blocks: 5 3 2
533
Blocks: 5 3 3
534
Blocks: 5 3 4
535
Blocks: 5 3 5
536
Blocks: 5 3 6
537
Blocks: 5 3 7
538
Blocks: 5 3 8
Number: 538 Blocks: 5 3 8
538
Blocks: 5 3 8
0 Blocks: 0 0 0
100
Blocks: 1 0 0
110
Blocks: 1 1 0
120
Blocks: 1 2 0
130
Blocks: 1 3 0
140
Blocks: 1 4 0
150
Blocks: 1 5 0
160
Blocks: 1 6 0
150
Blocks: 1 5 0
140
Blocks: 1 4 0
150
Blocks: 1 5 0
151
Blocks: 1 5 1
152
Blocks: 1 5 2
152
Blocks: 1 5 2
0 Blocks: 0 0 0
100
Blocks: 1 0 0
200
Blocks: 2 0 0
300
Blocks: 3 0 0
400
Blocks: 4 0 0
500
Blocks: 5 0 0
600
Blocks: 6 0 0
700
Blocks: 7 0 0
710
Blocks: 7 1 0
720
Blocks: 7 2 0
720
Blocks: 7 2 0
720
Blocks: 7 2 0
710
Blocks: 7 1 0
700
Blocks: 7 0 0
700
Blocks: 7 0 0
700
Blocks: 7 0 0
701
Blocks: 7 0 1
702
Blocks: 7 0 2
703
Blocks: 7 0 3
704
Blocks: 7 0 4
705
Blocks: 7 0 5
706
Blocks: 7 0 6
707
Blocks: 7 0 7
708
Blocks: 7 0 8
709
Blocks: 7 0 9
10:00:38
10:00:38
10:00:38
10:01:03
10:01:11
10:01:16
10:02:35
10:02:39
10:02:55
10:02:58
10:03:06
10:03:26
10:03:26
10:03:27
10:03:27
10:03:27
10:03:28
10:03:28
10:03:28
10:03:32
10:03:32
10:03:32
10:03:34
10:03:46
10:03:57
10:03:59
10:04:55
10:05:04
10:05:05
10:05:10
10:05:10
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
AM
Click: 1 on PickOne
Number: 710
Blocks: 7 0 10
Click: 1 on PickOne
Number: 711
Blocks: 7 0 11
Click: 1 on PickOne
Number: 712
Blocks: 7 0 12
Click: 1 on Show Number Number: 712
Blocks: 7 0 12
Click: 1 on Show Number Number: 712
Blocks: 7 0 12
Click: 1 on Show Number Number: 712
Blocks: 7 0 12
Click: cursor "net" on Regroup
Number: 712 Blocks:
Click: cursor "net" on
Number: 712
Blocks: 7 0 12
Click: 1 on Toolbox
Number: 712
Blocks: 7 1 2
Click: 1 on Restart
Number: 0 Blocks: 0 0 0
Click: 1 on Show Number Number: 0 Blocks: 0 0 0
Click: 1 on PickHun
Number: 100
Blocks: 1 0 0
Click: 1 on PickHun
Number: 200
Blocks: 2 0 0
Click: 1 on PickHun
Number: 300
Blocks: 3 0 0
Click: 1 on PickHun
Number: 400
Blocks: 4 0 0
Click: 1 on PickHun
Number: 500
Blocks: 5 0 0
Click: 1 on PickHun
Number: 600
Blocks: 6 0 0
Click: 1 on PickHun
Number: 700
Blocks: 7 0 0
Click: 1 on PickHun
Number: 800
Blocks: 8 0 0
Click: 1 on PickTen
Number: 810
Blocks: 8 1 0
Click: 1 on PickTen
Number: 820
Blocks: 8 2 0
Click: 1 on PickTen
Number: 830
Blocks: 8 3 0
Click: 1 on TakeTen
Number: 820
Blocks: 8 2 0
Click: 1 on Show Number Number: 820
Blocks: 8 2 0
Click: cursor "speak" on Speech
Number: 820 Blocks:
Click: cursor "speak" on hunDrop
Number: 820 Blocks:
Click: cursor "glue" on Show Ones Number: 820 Blocks:
Click: cursor "glue" on Show Ones Number: 820 Blocks:
Click: cursor "glue" on hunones
Number: 820 Blocks:
Menu Item Selected: , alias exit
Number: 820 Blocks:
Session Ended.
285
7 0 12
8
8
8
8
8
8
2
2
2
2
2
2
0
0
0
0
0
0
Appendix F – Results of The Year Two Diagnostic Net, Used to
Select Participants for the Main Study
Name
Phase C
Phase D
1 2 3 4 5 6 7 8 9 1 1 1 1 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1
0 1 2 3
0 1 2 3 4 5
John
Craig
Daniel
Amanda
Belinda
Rory
Simone
Yvonne
Student
Student
a
Student
Student
Student
Student
Student
Student
Nerida
Haydenb
Kelly
Terry
Amy
Clive
Michelle
Jeremy
Student
Student
Note. Students are ranked roughly in order of criteria achieved on the previous year’s Year 2 Net
(Queensland Department of Education, 1996). Numbered columns include criteria from Phases C & D
of the test. All students in this population achieved all criteria in Phases A & B. Cells with solid
shading indicate criterion was fully achieved. Cells with line shading indicate criterion was partially
achieved. All judgements about student achievement were made by the Year 2 class teachers at the
time.
a
Indicates unnamed student from the available population of Year 3 students, not selected for
inclusion in the study.
b
The results for Hayden were unavailable. Hayden’s Year 3 teacher assessed his mathematical
achievement, relative to other students, to be at the level indicated on this table.
287
Appendix G – List of Participants
Pseudonym
Amanda
Simone
Craig
John
Belinda
Yvonne
Daniel
Rory
Michelle
Nerida
Clive
Jeremy
Amy
Kelly
Hayden
Terry
Gender
Mathematics
Achievement
Level
F
F
M
M
F
F
M
M
F
F
M
M
F
F
M
M
High
High
High
High
High
High
High
High
Low
Low
Low
Low
Low
Low
Low
Low
289
Number
Age at start of
Representation study (yy:mm)
Blocks
Blocks
Blocks
Blocks
Computer
Computer
Computer
Computer
Blocks
Blocks
Blocks
Blocks
Computer
Computer
Computer
Computer
08:08
08:03
08:03
08:02
08:01
08:04
08:02
08:00
08:02
07:07
08:06
08:06
07:09
08:01
08:02
07:08
Appendix H - Main Study Teaching Program
TABLE H.1.
Overview of Teaching Program Tasks
Two-digit numbers
Introduce notation
Representations of numbers:
Concrete to Verbal
Concrete to Symbolic
Verbal to Concrete
Verbal to Symbolic
Symbolic to Verbal
Symbolic to Concrete
Regrouping
Use of numeral expander
Comparing 2 numbers
Ordering 3 or more numbers
Counting on and back by 1, 2
Counting on and back by 10
Counting on and back by 100
Addition
Subtraction
Grouping ones
Three-digit
numbers
24, 25, 26, 27
1
1
2
2
3
3
4, 5, 6
7
8, 9, 10
11, 12
13, 14, 15
16, 17
18, 19
20, 21
22, 23
28
28
29
29
30
30
31, 32, 33
34
35, 36, 37
38, 39
40
41, 42
43
44, 45
Introduction
[Instructions to participants:] For each task, show the numbers with the blocks. Discuss the
question and the answer with the others in your group. Make sure everyone in the group
agrees with each answer. Ask Mr Price if you need any help.
Task 1 - Representing numbers
Look at the blocks put out by Mr Price. Say what number the blocks show. Write the number
in your workbook. [Numbers were not printed on task cards provided to participants.]
a) 25
b) 61
c) 13
d) 40
Task 2 - Representing numbers
Listen to the number given by Mr Price. Show the number with the blocks. Write the number
in your workbook. [Numbers were not printed on task cards provided to participants.]
a) 28
b) 31
c) 19
d) 90
291
Task 3 - Representing numbers
Show each number with the blocks. Say the name of each number:
a) 38
b) 72
c) 15
d) 80
Task 4 - Regrouping
Show the number with the blocks. Now swap one of the tens for ones. How many ones do
you need? Record what you have done in your workbook.
a) 77
b) 23
c) 91
d) 58
Task 5 - Regrouping
Show the number with the blocks. Now swap all of the tens for ones. How many ones do you
need? Record what you have done in your workbook.
a) 21
b) 36
Task 6 - Regrouping
Show the number with the blocks. If you were to swap all the tens for ones, how many ones
would there be? Write your answer in your workbook.
a) 64
b) 89
Task 7 - Use of numeral expander
" Write the number on the numeral expander. Show the number with the blocks. Use the
expander to show the number in different ways. Write the number in two ways in your
workbook.
a) 34
b) 96
c) 52
Task 7 - Use of numeral expander
Show the number with the blocks. Turn on the numeral expander. Use the expander to
show the number in different ways. Write the number in two ways in your workbook.
a) 34
b) 96
c) 52
Task 8 - Comparing 2 numbers
Tommy and Billy were arguing about who had more marbles. Tommy had 48 marbles, and
Billy had 62 marbles.
a) Who had more marbles? Show the numbers with the blocks. Explain your answer
in your workbook.
292
Task 9 - Comparing 2 numbers
Suzie and Margie were collecting stickers. Suzie had 20 in one book and 9 in a packet.
Margie had 70 in one book and 3 in her pocket.
a) Who had more stickers? Show the numbers with the blocks. Explain your answer
in your workbook.
Task 10 - Comparing 2 numbers
GameBoys cost 51 dollars and basketballs cost 39 dollars.
a) Which is more expensive? Show the numbers with the blocks. Explain your answer
in your workbook.
Task 11 - Ordering 3 numbers
Kellie wanted to put her books in order of size on her bookshelf, so the book with the most
pages was first, then the middle one, and then the one with the least pages. One book had 82
pages, one had 37 pages, and one had 88 pages.
a) Show the numbers with the blocks. Show in your workbook how Kellie should put
the books on her bookshelf.
Task 12 - Ordering 3 numbers
Simon has 75 toy soldiers, 57 toy cowboys, and 54 toy animals.
a) Show the numbers with the blocks. If Simon puts the group of toys with the most
on the top shelf, the next group on the middle shelf, and the smallest group on the
bottom shelf, where should he put them?
b) Show your answer in your workbook.
Task 13 - Counting on by 1s
Penny is writing down the dates until her birthday on the 28th of the month. Today is the 4th.
a) What are the dates before her birthday?
b) Show the numbers with the blocks.
c) Say them, then write them in your workbook.
Task 14 - Counting back by 1s
The Sunny Surfboard Company has 75 boogie boards left.
a) If one is sold, how many are left?
b) Then how many if another is sold?
c) Say all the numbers in order from 75 back to 60. Show the numbers with the
blocks. Write them in your workbook.
Task 15 - Odd and even numbers
Fern Street has the even-numbered houses on one side, and the odd-numbered houses on the
other side.
a) The Smith family lives at number 30 Fern Street. What are the house numbers on
either side of their house?
b) The Jones family lives at number 71 Fern Street. What are the house numbers on
either side of their house?
c) Show the numbers with the blocks. Write your answers in your workbook.
293
Task 16 - Counting on by 10s
Michelle has a collection of 26 football cards.
a) If she buys another packet of 10 cards, how many will she have?
b) How many with another 10?
c) Show the numbers with the blocks. Keep adding tens until you reach ninety-six.
Write the numbers in your workbook.
Task 17 - Counting back by 10s
Mr Walker has made 82 bread rolls to sell in his shop.
a) He sells a packet of 10 rolls. How many rolls are there now? Show the numbers
with the blocks.
b) Then he sells another packet of 10 - how many are there now?
c) Keep taking away tens. Write the numbers in your workbook.
Task 18 - Addition
Classes 3L and 3M went in a bus to the zoo. There are 28 children in class 3L and 31 in class
3M.
a) How many children went on the bus? Show the numbers with the blocks.
b) Discuss how to work it out with your group. Show how you work it out in your
workbook.
Task 19 - Addition
A Space Race video game costs 75 dollars, and a set of batteries costs 19 dollars.
a) How much will the game and the batteries cost? Show the numbers with the
blocks.
b) Discuss how to work it out with your group. Show how you work it out in your
workbook.
Task 20 - Subtraction
There are 95 soldiers on parade. The sun is hot, and 23 soldiers faint.
a) How many soldiers are still standing? Show the numbers with the blocks.
b) Discuss how to work it out with your group. Show how you work it out in your
workbook.
Task 21 - Subtraction
Mrs Perry has 83 dollars in her purse. She buys a coat costing 48 dollars.
a) How much money is now in Mrs Perry’s purse? Show the numbers with the blocks.
b) Discuss how to work it out with your group. Show how you work it out in your
workbook.
Task 22 - Grouping ones
Pat wants to buy some mints. Mints are sold in packets of 10, or one by one.
a) Will Pat have more if she buys six packets, or 45 single mints?
b) Discuss how to work it out with your group. Show the numbers with the blocks.
Show how you work it out in your workbook.
294
Task 23 - Grouping ones
Trent’s mum is making lolly bags for Trent’s birthday party. She has 2 rolls of 10 toffees,
and 16 single toffees.
a) Show the number of toffees with the blocks. How many toffees are there?
b) Does Trent’s mum have enough toffees to give 4 guests 10 toffees each?
c) Discuss how to work it out with your group. Show how you work it out in your
workbook.
Task 24 - Hundreds place
Show the number 40 with the blocks.
a) Put out another ten, and say the number’s name.
b) Keep adding tens. Stop when you have 10 tens.
c) What is this number called? Can you trade 10 tens?
d) Keep adding tens. Stop when you reach two hundred.
e) Write the numbers you made in your workbook.
Task 25 - Hundreds place
Donna and James have picked 90 apples.
a) Show that number with the blocks.
b) Donna and James pick more apples, one by one. Show each number with the
blocks. Stop at 99.
c) Write the numbers 90 to 99 in your workbook.
d) What is the next number? Write it in your workbook.
e) Show the number with the blocks. Can you trade any blocks?
Task 26 - Notation
Write four numbers between 10 and 99 in your workbook.
a) Choose one number. Explain what each digit means.
b) Now write the number that Mr Price tells you. Explain what each digit means.
c) Show what the number means with the blocks.
Task 27 - Notation
Show the number 248 with blocks.
a) Write the number in your workbook.
b) Explain the meaning of the 2, the 4 and the 8.
Task 28 - Representing numbers
Look at the blocks put out by Mr Price. Say what number the blocks show. Write the number
in your workbook.
a) 369
b) 541
c) 215
d) 670
295
Task 29 - Representing numbers
Listen to the number given by Mr Price. Show the number with the blocks. Write the number
in your workbook.
a) 538
b) 152
c) 712
d) 820
Task 30 - Representing numbers
Show each number with the blocks. Say the name of each number:
a) 147
b) 394
c) 516
d) 470
Task 31 - Regrouping tens
Show the number with the blocks. Now swap one of the tens for ones. How many ones do
you need? Record what you have done in your workbook.
a) 255
b) 932
c) 314
Task 32 - Regrouping hundreds
Show the number with the blocks. Now swap one of the hundreds for tens. How many tens
do you need? Record what you have done in your workbook.
a) 340
b) 627
Task 33 - Regrouping tens and ones
" Put out a handful of tens and ones blocks.
a) Write the number of tens and ones you have in your workbook.
b) What number is shown by the blocks? Do you need to regroup any blocks?
c) Write the number shown by the blocks in your workbook.
Task 33 - Regrouping tens and ones
Put out more than 9 tens and more than 9 ones.
a) Write the number of tens and ones you have in your workbook.
b) What number is shown by the blocks? Do you need to regroup any blocks?
c) Write the number shown by the blocks in your workbook.
Task 34 - Numeral expander
" Write the number on the numeral expander. Show the number with the blocks. Use the
expander to show the number in different ways. Write the number in two ways in your
workbook.
a) 381
b) 419
c) 158
296
Task 34 - Numeral expander
Show the number with the blocks. Turn on the numeral expander. Use the expander to
show the number in different ways. Write the number in two ways in your workbook.
a) 381
b) 419
c) 158
Task 35 - Comparing 2 numbers
Mary and Harriet were arguing about who had more insects. Mary had 341 insects, and
Harriet had 289 insects.
a) Who had more insects? Show the numbers with the blocks. Explain your answer in
your workbook.
Task 36 - Comparing 2 numbers
Frank and Lenny were collecting stamps. Frank had 250 in one book and 7 in his pocket.
Lenny had 170 in one book and 34 in a packet.
a) Who had more stamps? Show the numbers with the blocks. Explain your answer in
your workbook.
Task 37 - Comparing 2 numbers
Super Fast computer games cost 432 dollars and Ultra mountain bikes cost 419 dollars.
a) Which is more expensive? Show the numbers with the blocks. Explain your answer
in your workbook.
Task 38 - Ordering 3 numbers
A zoo keeper wants to put three baby animals in three pens: a large pen, a middle sized pen,
and a small pen. The animals’ masses are shown in the table below:
Panda
Elephant
Giraffe
419 kilograms
823 kilograms
485 kilograms
a) Show the numbers with the blocks. Discuss with your group how the animals should be penned.
b) Show in your workbook how the keeper should put the animals in the pens.
Task 39 - Ordering 4 numbers
At a sports meeting, Fiji has 79 competitors, New Zealand has 607 competitors, Indonesia
has 398 competitors, and Australia has 624 competitors.
a) If the four teams march in order with the largest team first, and the smallest team
last, which order should they be in?
b) Show the numbers with the blocks. Show your answer in your workbook.
Task 40 - Counting on by 1s
As people arrive at the fun park, they are given a number that counts how many people have
arrived that day. Jenny is given the number 283.
a) Show the number with the blocks. What will the next number be? Which number is
next after that?
b) Keep adding numbers. Stop at three hundred and twenty-three.
c) Say the numbers, then write them in your workbook.
297
Task 41 - Counting on by 10s
At the cinema each person pays 10 dollars to see the film. There is 462 dollars in the cash
drawer.
a) Show the number with the blocks. How much money will there be in the drawer
after the next person pays their 10 dollars? Then how much after the next person?
b) Keep adding tens. Stop at 600.
c) Write the numbers in your workbook.
Task 42 - Counting back by 10s
The supermarket is selling Easter eggs in packets of 10. There are 380 eggs left, and one
more packet is sold.
a) Show the numbers with the blocks. How many eggs are there now? How many
when another packet is sold?
b) Keep taking away packets. Stop at 200 eggs.
c) Say the numbers, and write them in your workbook.
