____ THE
_____ MATHEMATICS ___
________ EDUCATOR _____
Volume 14 Number 2
Fall 2004
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor
Holly Garrett Anthony
Dear TME readers,
This has been a productive year for the editors of TME. We overhauled and updated our website,
broadened our reach nationally and internationally, and doubled our number of reviewers. Collectively,
the editors worked with almost 30 authors providing feedback and readying articles for publication.
We are now proud to present the final of two issues to be published in 2004. This issue showcases both
national and international research and commentary. We hope your reading of these articles will be
both educational and thought provoking.
David Clarke opens this issue with an editorial inviting you to read the research he, Margarita
Breed, and Sherry Fraser conducted in the early 1990s and published in this issue. In their research
article, they highlight some positive consequences of teaching with the Interactive Mathematics
Program (IMP), a problem-based curriculum that has gained attention in recent years. Clarke’s
editorial further asks mathematics education researchers to consider research methodology when
studying classroom learning. In so doing, he draws on the international comparative research of the
Learner’s Perspective Study.
Two studies presented in this issue examine the effectiveness of assessment items. Bates and
Wiest discuss the impact personalization of word problems can have on students’ performance on
mathematics assessments. Contrary to recent research they report no significant difference in students’
performance with personalized and non-personalized problems. Rueda and Sokolowski study the
effectiveness of their locally developed mathematics placement test at Merrimack College and show
that students who follow the recommendations for course enrollment based on their test scores perform
well in those classes.
Finally, Cyril Julie, a scholar in South Africa, invites readers to consider the development of
democratic competence in students within a newly formed democratic country and the role
mathematics might play in that development. He asks whether democratic competence can be realized
within Realistic Mathematics Education (RME), a curriculum developed in the Netherlands and
recently imported into South Africa. His question is important for consideration and his discussion is
stimulating.
As I close my final comments as the editor of TME, I encourage readers to support our journal by
submitting manuscripts, reviewing articles, or joining our editorial team. TME is growing in
recognition, and it is through the efforts put forth by all of us that it will continue to thrive.
Serving as the 2004 editor of TME has been truly rewarding. I was privileged to lead a team of
editors who worked well both together and independently. The publication of TME is a direct result of
their time and effort. I appreciate MESA allowing me the opportunity to do this work and I hope that
my efforts have been notable. I also extend my thanks to all of the other people who make TME
possible: reviewers, authors, peers, and faculty.
Associate Editors
Ginger Rhodes
Margaret Sloan
Erik Tillema
Publication
Stephen Bismarck
Dennis Hembree
Advisors
Denise S. Mewborn
Nicholas Oppong
James W. Wilson
MESA Officers
2004-2005
President
Zelha Tunç-Pekkan
Vice-President
Natasha Brewley
Secretary
Amy J. Hackenberg
Treasurer
Ginger Rhodes
NCTM
Representative
Angel Abney
Undergraduate
Representatives
Erin Bernstein
Erin Cain
Jessica Ivey
With Sincere Thanks,
Holly Garrett Anthony
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@coe.uga.edu
www.coe.uga.edu/tme
About the cover
Cover artwork by Tyler M. Ricks. Untitled, 2004.
For questions or comments, contact: tricks3@email.byu.edu
This piece is the culmination of years of experimentation and study with computer art. Starting with just a simple 3 by 3 grid in an
8th grade art class, the style blossomed into hundreds of different pieces using many different geometrical ideas. The process of
hand-drawing each line takes hours of work, but can produce extremely complex mathematical images. The piece featured on the
cover was created on the computer program GeoSketchpad™, and is part of a larger series using complex “string frames,” so
called because they resemble physical frames on which strings are tightly strung. Other series use complex grids, different
geometric shapes, or skewed frames to create intricate line drawings.
This publication is supported by the College of Education at The University of Georgia.
____________ THE ________________
___________ MATHEMATICS ________
______________ EDUCATOR ____________
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Fall 2004
Volume 14 Number 2
Table of Contents
2
Guest Editorial… Researching Classroom Learning and Learning Classroom
Research
DAVID CLARKE
7
The Consequences of a Problem-Based Mathematics Curriculum
DAVID CLARKE, MARGARITA BREED, & SHERRY FRASER
17 Impact of Personalization of Mathematical Word Problems on Student
Performance
ERIC T. BATES & LYNDA R. WIEST
27 Mathematics Placement Test: Helping Students Succeed
NORMA G. RUEDA & CAROLE SOKOLOWSKI
34 In Focus… Can the Ideal of the Development of Democratic Competence Be
Realized Within Realistic Mathematics Education? The Case of South Africa
CYRIL JULIE
38 Upcoming Conferences
39 Submissions Information
40 Subscription Form
© 2004 Mathematics Education Student Association.
All Rights Reserved
The Mathematics Educator
2004, Vol. 14, No. 2, 2–6
Guest Editorial…
Researching Classroom Learning and Learning Classroom
Research
David Clarke
One of the central goals of the mathematics
education research community is the identification of
classroom practice likely to facilitate student learning
of mathematics. In the paper by Clarke, Breed and
Fraser in this issue of The Mathematics Educator, the
results of an investigation into the outcomes of the
Interactive Mathematics Program (IMP) undertaken
back in the early 1990s are reported. Why is this
important? Because the focus of the analysis was an
expanded conception of the outcomes of classroom
practice that included both the cognitive and the
affective consequences of introducing a problem-based
mathematics program. The findings demonstrate that
the consequences of a particular curriculum and its
associated classroom practices cannot be adequately
characterized solely by the mathematical performance
of the students. Most importantly, the IMP classrooms
studied were most clearly distinguished from
conventional classrooms by affective rather than
cognitive outcomes. At the time, this was an attempt to
embrace a broader vision of valued classroom practice
and significant learning outcomes than could be
documented in an achievement test. The message of
this research has contemporary significance, but in the
time since that study was conducted our capacity to
investigate classroom practice and to connect it to
learning outcomes has increased considerably.
David Clarke is a Professor in the Faculty of Education at the
University of Melbourne and Director of the International
Centre for Classroom Research. His consistent interests have
been Assessment, Learning in Classrooms, and Teacher
Professional Development, and he has undertaken research
related to all of these areas. Recent publications include the
book “Perspectives on Practice and Meaning in Mathematics
and Science Classrooms” published by Kluwer Academic
Publishers in 2001, and the chapters on Assessment and
International Comparative Research in the 1996 and 2003
editions of the “International Handbook of Mathematics
Education.” He is currently directing the 14-country Learner's
Perspective Study.
2
The Participant’s Voice
I have argued consistently and persistently (Clarke,
1998, 2001, 2003) that since a classroom takes on
different aspects according to how you are positioned
within it or in relation to it, our research methodology
must be sufficiently sophisticated to accommodate and
represent the multiple perspectives of the many
participants in complex social settings such as
classrooms. Only by seeing classroom situations from
the perspectives of all participants can we come to an
understanding of the motivations and meanings that
underlie their participation. Our capacity to improve
classroom learning depends on such understanding.
The methodological challenge is how to document and
analyze the fundamental differences in how each
participant experiences any particular social
(classroom) situation. My colleagues, Sverker Lindblad
and Fritjof Sahlström (2002), argue that if early
researchers had access to the tools for data collection
and analysis that are available today, the general view
of classroom interaction would be quite different.
The most striking of these differences, and a very
important one from an education point of view,
concerns the role of students in classrooms. Thorsten
(2000) has made this point very clearly in relation to
the Third International Mathematics and Science Study
(TIMSS).
What is absent from nearly all the rhetoric and
variables of TIMSS pointing to the future needs of
the global economy is indeed this human side: the
notion that students themselves are agents.
(Thorsten, 2000, p. 71)
Single-camera and single-microphone approaches,
with a focus on the teacher, embody a view of the
passive, silent student at odds with contemporary
learning theory and classroom experience. Research
done with technologically more sophisticated
approaches has described a quite different classroom,
where different students are active in different ways,
contributing significantly to their own learning (cf.
Sahlström & Lindblad, 1998; Clarke, 2001).
Researching Classroom Learning
International Comparative Research
Further, classroom researchers have until recently
had limited opportunities for engaging in manageable
comparative research, where materials from different
countries and different periods of time can be accessed
and analyzed in feasible ways. At the International
Centre for Classroom Research at the University of
Melbourne
(http://www.edfac.unimelb.edu.au/DSME/ICCR/),
contemporary technology makes it possible to carry out
comparative analyses of an extensive database that
includes three-camera classroom video records of
lesson sequences, supplemented by post-lesson videostimulated interviews with students and teachers,
scanned samples of written work, and test and
questionnaire data, drawn from mathematics
classrooms as geographically distant as Sweden and
Australia and as culturally distant as Germany and
China.
Watanabe (2001) quotes White (1987) as writing
“we should hold Japan up as a mirror, not as a
blueprint.” This powerful and appealing metaphor can
serve as a general characterization of one of the major
uses of international comparative studies of classroom
practice. The agency for the interpretation and
adaptation of any documented practice resides with the
person looking in the mirror. There is no invocation of
absolute best practice – the judgement is a relativist
one, and an instructional activity with a high degree of
efficacy in Hong Kong may retain little effectiveness
when employed in a Swedish classroom, where
different cultural values inform and frame the actions
of all classroom participants. Most importantly, we are
encouraged to study Japanese (or South African or
German) classrooms not solely for the purposes of
mimicking their practices but for their capacity to
support us in our reflection on our own practice. The
mutuality of the potential benefit provides further
motivation for such research.
There is a small but growing body of research that
works at developing techniques of documenting
classroom interaction in ways that will facilitate highquality analysis of children’s learning. The transfer
from single-microphone audio (as in the early studies),
via single-camera video (as in many recent studies) to
multi-camera and multi-audio (as in the studies at the
technological forefront) is not primarily technologydriven, but rather motivated by the recent shifts in
education theories on learning, from a view of learning
as transfer to a view of learning as constructed in
action (see Sfard, 1998, for a discussion). Thus,
technological sophistication is a requirement of recent
David Clarke
theory, rather than a matter of sophisticated equipment
for technologically-minded project coordinators. This
is an essential point: Educational research, like
research in the physical and biological sciences, must
make optimal use of available technologies in
addressing the major problems of the field. But the
prime motivation must be “What are the big questions
and what tools do we need to address these questions?”
rather than “What questions can be addressed with
available tools?” Our research must be fuelled by a
need to answer important questions, not by a need to
use new tools. In addition, it is the first question that
will lead to recognition of the need for new tools and
provide the motivation for their development.
The Learner’s Perspective Study: Complementary
Accounts
Data collection in the Learner’s Perspective Study
(http://www.edfac.unimelb.edu.au/DSME/lps/)
involves a three-camera approach (Teacher camera,
Student camera, Whole Class camera) that includes the
onsite mixing of the Teacher and Student camera
images into a split-screen video record that is then used
to stimulate participant reconstructive accounts of
classroom events. So far, these data have been
collected for sequences of at least ten consecutive
lessons occurring in the “well-taught” eighth grade
mathematics classrooms of three teachers in each of
ten participating countries (Australia, Germany, Hong
Kong and mainland China, Israel, Japan, Korea, The
Philippines, South Africa, Sweden and the USA). This
combination of countries gives good representation to
European and Asian educational traditions, affluent
and less affluent school systems, and mono-cultural
and multi-cultural societies. Data collection will
commence next year in the Czech Republic, England
and Singapore.
Each participating country uses the same research
design to collect videotaped classroom data for at least
ten consecutive math lessons and post-lesson videostimulated interviews with at least twenty students in
each of three participating 8th grade classrooms. The
three mathematics teachers in each country are
identified for their locally-defined ‘teaching
competence’ and for their situation in demographically
diverse government schools in major urban settings. In
a major component of the post-lesson student
interviews, in which a split-screen video record is used
as stimulus for student reconstructions of classroom
events, students are given control of the video replay
and asked to identify and comment upon classroom
events of personal importance. Each teacher is
3
interviewed at least three times using a similar
protocol.
Goffman’s conception of a working consensus as a
transient convergence on a locally viable interpretation
(Goffman, 1959) is a particularly apt characterization
of the goal of the consensus process operating in many
interpretive research teams (e.g., Cobb & Bauersfeld,
1995; Stigler & Hiebert, 1999). The research in which I
have been involved (e.g., Clarke, 2001) problematizes
such consensus and attempts to synthesize portrayals of
practice from ‘complementary accounts’ provided by
researchers and the participants in the research setting
relating to a common body of data (rationale provided
in Clarke, 1998).
I would like to assert the inevitable existence of
multiple reflexivities between theory, research into
practice, and the practice of research. The argument is
predicated on three basic premises:
Shepard’s provocative question, “But what if learning
is not linear and is not acquired by assembling bits of
simpler learning” (Shepard, 1991, p. 7).
In the case of the Learner’s Perspective Study:
Research guided by a theory of learning that accords
significance to both individual subjectivities and to the
constraints of setting and community practice must
construct and frame its conclusions (and collect its
data) accordingly. Such a theory must accommodate
complementarity rather than require convergence and
accord both subjectivity and agency to individuals not
just to participate in social practice but to shape that
practice. Research that aims to apply such theories
must construct its methodologies accordingly and draw
from available technologies in ways that afford rather
than constrain the methodological ambitions of the
researcher.
1. The discourse of the classroom (for
example) acts to position participants in ways
that afford and constrain certain practices.
International comparative classroom research need
not appeal to a separate and distinct research paradigm
from that enacted in conventional classroom research,
although the methodological and theoretical
considerations are more complex than research within
a single culture. Part of the power of international
comparative research lies in its capacity to offer us the
opportunity to juxtapose, compare and contrast
documented practices drawn from settings that simply
would not pertain in our local culture. What form does
teaching competence take when confronted with a class
of 60 or more students (as is the case in the
Philippines)? How must we reconceive our notions of
effective instructional practice to accommodate
apparently successful classrooms in which students
seldom if ever speak to each other (as pertains in some
Asian classrooms)? How much more compelling must
our theories of learning become if they can be
demonstrated to accommodate and explain learning in
such disparate settings?
As new theories of learning and social interaction
develop, research techniques must have the capacity to
accommodate these new theories. All too often it is
forgotten that any use of technology in a research
setting implies the existence of an underlying theory on
which the type of data, the means of data collection,
and the anticipated method of analysis are all
predicated. Of all data sources currently available to
researchers in education, videotape data seems most
amenable to secondary analysis. Further, the potential
of videotape data to sustain secondary analysis carries
an associative potential for the synthesis of those
analyses.
2. The discourse of educational research acts
to position participants in ways that afford and
constrain certain interpretations.
3. The adoption of a theory of learning in
social situations will inevitably find its
reflection in the manner in which those
situations are researched.
These fundamental reflexivities are seldom
acknowledged. Since research activity constitutes a
form of learning or knowledge construction, the
processes by which a research project is conducted
should be in harmony with whatever theory of learning
structures the researcher’s analysis of data.
Consistency between methodology and theory should
be a matter of purposeful and deliberate design. Lorrie
Shepherd turns this argument delightfully on its head
in her paper “Psychometricians’ Beliefs About
Learning” (Shepard, 1991), where she contends that
the disputes of the testing community can be explained
in terms of differences in the beliefs about learning
held by the various educational measurement
specialists. In particular, Shepard argues that the
beliefs of many psychometricians derive from an
implicit behaviorist learning theory in flagrant
contradiction with evidence from cognitive
psychology. What Shepard does to good effect in her
paper is reverse engineer psychometricians’ learning
theories on the basis of their test instruments. The
fruitfulness of this approach is fully evident in
4
A Layered Vision
Researching Classroom Learning
Multi-site international research projects offer
access to a layered vision of practice, outcome and
theory development. It may help to illustrate this
stratification with examples from the Learner’s
Perspective Study.
Classroom Practice – Lesson Events
At the level of classroom practice, the challenge
has been to find a suitable instructional unit to provide
the basis for comparative analysis. Demonstration of
the inadequacy of “the lesson” to serve this role (at
least in the form of nationally characteristic lesson
“scripts” or “patterns”) has led to analyses focusing on
the “lesson events” from which each lesson is
constituted. Lesson events such as “Beginning the
Lesson,” “Learning Tasks,” “Guided Development”
(Whole class discussion), “Between Desks
Instruction,” and “Summing Up” have emerged as
internationally recognizable activities, differently and
distinctively employed and enacted in classrooms
around the world.
Patterns of Participation
In participating in each of the lesson events
identified above, teacher and students position
themselves and are positioned within the constraints
and affordances offered by the classroom setting and
its peculiar practices (peculiar here is used in all
possible senses). The consequences of this process of
social positioning are characteristic patterns of
participation accessible to classroom participants (and
co-constructed by them) in ways that reflect each
individual’s unique interaction with the classroom
setting and community.
The Distribution of Responsibility for Knowledge
Generation
Each classroom affords and constrains access to
various patterns of participation. Within the patterns of
participation characteristic of a classroom can be
found the “distribution of responsibility for knowledge
generation” – a much more useful characterization of
the classroom than a simplistic dichotomization into
teacher-centered and student-centered, and much more
revealing of the sociocultural nature of learning.
The use of video material supported by post-lesson
video-stimulated interviews provides a complex
database amenable to analysis at any and all of the
three levels indicated above. Complex databases,
configured in anticipation of multiple and
complementary analyses, offer our best chance to
match the complexity of social phenomena with an
David Clarke
appropriate sophistication of approach. Advances in
technology bring us ever closer to the realization of
this vision. The developmental pathway that has led us
from early attempts at classroom observation and
process-product studies to our present level of
sophistication represents an on-going attempt to
accommodate the complexity of social situations.
Eugene Ionescu is reputed to have said, “Only the
ephemeral is of lasting value.” Social interactions are
nothing if not ephemeral; and, since it is through social
interaction that we experience the world, the
understanding of social interactions must underlie any
attempts to improve the human condition. Our
difficulties in characterizing social interactions for the
purpose of theory building in education are
compounded by the fluid and transient nature of the
phenomena we seek to describe. Attempts to categorize
social behavior run the risk of sacrificing the
dynamism, contextual-dependence and variation that
constitute their essential attributes. This poses a
challenge both for methodology and for theory. The
ephemeral nature of social interactions is something
that must be honored in the methodology but
transcended in the analysis. Those of us who have
accepted the challenge of researching classroom
learning continue to learn how better to undertake
classroom research.
REFERENCES
Clarke, D. J. (1998). Studying the classroom negotiation of meaning:
Complementary accounts methodology. In A. Teppo (Ed.),
Qualitative research methods in mathematics education (Chapter 7,
pp. 98–111). Journal for Research in Mathematics Education,
Monograph No. 9. Reston, VA: National Council of Teachers of
Mathematics.
Clarke, D. J. (Ed.). (2001). Perspectives on practice and meaning in
mathematics and science classrooms. Dordrecht, Netherlands: Kluwer
Academic Press.
Clarke, D. J. (2003). International comparative studies in mathematics
education. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, &
F. K. S. Leung (Eds.), Second international handbook of mathematics
education (Chapter 5, pp. 145–186). Dordrecht, Netherhlands: Kluwer
Academic Publishers.
Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical
meaning: Interaction in classroom cultures. Hillsdale, NJ: Lawrence
Erlbaum.
Goffman, E. (1959). The presentation of self in everyday. New York:
Doubleday. [Cited in Krummheuer (1995)]
Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb &
H. Bauersfeld (Eds.), The emergence of mathematical meaning:
Interaction in classroom cultures (Chapter 7, pp. 229–269). Hillsdale,
NJ: Lawrence Erlbaum.
Lindblad, S., & Sahlström, F. (2002, May). From teaching to interaction:
On recent changes in the perspectives and approaches to classroom
research. Invited plenary lecture at the Current Issues in Classroom
Research: Practices, Praises and Perspectives Conference, Oslo,
Norway.
5
Sahlström, F., & Lindblad, S. (1998). Subtexts in the science classroom –
an exploration of the social construction of science lessons and school
careers. Learning and Instruction, 8(3), 195–214.
Thorsten, M. (2000). Once upon a TIMSS: American and Japanese
narrations of the Third International Mathematics and Science Study.
Education and Society, 18(3), 45–76.
Sfard, A. (1998). On two metaphors for learning and the danger of choosing
just one. Educational Researcher, 27(2), 3–14.
Watanabe, T. (2001). Content and organization of teacher’s manuals: An
analysis of Japanese elementary mathematics teacher’s manuals.
School Science and Mathematics, 101(4), 194–201.
Shepard, L. A. (1991). Psychometrician’s beliefs about learning.
Educational Researcher, 20(6), 2–16.
Stigler, J., & Hiebert, J. (1999). The teaching gap. New York: Free Press.
6
White, M. (1987). The Japanese educational challenge: A commitment to
children. New York: The Free Press. [Cited in Watanabe (2001)]
Researching Classroom Learning
The Mathematics Educator
2004, Vol. 14, No. 2, 7–16
The Consequences of a Problem-Based Mathematics
Curriculum
David Clarke, Margarita Breed, & Sherry Fraser
Implementation of a problem-based mathematics curriculum, the Interactive Mathematics Program (IMP), at
three high schools in California has been associated with more than just differences in student achievement. The
outcomes that distinguished students who participated in the IMP program from students who followed a
conventional algebra/geometry syllabus were the students’ perceptions of the discipline of mathematics, of
mathematical activity and the origins of mathematical ideas, of the mathematical nature of everyday activities,
and of school mathematics and themselves as mathematicians. A coherent and consistent picture has emerged of
the set of beliefs, perceptions and performances arising from such a program. Students who have participated in
the IMP program appear to be more confident than their peers in conventional classes; to subscribe to a view of
mathematics as having arisen to meet the needs of society, rather than as a set of arbitrary rules; to value
communication in mathematics learning more highly than students in conventional classes; and to be more
likely than their conventionally-taught peers to see a mathematical element in everyday activity. These
outcomes occurred while the IMP students maintained performance levels on the mathematics portion of the
SAT at or above those of their peers in conventional classes. If student achievement outcomes are comparable,
the mathematics education community must decide whether it values these consequences of a problem-based
curriculum.
Among the debates engaging the energies of the
mathematics education community, one of the more
energetic has concerned the role of problem solving in
mathematics instruction. This debate has encompassed
issues from what constitutes a problem to whether
problem solving should be the medium or the message
of the mathematics curriculum (cf., Clarke &
McDonough, 1989; Lawson, 1990; Owen & Sweller,
1989; Schoenfeld, 1985). Claims and counter-claims
David Clarke is a Professor in the Faculty of Education at the
University of Melbourne and Director of the International
Centre for Classroom Research. He is currently directing the
14-country Learner's Perspective Study.
Margarita Breed is currently a fulltime Ph. D. student funded
under the APAI (Australian Postgraduate Award Industry)
Scheme for the Scaffolding Numeracy in the Middle Years
Research Project at RMIT University. Her background has
been in primary teaching and as a Middle Years Numeracy
Leader for Eastern Metropolitan Region and she is particularly
dedicated to students in the Middle Years. Whilst completing
her Master of Education (Research) she was Research
Assistant for the Mathematics Teaching and Learning Centre at
the Australian Catholic University.
Sherry Fraser is currently the Director of the IMPlementation
Center for the Interactive Mathematics Program. She continues
to be interested in providing both students and teachers access
to rich secondary mathematics materials.
Acknowledgements
The cooperation of the teachers whose pupils participated in
this study is gratefully acknowledged. The comments of Barry
McCrae, Kevin Olssen, Diane Resek and Peter Sullivan on
early drafts of this paper are also gratefully acknowledged.
Clarke, Breed, & Fraser
have been made regarding the advisability and the
feasibility of basing a mathematics syllabus on nonroutine mathematics tasks. Attempts to evaluate the
success of such curricula have typically employed
achievement tests to distinguish student outcomes.
The authors of this study felt that a problem-based
curriculum would be characterized more
appropriately by the belief systems which the
instructional program engendered in participating
students than by the students’ achievement on
conventional mathematical tasks.
It is students’ belief systems that are likely to
influence the students’ subsequent participation in
the study of mathematics, to structure their
consequent learning of mathematics, and to guide
and facilitate the application of mathematical skills
to everyday contexts. If it could be demonstrated
that student achievement on conventional
mathematics tasks was enhanced by a problembased program, and if student performance on nonroutine problem-solving tasks was heightened by
such a program, the ultimate value of the instruction
would depend still on whether the student chooses
to continue to study mathematics, develops a set of
beliefs which supports and empowers further
learning, and sees any relevance in the skills
acquired in class for situations encountered in the
world beyond the classroom. Conventional
instruction does little to address such concerns, and
research has commonly ignored such outcomes.
7
The evaluation of teaching experiments currently
in progress must address these other consequences of
instruction. In discussing their work on “one-on-one
constructivist teaching,” Cobb, Wood, and Yackel
(1990) drew attention to non-conventional learning
outcomes. This instructional approach provides
opportunities for the children to construct
mathematical knowledge not found in traditional
classrooms. The difficulty for researchers evaluating
innovative classroom practices is that many of the
conventional research tools are insensitive to the
behaviors and the knowledge that distinguishes such
instruction. This concern is also relevant where the
goals of the program are affective as well as cognitive.
Since studies such as that of Erlwanger (1975) drew
attention to the significance of a student’s belief
system regarding mathematics and mathematical
behavior, research into effective teaching practice has
had an obligation to address student belief outcomes.
This obligation is linked to the recognition of
“cognition as socially situated activity” (Lave, 1988,
p.43). While the subject of student beliefs has been
discussed usefully in a variety of forums (for instance,
Clarke, 1986; Cobb, 1986), research studies have still
to accept a responsibility to address student belief and
perception outcomes routinely in the evaluation of
instructional programs. The study reported here is one
attempt to do so.
The Instructional Program
In 1989, the California Postsecondary Education
Commission (CPEC) released a request for proposals
that would drastically revamp the Algebra I-GeometryAlgebra II sequence. The curriculum envisioned in the
guidelines would set “problem solving, reasoning and
communication as major goals; include such areas as
statistics and discrete mathematics; and make
important use of technology” (CPEC, 1989, p. 4). The
Interactive Mathematics Project (IMP) Curriculum
Development Program obtained funding to develop
and field test three years of problem-based
mathematics that would satisfy six of the University of
California requirements for high school mathematics.
Program Goals
The goals of IMP were to:
•
broaden who learns mathematics, by making the
learning of core mathematics accessible to groups
previously underrepresented in college
mathematics classes;
•
expand what mathematics was learned, consistent
with the recommendations of the Curriculum and
8
Evaluation Standards (NCTM, 1989),
emphasizing problem solving and the
communication of mathematical ideas;
•
change mathematics instruction, by requiring
students to be active learners and investigators,
by integrating the study of mathematical
domains, such as algebra, geometry and
statistics, with each other and with areas of
application, and by making use of current
technology;
•
change how teachers perceive their roles, by
emphasizing the role of the teacher as guide and
model learner and by changing dominant modes
of classroom communication from teacher
explanation to student interaction;
•
change how mathematics learning is assessed,
by assessing students’ use of mathematical
knowledge to solve complex problems, and by
diversifying assessment strategies to include
student portfolios, self-assessment, teacher
observations, oral presentations, and group
projects, as well as written homework and tests.
Pupil Selection
Methods of selection of pupils for participation
in the IMP classes varied. The principal criterion
was student self-nomination. One high school
collected information on student performance,
instructional preferences, and academic history and
then selected “60% of the group who would have
been placed in Algebra and 40% from those below.”
It was the opinion of the various school
administrations that the academic standing of the
sample of IMP students arising from the various
selection criteria was certainly no higher than that
of the students in conventional Algebra classes. In
fact, in the case of the high school just mentioned,
the overall academic standing of students
commencing IMP was almost certainly lower than
that of commencing Algebra students.
Teacher Selection
Teachers were also self-nominated.
The IMP Materials
The IMP materials consist of modular units,
each requiring approximately five weeks of
instructional time. These units employ historical,
literary, scientific and other contexts to provide a
thematic coherence to the pupils’ exploration of
mathematics. For instance, in one unit the Edgar
Problem-Based Mathematics Curriculum
Allan Poe short story The Pit and the Pendulum is used
to facilitate student investigation of variation,
measurement uncertainty, normal distribution,
graphing, mathematical modeling, and non-linear
functions. The instructional sequence of each unit
addresses mathematical concepts and skills and
mathematical problem solving in a context that
provides both the rationale for the skills being acquired
and a means of integrating newly acquired knowledge
within a coherent structure.