Task 43 - Counting on by 100s
Helen has 264 stamps in her collection. Her mother gives her a packet of 100 stamps.
a) How many stamps does she have now? Show the numbers with the blocks.
b) Keep adding hundreds. Stop at 900.
c) Say the numbers, and write them in your workbook.
Task 44 - Addition
Children from two schools compete in a sports day competition. One school has 274
students, and the other has 315 students.
a) How many students are at the sports day? Show the numbers with the blocks. Show
how you work out your answer in your workbook.
Task 45 - Addition
A cattle farmer has 624 cattle in one paddock, and 193 in another.
How many cattle are in the two paddocks? Show the numbers with the blocks. Show how you work
out your answer in your workbook.
298
Appendix I - Main Study Interview 1 Instrument
Year 3 Place Value Interview 1
Name:........ ..................................................... Class: ..............................
Date:........./ ........ / ........
Number Representation
Qu. 1 Place base-ten blocks on the desk in front of student, randomly arranged. Ask
for the number that is shown by the blocks.
a) 3 t & 8 ones.
b) 4 t & 12 ones.
c) 2 h, 16 t, & 1 one.
Qu. 2 Place base-ten blocks on the desk in front of the student, randomly arranged.
Ask the student to show the number with blocks. Then ask the student to show the
number another way.
a) 5 t & 30 ones: show 28
b) 3 h, 16 t, & 60 ones: show 134.
Qu. 3 Show the student the written symbol for 136. Ask student to look at the
following block representations in turn, and to say whether it equals 136:
a) 1 hundred, 2 tens, & 16 ones.
b) 13 tens & 6 ones.
c) 1 ten, 3 hundreds, & 6 ones.
Counting
Qu. 4 Ask the student to count on or back in ones or tens. In each case continue
beyond the next necessary regrouping.
a)
b)
c)
d)
74 - 1 etc
65 + 10 etc.
342 + 10 etc.
496 - 10 etc.
Number Relationships
Qu. 5 Ask student for numbers, and for explanations:
a)
b)
c)
d)
a little bigger than 56.
much bigger than 56.
a little smaller than 56.
much smaller than 56.
299
Qu. 6 Show the student the numbers written on paper. Ask student to point to the
larger number, and explain.
a) 27; 42
b) 174; 147
Digit Correspondence
Qu. 7 Show the student 24 bundling sticks.
a) Ask the student to count them (correct if necessary), and to write the symbol for the
number.
b) Circle first the “4,” and then the “2,” and ask “Does this part of your 24 have
anything to do with how many sticks you have? Please show me. How do you
know?”
Qu. 8 Show the student 13 lollies.
a) Ask the student to count them (correct if necessary), and to write the symbol for the
number. Ask the student to share the lollies evenly among four cups.
b) Circle first the “3,” and then the “1,” and ask “Does this part of your 13 have
anything to do with how many lollies you have? Please show me. How do you
know?”
Novel Tens Grouping
Qu. 9 Show the student some packets of chewing gum. Tell the student that each
packet contains 10 sticks of gum. Ask the following questions:
a) If Carla has 6 packets and 4 other pieces of gum, how much chewing gum does she
have altogether?
b) If Bruce has 3 packets and 17 other pieces of gum, how much chewing gum does
he have altogether?
c) If Sam buys 5 packets and eats 8 pieces of gum, how many pieces does he have
left?
300
Appendix J - Main Study Interview 2 Instrument
Year 3 Place Value Interview 2
Name:........ ..................................................... Class: ..............................
Date:........./ ........ / ........
Number Representation
Qu. 1 Place base-ten blocks on the desk in front of student, randomly arranged. Ask
for the number that is shown by the blocks.
a) 6 t & 7 ones.
b) 3 t & 16 ones.
c) 5 h, 13 t, & 2 ones.
Qu. 2 Place base-ten blocks on the desk in front of the student, randomly arranged.
Ask the student to show the number with blocks.
a) 5 t & 30 ones: show 38.
b) 3 h, 16 t, & 60 ones: show 261.
Qu. 3 Show the student the written symbol for 172. Ask student to look at the
following block representations in turn, and to say whether it equals 172:
a) 1 hundred, 6 tens, & 12 ones
b) 17 tens & 2 ones
c) 1 ten, 7 hundreds, & 2 ones
Counting
Qu. 4 Ask the student to count on or back in ones or tens. In each case continue
beyond the next necessary regrouping.
a) 96 - 1 etc.
b) 42 + 10 etc.
c) 263 + 10 etc.
d) 681 - 10 etc.
Number Relationships
Qu. 5 Ask student for numbers, and for explanations:
a) a little bigger than 73
b) much bigger than 73
a) a little smaller than 73
b) much smaller than 73
Qu. 6 Show the student the numbers written on paper. Ask student to point to the
larger number, and explain.
a) 61; 38
b) 259; 295
301
Digit Correspondence
Qu. 7 Show the student 37 bundling sticks.
a) Ask the student to count them (correct if necessary), and to write the symbol
for the number.
b) Circle first the “7,” and then the “3,” and ask “Does this part of your 37
have anything to do with how many sticks you have? Please show me. How
do you know?”
Qu. 8 Show the student 26 counters.
a) Ask the student to count them (correct if necessary), and to write the symbol
for the number. Ask the student to stack the counters evenly on six circles
on a card.
b) Circle first the “6,” and then the “2,” and ask “Does this part of your 26
have anything to do with how many counters you have? Please show me.
How do you know?”
Novel Tens Grouping
Qu. 9 Show the student some packets of clothes pegs. Tell the student that each
packet contains 10 clothes pegs. Ask the following questions:
a) If Julie has 4 packets and 8 other clothes pegs, how many pegs does she
have altogether?
b) If Frank has 5 packets and 13 other clothes pegs, how many pegs does he
have altogether?
c) If Sarah buys 7 packets and loses 6 clothes pegs, how many pegs does she
have left?
302
Appendix K – Letter Requesting Consent by Parents or
Guardians of Prospective Participants
13 May, 1997
Dear Mr/Mrs _______
re: MATHEMATICS RESEARCH STUDY
I am a PhD student and part-time lecturer in mathematics education at QUT.
Your son/daughter, _______, has been selected from the year 3 students at
___________ State School to take part in a research study being conducted by
myself, from 2nd to 20th June 1997. Each child will be needed for 12 teaching
sessions of approximately 20 minutes each, during school time, in a room separate
from the classroom. The children will be learning about 2- and 3-digit numbers.
The school principal, Mr ________, has given his approval for the study to be
conducted in the school. Anonymity of the students will be maintained in all reports
of the study, except in reporting results of the study to the class teachers.
Please indicate your agreement for your child to take part in the study, or otherwise,
on the pro forma below, and return it to the school as soon as convenient.
Yours faithfully
Peter Price
PhD Student/Lecturer
QUT
-------------------------------------------------------------------------------------------------------------Mathematics Research Study __________ State School, Semester 1, 1997
I hereby {give / do not give}* my permission for my son/daughter ________ to be
involved in the above study. I understand that his/her anonymity will be protected,
and that he/she may leave the study at any time.
Signed: .........................................................
Parent/Guardian
* Cross out whichever does not apply
303
Appendix L – Coding Teaching Session Transcripts for
Feedback
Coding of transcripts for feedback.
It is necessary at this point to describe how categories of feedback were
developed and how decisions were made about how to categorise each potential
incident involving feedback. When the videotapes from each session were
transcribed, initial readings of the transcripts revealed a large number of categories of
participant responses. It gradually emerged that one defining difference between the
two representational formats was the ways in which participants were able to find out
whether or not their answers were correct using each type of representational
material. In light of this finding, the transcripts were re-analysed in more detail,
looking specifically for incidents of feedback. A computer-based database file
designed by the author was used to record and categorise each incident indicating the
occurrence of feedback in the 40 teaching sessions (see Figure L.1). Feedback was
defined for the purposes of this thesis in this way:
Feedback is considered to have taken place if a participant received
information from another source indicating whether the participant’s thinking
about numbers was correct.
It is important to note that feedback is considered to have occurred whether or
not the recipient of the feedback requested it: The primary consideration is to note
incidents in which participants received information about their answers. It is
presumed that feedback helped participants to decide whether their ideas were
correct, and whether or not they should change their answers.
305
Figure L.1. Data entry screen for feedback database.
As the analysis of feedback was conducted, a number of aspects of each
incident were coded for later analysis, as illustrated in Figure L.1. Firstly, details of
the participant, session number, and task were noted. The feedback itself was
categorised according to its source, the effect of the feedback for the participant
receiving it, and the response of the feedback recipient. Added to this was an
assessment of the status of the answer for which the participant was receiving
feedback: in other words, whether or not the recipient of the feedback already had an
answer, and whether or not the answer was correct, at the time the feedback was
provided. Finally a note was made of the transcript reference that referred to the
same incident, and a text comment was added briefly describing the incident.
Aspects of feedback that emerged from the data analysis—sources of feedback,
effects of feedback, and responses to feedback—are listed in Tables L.1, L.2, and
L.3.
306
TABLE L.1.
Source of Feedback
Feedback Description
Teacher feedback
Peer feedback
Check peer writing
Count on fingers
Mental computation
Check own writing
Count/recount blocks
Count computer blocks
Check computer counter
Check computer symbol
Check computer verbal number name
Check peer computer
Source
Teacher
Peer
Peer
Self
Self
Self
Materials
Materials
Electronic
Electronic
Electronic
Electronic
Table L.1 reveals the range of feedback types available to participants. At
least 12 types of feedback were observed in the data, including 4 types of electronic
feedback provided by the software. It is interesting that the “count on fingers”
category was used by participants in computer groups only, though only twice per
group. Note that feedback accessed by counting computer blocks is considered to
have “Materials” as its source, because it involves merely looking at pictorial
representations of blocks, much as physical blocks may be counted. Other electronic
feedback requires the computer’s computational facility to provide feedback
information.
TABLE L.2.
Effects of Feedback
Effect of feedback
Confirm answer
Contradict answer
Explain a wrong answer
Ask a question in return
Provide an answer
Give directions
Counter-suggestion
No effect
Quality
Positive
Negative
Negative
Neutral
Neutral
Neutral
Neutral
Neutral
Table L.2 lists the evident effects of feedback observed during this analysis,
and determinations made of the quality of each effect: This is defined as the likely
effect that feedback would have on the participant receiving it, with regards to the
recipient’s confidence in the answer. In other words, if the feedback is likely to have
307
encouraged a participant to retain the answer, whether correct or not, then the
feedback is said to be positive. If, on the other hand, the feedback is considered
likely to have encouraged its recipient to reject the answer, then it is said to be
negative in quality.
TABLE L.3.
Responses to Feedback
Responses to feedback
No visible response
Change answer
Reject feedback
Seek further feedback
Express satisfaction
Reconsider question
Laugh or smile
Repeat answer
Write or represent answer
State answer
Explain answer
Follow directions
Table L.3 lists the various responses to feedback observed in this phase of
analysis. The purpose of descriptions of responses to feedback is to consider the
effect of each incident of feedback on the recipient’s actions. It is important to
consider how likely children are to accept feedback provided by either blocks or
software, based on their actions after receiving such feedback.
The following steps describe the method used to analyse videotapes for
incidents of feedback:
1.
An incident in which a participant received information regarding an
answer was noted. This included occasions in which a participant
received an answer from another source, such as another participant or
from the computer column counters.
2.
The status of the participant’s answer prior to the feedback was
determined: was the answer correct, incorrect, incomplete, or nonexistent? In some cases, the status had to be coded as “unknown,” as it
could not be determined from the data available.
308
3.
The source of the feedback was determined. In most cases, this was
obvious once the identification of a feedback incident had been made,
as the source of the feedback is a necessary part of the incident itself.
4.
The effect of the feedback on the participant was determined. Again,
this was quite simple once an incident had been identified, as the effect
that the feedback had on the participant was generally clear from the
feedback itself. For example: Did the feedback confirm what the
participant had already stated or otherwise demonstrated as the answer?
Did it contradict the previous answer?
5.
The main response the participant made to the feedback, if any, was
noted. If a participant responded in two different ways, the incident was
coded as a single incident, and the principal response of the recipient
noted. The category “no visible response” had to be used in incidents
where the response of the recipient of the feedback could not be
determined.
6.
Note that if two participants each contradicted the other, then two
incidents of feedback were coded: one for each participant receiving the
feedback. Similarly, if the researcher gave feedback to an entire group
or to more than one participant, then that feedback was coded separately
for each recipient of the feedback.
As the analysis of feedback was conducted, it was discovered that participants
received feedback many times, and often with great frequency. Over the 40 teaching
sessions, lasting a total of approximately 1000 minutes, 1134 incidents of feedback
were identified. Whereas this indicates an average of little more than one incident per
minute, there were periods of time in which little feedback was evident, and others in
which feedback occurred rapidly over a short period. This is demonstrated by the
transcript excerpt in Appendix U, showing a short period of time in which several
incidents of feedback occurred, as the low/blocks group attempted to work out
75 + 19.
Clearly coding for feedback is not an exact process, as it requires subjective
judgements to be made about what feedback a participant received, what effect the
feedback had on the participant, and what the participant’s response was to the
feedback. Nevertheless, it is clear that incidents of feedback did occur during the
teaching sessions, and the author contends that its occurrence can be described
309
reasonably accurately using the methods detailed here. It is important to point out
that some incidents of feedback must necessarily be missed, no matter what method
is used to identify them. At times during teaching sessions participants were almost
certainly thinking about the information presented to them without any outward sign
of the character of that thinking. The incidents of feedback described here can
include only incidents in which participants made some outward sign of receiving
some feedback from their environment. During periods when participants were not
obviously receiving feedback or responding to their environment, it was difficult to
decide if feedback was occurring. For example, participants using the computer spent
much of the sessions looking at the computer screens; similarly, participants using
blocks spent much time manipulating and looking at the blocks. Though the learning
environments for all participants clearly provided almost continuous feedback of
various types, it is only possible to identify aspects of incidents that are visible on the
videotapes. However, by careful and consistent use of observation techniques it is
asserted that valid and reliable descriptions of numbers and types of feedback have
been made that can be compared among participants and among groups.
An attempt was made to check the reliability of the identification and coding
of feedback incidents. A second experienced teacher was asked to view a selection of
four videotapes of teaching sessions, one of each group, and compare events
observed on the videotapes with the coding already carried out by the researcher. The
second observer confirmed every single incident coded by the researcher, but also
felt that other incidents were evident on the tapes that had not been coded. As the
reliability of the feedback identification and coding is not high when considered by a
second observer, it must be considered as just an indication of feedback activity that
took place rather than an objective measure of it. As in much qualitative research,
judgement about feedback incidents is necessarily subjective, relying on
interpretations of many subtle interactions that took place among participants, the
researcher and the materials. The identification of feedback incidents reported here is
the interpretation of the researcher that would be certain to differ from interpretations
made by other observers of the same data.
310
Appendix M – Descriptions of Numeration Skills Targeted by
Interview Questions and Criteria for Their Assessment
No.
Skill
1a
State number Two-digit canonical
represented by block representation
blocks
Two-digit noncanonical block
representation
Three-digit noncanonical block
representation
Show blocks Two-digit number
to represent
given number
Three-digit number
1b
1c
2a
2b
3a
3b
3c
State whether
block
representation
matches
numerical
symbol
Sub-Skill Description
Interview
Question(s)
(Q 1a)
(Q 1b)
(Q 1c)
(Q 2a)
(Q 2b)
Three-digit non(Q 3a)
canonical block
representation with >9
ones
Three-digit non(Q 3b)
canonical block
representation with 0
hundreds, >9 tens
Reject incorrect three- (Q 3c)
digit face-value block
representation
311
Assessment Criteria
Correct number name
stated; single miscount
allowed.
Correct number name
stated; single miscount
allowed.
Correct number name
stated; single miscount
allowed.
Correct number of blocks
shown, non-canonical
representation allowed.
Correct number of blocks
shown, non-canonical
representation allowed.
Correct number name
stated; single miscount
allowed.
Correct number name
stated; single miscount
allowed.
Blocks counted correctly;
statement that blocks
represented the stated
number rejected.
4a
4b
4c
4d
5a
5b
6a
6b
7a
7b
Number sequence
correctly counting past
change of decade; no
omissions or insertions.
Count on in tens from (Q 4b) Number sequence
two-digit number, past
correctly counting past
100
120; no omissions or
insertions.
Count on in tens
(Q 4c)
Number sequence
through three-digit
correctly counting past
numbers
change of hundred; no
omissions or insertions.
Count back in tens
(Q 4d) Number sequence
through three-digit
correctly counting past
numbers
change of hundred; no
omissions or insertions.
Nominate
Nominate numbers
(Q 5a, c) Numbers stated that are
numbers
close to a given twoeach within 10 of the
relative to a
digit number
given number, or such that
single given
difference is no more than
number
25% of difference in
related “far” example.
Nominate numbers far (Q 5b, d) Numbers stated that are
from a given two-digit
each at least 30 away from
number
the given number, or such
that difference is more
than 4 times the difference
in related “near” example.
State which of Two-digit numbers,
(Q 6a)
Correct number stated as
two written
ones digit of smaller
greater, incorrect countersymbols
number > either digit
suggestion(s), if any, not
represents the of larger number
accepted.
greater
number
Correct number stated as
Three-digit numbers, (Q 6b)
greater, incorrect counterrespective tens and
suggestion(s), if any, not
ones digits reversed
accepted.
Count
Two-digit number
(Q 7)
Correct number of objects
collection of
shown as referent for each
10-40 objects,
digit, incorrect countershow referent
suggestion(s) rejected.
for each digit
Two-digit number with (Q 8)
Correct number of objects
misleading visual cues
shown as referent for each
digit, incorrect countersuggestion(s) rejected.
Recite verbal
number
sequence
Count back in ones
through two-digit
numbers
312
(Q 4a)
8a
8b
8c
Mental
Add a number of tens
computation in and a number of ones
a novel tens
grouping
situation
Add a number of tens
and 11-19 ones
(Q 9a)
Correct answer stated; unprompted self-correction
allowed.
(Q 9b)
Correct answer stated; unprompted self-correction
allowed.
Correct answer stated; unprompted self-correction
allowed.
Subtract a number of (Q 9c)
ones from a number of
tens
313
Appendix N – Transcript of Interview 1 Question 6 (a) with
Terry
Qu. 6 (a)
Show the student the numbers ‘27’ and ‘42’ written on paper. Ask
student to point to the larger number, and explain.