An IMP Classroom
Class size averaged around 32 students. Classroom
activities were typified by group work, writing, and
oral presentations. Graphing calculators were available
at all times. The characteristics of IMP and Algebra
classrooms, as perceived by the pupils, were
documented in the course of this study, and are
detailed in the results presented later in this paper.
Assessment Practices
Priority was given in IMP classrooms to a diversity
of assessment strategies, consistent with the program
goals. For example, in one IMP class, grades were
calculated from student performance on homework
(30%), classwork and class participation (30%),
problems of the week (30%), and unit assessments
(10%). It appeared that most assessing of Algebra
students was through weekly quizzes and chapter tests.
Method
report their perceptions of those valued activities,
which, in their opinion, assisted their learning of
mathematics, in addition to their perceptions of
what constituted typical classroom activities in
mathematics and their attitudes towards
mathematics. The Mathematics World questionnaire
required students to identify the extent to which
specific everyday activities were mathematical. At
the time of administration of the questionnaires,
IMP students had completed almost one year in the
program.
In addition, the next fall, the Mathematics
Scholastic Aptitude Test (SAT) was administered to
the school populations, facilitating comparison of
the mathematics performance of IMP students with
their peers in conventional classes.
Mathematics belief. The mathematics belief
questionnaire was adapted from an instrument
employed to measure the student belief outcomes of
an innovative program employing student journals
(Clarke, Stephens & Waywood, 1992; Clarke,
Waywood, & Stephens, 1994). Every item was
validated through interviews with students. Minor
changes in phrasing were made for administration in
American schools. Some sample items were:
1.
If I had to give myself a score out of 10 to
show, honestly, how good I think I am at math,
the score I would give myself would be…
3.
The ideas of mathematics:
A. Have always been true and will always be
true.
Agree
Disagree
Subjects
The subjects of this study were 182 students at
three Californian high schools participating in the IMP
program outlined above. In addition, matching data
were collected on 74 Algebra 2 students and 143
Algebra 4 students from the same schools. Data on an
additional 52 Algebra 2 students were collected from a
fourth high school to provide a comparable sample of
students at the same level as the IMP pupils.
Procedures and Measures
During June, towards the end of the academic year,
all students completed two questionnaires. The student
questionnaire was constructed in large part by
combining items developed and tested in a study of
student mathematics journal use and a further study of
student self-assessment.
The Mathematics Belief questionnaire examined
student perceptions of their mathematical competence,
and student beliefs about mathematical activity and the
origins of mathematical ideas. Students were asked to
Clarke, Breed, & Fraser
D. Developed as people needed them in
everyday life.
Agree
Disagree
F. Are most clearly explained using numbers.
Agree
Disagree
5.
When I am doing mathematics at school, I am
likely to be:
A. Talking
Always Often Sometimes Seldom Never
C. Writing words
Always Often Sometimes Seldom Never
F. Working with a friend
Always Often Sometimes Seldom Never
I. Listening to other students
Always Often Sometimes Seldom Never
K. Working from a textbook
Always Often Sometimes Seldom Never
9
7.
An adaptation of the IMPACT instrument (Clarke,
1987) was included as item 7, including such subitems as:
Write down one new problem that you can now do.
How could math classes be improved?
Student attitudes towards mathematics classes were
measured explicitly through the sub-item:
How do you feel in math classes at the moment?
(circle the words which apply to you.)
A. Interested
B. Relaxed
C. Worried
D. Successful
E. Confused
F. Clever
G. Happy
H. Bored
I. Rushed
J. (Write one word of your own) _____________
The response alternatives provided in this sub-item
arose from extensive interviewing of high school
students in the course of a study of student
mathematical behavior at the point of transition from
primary school (elementary school) mathematics to
high school mathematics (Clarke, 1985, 1992). The
IMPACT instrument, from which the sub-item was
drawn, was extensively field-tested with 753 grade 7
students over a period of one year (Clarke, 1987).
Mathematics world. The mathematics world
questionnaire was adapted for American administration
from an instrument employed in a study of community
perceptions of mathematical activity (Clarke &
Wallbridge, 1989; Wallbridge, 1992). In this
questionnaire, students were asked to indicate whether
they thought specific everyday activities were highly
mathematical, quite mathematical, slightly
mathematical, barely mathematical, or not
mathematical. The activities listed included:
4.
7.
9.
Cooking a meal using a recipe
Playing a musical instrument
Buying clothing at a sale
A complete listing of all questionnaire items is
available in Clarke, Wallbridge, and Fraser (1992).
Results
The results that follow make reference to three
groups of students to whom questionnaires were
administered:
i. 180 IMP students – mean age 15.3 years
ii. 126 Algebra 2 students – mean age 15.4 years
iii. 137 Algebra 4 students – mean age 16.9 years
Comparing the Algebra 2 and Algebra 4 Samples
differ significantly on any of the 48 measures
except the use of worksheets, for which the Algebra
4 students recorded an even lower incidence than
did the Algebra 2 students, and the importance
accorded to the teacher's explanations: Algebra 4
students attached lower importance to these than did
the Algebra 2 students. It seems reasonable to
summarize these findings by observing that, with
respect to the beliefs documented here,
conventionally-taught students adhere to a specific
set of beliefs with a high level of stability over time.
These beliefs and the associated perceptions of
classroom practice were sufficiently distinct from
those held by IMP students to clarify the
characteristics of both class types. Results are given
as comparisons between IMP and Algebra 2
students since these represent the most similar
sample populations.
In each table where comparisons are made
between groups the corresponding p value is given.
Differences between groups which achieved
statistical significance are asterisked.
Student Mathematics Achievement on Conventional
Tests
Where comparison was possible between IMP
and Algebra students at the same school, mean SAT
scores for IMP classes were higher than mean SAT
scores for traditional Algebra/Geometry classes.
Pair-wise comparison of group means (t test) was
used to identify any statistically significant
difference under a conventional null hypothesis
assumption. At one high school the difference in
performance was statistically significant. These
results are documented in Table 1.
Table 1
SAT Scores for Algebra and IMP Students on Two School
Sites
School
A
B
Class
type
Mean SAT
score
SD
Algebra
(n = 83)
420.48
82.96
IMP
(n = 74)
443.37
77.21
Algebra
(n = 86)
367.56
57.02
IMP
(n = 67)
373.88
60.95
p value
.0372*
.1003
In all, 48 student measures were generated through
the two questionnaires. Algebra 2 and Algebra 4
samples (n = 126 and n = 137, respectively) did not
10
Problem-Based Mathematics Curriculum
Student Perceptions of Their Mathematics Competence
IMP students were significantly more likely to rate
themselves highly on how good they were at
mathematics than were Algebra 2 students (Table 2).
Sample Item:
If I had to give myself a score out of 10 to show,
honestly, how good I think I am at math, the score I
would give myself would be:
Two comments should be made concerning this
higher self-rating by IMP students. First, SAT scores
indicated that where comparison was possible IMP
students tended to be more capable at conventional
mathematics tasks than were their peers in Algebra
classes, which suggests that these self-ratings had some
basis in fact. Second, the difference in self-ratings can
also be interpreted as a difference in confidence. We
would suggest that heightened self-confidence in
mathematics is likely to lead to increased participation
in further mathematics, and a greater likelihood that the
student will make use of the mathematical skills
acquired. Both are desirable outcomes.
Table 2
Self-rating Scores for IMP and Algebra 2 Students
Class type
Mean
SD
Algebra 2
(n = 125)
6.86
1.2
IMP
(n = 173)
7.5
1.38
p value
Table 3
Student Attitude Index for IMP and Algebra 2 Students
Class type
Mean
SD
Algebra 2
(n = 126)
–.52
1.85
IMP
(n = 174)
.97
Student Attitude Toward Mathematics Classes
IMP students were significantly more likely to feel
positive about mathematics classes (Table 3).
A. Interested
B. Relaxed
C. Worried
D. Successful
E. Confused
F. Clever
G. Happy
H. Bored
I. Rushed
J. (Write one word of your own) _____________
A student attitude index was calculated by scoring each
positive response +1 and each negative response –1,
and summing for each student.
2.14
The distinguishing characteristic between the
problem solving students and the Algebra 2 students
was the degree to which they perceived
mathematics to be a mental activity (Table 4).
Sample Item:
Mathematics is something I do (circle one or
more):
A. Every day as a natural part of living
B. Mostly at school
C. With a pencil and paper
D. Mostly in my head
E. With numbers
Table 4
Student Perceptions of Mathematical Activity
Class type
Proportion
(%)
Every day as a natural
part of living
Algebra 2
IMP
49
52
Mostly at school
Algebra 2
IMP
63
64
With a pencil and paper
Algebra 2
IMP
41
42
Mostly in my head
Algebra 2
IMP
27
39
With numbers
Algebra 2
IMP
51
46
Sample item:
How do you feel in math classes at the moment? (circle
the words which apply to you.)
.0001*
Student Perceptions of Mathematical Activity
Response
alternatives
.0012*
p value
Table 4 is significant in the context of this paper
in that it was only in these perceptions of
mathematical activity that the IMP and Algebra
students responded in a similar fashion. The marked
differences in beliefs and perceptions reported by
the two groups, which constitute the essential
findings of this study, are only evident in Table 4 in
the significantly greater inclination for IMP students
to report mathematics as being a mental activity.
Student Perceptions of Mathematical Ideas
IMP students were more likely to agree that
mathematical ideas could be clearly explained using
Clarke, Breed, & Fraser
11
every day words that anyone could understand, than
were Algebra 2 students. IMP students were also less
likely to view the ideas of mathematics as ones that can
only be explained using numbers and language specific
to mathematics. The IMP students were more likely to
view mathematics as having developed in response to
people’s needs. The IMP students were also less likely
than the Algebra 2 students to view mathematics as
having been invented by mathematicians or to hold that
the ideas of mathematics have always and will always
be true. Figure 1 and Table 5 document these
differences.
Sample Item:
The ideas of mathematics
A. Have always been true and will always be true.
Agree
Disagree
B. Were invented by mathematicians.
Agree
Disagree
C. Were discovered by mathematicians.
Agree
Disagree
D. Were developed as people needed them in daily
life.
Agree
Disagree
E. Have very little to do with the real world.
Agree
Disagree
F. Are most clearly explained using numbers.
Agree
Disagree
G. Can only be explained using mathematical
language and special terms.
Agree
Disagree
H. Can be explained in everyday words that
anyone can understand.
Agree
Disagree
In summary: IMP students were more likely to
hold a socially-oriented view of the origins and
character of mathematical ideas rather than a Platonist
belief in the existence of mathematical absolutes
awaiting discovery.
12
1.00
0.80
0.60
0.40
0.20
0.00
-0.20
-0.40
-0.60
-0.80
-1.00
Always
True
Discovered
Invented
Unreal
Developed
Special
Terms
Numbers
Everyday
Words
Algebra students = light bars, IMP students = dark bars;
a positive mean value indicates agreement;
a negative mean value indicates disagreement
Figure 1. Students’ perceptions of the ideas of
mathematics.
Table 5
Students’ Perceptions of the Ideas of Mathematics
Class
Type
Sub-items
Mean
Have always been true and will
always be true.
Algebra 2
IMP
.02
–.28
Were invented by mathematicians.
Algebra 2
IMP
–.13
–.36
Were discovered by
mathematicians.
Algebra 2
IMP
.18
.01
Developed as people needed them
in daily life.
Algebra 2
IMP
.57
.77
Have very little to do with the real
world.
Algebra 2
IMP
–.72
–.82
Are most clearly explained using
numbers.
Algebra 2
IMP
.26
–.02
Can only be explained using
mathematical language and special
terms.
Algebra 2
IMP
–.41
–.69
Can be explained in everyday
words that anyone can understand.
Algebra 2
IMP
.21
.63
Problem-Based Mathematics Curriculum
Student Perceptions of School Mathematics
The IMP students were significantly more likely to
agree that writing was important in helping them to
understand mathematics. The IMP students were also
more likely to see value in talking to other students
than were the Algebra 2 students. The IMP students
were significantly less likely than the Algebra 2
students to view drill and practice as the best way to
learn mathematics.
•
•
•
Sample Item:
Circle the alternative which best describes how true
you think each statement is (SA = Strongly Agree,
A = Agree, D = Disagree, and SD = Strongly
Disagree):
1. Explaining ideas clearly is an important part of
mathematics.
SA
A
D
SD
2. Mathematics does not require a person to use
very many words.
SA
A
D
SD
3. Writing is an important way for me to sort out
my ideas in mathematics.
SA
A
D
SD
4. Talking to other students about the mathematics
we are doing helps me to understand.
SA
A
D
SD
5. Drill and practice is the best way to learn
mathematics.
SA
A
D
SD
The distinguishing characteristics between the
IMP students and the Algebra 2 students were:
•
•
•
the importance attached by IMP students to writing
in mathematics (p = .04*)
the degree to which IMP students perceived talking
to other students as useful in helping them to
understand mathematics (p = .0005*)
the relative importance attached to drill and
practice by the Algebra 2 students (p = .0001*)
Student Perceptions of Mathematical Activity at School
The greatest degree of difference between IMP
students and the Algebra 2 students was evident in
their perceptions of mathematical activity at school.
Table 6 illustrates the differences in student
perceptions of their mathematics classrooms. In these
statistics, the differences between the two class types
are most clearly illustrated. Key differences between
IMP and Algebra 2 classes can be summarized as
follows:
Clarke, Breed, & Fraser
•
IMP students were significantly more likely to
be writing words and drawing diagrams, and
less likely to be writing numbers.
IMP students were significantly more likely to
be working with a friend or with a group, and
less likely to be working on their own.
While there was no difference between IMP and
Algebra 2 classes in the relative frequency of
listening to the teacher, IMP students were
significantly more likely to be listening to other
students than were students in Algebra 2
classes.
IMP students were significantly more likely to
be working from a worksheet and less likely to
be copying from the board or working from a
textbook.
Students were asked to respond on a four-point
scale to the cue “When doing mathematics at
school, I am likely to be...” The mean values in
Table 6 should be read as students’ perceptions of
the relative frequency (on a 5-point scale) with
which they engaged in each of the listed activities.
Table 6
Mean Relative Frequency of Student Engagement
Class
type
Mean
SD
p
value
Talking
Algebra 2
IMP
2.25
2.59
1.0
.88
.002*
Writing numbers
Algebra 2
IMP
3.13
2.66
.86
.83
.0001*
Writing words
Algebra 2
IMP
1.70
2.68
.93
.94
.0001*
Drawing diagrams
Algebra 2
IMP
1.85
2.70
.82
.83
.0001*
Working on my
own
Algebra 2
IMP
2.59
1.91
.90
.87
.0001*
Working with a
friend
Algebra 2
IMP
2.10
2.69
.90
.80
.0001*
Working with a
group
Algebra 2
IMP
1.96
3.17
1.0
.80
.0001*
Listening to the
teacher
Algebra 2
IMP
2.80
2.75
.98
.96
.63
Listening to other
students
Algebra 2
IMP
2.19
2.79
.96
.83
.0001*
Copying from the
board
Algebra 2
IMP
2.40
1.91
1.02
1.04
.0001*
Working from a
textbook
Algebra 2
IMP
3.15
0.25
.93
.70
.0001*
Working from a
worksheet
Algebra 2
IMP
2.05
3.32
1.06
.97
.0001*
Sub-items
13
Student Perceptions of the Relative Importance of
Course Components
IMP students placed more value on working with
others
than
did
Algebra
2
students
(p = .0001*). By contrast, Algebra 2 students valued
the teacher’s explanations (p = .0005), and the
textbook (p = .0001) more than did IMP students.
Student Perceptions of Mathematics in Everyday
Activity
IMP students were significantly more likely to
identify a mathematical component in everyday
activities than were Algebra 2 students. This result is
evident in Table 7.
Table 7
Mean Math World Index for Algebra 2 and IMP Students
(Incomplete responses from some students led to a slightly
smaller sample size for both groups.)
Class type
Mean
SD
Algebra 2
(n = 113)
19.292
5.591
IMP
(n = 172)
•
•
•
•
using a calculator to work out interest paid on a
housing loan over 20 years (p = .003*)
planning a family’s two week holiday (p = .006*)
chopping down a pine tree (p = .007*)
buying clothing at a sale (p = .03*)
painting the house (p = .0001*)
Gender Differences
Comparison was made in this study of the attitudes
to mathematics of boys and girls in IMP and Algebra
classes, and of the boys’ and girls’ self-ratings of their
mathematics competence. These results are shown in
Table 8.
Girls in both class types were less likely than boys
to rate highly their own mathematical competence.
However this difference was only statistically
significant for students in Algebra classes. Both boys
and girls in IMP classes had similar positive attitudes
towards mathematics. In Algebra classes, both male
and female students felt negatively towards
mathematics, however boys’ attitudes were less
negative than those of girls. On the basis of these
findings, it appears that the IMP program was of
particular value to female students. The statistical
14
Class type &
Measure
Algebra 2
Self-rating
IMP
Self-rating
Algebra 2
Attitude
5.598
In particular, the IMP students were more likely to
view as mathematical:
•
Table 8
Gender Comparison of Self-ratings and Attitude
Measures for Algebra 2 and IMP Classes
p value
.0014*
21.477
significance of the direct comparison of Algebra 2
girls with IMP girls is quite clear from Table 8,
where the difference in mean attitude and self-rating
for the two groups of girls is even more striking
than in the comparison of the Algebra and IMP
cohorts reported in Tables 2 and 3.
IMP
Attitude
Gender
Mean
SD
Male
(n = 58)
7.333
1.875
Female
(n = 67)
6.433
1.994
Male
(n = 77)
7.636
1.297
Female
(n = 96)
7.271
1.410
Male
(n = 58)
–0.345
1.821
Female
(n = 68)
–0.676
1.872
Male
(n = 78)
0.756
2.021
Female
(n = 96)
1.146
p value
.0101*
.081
.3176
.2344
2.234
Conclusions
For the purpose of drawing conclusions from
the findings reported here, the inclusion of the
Algebra 4 sample in the study encourages the
extrapolation of conclusions from comparisons of
class types at a specific grade level to more general
conclusions comparing problem-based and
conventional instruction for high school
mathematics classes.
The conclusions that follow, however, relate
specifically to the study sample.
The Students as Learners
1. IMP students rated themselves as significantly
more mathematically able than did the Algebra
students.
2. IMP students held a significantly more positive
attitude towards their mathematics classes than
did the Algebra students.
3. On school sites where comparison was possible,
IMP students averaged higher SAT scores than
did pupils of conventional classes.
Problem-Based Mathematics Curriculum
4. IMP classes appeared to have less negative
outcomes for girls than did conventional Algebra
classes.
Student Perceptions of Mathematics
5. IMP students were significantly more likely to
perceive mathematics as a mental activity.
6. IMP students held beliefs consistent with a view of
mathematics as arising from individual and societal
need; while Algebra students were more likely to
view mathematical ideas as having an independent,
absolute and unvarying existence.
7. The IMP students were significantly more likely to
perceive mathematics as having applications in
daily use.
8. IMP students were significantly more likely than
Algebra students to believe that mathematical ideas
can be expressed “in everyday words that anyone
can understand.”
Instructional Alternatives
9. IMP students attached significantly more value to
interactive learning situations; whereas Algebra
students valued “the teacher’s explanations” and
“the textbook.”
10. IMP students valued writing and talking to other
students as assisting their learning. Algebra
students were significantly more likely to value
“drill and practice.”
11. (a) It is possible to identify a coherent and
consistent set of classroom practices which can be
associated with conventional instruction (cf.
Clarke, 1984).
11 (b) It is similarly possible to identify a set of
classroom practices which identify, in the students’
view, the characteristics of the IMP classroom.
11 (c) The characteristics of these two instructional
models are sufficiently distinct to represent clear
alternatives.
In conclusion, the classroom practices of the IMP
program, as reported by the students, placed greater
emphasis on a variety of modes of communication and
on facilitating student-student interaction than was the
case with conventional instruction. By contrast,
conventional instruction was perceived as solitary,
text-driven, and typically expressed through special
terms and numbers.
To what aspect of the IMP experience might we
attribute the student beliefs documented in this study?
The small-group, interactive classroom and the
Clarke, Breed, & Fraser
problem-based mathematics curriculum represent
two key characteristics of the Interactive
Mathematics Project. Whether such belief systems
would arise in interactive classrooms lacking a
problem-based emphasis or in more conventionally
taught, problem-based classrooms is a matter for
further research.
Certainly the IMP program has provided
students with significantly different experiences
from those found in conventional mathematics
classes, and these experiences appear to have led to
demonstrably different beliefs about mathematical
activity, mathematics learning, school mathematics,
and the mathematics evident in everyday activity.
The findings of other studies suggest that students
whose instruction has included experience with
open-ended tasks can be expected to perform more
successfully on both conventional and non-routine
tasks than students lacking that experience (for
instance, Sweller, Mawer & Ward, 1983). In
combination, this research suggests that a problembased curriculum is capable of developing
traditional mathematical skills at least as
successfully as conventional instruction, while
simultaneously developing non-traditional
mathematical skills and engendering measurably
different belief systems in participating students.
The nature of these different beliefs has formed the
basis of this study.
REFERENCES
Baird, J. R., & Mitchell, J. (Eds.). (1986). Improving the quality
of teaching and learning: An Australian case study – the
PEEL project. Melbourne: Monash University Printery.
Bishop, A. J. (1985). The social construction of meaning – a
significant development for mathematics education. For
the Learning of Mathematics, 5(1), 24–28.
California Post-Secondary Education Commission (CPEC).
(1989). Request for program proposals under the
Education for Economic Security Act, Federal Public Law
98-377 Fourth Year, San Francisco: California PostSecondary Education Commission and California State
Department of Education.
Clarke, D. J. (1984). Secondary mathematics teaching: Towards
a critical appraisal of current practice. Vinculum, 21(4),
16–21.
Clarke, D. J. (1985). The impact of secondary schooling and
secondary mathematics on student mathematical
behaviour. Educational Studies in Mathematics, 16(3),
231–251.
Clarke, D. J. (1986). Conceptions of mathematical competence.
Research in Mathematics Education in Australia, 2,
17–23.
Clarke, D. J. (1987). The interactive monitoring of children’s
learning of mathematics. For the Learning of
Mathematics, 7(1), 2–6.
15
Clarke, D. J. (1992). The role of assessment in determining
mathematical performance. In G. Leder (Ed.), Assessement
and learning of mathematics (Chapter 7, pp. 145–168).
Hawthorn: Australian Council for Educational Research.
teaching and learning of mathematics (pp. 125–146).
Journal for Research in Mathematics Education,
Monograph No. 4. Reston, VA: National Council of
Teachers of Mathematics.
Clarke, D. J., & McDonough, A. (1989). The problems of the
problem solving classroom. The Australian Mathematics
Teacher, 45(2), 20–24.
Erlwanger, S. H. (1975). Case studies of children’s conceptions
of mathematics – part 1. Journal of Children’s
Mathematical Behaviour, 1, 157–283.
Clarke, D. J., & Sullivan, P. (1990). Is a question the best answer?
The Australian Mathematics Teacher, 46(3), 30–33.
Lave, J. (1988). Cognition in practice. Cambridge: Cambridge
University Press.
Clarke, D. J., Stephens, W. M., & Waywood, A. (1992).
Communication and the learning of mathematics. In T. A.
Romberg (Ed.), Mathematics assessment and evaluation:
Imperatives for mathematics educators (Chapter 10). Albany,
New York: The State University of New York (SUNY) Press.
Lawson, M. J. (1990). The case for instruction in the use of
general problem-solving strategies in mathematics
teaching: A comment on Owen and Sweller. Journal for
Research in Mathematics Education, 21(5), 403–410.
Clarke, D. J., & Wallbridge, M. (1989). How many mathematics
are there? In B. Doig (Ed.), Everyone counts. Parkville:
Mathematical Association of Victoria.
Meece, J. L., Parsons, J. E., Kaczala, C. M., Goff, S. B., &
Futtennan, R. (1982). Sex differences in math
achievement: Toward a model of academic choice.
Psychological Bulletin, 91(2), 324–348.
Clarke, D. J., Wallbridge, M., & Fraser, S. (1992). The other
consequences of a problem-based mathematics curriculum.
Research Report No. 3. Mathematics Teaching and Learning
Centre, Australian Catholic University (Victoria). Oakleigh:
Mathematics Teaching and Learning Centre.
Owen, E., & Sweller, 1. (1989). Should problem-solving be
used as a learning device in mathematics? Journal for
Research in Mathematics Education, 20, 322–328.
Clarke, D. J., Waywood, A., & Stephens, W. M. (1994). Probing
the structure of mathematical writing. Educational Studies in
Mathematics, 25(3), 235–250.
Sullivan, P., & Clarke, D. J. (1991). Catering to all abilities
through the use of “good” questions. Arithmetic Teacher,
39(2), 14–21.
Cobb, P. (1986). Contexts, goals, beliefs, and learning
mathematics. For the Learning of Mathematics, 6(2), 2–9.
Wallbridge, M. (1992). Community perceptions of
mathematical activity. Unpublished master’s thesis,
Australian Catholic University (Victoria) – Christ Campus,
Oakleigh, Victoria, Australia.
Cobb, P., Wood, T., & Yackel, E. (1990). Classrooms as learning
environments for teachers and researchers. In R. B. Davis, C.
A. Maher, & N. Noddings (Eds.), Constructivist views on the
16
Schoenfeld, A. (1985). Mathematical problem-solving.
Orlando, FL: Academic Press.
Problem-Based Mathematics Curriculum
The Mathematics Educator
2004, Vol. 14, No. 2, 17–26
Impact of Personalization of Mathematical Word Problems on
Student Performance
Eric T. Bates & Lynda R. Wiest
This research investigated the impact of personalizing mathematical word problems using individual student
interests on student problem-solving performance. Ten word problems were selected randomly from a
mathematics textbook to create a series of two assessments. Both assessments contained problems exactly as
they appeared in the textbook and problems that were personalized using student interests based on studentcompleted interest inventories. Fourth-grade students’ scores on the non-personalized and personalized problems
were compared to investigate potential achievement differences. The scores were then disaggregated to examine
the impact of reading ability and problem type on the treatment outcomes. The results showed no significant
increase in student achievement when the personalization treatment was used regardless of student reading
ability or word problem type (t = –.10, p = .46).
.
“Problem solving is the cornerstone of school
mathematics. Unless students can solve problems, the
facts, concepts, and procedures they know are of little
use” (National Council of Teachers of Mathematics,
2000, p. 181). Students can learn mathematical
procedures, but without real-world applications, these
skills are rendered meaningless and are forgotten
readily. In the school curriculum, word problems allow
one means by which students can work toward
developing problem-solving skills within
contextualized settings that do not require application
of rote procedures. However, research has shown that
students have difficulty solving word problems (Hart,
1996).
At least three reasons have been proposed for why
students have little success solving word problems:
limited experience with word problems (Bailey, 2002),
lack of motivation to solve word problems (Hart,
1996), and irrelevance of word problems to students’
lives (Ensign, 1997). These factors should be addressed
in an effort to improve student performance on word
problems, a fundamental component of mathematics
education. Personalizing word problems—replacing
selected information with students’ personal
information—can address the latter two, motivation
and relevance, which may in turn lead to the first,
greater experience with word problems.
Eric T. Bates is a fourth-grade teacher at Sunnycrest
Elementary School in Lake Stevens, Washington. He holds a
Master’s degree in elementary education with a concentration
in technology in mathematics education.
Lynda R. Wiest is an Associate Professor of Education at the
University of Nevada, Reno. Her professional interests include
K–8 mathematics education, educational equity, and teacher
education.
Bates & Wiest
The purpose of this study was to investigate the
impact of personalizing word problems on fourth-grade
students’ problem-solving performance. Results of this
research, conducted at Copper Flats Elementary
School1 in Northern Nevada, were disaggregated to
examine how reading ability and problem type might
influence scores in solving personalized versus nonpersonalized problems.