Terry:
I’m still … Oh! … I can tell by the even numbers. This one [‘42’] is bigger
because it’s even and this one [‘27’] is smaller because it’s odd.
Interviewer: OK what is this number [‘42’] here?
Terry:
42.
Interviewer: And this one?
Terry:
27.
Interviewer: And 42 is bigger because it’s even?
Terry:
Yup.
Interviewer: And that’s [‘27’] smaller because it’s odd?
Terry:
Yup.
Interviewer: Supposing this one here was … 57.
Terry:
Yeah.
Interviewer: That’s still odd.
Terry:
Yeah.
Interviewer: Would it still be smaller than that?
Terry:
Yeah.
Interviewer: OK. So all even numbers are bigger than all odd numbers?
Terry:
Yup.
Interviewer: How do you know that’s even?
Terry:
Because it’s got a ‘2’ at the end.
Interviewer: Uh-huh. And how do you know this one’s odd?
Terry:
Because you got a ‘7’ at the end.
Interviewer: All right, what if they were both even? Supposing this one [‘27’] was 26,
which would be bigger then?
Terry:
This one. [‘26’] ‘Cos it would be 26.
Interviewer: 26 would be bigger? Why is that?
Terry:
‘Cos they are both even and this one is only 42 and the other one’s 26.
315
Interviewer: OK, and why would 26 be bigger?
Terry:
‘Cos it’s got a ‘6’ at the end.
Interviewer: All right, and the ‘6’ is bigger than …?
Terry:
Oops! It’s this one [‘42’] because this one is away from 20.
Interviewer: I think we’d better write these down.
Terry:
Ah.
Interviewer: Because we just want to be sure we both know what we are talking about.
[writes ‘26’ and ‘42’] 26 and 42. Which one is bigger?
Terry:
42.
Interviewer: Uh-huh. Why?
Terry:
‘Cos … ‘cos it’s one way [pause] it’s … like 46 is in the 20s …
Interviewer: 26.
Terry:
26 [correcting himself] is in the 20s, and 42 is in the 40s, and it’s all missing
the 30s so … so 42 is bigger?
Interviewer: How do you know the 40s are bigger than the 20s?
Terry:
‘Cos if you count in 10, 20, 30, 40, it’s a long way away from 20.
Interviewer: Comes after it you mean?
Terry:
Comes after it.
Interviewer: OK, that’s a good answer, but with this one [‘27’] because that’s odd …
Terry:
Yeah.
Interviewer: That’s smaller because it’s odd?
Terry:
Yup.
Interviewer: All right. [pause] Right, right, right, right. What about … all right let me ask
you this one then. [writes ‘57’ and ‘42’] Look at those two numbers. What’s
this one here now? [‘57’]
Terry:
57.
Interviewer: And? [‘42’]
Terry:
42.
Interviewer: Which one is bigger now?
Terry:
Oh, you just gave me an odd number. This one [‘42’] because it’s even.
Interviewer: 42 is bigger because it’s even?
Terry:
Yep.
316
Interviewer: Uh-huh. But this [‘57’] is in the 50s isn’t it?
Terry:
Yeah.
Interviewer: Don’t the 50s come after the 40s?
Terry:
Yeah … oh yeah?! So this one is actually even … um … more ‘cos it’s after?
Interviewer: Right.
Terry:
So are we actually talking in after and not before?
Interviewer: [Laughs] We’re talking about which one’s bigger. Which one means the
bigger amount.
Terry:
Oh.
Interviewer: 57 or 42.
Terry:
57.
Interviewer: Because …
Terry:
Because it’s got … it’s one way away from … it’s right after 40.
317
Appendix O – Transcript of Interview 2 Question 6 (a) with
Hayden
Qu. 6 (a)
Show the student the numbers ‘61’ and ‘38’ written on a card. Ask
the student to point to the larger number, and to explain.
Interviewer: Can you tell me which of these numbers is larger?
Hayden:
[Points to ‘61’]
Interviewer: What number is that?
Hayden:
61.
Interviewer: Uh-huh, and how do you know that’s bigger?
Hayden:
Because it’s 6 … 38 takes shorter and 61 takes longer.
Interviewer: If you’re counting you mean?
Hayden:
Yeah.
Interviewer: OK, what about the numbers that are in it? Does that tell you anything?
Hayden:
No, it doesn’t … like it still doesn’t mean that it’s got an ‘8’ on the end and
it’s got a ‘1’ on the end [points to respective digits] …
Interviewer: Uh-huh.
Hayden:
… because that’s um … like that … like if you get 1, 2, 3, 4 … like 10, 20, 30,
40… no 10, 20, 30 and you just count to 8 …
Interviewer: Uh-huh.
Hayden:
…in the 30s, like it’s only the 38.
Interviewer: Uh-huh.
Hayden:
And if you count the 61 it’s a 60 one.
Interviewer: All right, so if you are counting to 60 it would …
Hayden:
… like take longer.
Interviewer: Uh-huh, that’s a good answer.
319
Appendix P – Transcript of Low/Blocks Group Answering
Task 28 (a)
Task 28 (a) Look at the blocks put out by Mr Price [369]. Say what number the
blocks show. Write the number in your workbook.
Michelle:
[Puts one hand on top of the hundreds blocks, says immediately] 300.
Teacher
Please don’t say them aloud, Michelle.
Clive:
[Looks dejected again, is not moving to touch or count the blocks.]
Teacher:
Girls, please keep your hands off blocks, so that boys can see them.
Michelle:
[Puts her hand on the hundreds again.] 300 …
Teacher:
Work out the whole number, Michelle.
Clive:
[Moves hundreds so he can see the tens and ones. Frowns, apparently counting
the blocks.]
Teacher:
Girls, please take your hands off the blocks. (It appears that the girls were
finding this too difficult, that they needed to put their hands on the blocks in
order to keep track of their count.) [Puts out another copy of the block
representation for the boys.]
Nerida:
[Whispering, as she counts the ones] … 362, 363, 366, 367, 368 …
Michelle:
[Does her own counting.] 3 hundreds… [She puts her hand to her forehead,
and apparently finding the next step difficult.] 300, 400, 500, 600, 700, 800,
900, 101, 102, 103, 104, 105, 106, 107, 108, 109. 109. 109. [To teacher,
quietly] We got 109.
Clive:
[Counts hundreds, then tens. He starts to count ones, stops and frowns, moves
hundreds] That’s 300 … [He counts the tens quietly aloud by tens, then counts
on by ones to 69.] 369 [Quietly; writes in his workbook. He writes ‘3,’ counts
tens again, writes ‘69.’]
Teacher:
Jeremy, what is the number?
Nerida:
[Finishes counting the blocks under her breath, reaching ‘359.’ Writes her
answers straight away in her workbook, apparently before she forgets what the
number is.]
Michelle:
[Looks surreptitiously at Nerida’s book. She starts to re-count the blocks,
apparently confused about what she should count after 300.] 300 [Very
quietly; looks away. She starts counting the ten-blocks, adding them to the
321
hundreds, counting by ones. It is very difficult to hear the numbers that she
stands, but she includes the numbers “7, 8, 9, 22, 23, 24.” She stops.] Oh, no!
Jeremy:
[Looks at blocks for a while.] 300 … and … [counts, while nodding his head,
without touching blocks. Stops for a long while.] 30 … thirty one hundred …
thirty one hundred … thirty one hundred and …
Michelle:
[Distracted by someone entering the room, stops counting.]
Teacher:
Children, what are your answers?
Michelle:
30 …
Nerida:
359.
Clive:
[Confidently] It means 3 hundreds, and 6 tens and 9 ones.
Girls:
[Say they have a different number.]
Teacher:
[Asks children to confirm that there are 3 hundred, 6 tens and 9 ones. The
boys agree, but Nerida shakes her head, and Michelle moves her head slightly,
in a nondescript sort of way, neither shaking it nor nodding. The teacher asks
what they disagree with.]
Nerida:
We have a different number of tens. [The girls recount their ten-blocks by
ten.] There are 60 tens.
Michelle:
Now, do you …
Michelle:
60.
Nerida:
60.
Teacher:
Do you mean 60, or 6?
Michelle:
6 tens.
Nerida:
There are 6 tens.
Teacher:
What is the number altogether?
Clive:
[Loudly] 369.
Nerida:
369.
Michelle:
[Shakes her head.] I don’t reckon.
Teacher:
Michelle, what do you think the number is?
Michelle:
369.
Teacher:
Girls, don’t just change your minds just to agree with Clive.
Michelle:
[Is having trouble reading the blocks. She starts to count on from 300:] 300,
40 …
Teacher:
[Stops her.]
322
Michelle:
No, 140, 150, …
Teacher:
[Stops her again] It is not 30, but 300.
Michelle:
300, 4 … [She counts the tens alone, then counts on ones, to 69.]
Nerida:
[To Michelle] It is 369.
Michelle:
[Writes in her workbook. She repeats the first place of the number several
times as she starts. It appears that Michelle could not keep the three numbers
in her mind at once. Clive also appeared to have this same difficulty.]
Teacher:
Now, write the number 369, rather than just the values in each place.
323
Appendix Q – Transcripts of Task 4 (a)
from 4 groups
Task 4 (a)
Show the number 77 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
High/computer:
Yvonne:
[Reads card]
Belinda:
I don’t understand that. [Smiles]
Teacher:
Do it one step at a time.
Belinda:
[Reading card again, holding mouse] “Show the number with the blocks.”
Yvonne:
Can I just use your pencil for a minute? [Borrows Belinda’s pencil to draw a
line in her workbook.]
Yvonne:?
77
Daniel:
77
Belinda:
[Adding ten-blocks] 1, 2, 3, 4, 5, 6, 7
Yvonne:
The first one … [indistinct] [Watches both computers, shifting gaze from one
to the other as Rory uses mouse]
Belinda:
[Reading card again] Now swap one. Um, 70.
Yvonne:
Task …
Belinda:
[Adding one-blocks] 1, 2, 3, 4, 5, 6, 7
Computer1:
77
Daniel:
[Pointing to screen] OK, put … [indistinct]
Belinda:
[Reading card again] “Now swap one of the tens for ones.”
Teacher:
Don’t start again, Rory, you haven’t finished.
Belinda:
“How many ones do you need? Record what you …” I don’t get that. Now, go
to “Take away” [Looks back at teacher] mmm …
Yvonne:
Take away one from the ones. … Don’cha? [Looks back at teacher]
Belinda:
Oh, no.
Rory:
[Clicks on “Start again,” then starts to add ten-blocks again]
Teacher:
No, no. Don’t take the number away. Put the 77 back.
Rory:
[Puts blocks back]
Yvonne:
Swap one of the tens for ones.
325
Teacher:
OK, do you know how to do that?
Belinda:
Nope
Yvonne:
Nope… We done that before, didn’t we?
Daniel:
Can we put only 3 on?
Yvonne:
[Points at screen] Look!
Belinda:
Oh! [understands] “remove block.”
Yvonne:
No, you can’t. It’s only got 7.
Belinda:
Oh.
All:
[Unsure of how to continue. All look at teacher]
Teacher:
You know what “swap” means, don’t you?
All:
Yeah.
Teacher:
OK, well you have to swap.
Daniel:
It’s like trading.
Teacher:
Yes. Swap one of the tens for ones.
Yvonne:
Oh!
Belinda:
I don’t get that.
Teacher:
Make a swap. Swap a ten for …
Yvonne:
[Points and taps pencil on screen on buttons, then on blocks]
Belinda:
Huh?
Yvonne:
[Points at screen in two places again] For ones [indistinct]
Belinda:
Yeah. [Still looks puzzled]
Daniel:
Do you add 3 onto the ones?
Teacher:
No, we don’t want to make it into ten ones, we’re going to take one of the tens
and swap it. [Computer says there are too many blocks on screen, which will
be cleared] Oh, no. Too many blocks on the page. Click “OK.” I think we’ll
start yours again. [restarts program]
Daniel:
[Points at screen] OK, use the saw. Saw one of them big ones.
Belinda:
We need to put a bigger number on.
Yvonne:
I know.
Rory:
No.
Daniel:
You have to. [Nods]
Rory:
[Clicks saw on ten-block]
326
Yvonne:
See if you can … [Makes buzzing noise with mouth]
Daniel:
We’ve done ours. [Looks at teacher]
Computer2:
[Saw sound]
Belinda:
Done it.
Teacher:
OK, can we have the sound turned off if you don’t mind? Just not to disturb
the other class. Is that what the instructions were?
Belinda:
Yep, swap one.
Teacher:
Mmm? Read your instructions here.
Yvonne:
[Looks at teacher] Nope, nope [shakes head] nope.
Daniel:
Swap one, yep.
Belinda:
[Looking at screen] 77! There’s still 77! Cool.
Yvonne:
No, there’s sixty …
Rory:
It’s 77.
Yvonne:
Oh, yeah it is [laughs].
Computer:
77.
Belinda:
It sounds like Mr Price!
Yvonne:
So do we have to write 77 in here?
Teacher:
Well, you’ve got to answer this question here, now. Have you done what it
says “now swap one of the tens for ones”?
Yvonne:
Yep.
Belinda:
Yep.
Teacher:
“How many ones do you need?”
Belinda:
I don’t get that.
Teacher:
[Under breath] Neither do I. Who wrote that? [laughs]
Belinda:
Who writ that?
Teacher:
When you swap a ten for ones, how many tens do you swap it for?
Daniel:
10.
Belinda:
10.
Teacher:
Yeah. The computer does it for you.
Belinda:
Cool.
Teacher:
The other group only use the MABs and they have to work it out for
themselves.
327
Belinda:
So it’s going to be 10.
Teacher:
— OK. Record what you have done in your workbook.
Belinda:
So, you’ve got to record the 10.
Daniel:
So I’ve done …[indistinct]
Teacher:
What did you have to start with?
Belinda:
77.
Teacher:
Right, and what have you got now?
Yvonne:
[To Daniel, looking at his book] Why did you write 77?
Belinda:
77!
Teacher:
So what’s the difference?
Belinda:
Nothing.
Rory:
Nothing.
Teacher:
Well, there’s a difference in the blocks, isn’t there?
Belinda:
[Starts to write in workbook] Oh, 6.
Teacher:
I want you to write it down somehow, to show how it’s different now from
what it was before.
Daniel:
[Looking at teacher] Write ‘77’?
Belinda:
Write 60 … 6 …
Teacher:
Well, what’s the difference on the screen?
Belinda:
6 blocks …
Daniel:
There’s 6 tens and 17 ones.
Yvonne:
Yeah.
Teacher:
And what did you have before?
Yvonne:
77 one …
All:
7 tens and 7 ones.
Teacher:
OK. Can you write that down so it makes sense, that you had 7 tens and 7
ones, and now you’ve got 6 tens and 17 ones?
Belinda:
How would you write that? You could write “six plus seventeen.”
Teacher:
It’s not just 6, is it? It’s 6 tens plus 17. You could do that. Write down what
you had before, though, first.
Belinda:
Oh. [Rubs out previous writing]
Yvonne:?
Aw, now it won’t … [indistinct]
328
Belinda:
Or 6 plus ten.
Yvonne:
[To Belinda] Can I swap pencils? Mine’s not sharp.
Belinda:
No.
Yvonne:
Oh. [Keeps writing]
Belinda:
[Writing] 77, full stop.
Daniel:
[To Yvonne] Mine’s the same with yours. Mine’s the same.
Belinda:
[Writing] Seven tens [indistinct] equals 77.
Yvonne:
[Looking at Belinda’s workbook] 77, zero six nine [indistinct].
Belinda:
That’s a full stop.
Yvonne:
Oh, is it?
Belinda:
6 plus 17 equals 7, 77!
[Lots of indistinct speech from several children]
Teacher:
[to Belinda, pointing to her workbook] Now, is that right?
Belinda:
[Shrugs shoulders] I don’t know.
Teacher:
Six plus 17 - is that 77?
Belinda:
Uh-huh [confirming correct] ‘cos I checked it.
Yvonne:
[Looks at Belinda]
Belinda:
[Pause for 6 seconds] [Not so sure] I think. [Clicks on mouse to read number.]
Teacher:
OK. Now write down what you did.
Computer:
77
Belinda:
Yep. It’s right.
Daniel:?
We … oh.
Teacher:
You don’t have to write it all in a sentence, but write down what blocks you
had, and what you have now.
Yvonne:
[sighs]
Belinda:
I shouldn’t write these things. I reckon it looks too silly. [Uses rubber]
Yvonne:
[Watching Belinda] Here, I need one.
Teacher:
OK. Now what have you written? I want you to show each other and see that
you all agree with what you’ve written, because this is a group question. Well,
they’re all group questions.
Yvonne:
Well, shouldn’t we all do it the same?
Belinda:
[Looking at Yvonne’s workbook] We should be all … nuh.
329
Teacher:
Well, you all have to think for yourselves. And you have to check to see
whether you all agree.
Belinda:
[Swapping two workbooks] You check my answers, and I check your
answers.
Teacher:
No, no, no, I don’t mean that. I mean just show it to her. You’re not going to
mark it. I’ll mark it when I go home.
Yvonne:
[Laughs]
Belinda:
Aw! [disappointed]
Belinda:
[Comparing two girls’ workbooks] Yeah, that’s good. OK, that’s good.
Daniel:
I agreed with yours, Rory.
Teacher:
That’s very good.
Daniel:
He’s got that. [Pointing at Rory’s book]
Teacher:
He’s writing it in a sentence - that’s a good way of doing it.
Yvonne:
[Looks bored, flops in chair] Do you have to start again?
Teacher:
OK, well I’m going to have to challenge you children, because I don’t think
that 6 plus 17 is 77.
Belinda:?
I do.
Teacher:
I think 6 plus 17 is 23.
Yvonne:
Hey?
Teacher:
[Counting on fingers] 17, 18, 19, 20, 21, 22, 23. It’s not 77.
Yvonne:
Oh. [seeing problem with answer]
Belinda:
Aw. [disappointed] Oh, yeah but it says 6 plus 17.
Teacher:
It doesn’t say 6 plus 17.
Daniel:
Oh no, I’ll do it the same way as Rory now. [laughs] ‘Cos that’ll be the same
as Rory.
Teacher:
Well, the idea isn’t just to be the same as Rory, it’s to make sense of what
we’re doing. You see, the whole point of this is, Do you understand the
numbers? Do you understand what the blocks are showing?
Yvonne:
I need your rubber.