Review of Related Literature
The Role of Word Problems
Conventional word problems, despite their
artificial nature, are likely to “stick around” in school
mathematics (D. Brummett, personal communication,
February 29, 1996; Sowder, 1995; J. Stephens,
personal communication, February 29, 1996). This
may be due to their strong grounding in tradition, their
potential for fostering mathematical thinking, their ease
of use (e.g., conciseness and practicality within the
confines of school walls), and a lack of abundant and
pragmatic alternatives. Word problems may, in fact,
serve several important functions in the mathematics
classroom: They provide questions that challenge
students to apply mathematical thinking to various
situations, and they may be an efficient means of
relating this thinking to the real world. Practically
speaking, word problems are either readily available in
mathematics texts or can be written in a short period of
time, which makes them useful to time-conscious
teachers (Fairbairn, 1993).
Personalization and Student Interest
The idea of individualizing instruction certainly is
not new. Almost a quarter of a century ago, Horak
(1981) stated, “Meeting the educational needs of the
17
individual student has long been a concern of
professional educators” (p. 249). Personalizing
instruction to student experiences and interests is one
way to individualize instruction that may be important
for mathematics learning (Ensign, 1997). In particular,
it can enhance interest and motivation, which are
critically important factors in teaching and learning.
Mathematical word problems have been targeted
for personalization. Students “don’t care how many
apples Bob gave to Suzy. They’re much more
interested in things like music, video games, movies,
trading cards, money, and friends” (Bailey, 2002, p.
61). Giordano (1990) adds, “student fascination with
problems can be enhanced when names, locations, and
events are changed to personal referents” (p. 25). It is
important that word problems appeal to students in
order to generate interest in and motivation for solving
a problem (Fairbairn, 1993; Hart, 1996) However, in
practice, classroom mathematics rarely links to
students’ life experiences (Ensign, 1997).
Research on Personalized Word Problems
Numerous studies have investigated the impact of
personalizing problems—inserting individual students’
names and/or information from their background
experiences into the problems they solve—on student
interest/motivation and problem-solving success.
Personalized problems have been computer-generated
in some cases. Most of these studies found positive
effects on the three major variables investigated—
interest, understanding, and achievement (Anand &
Ross, 1987; d’Ailly, Simpson, & MacKinnon, 1997;
Davis-Dorsey, Ross, & Morrison, 1991; Hart, 1996;
Ku & Sullivan, 2002; López & Sullivan, 1991, 1992;
Ross & Anand, 1987; Ross, McCormick, & Krisak,
1985; Ross, McCormick, Krisak, & Anand, 1985).
Several researchers and educators credit
personalization of word problems with positively
influencing student affect, such as interest and
motivation. Hart (1996) notes, “Most students are
energized by these problems and are motivated to work
on them” (p. 505). Davis-Dorsey et al. (1991) say
personalization fosters and maintains attentiveness to
problems, and Jones (1983) claims that personalized
problems invest students in wanting to solve them
correctly.
López and Sullivan’s (1992) research found
individual personalization (tailoring problems to
individual rather than whole-class interests) to be
particularly effective in fostering positive attitudes
toward word problems. However, Ku and Sullivan’s
(2002) study involving 136 fourth-grade Taiwanese
18
students and their teachers also found group
personalization to have a positive impact. Both
students and teachers using personalized problems
showed better attitudes toward the program than those
using non-personalized word problems. Ku and
Sullivan argue that familiarity (reduced cognitive load)
and interest are the major factors that lead to greater
success solving personalized versus non-personalized
problems.
Another major area where personalization of word
problems has yielded favorable results is student
understanding. Davis-Dorsey et al. (1991) say
personalization supports development of meaningful
mental representations of problems and their
connections to existing schemata, and that it creates
strong encoding that aids retrieval of knowledge.
Personalized word problems may be more meaningful
in general and make contexts more concrete and more
familiar (López & Sullivan, 1992). Familiar people and
situations in personalized problems can aid
understanding (Davis-Dorsey et al., 1991; López &
Sullivan, 1992).
In their research, d’Ailly et al. (1997) employed a
type of personalization known as self-referencing. A
variety of problems were taken from a standard
mathematics text and some of the character names
were replaced with the word you. One hundred
students in grades three, four, and five were asked to
solve the problems within a mix of self-referencing and
non-self-referencing problems. The researchers found,
“When a you word was involved in the problem,
children asked for fewer repeats for the problems, and
could solve the problems in a shorter amount of time
and with a higher accuracy” (p. 566).
As noted, d’Ailly et al.’s (1997) study found that
personalized word problems (specifically, those using
self-referencing) positively impacted student
achievement—the third main area where word problem
personalization can benefit students. Numerous other
researchers have attained similar results in this area,
although some findings demonstrate positive effects in
some cases but not others, as some of the following
studies show.
For their study, Ku and Sullivan (2002)
personalized problems using the most popular
items—as determined by a completed interest
survey—for students as a whole. Students attained
higher problem-solving scores on personalized
problems both on the pretest and on the posttest (i.e.,
before and after instruction). The 53-minute interim
instruction and review used either personalized or nonpersonalized problems. Students who worked with
Personalization of Mathematical Word Problems
personalized problems performed better on both
personalized and non-personalized problems than those
who received non-personalized instruction, suggesting
that transfer of learning had occurred from the
personalized to non-personalized problems.
Davis-Dorsey et al. (1991) studied the effects of
personalizing standard textbook word problems on 68
second-grade students and 59 fifth-grade students.
Prior to the treatment, all of the students completed a
biographical questionnaire that was later used to
develop the personalized problems. Personalization
proved to be highly beneficial to the fifth graders, but it
did not positively impact the second-grade students’
test scores.
Wright and Wright (1986) researched the use of
personalized word problems with 99 fourth-grade
students. They examined both the processes used to
solve the problems and the accuracy of the answers.
Interestingly, the researchers found that a correct
process was chosen more often when the problems
were personalized, but correct and incorrect answers
were given equally on personalized and nonpersonalized problems.
López and Sullivan (1992) found significant
differences favoring personalization on problemsolving scores for two-step but not for one-step
problems, although the seventh graders in their study
also scored higher on the latter in comparison with
non-personalized problems. The researchers say
personalization may be particularly important for more
demanding (e.g., unfamiliar or mathematically
complex) cognitive tasks. They found personalization
to be effective on a group basis—personalizing
problems using dominant interests of a group of
students—as well as on an individual
basis—personalizing problems for each student using
individual interests—in relation to students’ problemsolving scores.
Most evidence indicates that personalizing word
problems can be an effective technique in teaching and
understanding mathematical word problems.
Nevertheless, some research data suggest caution in
assuming that personalization of word problems
always yields positive results. As noted, López and
Sullivan (1992) found significant differences favoring
personalization for two-step but not one-step problems,
and the Wright and Wright (1986) study showed no
significant improvement in student achievement on
personalized word problems, even though students
more often chose appropriate solution strategies for
personalized problems. Ross, Anand, and Morrison
(1988) raise other issues for consideration. They
Bates & Wiest
suggest that the effectiveness of the personalized
treatment may wear off over time. The researchers
express concern that the higher scores on personalized
tests could be due, in part, to the novelty of the
personalization and that the novelty might dissipate if
the treatment were used often. They also point out that
preparing individualized materials could limit its use in
the classroom due to time constraints. Finally, in their
research with 11-year-olds, Renninger, Ewen, and
Lasher (2002) found that personalized contexts based
on individual interests can have a differential effect on
students. For the most part, these contexts encourage
students to connect with the meaning of problems. This
leads some students to consider a task more carefully
to be sure they understand it. However, it leads others
to assume falsely that they have answered a problem
correctly, which hinders a “healthy skepticism” that
encourages problem solvers to check their work after
completing problems.
More research is needed to address how problem
type interacts with word problem personalization,
where personalization has its greatest impact—student
attitude, understanding, or achievement, grade levels
and types of students that are most responsive to
personalization, the long-term effects of
personalization, and the potentially differential impact
of individual versus group personalization. Ku and
Sullivan (2002) also call for future research on tapping
technology’s potential for creating personalized
problems and on investigating the implications of using
personalized problems for assessment.
Research Purpose
The purpose of this exploratory study was to
investigate if the predominantly positive research
results concerning personalization of mathematical
word problems would apply to elementary school
students regardless of reading ability or word problem
type. The intent was to contribute to the body of
knowledge about the impact of personalizing word
problems and to extend previous explorations by
considering particular student subgroups and problem
types (simple translation and process, discussed under
Instrumentation). If the benefits of personalization
were to outweigh the time constraints of planning and
preparing for this type of activity, the use of
personalized mathematical word problems could be an
effective tool for elementary teachers working with
students who struggle to understand and solve word
problems.
19
Research Method
Sample
Participants in this study were fourth-grade
students at Copper Flats Elementary School. Copper
Flats is a small desert community in rural Northern
Nevada. The school receives federal Title 1 money,
reflecting the fact that Copper Flats Elementary serves
students from a lower-income community. The school
houses four fourth-grade classrooms. Students who
returned parental consent forms in all four of these
classrooms were selected for the present study.
Ninety-seven parental consent forms were
distributed. Of those, 42 were returned in time for the
study. Therefore, the sample included 42 students— 22
boys and 20 girls. Students not participating in the
study worked on classroom assignments given by their
regular teacher, while the participants completed the
assessments. All participants in this project were
present for the two data-collection sessions. By reading
ability, 20 participants ranked high, 8 ranked medium,
and 14 ranked low.
Research Design
This study was a quantitative analysis of the effects
of personalizing word problems on fourth-grade
students’ achievement in solving the problems. In the
fall of 2002, participants completed an interest
inventory to provide individual information for
personalizing the assessments. One week later,
participants were administered an instrument
containing 10 word problems to solve. On that
assessment, 5 problems were personalized and 5 were
not. Two weeks later, participants were given a similar
10-item instrument. On this second and final
assessment, parallel versions of the 5 problems that had
been non-personalized on the initial instrument were
personalized, and vice versa. Therefore, all
participants—across the two test administrations in
which they took part—answered 20 problems, 10
personalized problems and 10 similar problems that
were not personalized. The two-week period between
the two tests provided necessary time to reduce threats
to validity due to repeated testing of participants on
similar test items (Parsons & Brown, 2002).
During each test administration, each participant
was given an instrument and a blank sheet of paper on
which to solve the problems. Participants were allowed
15 minutes to complete each assessment. All
participants finished within the allotted time.
Teacher-reported scores on the Developmental
Reading Assessment (DRA) (Beaver, 2001)
20
established participants’ reading level for the purpose
of comparing achievement on personalized versus nonpersonalized problems in relation to reading ability.
DRA levels 30 to 38 are considered to be third-grade
reading ability, DRA level 40 is fourth-grade reading
ability, and DRA level 44 is fifth-grade reading ability.
For this study, participants with DRA scores higher
than 40 were considered high readers, or above grade
level. Participants with DRA scores at 40 were
considered m e d i u m readers, or at grade level.
Participants with DRA scores below 40 were
considered low readers, or below grade level.
This research was not designed to include a
qualitative component. However, student comments
were recorded as field notes on the few occasions
where students made relevant, unsolicited remarks.
Instrumentation
Ten problems were randomly selected from
Mathematics: The Path to Math Success (Fennell et al.,
1999), the third-grade mathematics text, for use in
developing the assessment instruments (see
Appendices B and C). This text was chosen because it
was the text used for teaching third-grade mathematics
at Copper Flats Elementary School; therefore, the
participants were familiar with the format of the
problems. The problems were selected by scanning
every third page of the text that contained word
problems. Of the word problems selected from those
pages, five of each of the two problem types described
below were drawn from a basket and incorporated into
the assessments.
The problems selected for the assessments were
differentiated by problem type. Five of the problems
selected for the assessments were simple translation
problems and five were process problems (L. R. Wiest,
personal communication, August 27, 2002). Simple
translation problems can be solved using a one-step
mathematical algorithm. An example of a simple
translation problem is: “There are 7 seats in each of 6
vans. How many seats are there in all?” (Fennell, et al.,
1999, p. 360). Process problems typically are not
solved through direct application of an algorithm.
Another strategy is generally sought and chosen, such
as working backward, drawing a picture or diagram, or
using guess-and-check. A sample process problem is:
“Jen is older than Arnie. Paul is older than Jen. Who is
the oldest?” (p. 313).
An interest inventory (see Appendix A) was
created to determine selected participant preferences.
Inventory items included students’ name, favorite toy,
favorite store, something to buy at that store, names of
Personalization of Mathematical Word Problems
friends, something they like to make, name of a game,
and favorite type of vehicle. Each inventory was used
to personalize the original textbook word problems.
Two assessments were developed from the word
problems taken from the mathematics text. Items from
the interest inventory replaced the original characters,
objects, and situations in order to personalize the
problems for each individual student on five of the ten
problems on each of the two instruments. On one
assessment, the odd-numbered problems were
personalized (see Appendix B). On the other, the evennumbered problems were personalized (see Appendix
C). Participants randomly received one instrument on
the first administration and the other instrument on the
second administration two weeks later. This method of
alternating personalized and non-personalized items on
the assessments was shown to be an effective
technique used in other research on this topic (DavisDorsey et al., 1991; d’Ailly, Simpson, & MacKinnon,
1997).
Data Analysis
A paired-samples t test (available online at
http://faculty.vassar.edu/lowry/tu.html) was performed
to compare the number of correct answers on
personalized versus non-personalized problems. Mean
scores and standard deviations were calculated and
significance was tested at the .05 level using a onetailed test. This analysis included 42 pairs of scores.
An additional paired-samples t test was performed
to compare the number of correct responses on
personalized versus non-personalized items
disaggregated by participants’ predetermined reading
levels. Again, mean scores and standard deviations
were calculated and significance was tested for at the
.05 level using a one-tailed test. There were 20 pairs of
scores at the high level, 8 pairs of scores at the medium
level, and 14 pairs of scores at the low level.
Two final paired-samples t tests were performed to
compare the number of correct responses on
personalized versus non-personalized test items
disaggregated by problem type. Each assessment
contained five simple translation problems and five
process problems. On the first of these two pairedsamples t tests, correct responses on the personalized
simple translation problems were compared to correct
responses on the non-personalized simple translation
problems. On the second of the two paired-samples t
tests, correct responses on the personalized process
problems were compared to correct responses on the
non-personalized process problems. Through these
statistical methods, mean scores and standard
Bates & Wiest
deviations were calculated with significance tested at
the .05 level using a one-tailed test. Both of these
analyses included 42 pairs of scores.