Teacher:
[Laughs]
Belinda:
It’s 14.
Teacher:
I’m sorry; what’s 14?
Belinda:
[Points at screen] That.
330
Daniel:
Can I borrow the rubber after you?
Rory:?
I’m going to write 36.
Yvonne:
I’m going to write 23.
Teacher:
I think you’d better talk about it between yourselves. I don’t want to tell you
the answer, unless I have to. I want you children to work it out yourselves.
Belinda:
I want my rubber. [Reaches across to Daniel]
Yvonne:?
I just changed mine to 23.
Teacher:
Children, I need you to discuss the question, and work out what you’re going
to do.
Teacher:
You need to talk about it with the others. Don’t just write down something on
your own. I want you to make sense of this, otherwise there’s no point going
on to hundreds, if we’re having trouble with the tens and ones.
Yvonne:
What is it? [Reads card] “Show the number with the blocks.”
Belinda:
It does not equal … something. It does not equal something …
Teacher:
Well, first of all are the blocks showing 77?
Yvonne:
Yes.
Daniel:
Yes, they still are.
Teacher:
They still are.
??
mmm [confirming]
Teacher:
So, when we did that swap, it’s still 77?
Daniel:
Yeah.
Belinda:
Yeah! [sounding surprised that it should be questioned]
Teacher:
You agree with the computer, that it’s still 77?
Daniel:
Yes.
Teacher:
You can see the 77 still there?
Daniel:
Yeah.
Belinda:
Hang on, hang on. Six…ty
Teacher:
You explain to me how that’s 77, ‘cos I don’t see 70, and I don’t see 7.
Belinda:
I do. ‘Cos just that’s one… There’s a ten there, and there’s another 7, and
that’s 60, 7, 7! So it’s right.
Teacher:
OK. Can you write that down so it makes sense? Look, I tell you what. Let’s
go backwards. We’ll get these back to 7 tens and 7 ones again, by regrouping
331
with the net, OK? [Uses “net” tool to regroup ten ones on each computer]
Right, that’s where we started.
Belinda:
Yeah.
Teacher:
OK. Now do the swap again, the trade, with the saw.
Daniel:
[Both boys go to use mouse; Daniel continues] Can I do it, this one?
Belinda:
Depose [sic].
Teacher:
Now watch, ‘cos I want you to see what happens.
Belinda:
Look, it’s right. It’s right.
Yvonne:
77 again?
Belinda:
No, six… Yeah, it’s right. It is. [She appears to feel intuitively that it is still
77, but is unable to explain it to Yvonne.]
Yvonne:
[Pointing at computer] Yeah, but it hasn’t got the zero! [Looks at other
computer also.]
Belinda:
Yeah!
Teacher:
Why doesn’t it have the zero?
Belinda:
It’s, it’s… I, I think it’s … Hang on, 17, 18, 19, 20. 21, 22, 23. 23!
Yvonne:
[Laughs]
Belinda:
Mmmm. No, but that’s 60. [Looks at teacher]
Yvonne:
It’s not - it hasn’t got the zero. It’s supposed to have the zero.
Belinda:
[Pointing at ten-blocks] No - 10, 20, 30, 40, 50, 60.
Yvonne:
[Looks at teacher] Yeah!
Daniel:
Oh, yeah! [understands; looks at teacher]
Teacher:
Don’t keep looking at me. Is that right? Does it make sense?
Daniel:
Yeah.
Belinda:
Yes. [Definite]
Yvonne:
Mmmm. [Confirming agreement]
Daniel:
But what does it all mean, though? [Raises hands palm up.]
Belinda:
See.
Teacher:
Well, what does the 17 mean?
Belinda:
17. No, that’s 60, 17. [Pointing at screen.] Group one, put it there, and it’s 77!
Yvonne:
Yeah. [Understands; smiles]
Teacher:
You look a bit confused, Rory. Is that alright?
332
Rory:
[No verbal response.]
Teacher:
You are confused?
Belinda:
I aren’t.
Teacher:
Girls, can you explain it to Rory?
Belinda:
[Stands up] Well, …
Yvonne:
Here, use it …
Daniel:
I’m confused as well. I don’t understand.
Teacher:
Well, explain it to Daniel as well. No, don’t get up. Stay there.
Belinda:
There’s 60 there, because there’s 6 tens there. That equals 60. Then there’s a
ten over here, which, that you put that back there, which makes 70, and then
there’s 7 there. [laughs]
Yvonne:
I don’t think he understands.
Daniel:
Oh, what does 6 plus 6 [indistinct]
Teacher:
[Uses mouse for computer 1] Here. Here, here, here. This label at the top says
‘6,’ doesn’t it?
Belinda:
Yeah.
Teacher:
But it’s 6 tens. What are these 6 tens showing?
Rory:
60.
Daniel:
60.
Teacher:
It’s showing 60, isn’t it?
Belinda:
And there’s another ten there. [Points at own screen]
Teacher:
Tell you what, if you click on the “Show as ones” button, it will show you all
the ones. [Clicks on “Show as ones”] OK, can you see the 77 ones?
??
Yep.
Daniel:
Oh yeah. [understands]
Teacher:
All these here, and all those there make 77. That’s 60, and these here are 17.
Belinda:
[Looking at her workbook] I’m right. Excellent!
Teacher:
You need to write that down somehow in your book.
Belinda:
Well, if I have … [indistinct]
Teacher:
OK, you’re on the right track. We started with 7 tens and 7 ones, didn’t we?
Belinda:
Oh! [understands] 60, plus 17, is 77!
Teacher:
Now it makes sense, doesn’t it?
333
Yvonne:?
Mmm. [confirming]
Teacher:
But what we had before was different, wasn’t it?
Yvonne:
Yep.
Daniel:
Yeah.
Belinda:
But then I [indistinct]
Teacher:
OK. Write down what we’ve got now.
Daniel:
Oh, yeah, 6 [indistinct]
Belinda:
I’m so clever. Can we put, click the net and [motions with two hands] put it
back to 77?
Teacher:
If you wish.
Belinda:
Wh-who [pleased]
Teacher:
You’re just about to start again with this one, so you can put the net and make
it go back to 7 tens and 7 ones.
Belinda:
Whooo
Teacher:
No, don’t drag it, just click it, Rory. Do it again, click on the net, and don’t
drag it. That’s it. Oh, it didn’t work. Sorry - do that again. [Uses mouse]
Sometimes the net doesn’t show up. There.
Daniel:
There you go.
(h/c S1, T 4a)
Low/computer:
Amy:
[Reads card]
Hayden:
[Helps her to read unknown words]
Teacher:
First thing: “Show the number with the blocks.”
Hayden:
[Starts to use mouse]
Amy:
[Starts to use mouse] 77.
Kelly:
No we did, we did … oh.
Hayden:
[Using mouse, looking at screen] Do we press that?
Teacher:
No, that’s the hundreds. You wanted ten, Amy. Make the number 77. 77.
Terry:
I know how to do that.
Hayden:
[Checks card, returns to using mouse]
Terry:
[Pointing to screen with pencil] Keep on doing it until it’s 7. No, put, do this
to help ya’.
Hayden:
What?
334
Terry:
You can do that and it’ll tell you what number you’re on to. [Display number
window, presumably] 5, 6, 7. Now you need 7 ones.
Hayden:
[Puts out too many one-blocks] Whoops! [Laughs]
Terry:
Seven! You went up to 8. Get the bomb out. [Hears Kelly say “Take one
away”] Oh, take one away! I forgot that!
Hayden:
[Laughs]
Terry:
[Laughs] Oh, man!
Hayden:
[To teacher] I made 77. Oh, I’ll have to check it [Starts to use mouse].
Terry:
I made … 78.
Computer:
77.
Terry:
Yep. I believe it’s 77.
Kelly:
[Watching screen as Amy works, giving vocal encouragement. Moves her
head as Amy does something on screen] It’s not working. 2 … I’ll count them.
2, 3, [shakes head] um 2, 3, 4, 5, 6, 7.
Amy:
[Intake of breath; apparently made a mistake]
Kelly:
No, just take one away. Er-er. No, not one of them. Now you’ve got six.
Amy:
No, er …
Kelly:
Now, put one of them [points at screen] back on.
Computer:
77.
Kelly:
[Moves hand toward mouse]
Amy:
Now, I’m pressing it.
Teacher:
Kelly, you can do the next one.
Terry:
[Looking at girls’ computer] Wow! They’ve got 78!
Hayden:
77.
Terry:
Hah. I thought it was 78.
Computer2:
77.
Teacher:
Now what’s the next thing you have to do?
Kelly:
— OK. [Starts to use mouse]
Teacher:
You remember how you can do that? Just a minute, Kelly. Stop. You can get
the computer to do it for you in one go.
Terry:
[Intake of breath] Oh yeah, [points to screen] the saw! The saw!
Computer:
[Saw sound]
335
Hayden:
[Looks at Terry and smiles]
Terry:
[Laughs] That’s easy! We’ve done it already! Now put 6 …
Amy:
[Pointing to screen] There, there, there.
Teacher:
I want you to swap one of the tens.
Kelly:
Oh, one.
Teacher:
OK, now you have to write down what you’ve done in your book.
Terry:
OK. [He and Hayden start to write] 6, 17.
Amy:
[Starts to use mouse straight away] I know how to …
Teacher:
[Starts to use mouse, with hand on top of Amy’s hand] You’re clicking in the
wrong places. Right. Try again? Now, how are you going to swap a ten for ten
ones?
Amy:
[Looking at screen] There.
Teacher:
No, that doesn’t help.
Terry:
[Watching girls’ computer] Nuh, it’s a saw. I bet it is. We already done it.
Hayden:
[Watches girls’ computer]
Amy:
Take …
Teacher:
No, no. Amy, click on the saw.
Amy:
[She does so]
Kelly:
[Nodding] Now, press it.
Computer:
[Sawing sound]
Amy:
Now press “OK.”
Teacher:
No, now you have to write something down in your book, to show what
you’ve done.
All:
[Write in books]
Kelly:
Mmmm, do you cut ten up? [Looks at teacher]
Terry:
[Writing in book, stops to talk to Hayden] I pressed the saw! [Laughs]
Hayden:
I pressed it! [Reads from workbook] We swapped the ten for a one.
Teacher:
[To Hayden] OK, now can you write down the numbers of blocks you’ve put
out?
Terry:
I already have: 6, 17.
Teacher:
What do you mean, “6, 17”?
Terry:
Uh, ooh, forgot to add it up!
336
Hayden:
[Laughs]
Terry:
Can we add it up?
Teacher:
Yes.
Hayden:
[Pointing to screen as he counts] 10, 20, 30, 40, 50, 60, … [Puts up fingers on
his left hand as he continues] 61, 62, 63, [Goes back to pointing to screen] 64,
65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77.
Terry:
[As Hayden gets to 70] Hey, no! Why didn’t you ask … [To teacher] There’s
an easier way to do it.
Hayden:
[Laughs] Oh, yeah. [Apparently realising that the computer will read the
number]
Terry:
No, [points to screen] there’s an easier way to do it.
Hayden:
[Starts to use mouse]
Computer1:
77.
Hayden:
[To Terry, with surprised look] 77!
Terry:
Oh! We’ve still got … Oh, cool, that’s easy! [Writes in workbook] Seventy …
77! [To teacher] How does it do that? It’s still got 77. [Teacher looks at him,
but does not respond] Oh yeah! [Understands; bangs himself on his head with
his hand]
Hayden:
[points to screen] It’s still … You cut it up, and it’s still 77! [Looks at Terry]
Terry:
Mmm. [Pencil in mouth, apparently thinking]
Amy:
Cut … cut …
Kelly:
We cut ten up [Indistinct]
Amy:
Blocks … with …
Kelly:
[Shows workbook to teacher]
Teacher:
OK, now I want you to write down how many blocks there are.
Kelly:
[Looking at screen] Um …
Amy:
[Quietly] Six …
Kelly:
I know! [Starts to use mouse]
Amy:
176.
Kelly:
Which one?
Amy:
It’s … [points to screen]
Kelly:
Oh … [Turns to teacher] Do you press that, and ask?
Computer:
77.
337
Kelly:
77. Do you write how much you have?
Amy:
[Puts book down] Now this is my go.
Teacher:
No, just a minute. We haven’t finished.
Amy:
I know that!
Terry:
Start again.
Hayden:
[Uses mouse to restart]
Computer1:
[Reveille]
Teacher:
No, I don’t want you to start again. I want to talk about this one a bit longer.
[Uses mouse to reset representation]
Hayden:
Whoops! [puts hand to mouth]
Teacher:
Alright. Girls, look up here at this one please. This is how it started off,
alright, with 7 tens and 7 ones. And you know that’s 77, don’t you?
Amy:
Yeah.
Kelly:
Yes.
Teacher:
When you cut one up …
Computer1:
[Sawing sound]
Hayden:
… it’s still 77 [turns to look at teacher].
Teacher:
It’s still 77.
Computer1:
77.
Teacher:
If you show the number here, it still says ‘77.’ Now, how can that still be 77?
Because we’ve only …
Amy:
‘Cos we’ve just cut the same ones up again. Just put the …
Teacher:
Mmm-mmm. So if we have 6 tens and 17 ones, can you see that that’s still 77?
Amy:
Yeah.
Hayden:
[Nods]
Kelly:
Yeah, bec…
Teacher:
I want you to write that down in your book, that you’ve got 6 tens and 17
ones.
Kelly:
I did.
Amy:
Sixty …
Kelly:
Write beside that? [Indistinct]
Teacher:
Write it underneath. Oh, it doesn’t matter. You can write it there.
338
Amy:
17 ones. [Puts down book] Finished.
Terry:
[Writing in workbook] 6 tens and 17 ones.
Teacher:
And what does that equal? Don’t keep putting your book down. You haven’t
finished, Amy. That equals how much?
Terry:
77.
Amy:
[Looking at what she has written] One hundred and … Oh no, 77!
Teacher:
Well write it down: “equals 77.”
Amy:
“e,” “q,” “q.”
Teacher:
Just an “equals” sign: two lines.
Amy:
[Writing] equals … equals 77.
Terry:
Whoops. Equals?
Teacher:
Equals. Now let’s see what you’ve got
Terry:
[Shows book to teacher] 6 tens, 17 ones.
Teacher:
Yeah, you’ll have to write “tens,” though, as a word. It looks funny if you just
write “6 t 17 ones.”
Kelly:
[Shows book to Amy] [Indistinct]
Amy:
Put them … [Indistinct] Oh, I took that as well.
Teacher:
OK, I asked you “Did it make sense?,” someone said “No,” before. [Boys are
looking as Terry writes in his book, and not really listening]
Amy:
It did.
Kelly:
Well, what do you mean by that?
Teacher:
Does that make sense to you, that that’s 77?
Amy:
Yes.
Kelly:
Well, yes, because it um, if you add them up together, it makes 77. ‘Cos 17,
16, …
(l/c S2, T 4a)
339
High/blocks:
John:
[Reads Task 4 card]. Huh? Don’t make sense.
Teacher:
OK. Do it one step at a time. “Show the number with the blocks.”
Simone:
[Starts to count ten-blocks. Moves a ten-block at same time as Amanda moves
7 blocks]
Amanda:
[Counts ten-blocks without counting them. Pushes Simone’s ten-block away,
counts out 7 one-blocks in a group, preventing Simone from contributing any.]
Simone:
[Puts hand on top of one-blocks, moves them from side to side]
Amanda:
77. [Stops and watches Craig.]
Simone:
[Watches Craig]
Craig:
[Starts to put out ten-blocks] 70. 1, 2, 3, 4, 5, 6.
John:
[Adds a ten-block to those already out, then another.]
Craig:
[indistinct] [Removes John’s second ten-block, starts adding one-blocks] 61,
62, 63, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, …
Amanda:
What are you doing, Craig?
Craig:
5, 6, 7, 8, 9, 10. [He had put out 6 tens and 20 ones altogether.]
John:
Mmm? That’s 80.
Craig:
No, 60 … 10, 20, 30, 40, 50, 60. 77.
Teacher:
You’re trying to do too much at once, Craig. Do it one step at a time. Show 77
with the blocks first.
Craig:
OK.
John:
[Starts to count one-blocks from table into hand] Oh [understands]. [Picks up
a ten-block, pushes some one-blocks away] Move all these.
Craig:
OK.
Teacher:
Girls, you can move onto the next part, if you’re ready. [see later part of
transcript]
John:
Put that there. [Counts ten-blocks, throws one away. Counts ten-blocks again there are 6. Starts to count one-blocks, loses count. Recounts, makes sure there
are 7 ones] 77.
Craig:
[Starts to remove a ten]
Teacher:
[Stops him, turns ten-blocks around so they are in vertical orientation] No, just
put those down. That’s 77.
John:
Yes.
340
Teacher:
Now do the second part - “Now swap one of the tens for ones.”
Craig:
[Makes silly noise with mouth, moves ten-block away, adds the one-blocks
that were previously removed - making 80 altogether again] OK.
John:
No. [Counts one-blocks carefully, until he has 17 in his hand, leaving 3 on the
table.] There. There’s 17 there [pointing to ones] 60 there [pointing to tens]
Teacher:
What about these 3 here? They’re extras? [Removes 3 ones]
John:
There’s 60 there [pointing to tens], and 17 there [pointing to ones].
Craig:
[Using silly voice, nodding] Yes. Yes.
John:
[Copying Craig] Yes.
Amanda:
[Moves some ten and one-blocks to leave 2 tens and 3 ones] 23 [sounds bored,
has head on hands]
Teacher:
No, no, no, we’re still on 77. Do this part - “Now swap one of the tens for
ones.”
Amanda:
Oh.
Simone:
[Starts to add 4 tens]
Amanda:
[Reaches across to move ones] No, put the 7s behind.
Simone:
[Starts to move a ten]
Amanda:
[Counts ten-blocks, takes ten out of Simone’s hand and puts it back. Removes
3 ones, counts out new 7 one-blocks under her breath. Adds them to 7 tens]
Simone:
[Picks up a ten, takes it away] [indistinct] … for ones. For 7 ones. [Adds 7
ones]
Teacher:
[Taps on card to remind girls to go on to next step]
Amanda:
[Counts tens and one-blocks] D’we have to put these [one-blocks] for ten?
Teacher:
No, no.
John:
Oh, well then we got it wrong.
Teacher:
Read what it says there: “How many ones do you need?” How many ones did
you need when you did the swap?
John:
17.
Craig:
17.
Amanda:
7.
Teacher:
How many ones did you swap the ten for?