Results
Mean scores for the number of items answered
correctly out of ten showed a difference of .03 points
between the personalized and non-personalized
problems (see Table 1). This difference was not
statistically significant (t = –.10, p = .46).
Table 1
Paired-Samples t Test for Personalized and NonPersonalized Problems
Context
n
Mean
SD
t
p
Personalized
Non-personalized
42
42
5.26
5.29
2.07
2.28
–.10
,46
Table 2 provides mean scores for the number of
problems answered correctly out of ten, separated by
student reading level. The high-reader scores for nonpersonalized problems were .10 points higher than for
personalized problems, a nonsignificant difference (t =
–.26, p = .39). The medium-reader scores for nonpersonalized and personalized problems differed by .50
points, also favoring non-personalized problems. A
paired-samples t-test indicated that this difference was
not significant (t = –1.08, p = .15). The low-reader
scores were .35 points higher for personalized
problems than for non-personalized problems. This
was the only group who attained better scores on
personalized problems, although the scores were not
significantly higher (t = –.84, p = .20).
Table 2
Paired-Samples t Test for Personalized and NonPersonalized Scores by Reading Ability
Personalized
NonPersonalized
Reading
Group
n
Mean
SD
Mean
SD
t
p
High
Medium
Low
20
8
14
5.90
5.50
4.21
1.77
2.07
2.19
6.00
6.00
3.85
2.15
2.14
1.96
–.26
–1.08
–.84
.39
.15
.20
Mean scores for the number of problems answered
correctly out of 5 were separated by problem type (see
Table 3). Scores for simple translation problem means
were 0.16 points higher for non-personalized than for
personalized problems. A paired-samples t-test showed
that these differences were not significant (t = –.84, p =
.20). Process problem means showed a difference of
0.1 points between the personalized and non21
personalized problems, favoring the former. Again, this
difference was not significant (t = .45, p = .32).
Table 3
Paired-Samples t Test for Personalized and NonPersonalized Scores by Problem Type
Personalized
NonPersonalized
Problem
Type
n
Mean
SD
Mean
SD
t
p
Simple
Trans.
42
2.41
1.56
2.57
1.74
–.84
.20
Process
42
2.86
0.98
2.76
1.12
.45
.32
Discussion
The results of this study suggest that students are
no more successful answering word problems when the
word problems are personalized and reflect their areas
of interest than when the problems are taken verbatim
from a mathematics textbook. Only in the subgroup of
low-reading-level students and the subcategory of
process problems did the personalized problem scores
improve slightly, although statistically significant
differences were not found in either case. The mean
scores in each other subgroup and subcategory were
somewhat lower on the personalized versions of the
word problems than on the non-personalized versions.
These research results point to a different
conclusion than many previous studies on this topic.
However, given the rather substantial amount of
previous research weighted toward positive effects of
personalizing word problems and the reasons explained
below, it is still quite possible that personalized word
problems can be a beneficial part of school
mathematics programs. Several factors may have
caused the lack of positive findings in this study. First,
the personalized problems may not have adequately
addressed the three aforementioned reasons students
fail at mathematical word problems. Second, the age of
the students may have been a factor in the treatment’s
lack of success. Third, this study looked only at
comparisons of personalized and non-personalized
problems on assessments. No attempt was made to
introduce personalization as an instructional practice.
The three reasons offered earlier for why students
fail at solving mathematical word problems were
limited student experience with word problems
(Bailey, 2002), lack of motivation to solve the word
problems (Hart, 1996), and irrelevance of word
problems to students’ lives (Ensign, 1997). The format
of this study could not—and did not intend to—have
22
much impact on student experience with word
problems. By simply taking two 10-problem
assessments, student exposure to word problems was
not greatly increased. Increased motivation was
noticed, however, when students saw their names or
favorite things included in a problem. On several
occasions while completing the assessments, students
made comments such as, “Hey, this has my name,” or
“These problems are fun ones.” This acknowledgement
and the smiles that followed were taken as signs of
increased student motivation. It was anticipated that by
utilizing student names and other referents to student
lives, relevance would be increased. This may have
been the case to an extent, but just seeing their names
and favorite things may not have given the problems
enough personal context to encourage correct answers.
In effect, the ability of this study to address the three
major reasons students fail at solving word problems
was not substantial or sustained enough to help
distinguish performance on the two problem contexts.
Personalized problems per se might not be
advantageous unless they are an integral part of a
larger instructional effort.
The young age of the students may also have
contributed to the results of the present study. These
students fall at the lower end of the grade levels
previously researched on this topic. Most studies that
found positive results for personalized problems took
place at upper elementary or middle grades (Anand &
Ross, 1987; d’Ailly et al., 1997; Davis-Dorsey et al.,
1991; Hart, 1996; Ku & Sullivan, 2002; López &
Sullivan, 1991, 1992; Renninger et al., 2002; Ross &
Anand, 1987; Wright & Wright, 1986). Only two
studies included younger grades—second and
third—among the older grades they investigated
(Davis-Dorsey et al., 1991; d’Ailly, et al., 1997). The
present study dealt exclusively with fourth-grade
students and found no relationship between
personalization and student scores. Perhaps fourth
grade is somewhat early for the personalization
treatment to be effective. Interest in problem contexts
may become more important across the many years in
which students encounter school word problems. In
relation to their study involving the impact of word
problem context, Parker and Lepper (1992) state that it
is “clear that the need for techniques to enhance
student interest in traditional educational materials may
actually increase with age” (p. 632). Advancing grade
levels also deal with increasingly difficult mathematics
problems, the complexity of which may allow for a
factor such as personalization to influence student
performance. As noted earlier, López and Sullivan
Personalization of Mathematical Word Problems
(1992) found personalization to have a positive impact
on two-step but not one-step problems, leading them to
conclude that personalization may be particularly
important for more difficult problems. Use of thirdgrade problems in this study may also have reduced the
cognitive demands of this research task, creating less
sensitivity to or discrimination among problem
variations.
Several previous studies found a significant
increase in correct answers on mathematical word
problems when students were taught with the
personalized format (e.g., Anand & Ross, 1987; López
& Sullivan, 1992). After the instructional period,
participants in these studies were assessed using
standard word problems. The present study sought to
discover the effects of the personalized format on
student achievement on the test items themselves
without prior instruction using these types of problems.
Perhaps these two approaches yield different results.
Students may need time to adjust to the new problem
contexts.
One benefit that did appear in using this treatment
was student excitement. Similar to the Ross,
McCormick, and Krisak study (1985), many
participants were visibly and audibly excited to
discover the personalized problems. In informal
discussions after the test administrations, participants
reported that they really liked reading about themselves
and their friends. They enjoyed seeing familiar stores
and games they like to play in this testing situation.
This affirms Hart’s (1996) reference to the
personalized treatment that “students are energized by
these problems” (p. 505). It must be recalled, however,
that interest in problems can be detrimental to some
students, who may incorrectly assume that they have
attained correct answers (Renninger et al., 2002). Also,
too much interest in a problem context can distract
some students, particularly girls (Boaler, 1994; Parker
& Lepper, 1992). If these potential negative effects
took place in this study, they might have countered and
thus masked potentially positive effects in the overall
results.
Limitations of the Study
The two major limitations of this study were the
sample size and the somewhat simplistic nature of the
research design. The sample size was reduced due to
the small number of parental consent forms that were
signed and returned so that students could participate
in the study. Ninety-seven consent forms were
distributed, but only 42 (43%) were signed and
returned in time for the first test administration. (Time
Bates & Wiest
constraints prohibited a second distribution of consent
forms, which might have raised the return rate.) This
greatly reduced the sample size, thus limiting the
power of the data used to determine the effectiveness
of the treatment.
This study was also limited by its lack of
complexity. Merely assessing student performance
based on two test administrations was restrictive. It
only gave a look into the results of those two tests. It
would be interesting to discover how students might
perform on word problems when they were taught with
the personalized format. Time and other constraints did
not allow for this additional research component.
Analysis of solution strategies might have yielded
further information. It is also difficult to know what
long-term impact the motivational aspects of these
problems may have.
Implications for Further Research
This study, in conjunction with the professional
literature discussed earlier, yields at least three major
implications for future research.
•
•
•
The potential of personalization of word problems
as an instructional method should be studied.
Alternative technologies should be explored to
decrease the time-intensive nature of preparing
individualized word problems.
Longitudinal research should be conducted on the
impact of personalizing mathematical word
problems.
As a teaching strategy, personalization of
mathematical word problems has been shown to
increase student achievement, particularly in the upper
elementary and middle grades (e.g., Anand & Ross,
1987; López & Sullivan, 1992). While this study did
not find such results for assessment problems, which
may be due to the mitigating factors discussed earlier,
it did find some anecdotal evidence that supported
other research findings of increased interest in these
problems. This might be an important underpinning of
mathematics learning. Personalization as an
instructional strategy could be implemented at various
grade levels and studied to assess its effectiveness for
students of those ages. Rather than comparing test
items only, as the present study did, students could be
taught with the personalization treatment and assessed
on standard textbook word problems to determine the
level of transferability of any possible positive effects.
This instructional method may increase student
motivation and interest when learning how to solve
problems in mathematics, thereby increasing their
23
comprehension of the material and increasing their
scores on textbook and other assessments.
In order to employ personalization as a teaching
strategy—based on the assumption that it may yield
positive results in affect, understanding, and/or
achievement—alternative methods of personalizing
word problems would be needed to decrease the
amount of time researchers and educators spend
creating personalized problems. One such method
might utilize the Internet. The capability of the Internet
to deliver individualized materials immediately and
simultaneously to a large population of students
remains untapped. A researcher could develop a web
site that allows students to complete an interest
inventory online and then submit the inventory to the
server. The server would then apply that information to
an existing word problem template document, updating
the characters and other referents to individualize the
problems for each student. This process would take
only seconds and would eliminate researchers’ (and
later teachers’) time investment in personalizing
individual worksheets and tests. Students could then
either print the problems or complete them on the
computer screen. The preparation time would be
greatly reduced and the number of participants could
be increased significantly. This technique would allow
researchers to create countless individualized word
problems for student instruction, practice, and
assessment. Such research should include attention to
what types of problems lend themselves well to this
type of problem generation. In the research reported in
this paper, for example, problems using names were
the easiest to personalize, with difficulty increasing
where gender-specific pronouns were included. The
process problems seemed to require greater attention
than the simple translation problems in preparing
personalized problems, mirroring the greater
mathematical complexity of the former compared with
the latter.
The present study and similar earlier studies
discussed here have been shorter than three months in
duration. Long-term effects of the personalizing
strategy have not been determined at any educational
level. Researchers might look at the use of
personalized word problems in a classroom over the
course of a school year and its relationship to word
problem achievement on standardized tests.
Closing Thoughts
Personalization of mathematical word problems
may not be an efficacious approach in fourth-grade
classrooms due to the age of the students and the
24
simplistic nature of the word problems the students are
required to complete. This should not, however,
discount other research on the personalization of
educational materials. Other researchers have shown
personalization to be an effective method in teaching
older students to solve mathematical word problems.
Excitement and interest tend to be rare when
students are working on word problems. Fairbairn
(1993) suggested that the terms story problems and
word problems can invoke uncomfortable memories
for many people. This may be due to the fact that word
problems can be boring and tedious to solve.
Unfortunately, student motivation is difficult to
quantify. In the present study, as well as in others, the
excitement level of individual students visibly and
audibly rose when personalized problems were
presented. At the very least, personalization could be
used as an instructional strategy to break the monotony
of word problems containing unknown people, dealing
with unfamiliar situations, asking uninspiring
questions.
REFERENCES
Anand, P. D., & Ross, S. M. (1987). Using computer-assisted
instruction to personalize arithmetic materials for elementary
school students. Journal of Educational Psychology, 79(1),
72–78.
Bailey, T. (2002). Taking the problems out of word problems.
Teaching PreK–8, 32(4), 60–61.
Beaver, J. (2001). Developmental Reading Assessment. Upper
Arlington, OH: Celebration Press.
Boaler, J. (1994). When do girls prefer football to fashion? An
analysis of female underachievement in relation to ‘realistic’
mathematics contexts. British Educational Research Journal,
20, 551–564.
d’Ailly, H. H., & Simpson, J. (1997). Where should ‘you’ go in a
math compare problem? Journal of Educational Psychology,
89(3), 567–567.
Davis-Dorsey, J., Ross, S. M., & Morrison, G. R. (1991). The role
of rewording and context personalization in the solving of
mathematical word problems. Journal of Educational
Psychology, 83(1), 61–68.
Ensign, J. (1997, March). Linking life experiences to classroom
math. Paper presented at the Annual Meeting of the American
Educational Research Association, Chicago, IL.
Fairbairn, D. M. (1993). Creating story problems. Arithmetic
Teacher, 41(3), 140–142.
Fennell, F., Ferinni-Mundy, J., Ginsburg, H. P., Greenes, C.,
Murphy, S., Tate, W., et al. (1999). Mathematics: The path to
math success. Parsippany, NJ: Silver Burdett Ginn.
Giordano, G. (1990). Strategies that help learning-disabled students
solve verbal mathematical problems. Preventing School
Failure, 35(1), 24–28.
Hart, J. M. (1996). The effect of personalized word problems.
Teaching Children Mathematics, 2(8), 504–505.
Personalization of Mathematical Word Problems
Horak, V. M. (1981). A meta-analysis of research findings on
individualized instruction in mathematics. Journal of
Educational Research, 74(4), 249–253.
Jones, B. M. (1983). Put your students in the picture for better
problem solving. Arithmetic Teacher, 30(8), 30–33.
Ku, H. Y., & Sullivan, H. J. (2002). Student performance and
attitudes using personalized mathematics instruction.
Educational Technology Research and Development, 50(1),
21–34.
López, C. L., & Sullivan, H. J. (1991). Effects of personalized math
instruction for Hispanic students. Contemporary Educational
Psychology, 16, 95–100.
Appendix A: Interest Inventory
Favorite Toy
____________________
Name of a Store You Shop At
____________________
Something You Would Like to Buy at That Store ___________
Name Three Friends ________ ________ ________
Name a School Supply
____________________
Something You Like to Make
____________________
A Game You Like to Play With One Partner
Name a Type of Vehicle
________________
____________________
López, C. L., & Sullivan, H. J. (1992). Effect of personalization of
instructional context on the achievement and attitudes of
Hispanic students. Educational Technology Research &
Development, 40(4), 5–13.