Craig:
10.
Amanda:
7. She [Simone] swapped ‘em for 7.
341
Teacher:
Oh. [To Simone] Is that right?
Simone:
[Nods]
Amanda:
[Shakes head]
Craig:
[Quietly] 10. 10 for 10. 10, 10, 10. You swap it for 10.
Teacher:
You don’t think so? Why not?
Amanda:
You have to swap it for 10, ‘cos otherwise it’s not the same.
Teacher:
Is that right, Simone?
John:
Well, then it’d just be 17 … no, then it’d just be 70. You need 77. Is it?
[Looks at card] Yep.
Craig:
[Silly voice] 77.
Teacher:
Do you understand, Simone?
Simone:
[Nods]
Teacher:
When you swap a ten, you’ve got to swap it for 10. Do you do that all the
time, or can you swap it for other numbers?
Amanda:
No, we have to do it all the time.
Teacher:
What do you think, boys?
John:
Mmmm, I don’t know. [Shrugs]
Craig:
I don’t know neither. [Shrugs]
Amanda:
[Sure] Do it all the time.
Teacher:
You always swap it for ten?
Amanda:
[Nods]
Simone:
[Shakes head]
John:
I don’t.
Teacher:
What do you think, Simone?
Simone:
We can swap it for other numbers too.
Amanda:
[Shakes head]
Teacher:
Like what?
Simone:
Like um, you can swap it for 7s, and 9, and 10, and the other numbers.
Teacher:
Do you boys think that’s right?
Amanda:
[To Simone] No we don’t.
Craig:
[Looks unsure]
John:
[Looks unsure] Well, …
342
Teacher:
Do you agree with it? ‘Cos Amanda’s saying you have to swap it for 10,
Simone’s saying you can swap it for 7 or 8 or 9 or other numbers; what do you
think?
Amanda:
Uh-uh [Disagreeing].
Craig:
I agree with Amanda. [Looks at John]
John:
I agree with Amanda.
Teacher:
Why?
Amanda:
Otherwise it’s not the same.
Teacher:
The same as what?
Amanda:
As 10.
John:
Because if you swap it for 7, there’s 10 [picks up ten-block and places it down
on its own].
Teacher:
Why does it have to be the same?
Amanda:
Otherwise it won’t be the same number.
Teacher:
This is a good point, but why does it have to be the same? [Pause 3 seconds]
John:
No, I don’t get it. [Sits back in seat]
Amanda:
It has to be the same.
Teacher:
Let me just show you. If we have a 10 [puts down a ten-block], or we have 10
ones [puts a line of 10 ones, parallel with ten-block], we all agree that you can
swap that for that [moves hand over ten and ones in turn], don’t we? You can
swap it? When we’re doing some sort of maths, sometimes the teacher will
say “Righto, swap that for that,” OK. Craig, do you understand or not? You’re
looking a bit …
Craig:
Mmm, yeah, I understand a little bit.
Teacher:
You do? Alright. We know we can swap that for that [indicates ten and ten
ones with hands]. Now if I take, say, three of those away [removes 3 ones],
and just leave 7, can I swap that for those, 7?
All:
[Shake heads]
Amanda:
[Definite] No.
Craig:
No.
John:
No.
Teacher:
Why not? Simone?
Simone:
Um, because that 10’s more than those.
343
Teacher:
It is, isn’t it? It wouldn’t, you couldn’t make that a fair swap. I mean, if this
was money, and someone said I’ll give you a $10 bill [picks up ten-block] and
you can give me 7 $1 coins
Craig:
[Under breath, smiling] Cool.
Teacher:
You’d be silly to do it, wouldn’t you? Because you wouldn’t have as much.
You need, you must have the 10 ones, and we can put next to each other, and
then that’d be the same. So, are you happy with that now, Simone?
Simone:
[Nods]
Teacher:
So, it’s always a swap of ten for ten. Now you [to boys] did that just now,
didn’t you?
Craig:
[Nods] Yes.
Teacher:
So how many ones do you have now?
Amanda:
[Counts girls’ ones]
John:
[Straight away] 17.
Craig:
[Silly voice] 17. No, it’s only 6 … 17. [Looks at John]
Teacher:
Now stop being silly, Craig, and work out the answer, please.
Amanda:
17.
Craig:
17.
Teacher:
Are you just guessing, or … are you just copying John, or what?
John:
No, ‘cos we counted … [indistinct].
Teacher:
Is it really 17?
Craig:
[Counts blocks, looks at John] It’s 17.
Teacher:
OK, the last thing you have to do was stop after this one. This is what you
have to do: It says “Record what you have done in your workbook.” I want
you to write down what you had before we did the swap.
John:
Huh?
Amanda:
[Quietly] 77.
Craig:
Oh yeah, 77.
John:
77 [writes in workbook]
Teacher:
And then …
Amanda:
Do the next one.
Teacher:
… somehow I want you to come up with a way of doing it. I’m not going to
tell you how to do it. I want you to write down what we did to change it.
344
Craig:
[Looks at Amanda’s workbook a couple of times, changes what he has
written, looks at John]
John:
[Watches what Craig is writing]
Craig:
Mmmm … [puts hand on forehead] [indistinct] … change that? [writing in
workbook] … that … we changed it.
Teacher:
Well, how did you change it, Craig? It’s not enough just to say “We changed
it.” We want to know the numbers that you changed. Look up here. Girls, you
can look here too [indicates block representation for 77]. This is how we
started, we had 7 tens and 7 ones. And then you traded one of these for ten
ones and it turns out like that [indicates block representation showing 6 tens
and 17 ones below the first representation]: with only 6 tens, and 17 ones.
How can you write down what that change is?
Craig:
We changed the … no, no … [indistinct] … got it.
Teacher:
But what did you change it to, Craig?
Craig:
Mmm, I changed one ten … to …
John:
There [shows book to teacher] You supposed to do it like that?
Teacher:
That’s a good way of doing it.
John:
[Holds workbook up and shows it to others]
Teacher:
Tell the girls, ‘cos they won’t be able to read it.
John:
“6 tens and 17 ones.”
Teacher:
OK, can you write down 7 tens and 7 ones? ‘Cos that’s what you had the first
time. That’s right, isn’t it?
John:
Under here?
Teacher:
You can write it underneath, or you can write it over the top. The last question
I have to ask you before we must go back to your class, is: Are the two
amounts [indicates the two block representations for 77] the same?
Simone:
No.
Amanda:
No. Y … [Stops, seems unsure]
Craig:
No.
Simone:
No.
Teacher:
And I want you to discuss that with each other. I mean, you know what
number that is [7 tens and 7 ones]. Is that [6 tens and 17 ones] the same
number?
Simone:
No.
345
Amanda:
Yes.
John:
Yes.
Craig:
[Counts, nodding head]
Teacher:
… and how can you be sure? I want you four to talk about it.
Amanda:
[To others, definite] It’s the same.
Simone:
[Shakes head]
Craig:
[Still counting blocks]
Amanda:
… ‘cept for one.
Craig:
[Shakes head strongly] No.
Amanda:
Yeah, ‘cos those ones, just for ten. Still the same. Make ‘em for ten.
John:
[Looks carefully at both representations, points to both with pencil]
[indistinct] … they’re both the same.
Simone:
[Shakes head again]
Teacher:
Don’t talk to me, talk to each other. ‘Cos people are disagreeing.
Amanda:
[To Simone] They’re both the same.
Simone:
[Nods]
Teacher:
How can you prove they’re both the same?
John:
[Counts two sets of tens, touching with pencil. Looks puzzled]
Amanda:
Swap … ‘cos there’s ten, and you swap them for ten it’s still the same.
Craig:
[Counts ones under breath] … 10. 1, 2, 3, 4, 5, 6, 7. Oh yeah, they’re the
same.
Teacher:
[Separates 7 ones from others and tens] So what number is shown by these
blocks [6 tens and 17 ones]?
Craig:
Er …
Amanda:
77.
Craig:
… 77.
John:
70.
Amanda:
[To John] 77!
Craig:
[Whispers to John] 77.
John:
He said by these blocks [6 tens & 10 ones].
Teacher:
We all agree this [7 tens & 7 ones] is 77 … Oh, no, no. I mean those [7 ones]
as well. I mean those as well, John. Sorry. All of those.
346
John:
Oh. 77.
Teacher:
So we haven’t changed the number, have we?
Amanda:
No.
Teacher:
We still have 77. So 7 tens and 7 ones is really the same as 6 tens and 17 ones.
John:
[Nods] Yep.
(h/b S1, T 4a)
Low/blocks:
Michelle:
[Reads first part of task from card]
Teacher:
Let’s just do that part. Show the number with the blocks. The first one.
Clive:
[Pushes tens away]
Michelle:
[Whispering] 77. [Pushes some tens back to Clive] You get the tens, … [Starts
counting out ones]
Clive:
2, 4, um, I lost count [laughs]. 5, 6, 7, and 7 ones.
Michelle:
[Hiding ones in hand] I don’t have the ones!
Clive:
What’s in there?
Michelle:
[Laughs, adds ones, counts them] 1, 2, 3, 4, 5, 6, 8.
Clive:
8 [laughs].
Michelle:
[Laughs] We got 8!
Clive:
No, we got 7.
Michelle:
[laughs]
Nerida:
[Adds a ten, then adds ones] 2, 4, 6, … hang on. [Counts ones again] 1, 2, 3, 4,
5, 6, 7. [indistinct] … 77.
Teacher:
[To Nerida and Jeremy] You look like you’ve got too many over there.
Jeremy:
[Counts tens]
Nerida:
[Counts tens from opposite side from Jeremy, removes two of them]
Teacher:
OK.
Michelle:
Look what we done. We done 1, 2, 3, 4, …
Teacher:
OK. Next part. Alright, concentrate, ‘cos this is getting harder now. Are you
listening?
Clive:
[Pushes blocks away]
Teacher:
[To Clive] No, no, no, no, don’t put them away. We’re going to use those for
the next part.
Clive:
[Puts hands over face]
347
Teacher:
That part was easy. Now.
Nerida:
Do you write it down?
Teacher:
No, not yet. “Now swap one of the tens for ones.”
Michelle:
[indistinct] … one of these …?
Nerida:
[Picks up one of the tens] So you have to …
Jeremy:
I’ll get it. [Picks up a one, adds it to blocks] … for one. So that’s 6 … 66.
Nerida:
[Checks count, nods]
Jeremy:
66.
Michelle:
[Swaps a ten for a one]
Clive:
[making verbal noises]
Michelle:
[indistinct] ‘cept … [?]
Teacher:
[To Michelle] You had better concentrate, OK. You’re starting to be a bit
silly.
Clive:
I know [how to carry out instructions].
Michelle:
How could it be 66?
Teacher:
How could it be 66?
Clive:
[Counting tens] 2, 4, 6, …
Michelle:
Shh! [Stops Clive from counting]
Jeremy:
68.
Teacher:
68. [To Michelle and Clive] Have you got 68?
Michelle:
Yep.
Clive:
[Counting ones] 2, 4, 6, 8.
Teacher:
OK, well I’m going to have to ask you something. You’ve both done the same
thing, but you’re both wrong. [Removes a one, and puts back ten] Now,
there’s our 7 tens and 7 ones that we started with, and the instructions say
“Swap one of the tens for ones.”
Clive:
[Starts to push away blocks]
Teacher:
Now, let me do it, let me do it. We take one of the tens, and we’re going to do
a swap, for ones. Not for 1 one. [Holds up a one and a ten] Is that a fair swap?
Nerida:
[Shakes head]
Jeremy:
[Shakes head]
Michelle:
No. Oh, yeah. [Understands] [Puts a ten back with their blocks]
348
Nerida:
[Quietly picks up the one that was added earlier, starts counting ones to add to
it]
Teacher:
Oh. No, no, no. Don’t put that back. We’re gonna swap that one for ones. But
we want it to be a fair swap.
Clive:
Oh, ten ones.
Jeremy:
[To Nerida] Ten ones.
Nerida:
[Finishes counting, adds to representation]
Michelle:
[Counts ones, adds to the one previously used in the trade, then adds to
representation]
Clive:
[Pushes some ones toward Michelle’s collection, but she ignores them. When
she has got ten of her own, she pushes Clive’s blocks away again.]
Nerida:
[Pushes ten ones with others, counts them all. Then she counts the tens]
Jeremy:
My brother went to the sports.
Teacher:
Did he? Not now, we’re concentrating. [To Michelle and Clive] OK, don’t get
mixed up. Make sure you’re doing the right thing. Why did you say ten ones,
Clive?
Clive:
Um, because there’s one ten what all of them are glued, and there has to be …
Michelle:
20 there. [Puts hand on top of all ones in representation]
Clive:
… and another te … them ones for another ten.
Nerida:
[Finishes counting tens and ones] [Whispering, to Jeremy] 60, 17. [Looks at
Jeremy’s workbook] [Quietly, to teacher] 117.
Teacher:
OK. [To Nerida and Jeremy] Do you agree with Clive that you have to have
ten of these [pointing to ones] to make one of those [picks up ten]?
Nerida:
[Nods]
Teacher:
Jeremy?
Jeremy:
[Nods]
Clive:
Swap one of them.
Teacher:
When we do a swap, could you swap it for a different number? Could you
swap it for 8, or 7, or 9, or something?
Clive:
No.
Michelle:
No, you gotta swap ten.
Nerida:
It’s gotta be ten swap [indistinct].
349
Teacher:
… or 11, or 12? It’s got to be 10, hasn’t it? Always got to be 10. Next part.
Now this is where it gets tricky, ‘cos I want you to work out a way of doing it.
It says “Record what you have done in your workbook.”
Clive:
Record?
Teacher:
Write it down. Now I want you … Now let me put this out, because it’ll be
easier if I show you this. You started with that [puts out 7 tens and 7 ones
above Michelle and Clive’s representation], didn’t you?
Clive:
25.
Teacher:
You started … No, it wasn’t 25. 7 tens and 7 ones. Jeremy, pay attention. It’s
nearly time to go, I know. Just this last one. We started with that [puts hands
on top representation], and then you did a swap with one of the tens and we
finished with that [puts hands on second representation]. Now I want you to
write what you’ve done there, in your book. Somehow.
Clive:
Oh … Just …
Teacher:
You do it how you think, so it makes sense, to explain what you did.
Michelle:
So you gotta write it in words?
Teacher:
Not necessarily words. You could do numbers, ‘cos you’ve got 7, and there
are other numbers that you can write as numbers.
Michelle:
Oh, yeah.
Nerida:
Can I just … Can you write the answer?
Teacher:
Well, it’s not just the answer that I’m interested in. I want to know what
you’ve done, and what you’ve got in front of you.
Clive:
[Counting] … 60, [Counts tens, pointing with pencil] 1, 2, 4, 6, [Counts ones]
61, 62, 63, 64, 65, 66, 67, 68. [laughs] 68!
Michelle:
[Counts tens] 1, 2, 3, 4, 5, 6,…
Teacher:
Hang on, hang on. Write it over here.
Michelle:
Uh oh! Don’t worry, I …
Clive:
Now I lost me counting.
Michelle:
[Laughs] I just crossed that out.
Nerida:
[indistinct; about how to write answer in workbook]
Teacher:
OK, you can do that. That’s a good way of doing it.
Nerida:
[Writes in workbook]
Jeremy:
[Watches Nerida write]
350
Clive:
[Touching tens with pencil] 2 … twen … 2, 4, 6
Michelle:
6. [Starts to count ones. Moves Clive’s hand away, puts both hands on ones]
Clive:
[Waves pencil at Michelle in frustration]
Michelle:
[Laughing] Clive!
Teacher:
No, Clive wants to count them, and you keep moving them across.
Michelle:
[Counting under breath]
Clive:
[Watches Michelle for a bit, stops and puts hand under chin] I’ll just let the
girls do the work! [Laughs] [When Michelle finishes] What is it?
Michelle:
17.
Clive:
Wrong!
Nerida:
[indistinct] … to get 17. [Closes workbook] I done it.
Teacher:
Well, you better count them, Clive. Michelle’s saying it’s 17.
Clive:
2, 4, 6, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, … What
was that one?
Teacher:
76.
Clive:
76, 77! Huh.
Michelle:
Uh-uh. You’re supposed to count them [holds tens] in ones, and them [puts
hand over ones] in … ones.
Clive:
So we got double 77s. Mmm? That was tricky.
Teacher:
[To Nerida and Jeremy] What do you think? Have you got 77? ‘Cos Clive’s
saying that that [second representation] is 77, and that [first representation] is
77. Is that right?
Clive:
Mmm-mm [confirming]
Michelle:
[Writing in workbook, whispering] Oh! [Speaking aloud] Always get that
mixed up. Keep doing it. There - did it.
Teacher:
Does that make sense?
Nerida:
The top one’s 77 …
Teacher:
Right. And the Bottom one?
Nerida:
And the Bottom one … is …
Clive:
[Starts to recount blocks] 2, 4, 6, 61, 62. That’s 62 …
Teacher:
You just did that. We’ll have to cut it short, Clive, but you did count them all
before and you got 77, didn’t you?
Clive:
Mmm-mm.
351
Teacher:
Does that make sense? To have 77 again?
Nerida:
[Shakes head]
Clive:
Mmm-mm.
Michelle:
Mmm … Yeah.
Teacher:
After doing that swap? Look over here, and let me show you, ‘cos it is nearly
time to go, but I want you to see this before we finish. OK. If I put ten of the
ones together, like that [puts ten ones of second representation together in a
line], that looks like a ten again, doesn’t it?
Clive:
Mmmm
Teacher:
So that would look like our 7 tens and 7 ones. And we did a trade - one of
these for ten of those, and [jumbles up 17 ones] just push them all together.
Clive:
They broke.
Teacher:
Yeah, it’s like it got broken up. Now we’ve got 6 tens. Do you know how
many ones there are there? Without counting them?
Nerida:
17.
Michelle:
17 …
Teacher:
There are 17 - you counted them before, didn’t you?
Michelle:
Yep.
Nerida:
I never.
Teacher:
Well, is 6 tens and 17 ones the same as 7 tens and 7 ones?
Michelle:
No.
Nerida:
six hundred and 17
Teacher:
Does it make 77?
Michelle:
Yeah [not very confidently]
Nerida:
[Shaking head] No.
Michelle:
No!
Clive:
Yes.
Michelle:
No.
Clive:
Yes.
Michelle:
No!
Clive:
Mmm-mm.
Teacher:
Well, 10, 20, 30, 40, 50, 60. Six tens are 60, aren’t they? 61, 62, 63, 64, 65,
66, 67, 68, 69, 70. There’s our other ten that we used to have [pushes ten ones
352
into a line like a ten-block], but they’re now ten ones. 71, 72, 73, 74, 75, 76,
77.
Michelle:
Oh yeah, that’s right. [Looks at Nerida and Jeremy]
Clive:
See. Told ya. I’m a genius. [laughs] I came out of a lamp.
353
(l/b S1, T 4a)
Appendix R – Transcript Excerpts Showing Participants
Predicting Equivalence of Traded Blocks.
Task 31 (a) Show the number 255 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Teacher:
What will the number [255] be after one of the tens is regrouped?
Hayden:
555 still.
Terry:
255.
Teacher:
You mean 255?
Hayden:
Yeah, 255.
Teacher:
How do you know it will be the same number?
Hayden:
Because, if you cut up a ten, it’ll, um … yeah, cut up a ten, and you swap it for
a one, it’ll still be 255.
Teacher:
— Are you sure that the blocks still show 255?
Terry:
[Confidently] Yeah, because that’s [points to the ones] still a 10, and that’s
15, 5.
(l/c S10, T 31a)
Task 5 (a)
Show the number 21 with the blocks. Now swap all of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Teacher:
You’ve shown the number with the blocks, now do the next part. Amy, Amy,
look at it: “Now swap all of the tens for ones.” You haven’t done that.
Terry:
Oooh!
Amy:
One tens …
Kelly:
[Starts to use mouse]
Terry:
I know how to do that.
Amy:
Start again.
Terry:
So we’ve got … we’ve still got something, I know.
Amy:
21 …
Teacher:
Terry, read your instructions. You haven’t done it yet. “Now swap all of the
tens for ones.”
Terry:
Oh, all of the tens.
Kelly:
And we still have 21. So do we write that down?
355
Teacher:
Write down what you’ve done in your book.
Terry:
Now we’re starting to get into the heart of the stuff that I liked. … I still know
what it is, ‘cos you just told us. Now let’s check, that it’s right.
Amy:
21 … 1, 2.
Computer:
21.
Terry:
It disappeared!
Teacher:
Now you have to write in your book what you’ve done.
Terry:
Right.
Amy:
OK. [Starts to use mouse] Start again?
Terry:
I already have. Oh, no.
Amy:
Oh, I haven’t. It’s …
Kelly:
5 … task …
Terry:
chopped …
Kelly:
I wrote what I did.
Teacher:
Have you written down what you’ve got there on the screen now, that equals
21?
Terry:
… ten …
Kelly:
Um, I’ve got …
Terry:
I did something very easy. “I chopped the tens up.” Is that all you really have
to do?
Kelly:
“I got 21 ones. I cut 2 tens up, and I still got 21.”
Teacher:
Did you write down what you’ve got on the screen now?
Amy:
[To Kelly] Hey, [Indistinct] …like that, Kelly.
Terry:
Yep. [Reads from book] 2 tens, 1 one. 21.
Teacher:
[Points to screen] That’s not 2 tens and 1 one, though.
Terry:
What is it?
Teacher:
There aren’t any tens at all now.
Terry:
Oh yeah! [understands] I get you now.
Amy:
I cut … cut …
Teacher:
Write it on the next line.
Kelly:
Like this, Mr Price? [Shows her book]
Amy:
Cut … cut …
356
Teacher:
Mm-mmm. I’d like you to write down “21 equals,” and then write down how
many you’ve got there [points at screen].
Kelly:
Could I do it on the next one, ‘cos I haven’t any …
Teacher:
Of course. No, put “equals,” Terry. “Equals.” Now what have you got [points
at screen]? How many tens and how many ones?
Terry:
21!
Teacher:
21 what?
Terry:
21 ones.
Teacher:
That’s what you write down.
Kelly:
21 equals …
Amy:
I still have … have … [She writes in her book I kute the bloes I still have 21.
(I cut the blocks. I still have 21.)]
(l/c S3, T 5a)
Task 5 (b)
Show the number 36 with the blocks. Now swap all of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Kelly:
[Writes briefly in book. Starts to use mouse] 36 …
Terry:
Do we chop the other one up too?
Teacher:
It says “all of the tens,” doesn’t it?
Computer:
36.
Kelly:
Now I gotta chop up … Now I gotta chop ‘em up.
Amy:
Um, 36.
Kelly:
Tut.
Teacher:
You have to get the “saw” again, Kelly.
Kelly:
OK.
Amy:
[Whispering] 36 … equals … I like the saw the best. Oh, the bomb is the best!
[Watching Kelly] Do it. Oh no, the bomb takes away.
Kelly:
Whoa! Now that …
Amy:
It’s still 36, see. [points at screen]
Kelly:
Watch out! Must’ve pressed something wrong.
Computer:
36.
Kelly:
Yep.
Amy:
There we can use your rubber.
Kelly:
36. 36. 36 is …
357
Teacher:
Mmm, I see what you were thinking of the first time.
Amy:
[Writes in her workbook] We … split … blocks … up … We ended … ended
… ended with … thirty, 36 … again. [Writes: we cat the bloes pu we end with
36 one a gen.]
Terry:
split … the blocks … I split the blocks up … [Writes: 36 = 36 one. I splet the
blos up an I hed 36 one]
Kelly:
… cut the … blocks up … 36. [Writes: i cut 3 tens up and i till got 36] I fitted
all mine on.
Teacher:
How are you going? All finished? 36 what, Amy?
Amy:
36 ones. Ones.
Teacher:
Mm-mmm. So 36 can be 3 tens and 6 ones, or it could be 36 ones, couldn’t it?
Amy:
Yep.
Task 6 (a)
Show the number 64 with the blocks. If you were to swap all the tens
for ones, how many ones would there be? Write your answer in your
workbook.
Teacher:
Now, it says “If you were to swap all the tens for ones, how many ones would
(l/c S3, T 5b)
there be?” Now I don’t want you to swap them, but if we did, how many ones
would there be? All of the tens.
Amy:
Still the same. Um, it’d be 64 again! — Because, if you have the sa… ‘cos it’s
6 tens and 4 ones, and if you chopped all them up with the saw, it’d still be the
same, but they’d all be ones!
(l/c S3, T 6a)
Task 6 (b)
Show the number 89 with the blocks. If you were to swap all the tens
for ones, how many ones would there be? Write your answer in your
workbook.
Amy:
[Writing] 89 … [looks at screen] … 89 …
Kelly:
Equals … Yep, we’ve still got 89.
Amy:
… ones.
Kelly:
Yeah.
Amy:
[writing] 89 ones equals … [Writes: 89 = 89 ones]
358
(l/c S3, T 6b)
Task 24
Show the number 40 with the blocks. Put out another ten, and say the
number’s name. Keep adding tens. Stop when you have 10 tens. What
is this number called? Can you trade 10 tens? Keep adding tens. Stop
when you reach two hundred. Write the numbers you made in your
workbook.
Terry:
[To teacher] It’s a ten, so we can take it away and put a ten, hundred! [Terry
seemed to realise straight away that 10 tens could be treated the same as 10
ones, and regrouped for a larger block.]
Teacher:
[Reads next part of task:] “What is the number called [all participants say it is
100]; Can you trade 10 tens?”
Terry:
Yes!
Kelly:
Yes, but you’ll end up with none left.
Terry:
[To Kelly] Yes, you’ll end up with a hundred still!
(l/c S8, T 24)
Task 31 (a) Show the number 255 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Teacher:
[Talks the girls through the regrouping process.] How many tens and ones will
there be after the trade? Amy, how do you know it will be the same?
Amy:
You’ve only cut up one ten.
Kelly:
And it’s still the same.
(l/c S10, T 31a)
Task 31 (b) Show the number 932 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Kelly:
[Puts out the blocks to represent 932.]
Amy:
Now cut one up. Cut that one there.
Kelly:
It will be 12 there, wouldn’t it?
Amy:
[Points to the screen] It’ll still be the same number, though.
Kelly:
Yeah, it’ll be 12. Ten plus 2 is 12. [Uses the mouse to cut up a ten-block.]
(l/c S10, T 31b)
Task 31 (c) Show the number 314 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Teacher:
How many blocks will be in each column after the trade?
Amy:
There will be, 3 hundred, and 14 …
Kelly:
There’d be 14 ones, and zero tens, and that will equal 314, still. (l/c S10, T 31c)
359
Task 32 (a) Show the number 340 with the blocks. Now swap one of the hundreds
for tens. How many tens do you need? Record what you have done in
your workbook.
Teacher:
Stops Hayden, asks the boys to predict what the number of hundreds, tens and
ones will be after the trade.
Terry:
314.
Teacher:
How many blocks will there be in each column after the trade?
Hayden:
3 hundreds, and 1 ten, and 4 ones.
Terry:
14, 14… [indistinct] …
Hayden:
3 hundreds, no tens,… and, um …
Terry:
There still will be tens!
Teacher:
Are you sure?
Hayden:
There will be 14.
Teacher:
You both agree that there will be 14 ones, but how many tens will there be?
Hayden:
None.
Terry:
w… [he was apparently going to say “1”] none … no, 1, 1..
Teacher:
Will there? How many tens there will be a after the trade; 1, or 0?
Terry:
[Pointing to the screen] There still will be ten, ‘cos, but it will only be cut up
into here [ones column].
Teacher:
But how many tens blocks will there be?
Terry:
None.
Teacher:
[Asks the girls the same question, about how many blocks will be in each
column after the trade.]
Amy:
There will be, 3 hundred, and 14….
Kelly:
There’d be 14 ones, and zero tens, and that will equal 314, still. (l/c S10, T 31c)
Task 32 (a) Show the number 340 with the blocks. Now swap one of the hundreds
for tens. How many tens do you need? Record what you have done in
your workbook.
Terry:
[Uses the mouse to show the number 340 with the blocks. He has the
computer read the number.]
Computer:
340.
Terry:
340. Oh, I got a odd number. I did an odd number. [After a while …] 340, if
we cut one up there’ll still be 340!
360
Hayden:
Yeah!
Kelly:
Mr Price, d’you want me to cut up one of the hundreds?
Teacher:
Not yet.
Hayden:
[Uses the mouse to swap one of the tens for ten ones.]
Teacher:
Terry, stop. The card says to swap a hundred for tens.
Terry:
Oh! I didn’t do it.
Teacher:
How many tens will be swapped for 1 hundred?
Kelly:
100…
Amy:
Still the same number. You’ll get 3 hundred and for … 340.
(l/c S10, T 32a)
Task 32 (b) Show the number 627 with the blocks. Now swap one of the hundreds
for tens. How many tens do you need? Record what you have done in
your workbook.
Hayden:
[Looking at the screen] So, there will be … 12 tens?.
Teacher:
Terry, do you agree with Hayden?
Terry:
[Looks at the screen, apparently thinking.] 12 tens, if you cut one up … Yep! I
agree!
Teacher:
And how many hundreds will there be?
Hayden:
5.
Teacher:
And how many ones will there be?
Hayden:
7.
Terry:
7.
Hayden:
No, there’ll still be 6, um, hundreds, because there will be a hundred in the
tens [points at screen].
Teacher:
OK, but in the actual hundreds column …
Hayden:
Yeah, there will be 5.
Task 14
The Sunny Surfboard Company has 75 boogie boards left. If one is
sold, how many are left? Then how many if another is sold? Say all
the numbers in order from 75 back to 60. Show the numbers with the
blocks. Write them in your workbook.
Teacher:
Well, you’ve gone back to 70, now take away another one.
Belinda:
You can’t.
Teacher:
If you think about it, there’s a way to do it.
Daniel:
There is a way. Just can’t work it out. 70, …
(l/c S10, T 32b)
361
Belinda:
Put another one in! Can’t! Regroup! No, can’t regroup. Depose! Decompose.
‘Cos then, you’d have one, and you can take away. [Nods to Rory] Do it!
Rory:
But it’ll be, it’ll still be 7.
Belinda:
No it won’t. [Puts hand on top of Rory’s to use mouse, but he keeps using it.]
Rory:
Yes, it will. [Nods]
Belinda:
Here, I’ll show ya’. [Puts hand to left of mouse as if to take over, but again
Rory keeps control of it.]
Rory:
Look, it’ll still be 7. [Regroups a ten]
Belinda:
Now saw one. Now you can take away some, more. [points at screen]
Rory:
What do you mean?
Belinda:
[Puts hand by screen to stop others from seeing what they are doing.] Here, let
me do it. [Removes Rory’s hand, starts to use mouse] Now, you can take
away. [She takes away ones until 60 is left]
(h/c S4, T 14)
Task 24
Show the number 40 with the blocks. Put out another ten, and say the
number’s name. Keep adding tens. Stop when you have 10 tens. What
is this number called? Can you trade 10 tens? Keep adding tens. Stop
when you reach two hundred. Write the numbers you made in your
workbook.
Teacher:
[Reads question] Can you trade 10 tens? [Belinda and Rory nod, say that they
can.]
Belinda:
You could decompose one … that’s trading.
Teacher:
Mmm … Does that help?
Belinda:
Yeah. [Uses mouse to cut a ten into 10 ones.] It’s still a hundred. (h/c S7, T 24)
Task 31 (a) Show the number 255 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Belinda:
[Shows the number 255 without hesitation. Uses saw to cut up a 10, shows
number window] It’s still 2 … yep, I thought so.
(h/c S8, T 31a)
Task 31 (b) Show the number 932 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
All participants: [Show the number with the blocks.]
Teacher:
Can you say what the number of blocks will be after regrouping? Write in
your books.
362
Belinda:
Easy - You’ll still have 9 hundreds, but you’ll have 2 … tens, and … whoa! …
[Finishes writing, puts book down] I bet I’m right. I know everything like that
… I don’t know - it’s easy.
(h/c S8, T 31b)
363
Appendix S – Transcript of Task 4 (d)
from Low/Blocks Group
Task 4 (d)
Show the number 58 with the blocks. Now swap one of the tens for
ones. How many ones do you need? Record what you have done in
your workbook.
Clive:
[Counts ten-blocks] 2, 4, 5. [Puts tens down] 8! 58, 58. [Counts out ones] 2, 4,
6, 8. [Does a little “victory” gesture with arms. Writes in workbook] 58 equals
5 tens and 8 ones. I am a genie [genius]!
Teacher:
[Laughs] OK, do this. This [indicating blocks Clive has put out] is 58 now,
Jeremy. And Clive’s just doing the swap.
Clive:
[Swaps ten for ones, counts them in his hand] 2, 4, 6, whoops, 8, 10. [Puts
them on table] Now that means … [Counts ones] 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58. 58, again. [Pauses, blows air with
finger in mouth, looks briefly toward Nerida’s book, taps pencil, smiles,
pauses.] I need some help.
Teacher:
You’ve done 5 tens and 8 ones, which you’ve got to write down, Jeremy. How
many tens and ones do you have now, Clive?
Clive:
Ah, ooh. That’s what I missed. [Starts to count one-blocks]
Teacher:
Write the tens down first. You know how many tens there are.
Clive:
[Writes in book] 58 equals 4 tens and … [counts ones] 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17! 17 ones.
Nerida & Michelle
Nerida:
[Looks at card] 58. [Removes some tens, counts out ones to show 5 tens and 8
ones]
Michelle:
Finished!
Teacher:
Do the swap now, please, Michelle.
Michelle:
[Looking at book] We’ll have 4.
Nerida:
[Picks up several blocks, puts down ones and picks up a ten] One ten …
Michelle:
[Picks up some ones]
Nerida:
[Puts out two hands side by side] Put them in my hand.
Michelle:
Hang on! [Takes ones away and counts them] 5. There’s 5. [She starts to add
them to other blocks, Nerida puts her hand under them to take them in her
hand. Michelle starts to count ones on table
365
Nerida:
[Stops her by putting her hand on top of them] We’ll lose count!
Michelle:
[Counts on from 8 under Nerida’s hand] 9, 10, 11, 12, 13.
Nerida:
[Looks dissatisfied with this, keeps 8 ones separate and counts 5 added ones.
Then she continues to count as she adds more ones to make up to ten]
Michelle:
[Starts to count with Nerida, then sits back and folds her arms] Finished!
Nerida:
[Carefully re-counts added ten-blocks, then starts with original 8 ones, and
counts all ones to reach 18. She writes in her book]
Michelle:
[Watches what Nerida writes, then writes in her own book, then looks at
Nerida’s again. Nerida looks at her]
Nerida:
[Quietly] 4 tens and 18 ones.
Michelle:
[Writes in her book]
Teacher:
The boys and girls have two different answers again. Clive? You have
different answers again.
Nerida:
[Smiles at boys]
Michelle:
We have 18. [Laughs]
Clive:
Youse are wrong.
Michelle:
No, we’re right!
Teacher:
Well, explain it.
Nerida:
We put out …
Michelle:
[Touching ten-blocks] Five …
Nerida:
We had 5 …
Michelle:
5 tens.
Nerida:
… tens and 8 ones, and then …
Michelle:
We traded it for …
Nerida:
… for ten ones and we kept our 8 ones already there.
Teacher:
And would that make 18, or would that make 17?
Nerida:
18.
Clive:
[With arms folded; in the previous dialogue of the girls, he has not been
showing agreement with what they said, or any apparent willingness to listen]
17.
Michelle:
18. Look [starts to count blocks, starting with tens] 5, [continues with oneblocks] 6, …
Nerida:
[Stops her; touching tens] 4 …
366
Teacher:
You had 8 to start with, Clive. Hang on, girls. Can we do it without counting?
Can you work it out, and say what’s sensible? If you had 8 to start with, and
then you swapped and had another ten, what number would that make, without
counting?
Clive:
17.
Teacher:
Ten and 8?
Nerida:
[Shakes head] 18.
Clive:
18, I think. Think.
Michelle:
18.
Teacher:
What’s ten and another 8?
Michelle:
[Counts sub-vocally; smiles] 18!
Clive:
… 18.
Teacher:
It is 18, isn’t it?
Nerida:
Clive, Clive!
Clive:
Doh!
All:
[Laugh]
Teacher:
It’s easy to miscount one, it’s very easy.
Clive:
[Changes answer in workbook] No, it isn’t, it’s hard!
Nerida:
That’s why I do my counting twice.
(l/b S7, T 19)
367
Appendix T – Comparison Between Ross’s (1989) Model and a
Proposed Model for Categories of Responses to Digit
Correspondence Tasks
Four-Category Model of Responses to
Digit Correspondence Tasks
Category I: Face-value interpretation of
digits.