Appendix B: Sample Assessment
National Council of Teachers of Mathematics. (2001). Principles
and standards for school mathematics. Reston, VA: Author.
Parker, L. E., & Lepper, M. R. (1992). Effects of fantasy contexts
on children’s learning and motivation: Making learning more
fun. Journal of Personality and Social Psychology, 62,
625–633.
Parsons, R. D., & Brown, K. S. (2002). Teacher as reflective
practitioner and action researcher. Belmont, CA: Wadsworth.
Renninger, K. A., Ewen, L., & Lasher, A. K. (2002). Individual
interest as context in expository text and mathematical word
problems. Learning and Instruction, 12, 467–491.
(odd numbered problems personalized)
1.
Four students are collecting empty soda cans. Josh has more
than Jon but fewer than Warren. Robby has the same number
as Josh. Who has the greatest number of cans so far?
2.
Tom has a ball. He passes it to Wally, and Wally passes it to
Anne. Anne passes it back to Tom. If they continue in this
order, who will catch the ball on the 10th throw?
3.
Suppose 30 bottles of glue are shared equally among 6 classes.
How many bottles of glue would each class get?
It’s the grand opening of Futura Florists! Every day for 8 days
they give away 50 roses. How many roses in all do they give
away?
Ross, S. M., & Anand, P. G. (1987). A computer-based strategy for
personalizing verbal problems in teaching mathematics.
Educational Communication and Technology, 35(3), 151–162.
4.
Ross, S. M., Anand, P. G., & Morrison, G. R. (1988). Personalizing
math problems: A modern technology approach to an old idea.
Educational Technology, 28(5), 20–25.
5.
Josh read 67 pages of a book. Jon read 32 pages. How many
more pages did Josh read than Jon?
6.
Jordan, Nina, Amy, and Gia are practicing for a dance. They
take turns dancing in pairs. If each girl practices one dance
with each of the other girls, how many dances do they practice
in all?
7.
A toy maker can put together 1 Gameboy™ every 6 minutes.
How many Gameboys™ can he put together in 60 minutes?
There are 7 seats in each of 6 vans. How many seats are there
in all?
Ross, S. M., McCormick, D., & Krisak, N. (1985). Adapting the
thematic context of mathematical problems to students’
interests: Individual versus group-based strategies, Journal of
Educational Research, 79(1), 245–252.
Ross, S. M., McCormick, D., Krisak, N., & Anand, P. (1985).
Personalizing context in teaching mathematical concepts:
Teacher-managed and computer-assisted models. Educational
Communication and Technology, 33(3), 169–178.
Sowder, L. (1995). Addressing the story-problem problem. In J. T.
Sowder & B. P. Schappelle (Eds.), Providing a foundation for
teaching mathematics in the middle grades (pp. 121–142).
Albany: NY, State University of New York Press.
8.
9.
Josh is older than Jon. Warren is older than Josh. Who is the
oldest of the three?
10. Paula made first-aid kits to sell at the fair. She made 1 kit on
Monday, 2 kits on Tuesday, 3 kits on Wednesday, and so on,
until Saturday. How many kits did Paula make on Saturday?
Wright, J. P., & Wright, C. D. (1986). Personalized verbal
problems: An application of the language experience
approach. Journal of Educational Research, 79(6), 358–362.
1
Pseudonym.
Bates & Wiest
25
Appendix C: Sample Assessment
(even numbered problems personalized)
1.
2.
Four students are collecting empty soda cans. Meg has more
than Jo but fewer than Sid. Bart has the same number as Meg.
Who has the greatest number of cans so far?
Josh has a ball. Josh passes it to Jon, and Jon passes it to
Robby. Robby passes it back to Josh. If they continue in this
order, who will catch the ball on the 10th throw?
3.
Suppose 30 musical instruments are shared equally among 6
classes. How many instruments would each class get?
4.
It’s the grand opening of Winco! Every day for 8 days they
give away 50 chocolates. How many chocolates in all do they
give away?
5.
Wendy read 67 pages of a book. Ellie read 32 pages. How
many more pages did Wendy read than Ellie?
Josh, Jon, Robby, and Warren are playing Battleship™. They
take turns playing Battleship™ in pairs. If each kid plays one
game of Battleship™ with each of the other kids, how many
games do they play in all?
6.
7.
8.
A toy maker can put together 1 toy robot every 6 minutes.
How many toy robots can he put together in 60 minutes?
There are 7 seats in each of 6 Toyotas™. How many seats are
there in all?
9.
Jen is older than Arnie. Paul is older than Jen. Who is the
oldest of the three?
10. Josh made dented cars to sell at the fair. Josh made 1 on
Monday, 2 on Tuesday, 3 on Wednesday, and so on, until
Saturday. How many did Josh make on Saturday?
26
Personalization of Mathematical Word Problems
The Mathematics Educator
2004, Vol. 14, No. 2, 27–33
Mathematics Placement Test: Helping Students Succeed
Norma G. Rueda & Carole Sokolowski
A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took the
recommended course as determined by their mathematics placement exam score and those who did not follow
this recommendation. The goal was to decide whether the mathematics placement exam used at Merrimack
College was effective in placing students in the appropriate mathematics class. During five years, first-year
students who took a mathematics course in the fall semester were categorized into four groups: those who took
the recommended course, those who took an easier course than recommended, those who took a course more
difficult than recommended, and those who did not take the placement test. Chi-square tests showed a
statistically significant relationship between course grade (getting a C– or higher grade) and placement advice.
The results indicate that students who take the recommended course or an easier one do much better than those
who take a higher-level course or do not take the placement exam. With achievement in coursework as the
measure of success, we concluded that the placement test is an effective tool for making recommendations to
students about which courses they should take.
There is a widespread recognition of the need for
appropriate placement in the mathematics courses for
undergraduate freshmen. Many colleges and
universities around the nation have used the
Mathematical Association of America (MAA)
Placement Test; others have designed their own exams
or used a combination of placement exams and other
measurements, such as ACT or SAT mathematics
scores and high school GPAs. Since the MAA has
discontinued its placement test program, the
responsibility has been put on individual institutions to
develop their own placement exam. The purpose of our
study was to determine the effectiveness of Merrimack
College’s placement test by examining the connection
between students enrolling in recommended courses
and their success in those courses.
Literature Review
In this section, we investigate some of the specific
methods reported in the literature for placing
undergraduates in their first mathematics course. We
point to some of the assumptions in these reports and
suggest some of the drawbacks to the method of
placement.
Cederberg (1999) reported on the three placement
tests administered at St. Olaf College in Minnesota.
Norma G. Rueda is a professor in the Department of
Mathematics at Merrimack College, North Andover, MA. Her
research interests include mathematical programming and
applied statistics.
Carole Sokolowski is an Assistant Professor in the Department
of Mathematics at Merrimack College, North Andover, MA.
Her research area is undergraduate mathematics education.
Rueda & Sokolowski
She explained that the placement recommendations
were based on a large number of regression equations
that required considerable expertise in development
and periodic redefinition. The placement test also
required the coordination of numerous categories of
student data used in the equations. Approximately 85%
of students who enrolled in a calculus course based on
the recommendations from the placement test at St.
Olaf College received a grade of at least C–.
Cohen, Friedlander, Kelemen-Lohnas and Elmore
(1989) recommended a placement procedure that was
less technically sophisticated than St. Olaf’s, but still
required considerable background data about students.
They recommended multiple criteria methods, which
included a placement test customized to an institution’s
curriculum. They started with sixty variables, and
found the eight best predictors: high school graduation
status, number of hours employed, units planned, age,
high school grade point average, mathematics
placement test score, reading placement test score, and
English placement test score. Cohen et al.’s study was
based on (a) thousands of surveys completed by
students and faculty members at eight California
community colleges, (b) a comparison between student
scores on assessment tests and grades in different
courses, and (c) the relationship between student
characteristics and their grades.
Krawczyk and Toubassi (1999) described a simpler
placement procedure used by the University of
Arizona. The University of Arizona used two
placement tests adapted from the 1993 California
Mathematics Diagnostic Testing Project (see
http://mdtp.ucsd.edu/). Students chose which test they
felt was most appropriate for their ability and choice of
27
major. One test covered intermediate algebra skills
and placed students in one of three levels of algebra or
a liberal arts mathematics course. The second test
covered college algebra and trigonometry skills and
placed students in courses from finite mathematics
through Calculus I. In the fall of 1996, their data
indicated that approximately 17% of freshmen placed
in College Algebra through Calculus I failed or
withdrew from their respective courses, compared with
a 50% attrition rate in the early 1980’s before the
mandatory testing and placement. Apart from a test,
they also considered other factors, such as high school
GPA.
A number of studies have investigated the use of
standardized tests, such as ACT and SAT. Bridgeman
and Wendler (1989) found that the mathematics SAT
score was a relatively poor predictor of grades
compared to placement exams. Their results were
based on grades from freshman mathematics courses at
ten colleges. Odell and Schumacher (1995) showed
that a placement test used in conjunction with
mathematics SAT scores could be a better predictor
than SAT scores alone. Their conclusion was based on
data from a private business college in Rhode Island.
Callahan (1993) studied the criteria followed at Cottey
College in Missouri to place students in the appropriate
level course, as well as their placement success rates.
As with the studies mentioned above, Cottey College
used several variables to achieve their results – the
MAA Placement Test, ACT and SAT mathematics
scores, and years of high school mathematics taken.
Each of these studies assumed that the success rates
were based on students following the placement
advice. Mercer (1995), on the other hand, conducted a
study to compare pass rates in a college-level
mathematics class for mathematically unprepared
students who enrolled in developmental courses and
those who did not. The results of this study showed a
statistically significant relationship between passing
the course and following placement advice.
Background
Merrimack College, located in North Andover,
Massachusetts, is a small four-year Catholic college
offering programs in the liberal arts, business, the
sciences, and engineering. Among the college’s
distribution requirements, students must complete three
mathematics or science courses, with no more than two
courses from the same department. Most of the
students take at least one mathematics course. Most
liberal arts majors usually choose Basic Statistics,
Finite Mathematics, or Discrete Mathematics to satisfy
28
the mathematics/science requirement. During data
collection for this study, business administration
majors were required to take Applied College Algebra,
Calculus for Business, and one other mathematics
course. Students majoring in science or engineering
generally were required to take more mathematics; for
example, engineering students were required to take
three calculus courses and one course in differential
equations. They also took Precalculus if they did not
place out of this course on their placement exam.
Since some incoming freshmen are not prepared to
take a college-level mathematics course, a non-credit
developmental mathematics course, Math I, has been
offered at Merrimack College since the fall of 1994.
Before 1994 we administered a mathematics diagnostic
exam to incoming students, but were unable to
accommodate students who were not ready to take a
college-level mathematics course. Instead, they
enrolled in a mathematics course at a higher level than
their exam score indicated they should. There was a
high failure rate among these students.
Because students at Merrimack College often have
difficulty in other courses if they have not completed
Math I, proper mathematics placement has become
important to all of our departments. For example, the
Chemistry Department now requires the students who
place into Math I to complete the course before they
take several of their chemistry courses. Krawczyk and
Toubassi (1999) have found similar results at the
University of Arizona in which all chemistry students
and 90% of biology students—whose placement test
scores indicated they should be placed in intermediate
algebra or lower—received grades below C– in their
chemistry or biology courses.
This interest in correct mathematics placement
extends beyond the chemistry department as evidenced
by the many questions from the business and liberal
arts faculty, as well as science and engineering faculty
at the meetings in preparation for orientation advising.
A major reason for our concern about student success
is that successful students are more likely to remain in
their studies. A high level of student retention is not
only academically and socially desirable at a school,
but it makes sense economically.
We do not place students according to their prior
high school GPA or whether they have taken calculus.
We do not assume that these factors indicate whether
or not they need algebra. In fact, many students placed
into Math I have had four years of high school
mathematics, including precalculus (and occasionally
calculus), but according to our placement test they do
not appear to understand the basic concepts of algebra.
Mathematics Placement Test
Mathematics Placement Test
All incoming freshmen at Merrimack College are
expected to take a mathematics placement test
developed by members of the Mathematics
Department. There are two versions of the exam, one
for students who will major in Business Administration
or Liberal Arts, and one for students in Science and
Engineering. The version for Business Administration
and Liberal Arts consists of two parts, elementary and
intermediate algebra. The version for Science and
Engineering students contains a third part that tests
students’ understanding of functions and graphs.
Students are instructed not to use calculators. From
1994 until 1999 the placement exam was taken at
Merrimack College in June during orientation or at the
beginning of the academic year. Since 2000 the exam
has been mailed to students at home. A Scantron form
with the students’ answers is mailed back to
Merrimack College and graded. Students are informed
that using outside resources or calculators may result in
their being placed in a course for which they are
unprepared and may result in their failing or
withdrawing from the course. There is not a difference
between these mail-in results and the previous
monitored exam results with respect to the percentages
of students who place into the various mathematics
courses. Thus, we believe that most students heed our
warnings. The recommendations are available to
students and advisors during June orientation. See
Appendix A for some problems similar to those given
in the mathematics placement test.
Part I of the placement test consists of seventeen
questions on elementary algebra. If a student does not
answer at least fifty percent of these questions
correctly, then that student must take Math I. For
students who score above fifty percent on Part I, liberal
arts majors may take any mathematics elective course;
business administration majors are placed into a
college algebra or a business calculus course,
depending on their overall score; and science and
engineering students may be placed into a college
algebra, Precalculus or Calculus I course, depending on
their total score. The specific recommendations
resulting from the mathematics placement exam are as
follows:
Science and Engineering:
Score below 9 in Part I ⇒ Math I
For those who score above 8 in Part I:
•
Score 20 or lower in Parts I–III ⇒ Applied College
Algebra
•
Score between 21 and 34 in Parts I–III ⇒ Precalculus
•
Score of 35 or higher in Parts I–III ⇒ Calculus I
Rueda & Sokolowski
Business:
Score below 9 in Part I ⇒ Math I
For those who score above 8 in Part I:
•
Score below 30 in Parts I–II ⇒ Applied College Algebra
•
Score of 30 or higher in Parts I–II ⇒ Calculus for
Business
Liberal Arts:
Score below 9 in Part I ⇒ Math I
Any other score ⇒ Enroll in a mathematics elective course
Data Analysis
We have performed statistical analyses since 1997
to study whether there was a relationship between the
score on the placement test and how well first year
students did on the first mathematics course taken at
Merrimack College. In order to determine this
relationship, we first examined the correlation between
total score on the placement test and students’ grades.