Category I thinking was evidenced by a
participant’s statements that each digit
represented only its face value, and that
remaining objects in the set represented
by the two-digit symbol as a whole were
not represented by either digit.
Five-Stage Model of Children’s
Interpretations of Two-Digit Numeralsa
“Stage 3: face value
Students interpret each digit as
representing the number indicated by its
face value. The set of objects represented
by the tens digit, however, may be
different from the objects represented by
the ones digit. They may verbally label
as “tens” the objects that correspond to
the tens digit, but these objects do not
truly represent groups of ten units to
students in stage 3: students do not
recognize that the number represented by
the tens digit is a multiple of ten.”
“Stage 2: positional property
Pupils know that in a two-digit numeral
the digit on the right is in the ‘ones
place’ and the digit on the left is in the
‘tens place.’ Their knowledge of the
individual digits is limited, however, to
the position of the digits and does not
encompass the quantities to which each
corresponds.”
“Stage 1: whole numeral
Category II: No referents for individual
As pupils in our culture construct their
digits.
knowledge about quantities up to ninetyCategory II responses indicated that a
participant accepted the two-digit symbol nine and their symbolic representation as
two-digit numerals, their cognitive
as representing the entire set of objects,
construction of the whole comes first—
but rejected the idea that each digit has
separate referents, on the basis that some the numeral 52 represents the whole
amount. They assign no meaning to the
objects would be left out.
individual digits.”
“Stage 4: construction zone
Students know that the left digit in a twodigit numeral represents sets of ten
objects and that the right digit represents
the remaining single objects, but this
knowledge is tentative and is
characterized by unreliable task
performances.”
369
Four-Category Model of Responses to
Digit Correspondence Tasks
Category III: Correct referents for digits,
tens not explained.
A Category III response is one in which
the participant knew that the tens digit
represented the remaining objects, once
the referents for the ones digit were
removed, but could not explain why that
digit represented a number of objects
larger than its face value.
Category IV: Correct referents for digits,
tens place explicitly mentioned.
Category IV includes responses stating a
correct number of objects for each digit,
explaining that the tens digit represents
the number of groups of ten.
Five-Stage Model of Children’s
Interpretations of Two-Digit Numeralsa
“Stage 5: understanding
Students know that the individual digits
in a two-digit numeral represent a
partitioning of the whole quantity into a
tens part and a ones part. The quantity of
objects corresponding to each digit can
be determined even for collections that
have been partitioned in nonstandard
ways.”
Note. aRoss’s stage descriptions that are judged to be equivalent are placed adjacent to this author’s
category descriptions (section 4.5). From S. H. Ross, 1989, Parts, wholes and place value: A
developmental view. Arithmetic Teacher, 36, p. 49.
370
Appendix U – Sample Coding of Transcript for Feedback
Note that incidents of feedback, their presumed effects, and the responses of
recipients are noted in bold type inside square brackets.
Clive:
[Puts out 7 tens & 5 ones, adds a ten, then counts on from 85.] 85, 86, 87, 88,
89, … [He gets stuck at 89, apparently not knowing the next number.] … 100,
101, 102, 103, 104. [Count blocks/Provide answer]
Nerida:
You’re wrong. [Peer feedback/contradict answer] [She counts blocks again,
getting 94.] [Count blocks/Provide answer]
Clive:
[Does not listen to or watch Nerida as she counts.] [Reject feedback]
Jeremy:
[While Michelle reads, reaches over her arm to show 75.]
Michelle:
75, and 10 more, makes … [Picks up tens, counts in tens to 100. She puts a ten
back straight away.] No, don’t need 100.
Teacher:
Look at the card again. [Teacher feedback/Ask a question]
Michelle:
Oh, I’m wrong. [Puts tens back] [Change answer] [She keeps a ten, counts on
9 ones. She re-counts the ones by two, removes one or two ones. She puts the
tens and ones together, counts tens, and starts to count ones from “81.”]
[Count blocks/Provide answer]
Jeremy:
[Counts blocks from 91.] [Count blocks/Provide answer]
Michelle:
Hang on, Jeremy. [She re-counts tens, continues with ones to 94.]
Jeremy:
[Not satisfied] You missed some blocks. [Peer feedback/Contradict answer]
Michelle:
Clive, it’s 94. [Peer feedback/Contradict answer]
Teacher:
Have you all got the same answer? [Teacher feedback/Ask question]
Clive:
[Re-counts blocks, changing to “100” after “89,” getting “103.”] [Count
blocks/Provide answer]
Michelle:
[Laughs.]
Clive:
I can count straight, but you can’t! [Peer feedback/Contradict answer]
Nerida:
[Begins re-counting the blocks. Again Clive does not watch her.]
Teacher:
Clive and Nerida, count them together. [Teacher feedback/Give directions]
Clive & Nerida: [They do so, arguing after “89” about whether it is “90” or “100.”] [Peer
feedback/Contradict answer]
Teacher:
Is it 90, or 100? [Teacher feedback/Ask question]
Michelle:
90. [Peer feedback/Contradict answer]
371
Clive:
100. [Peer feedback/Contradict answer]
Teacher:
What number comes after 89? [Teacher feedback/Ask question] [Clive and
Michelle repeat their respective answers.] Do you have 9 tens, or 100, which
is 10 tens?
Clive:
8.
Teacher:
Make sure you include the ones. Is there an easier way to show the number,
since there are so many ones? [Teacher feedback/Ask question]
Clive:
You can swap for ten.
Nerida & Michelle: [Do so.]
Teacher:
Nerida, don’t swap all the ones, just 10 of them. [Teacher feedback/Give
Directions] OK, re-check how many there are. [They do so, and Clive agrees
that it is 94.] [Change answer]
372
References
Akpinar, Y., & Hartley, J. R. (1996). Designing interactive learning environments. Journal
of Computer Assisted Learning, 12, 33-46.
Alexander, P. A., White, C. S., Haensly, P. A., & Crimmins-Jeanes, M. (1987). Training in
analogical reasoning. American Educational Research Journal, 24, 387-404.
Alexander, P. A., Willson, V. L., White, C. S., Fuqua, J. D., Clark, G. D., Wilson, A. F., &
Kulikowich, J. M. (1989). Development of analogical reasoning in 4- and 5-year-old
children. Cognitive Development, 4, 65-88.
Alexander, P. A., Wilson, A. F., White, C. S., Willson, V. L., Tallent, M. K., & Shutes, R. E.
(1987). Effects of teacher training on children’s analogical reasoning performance.
Teaching and Teacher Education, 3, 275-285.
Australian Association of Mathematics Teachers. (1996). Statement on the use of calculators
and computers for mathematics in Australian schools. Adelaide, South Australia:
Author.
Australian Education Council. (1990). A national statement on mathematics for Australian
schools. Carlton, Victoria: Curriculum Corporation.
Babbitt, B. C., & Usnick, V. (1993). Hypermedia: A vehicle for connections. Arithmetic
Teacher, 40, 430-432.
Baggett, P., & Ehrenfeucht, A. (1992). What should be the role of calculators and computers
in mathematics education? Journal of Mathematical Behaviour, 11, 61-72.
Ball, S. (1988). Computers, concrete materials and teaching fractions. School Science and
Mathematics, 88, 470-475.
Baroody, A. J. (1989). Manipulatives don’t come with guarantees. Arithmetic Teacher,
37(2), 4-5.
Baroody, A. J. (1990). How and when should place-value concepts and skills be taught?
Journal for Research in Mathematics Education, 21, 281-286.
Baturo, A. R. (1998). Year 6 students' cognitive structures and mechanisms for processing
tenths and hundredths. Unpublished doctoral thesis, Queensland University of
Technology, Brisbane, Australia.
Becker, D. A., & Dwyer, M. M. (1994). Using hypermedia to provide learner control.
Journal of Educational Multimedia and Hypermedia, 3, 155-172.
Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school.
Educational Studies in Mathematics, 13, 33-57.
Bednarz, N., & Janvier, B. (1988). A constructivist approach to numeration in primary
school: Results of a three year intervention with the same group of children.
Educational Studies in Mathematics, 19, 299-331.
Beilin, H. (1984). Cognitive theory and mathematical cognition: Geometry and space. In B.
Gholson & T. L. Rosenthal (Eds.), Applications of cognitive-developmental theory
(pp. 49-93). Orlando, FL: Academic Press.
Bell, G. (1990). Language and counting: Some recent results. Mathematics Education
Research Journal, 2(1), 1-14.
Best, J. W., & Kahn, J. V. (1993). Research in education (7th ed.). Needham Heights, MA:
Allyn & Bacon.
Bisanz, J., Bisanz, G. L., & LeFevre, J. (1984). Interpretation of instructions: A source of
individual differences in analogical reasoning. Intelligence, 8, 161-177.
Bobis, J. F. (1991). Using a calculator to develop number sense. Arithmetic Teacher, 38(5),
42-45.
373
Booker, G., Briggs, J., Davey, G., & Nisbet, S. (1997). Teaching primary mathematics (2nd
ed.). Melbourne, Victoria: Longman.
Bottino, R. M., Chiappini, G., & Ferrari, P. L. (1994). A hypermedia system for interactive
problem solving in arithmetic. Journal of Educational Multimedia and Hypermedia,
3, 307-326.
Boulton-Lewis, G. M. (1993). An analysis of the relation between sequence counting and
knowledge of place value in the early years of school. Mathematics Education
Research Journal, 5(2), 94-106.
Boulton-Lewis, G., & Halford, G. (1992). The processing loads of young children’s and
teachers’ representations of place value and implications for teaching. Mathematics
Education Research Journal, 4(1), 1-23.
Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of
learning. Educational Researcher, 18(1), 32-42.
Bruner, J. S. (1966). Toward a theory of instruction. New York: Norton.
Burns, R. B. (1990). Introduction to research methods in education. Melbourne: Longman
Cheshire.
Burstall, C. (1978). The Matthew effect in the classroom. Educational Research, 21(1), 1925.
Carpenter, T. P., Fennema, E., & Franke, M. L. (1993). Cognitively guided instruction:
Base-ten number concepts and multidigit operations [Draft paper]. Unpublished
manuscript.
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1997). A
longitudinal study of invention and understanding in children’s multidigit addition
and subtraction. Journal for Research in Mathematics Education, 29, 3-20.
Champagne, A. B. & Rogalska-Saz, J. (1984). Computer-based numeration instruction. In V.
P. Hansen & M. J. Zweng (Eds.), Computers in mathematics education: 1984
yearbook (pp. 43-53). Reston, VA: NCTM.
Chinnappan, M., & English, L. (1995). Students’ mental models and schema activation
during geometric problem solving. In B. Atweh & S. Flavel (Eds.), MERGA 18:
Galtha. Proceedings of the 18th Annual Conference of the Mathematics Education
Research Group of Australasia (pp. 156-162). Mathematics Education Research
Group of Australasia.
Clark, R. E. (1994). Media and method. Educational Technology Research and
Development, 42(3), 7-10.
Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching
Children Mathematics, 2, 270-279.
Cobb, P. (1995). Cultural tools and mathematical learning: A case study. Journal for
Research in Mathematics Education, 26, 362-385.
Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder.
Journal for Research in Mathematics Education, 14(2), 83-94.
Cobb, P., & Wheatley, G. (1988). Children’s initial understandings of ten. Focus on
Learning Problems in Mathematics, 10(3), 1-28.
Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in
the context of developmental research. Educational Psychologist, 31, 175-190.
Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational
view of mind in mathematics education. Journal for Research in Mathematics
Education, 23, 2-33.
Cockcroft, W. H. (Chairman) (1982). Mathematics counts: Report of the committee of
inquiry into the teaching of mathematics in schools. London: HMSO.
374
Cohen, S., Chechile, R., Smith, G., Tsai, F., & Burns, G. (1994). A method for evaluating the
effectiveness of educational software. Behavior Research Methods, Instruments, and
Computers, 26, 236-241.
Confrey, J. & Lachance, A. (2000). Transformative teaching experiments through
conjecture-driven research design. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of
research design in mathematics and science education. Mahwah, NJ: Erlbaum.
Creswell, J. W. (1994). Research design: Qualitative and quantitative approaches. Thousand
Oaks, CA: Sage.
Cruikshank, D. E., & Sheffield, L. J. (1992). Teaching and learning elementary and middle
school mathematics (2nd ed.). New York: Macmillan.
Davis, R. B. (1992). Understanding “understanding”. Journal of Mathematical Behaviour,
11, 225-241.
De Laurentiis, E. C. (1993). How to recognize excellent educational software. New York:
Lifelong Software, Inc. (ERIC Document Reproduction Service No. ED 355 932)
Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson Education.
Duffin, J. (1991). Mathematics for the nineties: A calculator-aware number curriculum.
Mathematics Teaching, 134, 56-62.
Edwards, L. D. (1995). The design and analysis of a mathematical microworld. Journal for
Educational Computing Research, 12, 77-94.
English, L. D. (1993). Children’s strategies in solving two- and three-dimensional
combinatorial problems. Journal for Research in Mathematics Education, 24(3),
255-273.
English, L. D. (1997). Children’s reasoning processes in classifying and solving
computational word problems. In L. D. English (Ed.), Mathematical reasoning:
Analogies, metaphors, and images (pp. 191-220). Hillsdale, NJ: Erlbaum.
English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes.
Mahwah, NJ: Erlbaum.
Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in
solving verbal problems in multiplication and division. Journal for Research in
Mathematics Education, 16, 3-17.
Fletcher-Flinn, C. M., & Gravatt, B. (1995). The efficacy of computer assisted instruction
(CAI): A meta-analysis. Journal for Educational Computing Research, 12, 219-242.
Fox, M. E. (1988). A report on studies of motivation teaching and small group interaction
with special reference to computers and to the teaching and learning of arithmetic
(CITE Report No. 31). Bletchley, Bucks, England: Open University. (ERIC
Document Reproduction Service No. ED 327 385)
Fraenkel, J. R., & Wallen, N. E. (1993). How to design and evaluate research in education
(2nd ed.). New York: McGraw-Hill.
Fuson, K. C. (1990a). Conceptual structures for multiunit numbers: Implications for learning
and teaching multidigit addition, subtraction, and place value. Cognition and
Instruction, 7, 343-403.
Fuson, K. C. (1990b). Issues in place-value and multidigit addition and subtraction learning
and teaching. Journal for Research in Mathematics Education, 21, 273-280.
Fuson, K. C. (1992). Research on learning and teaching addition and subtraction of whole
numbers. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic
for mathematics teaching (pp. 53-187). Hillsdale, NJ: Erlbaum.
Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for
first- and second-grade place value and multidigit addition and subtraction. Journal
for Research in Mathematics Education, 21, 180-206.
375
Fuson, K. C., Fraivillig, J. L., & Burghardt, B. H. (1992). Relationships children construct
among English number words, multiunit base-ten blocks, and written multidigit
addition. In J. I. D. Campbell (ed.), The nature and origins of mathematical skills
(pp. 39-112). Amsterdam, The Netherlands: Elsevier Science.
Fuson, K. C., & Smith, S. T. (1995). Complexities in learning two-digit subtraction: A case
study of tutored learning. Mathematical Cognition, 1, 165-213.
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I.,
Carpenter, T. P., & Fennema, E. (1997). Children’s conceptual structures for
multidigit numbers and methods of multidigit addition and subtraction. Journal for
Research in Mathematics Education, 28, 130-162.
Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive
Science, 7, 155-170.
Gentner, D. (1988). Metaphor as structure mapping: The relational shift. Child Development,
59, 47-59.
Gentner, D. (1989). The mechanisms of analogical learning. In S. Vosniadou & A. Ortony
(Eds.), Similarity and analogical reasoning (pp. 199-241). Cambridge University
Press.
Gentner, D., & Toupin, C. (1986) Systematicity and surface similarity in the development of
analogy. Cognitive Science, 10, 277-300.
Gilbert, R. K., & Bush, W. S. (1988). Familiarity, availability, and use of manipulative
devices in mathematics at the primary level. School Science and Mathematics, 88,
459-469.
Glaser, R. (1982). Instructional psychology: Past, present, and future. American
Psychologist, 37, 292-305.
Gluck, D. H. (1991). Helping students understand place value. Arithmetic Teacher, 38(7),
10-13.
Gold, R. L. (1969). Roles in sociological field observations. In G. J. McCall & J. L.
Simmons (Eds.), Issues in participant observation: A text and reader (pp. 30-38).
Reading, MA: Addison-Wesley.
Goswami, U. (1992). Analogical reasoning in children. Hove, UK: Erlbaum.
Grandgenett, N. (1991). Using LOGO programming to teach analogical reasoning in science
and mathematics education. Journal of Computers in Mathematics and Science
Teaching, 10(3), 29-36.
Greene, J. C., Caracelli, V. J., & Graham, W. F. (1989). Toward a conceptual framework for
mixed-method evaluation designs. Educational Evaluation and Policy Analysis, 11,
255-274.
Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for
Research in Mathematics Education, 22, 170-218.
Guba, E. G., & Lincoln, Y. S. (1989). Fourth generation evaluation. Newbury Park, CA:
Sage.
Halford, G. S. (1993a). Children’s understanding: The development of mental models.
Hillsdale, NJ: Erlbaum.
Halford, G. S. (1993b, March). Experience and processing capacity in cognitive
development: A PDP approach. Paper presented at the Annual Conference of the
Society for Research in Child Development, New Orleans, LA. (ERIC Document
Reproduction Service No. ED 355 028)
Harley, S. (1993). Situated learning and classroom instruction. Educational Technology,
33(3), 46-51.
Hart, K. (1989). There is little connection. In P. Ernest (Ed.), Mathematics teaching: The
state of the art (pp. 138-142). Lewes, UK: Falmer Press.
376
Helms, C. W., & Helms, D. R. (1992, June). Multimedia in education. Association of Small
Computer Users in Education (ASCUE) Conference (pp. 34-37). Greencastle, IN:
ASCUE. (ERIC Document Reproduction Service No. ED 357 732)
Hereford, J., & Winn, W. (1994). Non-speech sound in human-computer interaction: A
review and design guidelines. Journal for Educational Computing Research, 11,
211-233.
Hiebert, J. (1984). Children’s mathematics learning: The struggle to link form and
understanding. The Elementary School Journal, 84, 496-513.
Hiebert, J. (1988). A theory of developing competence with written mathematical symbols.
Educational Studies in Mathematics, 19, 333-355.
Hiebert, J. (1989). The struggle to link symbols with understandings: An update. Arithmetic
Teacher, 36, 38-44.
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A.
Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 6597). New York: Macmillan.
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A.,
& Human, P. (1997). Making sense: Teaching and learning mathematics with
understanding. Portsmouth, NH: Heinemann.
Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with
understanding in first grade. Journal for Research in Mathematics Education, 23,
98-122.
Holt, J. (1964). How children fail. Hammondsworth, Middlesex, UK: Penguin.
Hope, J. A. (1987). A case study of a highly skilled mental calculator. Journal for Research
in Mathematics Education, 18, 331-342.
Howard, P., Perry, B., & Conroy, J. (1995, November). Manipulatives in K-6 mathematics
learning and teaching. In Directions: Yesterday, Today, Tomorrow: Annual
conference of the Australian Association for Research in Education, Hobart,
Tasmania.
Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36, 6-11.
Hughes, P. (1995, August). Difficulties with understanding place-value. In NZAMT4: The
many faces of mathematics education. Biennial conference of the New Zealand
Association of Mathematics Teachers, Auckland, New Zealand.
Hunting, R. P. (1983). Emerging methodologies for understanding internal processes
governing children’s mathematical behaviour. The Australian Journal of Education,
27, 45-61.
Hunting, R. P., & Lamon, S. (1995). A re-examination of the rôle of instructional materials
in mathematics education. Nordisk Matematik Didaktik [Nordic Studies in
Mathematics Education], 3(3), 45-64.
Hunting, R. P., Davis, G., & Pearn, C. A. (1996). Engaging whole-number knowledge for
rational-number learning using a computer-based tool. Journal for Research in
Mathematics Education, 27, 354-379.
Hunting, R. P., & Doig, B. A. (1992). The development of a clinical tool for initial
assessment of a student's mathematics learning. In M. Stephens & J. Izard (Eds.),
Reshaping assessment practices: Assessment in the mathematical sciences under
challenge (pp. 201-217). Hawthorn: ACER.
Irons, C. J., & Burnett, J. (1994). Mathematics from many cultures (Teachers’ notes).
Hawthorn, Victoria:: Mimosa.
Janvier, C. (1987). Representation and understanding: The notion of function as an example.
In C. Janvier (Ed.), Problems of representation in the teaching and learning of
mathematics (pp. 67-71). Hillsdale, NJ: Erlbaum.
377
Jonassen, D. H. (1995). Supporting communities of learners with technology: A vision for
integrating technology with learning in schools. Educational Technology, 35(4), 6063.
Jonassen, D. H., Campbell, J. P., & Davidson, M. E. (1994). Learning with media:
Restructuring the debate. Educational Technology Research and Development,
42(2), 31-39.
Jones, G. A., & Thornton, C. A. (1993a). Children’s understanding of place value: A
framework for curriculum development and assessment. Young children, 48(5), 1218.
Jones, G. A., & Thornton, C. A. (1993b). Vygotsky revisited: Nurturing young children’s
understanding of number. Focus on Learning Problems in Mathematics, 15(2 & 3),
18-28.
Jones, G. A., Thornton, C. A., & Putt, I. J. (1994). A model for nurturing and assessing
multidigit number sense among first grade children. Educational Studies in
Mathematics, 27, 117-143.
Jones, G. A., Thornton, C. A., & Van Zoest, L. R. (1992, April). First grade children’s
understanding of multi-digit numbers. Paper presented at the meeting of the
American Educational Research Association, San Francisco, CA. (ERIC Document
Reproduction Service No. ED 353 141)
Jones, K., Kershaw, L., & Sparrow, L. (1994). Number sense and computation in the
classroom. Perth, Western Australia: Mathematics, Science and Technology
Education Centre, Edith Cowan University.
Kamii, C., & Lewis, B. A. (1991). Achievement tests in primary mathematics: Perpetuating
lower-order thinking. Arithmetic Teacher, 38, 4-9.
Kamii, C., Lewis, B. A., & Livingston, S. J. (1993). Primary arithmetic: Children inventing
their own procedures. Arithmetic Teacher, 41, 200-203.
Kozma, R. B. (1991). Learning with media. Review of Educational Research, 61, 179-211.
Kozma, R. B. (1994a). A reply: Media and methods. Educational Technology Research and
Development, 42(3), 11-14.
Kozma, R. B. (1994b). Will media influence learning? Reframing the debate. Educational
Technology Research and Development, 42(2), 7-19.
Lobato, J. E. (1993). Making connections with estimation. Arithmetic Teacher, 40, 347-351.
Marchionini, G. (1988). Hypermedia and learning: Freedom and chaos. Educational
Technology, 28(11), 8-11.
Mason, J. H. (1987). What do symbols represent? In C. Janvier (Ed.), Problems of
representation in the teaching and learning of mathematics (pp. 73-81). Hillsdale,
NJ: Erlbaum.
Mason, S. A. (1993). Employing quantitative and qualitative methods in one study. British
Journal of Nursing, 2, 869-872.
Mayer, R. E. (1996). Learners as information processors: Legacies and limitations of
educational psychology’s second metaphor. Educational Psychologist, 31, 151-161.
McArthur, D. (1987). Developing computer tools to support performing and learning
complex cognitive skills. In D. E. Berger, K. Pezdek, & W. P. Banks (Eds.),
Applications of cognitive psychology: Problem solving, education, and computing
(pp. 183-200). Hillsdale, NJ: Erlbaum.
McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic
number sense. For the Learning of Mathematics, 12(3), 2-8.
378
McNeal, B. (1995, October). Fact families as socially constructed knowledge. Paper
presented at the Annual Meeting of the North American Chapter of the International
Group for the Psychology of Mathematics Education, Columbus, OH. (ERIC
Document Reproduction Service No. ED 389 563)
Merseth, K. K. (1978). Using materials and activities in teaching addition and subtraction
algorithms. In M. N. Suydam & R. E. Reys (Eds.), Developing computational skills:
1978 Yearbook (pp. 61-77). Reston, VA: NCTM.
Miles, M. B., & Huberman, A. M. (1984). Drawing valid meaning from qualitative data:
Toward a shared craft. Educational Researcher, 13(15), 20-30.
Misanchuk, E. R., & Schwier, R. A. (1992). Representing interactive multimedia and
hypermedia audit trails. Journal of Educational Multimedia and Hypermedia, 1, 355372.
Miura, I. T., & Okamoto, Y. (1989). Comparison of U.S. and Japanese first graders’
cognitive representation of number and understanding of place value. Journal of
Educational Psychology, 81, 109-113.
Miura, I. T., Okamoto, Y., Kim, C. C., Steere, M., & Fayol, M. (1993). First graders’
cognitive representation of number and understanding of place value: Cross-national
comparisons—France, Japan, Korea, Sweden, and the United States. Journal of
Educational Psychology, 85, 24-30.
Mullis, I. V. S., Martin, M. O., Beaton, A. E., Gonzales, E. J., Kelly, D. L., & Smith, T. A.
(1997). Mathematics achievement in the primary school years: IEA’s Third
International Mathematics and Science Study. Chestnut Hill, MA: TIMSS
International Study Centre.
Nagel, N. G., & Swingen, C. C. (1998). Students’ explanations of place value in addition and
subtraction. Teaching Children Mathematics, 5, 164-170.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards
for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching
mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1998). Principles and standards for school
mathematics: Discussion draft. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA: Author.
National Research Council. (1989). Everybody counts: A report to the nation on the future of
mathematics education. Washington, DC: National Academy Press.
Nevin, M. L. (1992). A language arts approach to mathematics. Arithmetic Teacher, 40, 142146.
Newby, T. J., Ertmer, P. A., & Stepich, D. A. (1995). Instructional analogies and the learning
of concepts. Educational Technology Research and Development, 43(1), 5-18.
Nitko, A. J., & Lane, S. (1990). Solving problems is not enough: Assessing and diagnosing
the ways in which students organize statistical concepts. (ERIC Document
Reproduction Service No. ED 329 597)
Ohlsson, S., Ernst, A. M., & Rees, E. (1992). The cognitive complexity of learning and
doing arithmetic. Journal for Research in Mathematics Education, 23, 441-467.
Parkes, A. P. (1994). A study of problem solving activities in a hypermedia representation.
Journal of Educational Multimedia and Hypermedia, 3, 197-223.
Patton, M. Q. (1990). Qualitative evaluation and research methods (2nd ed.). Newbury Park,
CA: Sage.
Payne, J., & Rathmell, E. (1975). Number and numeration. In J. N. Payne (Ed.),
Mathematics learning in early childhood (pp. 125-160). Reston, VA: NCTM.
379
Pea, R. D. (1985). Beyond amplification: Using the computer to reorganize mental
functioning. Educational Psychologist, 20, 167-182.
Perry, B., & Howard, P. (1994). Manipulatives - constraints on construction? In G. Bell, R.
Wright, N. Leeson, & J. Geake (Eds.), Challenges in mathematics education:
Constraints on construction. Proceedings of the 17th Annual Conference of the
Mathematics Education Research Group of Australasia (Vol. 2, pp. 487-495).
Mathematics Education Research Group of Australasia.
Peterson, S. K., Mercer, C. D., McLeod, P. D., & Hudson, P. J. (1989). Validating the
concrete to abstract instructional sequence for teaching place value to learning
disabled students (Monograph #20). Gainesville, FL: Florida University, Department
of Special Education. (ERIC Document Reproduction Service No. ED 355 728)
Peterson, S. K., Mercer, C. D., Tragash, J., & O’Shea, L. (1987). Comparing the concrete to
abstract teaching sequence to abstract instruction for initial place value skills
(Monograph #19). Gainesville, FL: Florida University, Department of Special
Education. (ERIC Document Reproduction Service No. ED 353 744)
Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in
high school mathematics. Educational Studies in Mathematics, 23, 595-610.
Price, P. S. (1996, July). The computer as a generator of analogues. Paper presented at the
19th annual conference of the Mathematics Education Research Group of
Australasia, Melbourne.
Price, P. S. (1997). Teacher presence as a variable in research into students’ mathematical
decision-making. In F. Biddulph & K. Carr (Eds.), People in mathematics education
(Proceedings of the 20th annual conference of the Mathematics Education Research
Group of Australasia, Rotorua, NZ, pp. 422-428).
Price, P. S., & Price, P. M. (1998). Hi-Flyer Maths - Decimals (Version 1.04) [Computer
software]. Brisbane, Queensland: Hi-Flyer Software.
Putnam, R. T., Lampert, M., & Peterson, P. L. (1990). Alternative perspectives on knowing
mathematics in elementary schools. Review of Research in Education, 16, 57-150.
Q.S.R. NUD*IST (Version 3.0.4) [computer software]. (1994). Melbourne, Victoria:
Qualitative Solutions & Research.
Queensland Department of Education. (1996). The Year 2 Diagnostic Net. Brisbane,
Queensland: Author.
Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg
(Ed.), The development of mathematical thinking (pp. 109-151). New York:
Academic Press.
Resnick, L. B. (1987). Education and learning to think. Washington, DC: National
Academy.
Resnick, L. B. (1988). Treating mathematics as an ill-structured discipline. I n R. I. Charles
& E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving
(Vol. 3) (pp. 32-60). Reston, VA: NCTM.
Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser
(Ed.), Advances in instructional psychology (Vol. 3) (pp. 41-95). Hillsdale, NJ:
Erlbaum.
Reys, R. E., & Yang, D. C. (1998). Relationships between computational performance and
number sense among sixth- and eighth-grade students in Taiwan. Journal for
Research in Mathematics Education, 29, 225-237.
Ross, R., & Kurtz, R. (1993). Making manipulatives work: A strategy for success. Arithmetic
Teacher, 40, 254-257.
Ross, S. H. (1989). Parts, wholes and place value: A developmental view. Arithmetic
Teacher, 36, 47-51.
380
Ross, S. H. (1990). Children’s acquisition of place-value numeration concepts: The roles of
cognitive development and instruction. Focus on Learning Problems in
Mathematics, 12(1), 1-17.
Ross, S. H., & Sunflower, E. (1996). Place-value: Problem-solving and written assessment
using digit-correspondence tasks. The Math Forum - Place-Value: Introduction [Online]. Available: http://forum.swarthmore.edu/mathed/nctm96/construct/ross/
Rossman, G. B., & Wilson, B. L. (1985). Numbers and words: Combining quantitative and
qualitative methods in a single large-scale evaluation study. Evaluation Review, 9,
627-643.
Rossman, G. B., & Wilson, B. L. (1991, March). Numbers and words revisited: Being
“shamelessly eclectic”. Paper presented at the Annual Meeting of the American
Educational Research Association, Chicago, IL. (ERIC Document Reproduction
Service No. ED 377 235)
Rutgers Math Construction Tools [computer software]. (1992). NJ: Rutgers University.
Salomon, G. (1988). AI in reverse: Computer tools that turn cognitive. Journal of
Educational Computing Research, 4(2), 123-134.
Salomon, G. (1998). Technology’s promises and dangers in a psychological and educational
context. Theory Into Practice, 37, 4-10.
Saxton, M., & Towse, J. N. (1998). Linguistic relativity: The case of place value in multidigit numbers. Journal of Experimental Child Psychology, 69, 66-79.
Schielack, J. F. (1991). Reaching young pupils with technology. Arithmetic Teacher, 38(6),
51-55.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook
of research on mathematics teaching and learning (pp. 334-370). New York:
Macmillan.
Schwier, R. A., & Misanchuk, E. R. (1990, June). Analytical tools and research issues for
interactive media. Paper presented at the Annual Meeting of the Association for
Media and Technology in Education in Canada, St John’s, Newfoundland, Canada.
(ERIC Document Reproduction Service No. ED 321 751)
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes
and objects as different sides of the same coin. Educational Studies in Mathematics,
22, 1-36.
Sfard, A., & Thompson, P. W. (1994). Problems of reification: Representations and
mathematical objects. In D. Kirshner (Ed.), Proceedings of the 16th Annual Meeting
of the North American Chapter of the International Group for the Psychology of
Mathematics Education (pp. 3-34). (ERIC Document Reproduction Service No. ED
383 533)
Shepard, R. N. (1978). The mental image. American Psychologist, 33, 125-137.
Silver, E. A. (1994). Treating estimation and mental computation as situated mathematical
processes. In R. E. Reys & N. Nohda (Eds.), Computational alternatives for the
twenty-first century: Cross-cultural perspectives from Japan and the United States
(pp. 147-160). Reston, VA: NCTM.
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist
perspective. Journal for Research in Mathematics Education, 26, 114-145.
Simons, P. R. J. (1984). Instructing with analogies. Journal of Educational Psychology, 76,
513-527.
Sinclair, A., & Scheuer, N. (1993). Understanding the written number system: 6 year-olds in
Argentina and Switzerland. Educational Studies in Mathematics, 24, 199-221.
Sinclair, K. E. (1993). The versatile computer: Both tutor and cognitive tool. Australian
Educational Computing, 8(2), 21-24.
381
Skemp, R. R. (1982). Communicating mathematics: Surface structures and deep structures.
Visible Language, 16, 281-288.
Sowder, J. T. (1988). Mental computation and number comparison: Their roles in the
development of number sense and computational estimation. In J. Hiebert & M. Behr
(Eds.), Number concepts and operations in the middle grades (pp. 182-197). Reston,
VA: NCTM.
Sowder, J. (1992). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of
research on mathematics teaching and learning (pp. 371-389). New York:
Macmillan.
Sowder, J. T. (1994). Cognitive and metacognitive processes in mental computation and
computational estimation. In R. E. Reys & N. Nohda (Eds.), Computational
alternatives for the twenty-first century: Cross-cultural perspectives from Japan and
the United States (pp. 139-146). Reston, VA: NCTM.
Sowder, J., & Schappelle, B. (1994). Number sense-making. Arithmetic Teacher, 41, 342345.
Sowell, E. J. (1989). Effects of manipulative materials in mathematics instruction. Journal
for Research in Mathematics Education, 20, 498-505.
Sparrow, L., Kershaw, L., & Jones, K. (1994). Calculators: Research and curriculum
implications. Perth, Western Australia: Mathematics, Science and Technology
Education Centre, Edith Cowan University.
Stedman, L. (1995). Technology in teaching. Unpublished manuscript, Queensland
University of Technology at Brisbane, Australia.
Steffe, L. P., Cobb, P., & von Glasersfeld, E. (1988). Construction of arithmetical meanings
and strategies. New York :Springer-Verlag.
Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures
and techniques. Newbury Park, CA: Sage.
Sweller, J. (1999). Instructional design in technical areas. Camberwell, Victoria: ACER.
Thompson, C. (1990). Place value and larger numbers. In J. N. Payne (Ed.), Mathematics for
the young child (pp. 89-108). Reston, VA: NCTM.
Thompson, P. W. (1992). Notations, conventions, and constraints: Contributions to effective
uses of concrete materials in elementary mathematics. Journal for Research in
Mathematics Education, 23, 123-147.
Thompson, P. W. (1994). Concrete materials and teaching for mathematical understanding.
Arithmetic Teacher, 41, 556-558.
Touger, H. E. (1986). Models: Help or hindrance? Arithmetic Teacher, 33(7), 36-37.
Trafton, P. R. (1992). Using number sense to develop mental computation and computational
estimation. In C. J. Irons (Ed.), Challenging Children To Think When They Compute
(pp. 78-92). Proceedings of Conference, Queensland University of Technology,
Brisbane.
Ullmer, E. J. (1994). Media and learning: Are there two kinds of truth? Educational
Technology Research and Development, 42(1), 21-32.
Van Engen, H. (1949). An analysis of meaning in arithmetic. The Elementary School
Journal, 49, 321-329, 395-400.
Vosniadou, S., & Ortony, A. (1989). Similarity and analogical reasoning: A synthesis. In S.
Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 1-17).
Cambridge University Press.
Wiersma, W. (1995). Research methods in education: An introduction (6th ed.). Needham
Heights, MA: Allyn & Bacon.
382
Williams, M. D., & Dodge, B. J. (1993, January). Tracking and analysing learner-computer
interaction. In Proceedings of Selected Research and Development Presentations at
the Convention of the Association for Educational Communications and
Technology, New Orleans, LA. (ERIC Document Reproduction Service No. ED 362
212)
Wilson, F., & Shield, M. (1993). Solving the problem of problem solving. The Australian
Mathematics Teacher, 49(3), 10-12.
Winn, W. (1993). Instructional design and situated learning: Paradox or partnership?
Educational Technology, 33(3), 16-21.
Yin, R. K. (1994). Case study research design and methods (2nd ed.). Thousand Oaks, CA:
Sage.
383