A preliminary study with n = 372 showed that the
correlation between the grade earned in a mathematics
course and the total score on the placement test was
0.466. For the same study, the correlation between
grades earned and SAT scores was 0.334. A multiple
regression model to estimate the final grade based on
the SAT score and the placement exam score gave the
equation: Final Grade = 0.80 + 0.00122(SAT) +
0.0641(Placement Exam). In addition, a t test for each
of the variables in the model indicated that the SAT
had a t value of 1.01 (p = 0.314) and the Placement
Exam had a t-value of 6.58 (p < 0.0005). Based on the t
test values and the p-values, we removed the SAT
variable from the model and concluded that the
placement exam was a better predictor of student
success.
Although we first used t-tests to compare the total
score on the placement test with SAT scores as
predictors of students’ final grades, we ultimately
decided against them as a means to further examine the
effectiveness of our placement test as a placement tool
for two reasons. First, there was not a true linear
relationship between the total score on the placement
test and placement. It was more like a step-function,
with a range of scores in each part of the test being
considered for placement. Second, there was a
relatively weak relationship between total score on the
placement test and final grades because students are
placed into so many different levels of courses. For
example, a student may have a very low placement test
score, be properly placed into an elementary algebra
class, and earn a high grade in that course. Thus, a
simple correlation between total score on the
placement test and grades earned was considered
inappropriate to determine the effectiveness of the
29
placement test, and therefore we decided to categorize
the data.
Each first-year student was categorized according
to the level of mathematics course taken: (1) the course
was easier than that recommended by the placement
test score; (2) the course was the recommended course;
(3) the course was more difficult than that
recommended (higher-level); or (4) the placement test
was not taken. Although the test was required, some
students—usually transfer students—were allowed to
take a mathematics course based on their mathematics
grade(s) at a previous institution. This policy has not
worked well and is being changed to require all
incoming students to complete the placement test.
Within these categories, students were counted
according to whether they (a) did well (received a
grade of C– or better) or (b) did poorly (received a
grade below C–). Chi-square tests were performed to
determine whether there was a relationship between the
level of the course taken and the grade received in the
class. Given the eight possible categories previously
described in this study, the chi-square test indicated
whether the percentage of students, say, who did well
in each category of course level taken was significantly
different from that in any of the other categories. The
use of the chi-square test assumes that a random
sample, representative of the population, was taken. In
this study, we used the entire population of freshmen
students who took mathematics in their entering fall
semester at our college for the years from 1997 to 2001
(n = 1710). The null hypothesis stated that there was no
association between the two variables. The alternative
hypothesis stated that the grade depended on whether
or not the student followed the placement test result
recommendation.
Results
We wanted to know whether there was an
association between the level of the course taken and
the grade earned. Table 1 shows the number and
percentage of students who did well in the class (C– or
higher) and the number and percentage of students who
did poorly (D+ or lower, or withdrew from the class)
from 1997 through 2001. It was generally accepted that
grades of D and F were unsatisfactory, as evidenced by
the fact that almost all comparable studies used cut-off
grades of C or C–. Those that used the C cut-off often
had a minimum grade requirement of C for a student to
move on to a subsequent course. Our department does
not have such a requirement, and thus the choice of C–
for this study was somewhat arbitrary. We felt that our
professors might be less likely to slightly inflate the
grade of C– to C than would those at schools with the
minimum requirement. As described above, students
were classified according to the level of the course
taken: easier than the one recommended, the one
recommended, or a higher-level course than the one
recommended. A fourth category was used in order to
include students who did not take the placement exam,
but took a mathematics course.
The data were analyzed using the chi-square test
(see Table 2). We found that there was a relationship
between the two variables for first year students who
took a mathematics course in the fall of 1997 [χ2 (3, n
= 369) = 24.66, p < 0.0005]. The same conclusion held
for the data corresponding to the fall of 1998
Table 1
Number and Percentage of Students Each Year Disaggregated by Grade and Course Category
1997
Course
Easier
Recom.
HigherLevel
No Exam
n
30
1998
1999
2000
2001
< D+
> C–
< D+
> C–
< D+
> C–
< D+
> C–
< D+
> C–
6
23
2
17
1
12
5
21
2
8
21%
79%
11%
89%
8%
92%
19%
81%
20%
80%
47
168
57
168
55
220
57
205
50
200
22%
78%
25%
75%
20%
80%
22%
78%
20%
80%
30
30
33
34
6
16
4
12
7
15
50%
50%
49%
51%
27%
73%
25%
75%
32%
68%
28
37
22
28
4
18
12
16
18
16
43%
57%
44%
56%
18%
82%
43%
57%
53%
47%
111
258
114
247
66
266
78
254
77
239
Mathematics Placement Test
Table 2
Contingency Table and Chi-Square Test (Expected counts are printed below observed counts; shaded cells indicate
expected counts less than 5.)
1997
Course
Easier
Recom.
HigherLevel
No Exam
n
< D+
1998
> C–
< D+
1999
> C–
< D+
2000
> C–
< D+
2001
> C–
< D+
> C–
6
23
2
17
1
12
5
21
2
8
8.72
20.28
6.00
13.00
2.58
10.42
6.11
19.89
2.44
7.56
47
168
57
168
55
220
57
205
50
200
64.67
150.33
71.05
153.95
54.67
220.33
61.55
200.45
60.92
189.08
30
30
33
34
6
16
4
12
7
15
18.05
41.95
21.16
45.84
4.37
17.63
3.76
12.24
5.36
16.64
28
37
22
28
4
18
12
16
18
16
19.55
45.45
15.79
34.21
4.37
17.63
6.58
21.42
8.28
25.72
111
258
114
247
66
266
78
254
77
239
Chi-Square
test
24.66
21.22
N/A
6.56
18.42
p value
<0.0005
<0.0005
N/A
0.087
<0.0005
[χ2 (3, n = 361) = 21.22, p < 0.0005]. We did not
perform a chi-square test for the data corresponding to
1999 because there were 3 cells with expected counts
less than 5 (see shaded cells in Table 2) and Moore
(2001) does not recommend the use of the chi-square
test when more than 20% of the expected counts are
less than 5. The relationship was not significant for the
fall of 2000 at an alpha level of 0.05 [χ2 (3, n = 332) =
6.56, p = 0.087] when taking p < 0.05 to be statistically
significant. The relationship was again significant for
the data corresponding to the fall of 2001 [χ2 (3, n =
316) = 18.42, p < 0.0005]. In sum, the percentage of
students who did well in their first undergraduate
mathematics course was higher for those students who
followed the advice or took an easier course than the
one recommended based on their placement test score.
The chi-square test only showed evidence of some
association between the variables. We then looked at
the tables to determine the nature of the relationship or
association (Moore 2001). Table 3 shows the number
of students who took the recommended course, an
easier one, a higher-level course, as well as the number
of students who did not take the placement exam, and
the mean and median grades on a 4.0 scale for each
group from 1997 to 2001. The following
correspondence between letter-grades and numbergrades was used at Merrimack:
A
4.0
B
3.0
C
2.0
D
1.0
A– 3.7
B– 2.7
C–
1.7
D– 0.7
B+ 3.3
C+ 2.3
D+
1.3
F
Rueda & Sokolowski
0.0
In addition, students who withdrew from a course were
counted and were assigned a number grade of 0 for this
study.
Discussion and Conclusion
From Table 1 we saw that students who took the
recommended course or an easier one did much better
than those who took a higher-level course or did not
take the placement exam. The same conclusion can be
drawn from Table 3, with the exception of the year
2000. In 2000 there was no significant difference
among the average grades received by those students
who took the recommended course and those who took
a higher-level course. An explanation for this may be
that the proportion of students who took a higher-level
course dropped from 19% in 1998 to 7% in 1999, and
to 5% in 2000. The few students who took a higherlevel course seemed to know that they would be able to
succeed. In addition, the percentage of students placed
into our developmental course, Math I, has been
decreasing. Twenty five percent of the 521 who took
the mathematics placement exam in 1998 were
recommended to take Math I. Twenty percent of the
532 who took the mathematics placement exam in
1999 were in that category. Those figures went down
to 14% out of 525 students in 2000 and 16% out of 517
in 2001. One of the reasons for this decrease was that
students were allowed to retake the placement exam
and to place out of our developmental mathematics
course. Even though that possibility was available to
students before, more effort has been made in the last
31
Table 3
Mean and Median Grade (on a 4.0 Scale) by Year and Course Category
3.1
3.3
26
2.4
2.5
(%)
10
Median
13
n
Mean
2.7
(%)
Median
2.7
n
2001
Mean
19
(%)
Median
3.0
n
2000
Mean
2.8
(%)
Median
29
n
1999
Mean
(%)
Median
n
Course
1998
Mean
1997
2.4
2.7
2.6
3.0
1.9
1.7
1.6
1.2
Easier
8%
215
Recom.
HigherLevel
n
2.4
2.7
58%
60
225
4%
2.4
3.0
62%
1.7
1.5
16%
65
No Exam
5%
67
2.0
50
1.6
1.7
2.7
22
1.7
2.0
22
262
3%
2.5
2.7
79%
2.2
2.0
7%
16
79%
2.4
2.5
5%
2.4
2.2
28
250
22
7%
1.8
2.2
34
18%
14%
7%
8%
11%
369
361
332
332
316
few years to avoid improperly placing students in a
non-credit course (Math I).
With the exception of 1999, there is no significant
difference, using z tests, between the proportions of
students who did well or poorly among those students
who did not take the mathematics placement exam.
It is not surprising that students who enrolled in the
recommended course or an easier course performed
better than did students who enrolled in a higher-level
course than the one recommended. What is important
here is that approximately 80% of these students who
took the recommended or easier course succeeded with
a grade of C– or higher.
We have found the mathematics placement exam
to be a useful tool to place students in the appropriate
mathematics course, and we have been successful in
convincing most of our students to follow our advice
with respect to which courses to take. A challenge for
us has been persuading students to take Math I, the
non-credit class, when they are not ready for a college
level course. While there is no perfect placement
method, we have found that our test is better than SAT
scores in placing students into the appropriate course.
In addition, our multiple-choice test is easier to
administer than methods used at other schools
mentioned in this paper.
From our experience, a well-designed in-house
placement test geared towards our curriculum is a
simple and powerful tool for placing incoming students
in an appropriate mathematics course. Keeping
32
2.5
83%
19%
1.8
275
8%
adequate records and analyzing them with regard to the
placement test’s effectiveness, as we have done in this
study, is a key component in maintaining the validity
and reliability of the test itself. A number of years ago,
the placement test score was viewed as the basis for a
“recommended” mathematics course for each student,
to be followed or not, as the student chose. Today, the
entire Merrimack community appears to view the test
score with increased respect because of the results
presented in this paper. These ongoing statistical
validations of the connections between proper
placement and successful achievement have served to
legitimize the placement test as part of a larger effort to
increase retention on our campus.
REFERENCES
Bridgeman, B., & Wendler, C. (1989). Prediction of grades in
college mathematics courses as a component of the placement
validity of SAT-mathematics scores. (College Board Report
No. 89-9). New York, NY: College Entrance Examination
Board.
Callahan, S. (1993, November). Mathematics placement at Cottey
College. Paper presented at the Annual Conference of the
American Mathematical Association of Two-Year Colleges,
Boston. (ERIC Document Reproduction Service No. ED
373813)
Cederberg, J. N. (1999). Administering a placement test: St. Olaf
College. In B. Gold, S. Keith, & W. Marion (Eds.),
Assessment practices in undergraduate mathematics (pp.
178−180). Washington, DC: Mathematics Association of
America.
Mathematics Placement
Cohen, E., Friedlander, J., Kelemen-Lohnas, E., & Elmore, R.
(1989). Approaches to predicting student success: Findings
and recommendations from a study of California Community
Colleges. Santa Barbara, CA: Chancellor’s Office of the
California Community Colleges. (ERIC Document
Reproduction Service No. ED 310808)
Krawczyk, D., & Toubassi, E. (1999). A mathematics placement
and advising program. In B. Gold, S. Keith, & W. Marion
(Eds.), Assessment practices in undergraduate mathematics
(pp. 181−183). Washington, DC: Mathematics Association of
America.
Mercer, B. (1995). A comparison of students who followed
mathematics advisement recommendations and students who
did not at Rochester Community College. Practicum prepared
for Nova Southeastern University, Ft. Lauderdale, FL. (ERIC
Document Reproduction Service No. ED 400014)
Moore, D. (2001). Statistics: Concepts and controversies (5th ed.).
New York: W. H. Freeman and Company.
Odell, P., & Schumacher, P. (1995). Placement in first-year college
mathematics courses using scores on SAT-math and a test of
algebra skills. PRIMUS, 5, 61−72.
Appendix A: Sample Problems
The following problems are similar to the ones given on the actual exam, but the format is different. These sample problems are free response.
The actual exam has a multiple choice format, in which several answers are provided to each problem, and only one of them is correct.
1.
Without a calculator, evaluate:
a. |5 – 9|
2.
b.
3
−8
a. 12 x + 3x
3x
b.
− 2x3
(−2 x) 3
e. 5x + 18 − 4(x + 7)
€
c. 3.42
d. 8 − 5
.02
9 12
Simplify the following:
€
2
3.
The following problems are representative of the additional section
of the Placement Test for Science and Engineering majors.
c. (25 x 4 y 8 ) −1 / 2
3 2
(3 x )
d.
a
4
+
a+2 a−3
f. log( x 2 − 1) − log( x + 2) + log x
6.
Let f (x) =
7.
Find the zeros of the function f ( x) = 2 x − 3 .
€
1 . Find the domain of f.
x −1
x +1
−1
( x), if f ( x) = 3 x + 2.
8.
Find the inverse function, f
9.
Which of the following points is not on the graph of y = e x
2
−1
?
Solve the following for x:
a. ax − b = 5
b. | 2 x + 1 | = 5
c. x = 2 x + 15
d. 1 = 64
4x
e. x + 2 y = 5
x + 4y = 7
f. x 2 + x < 6
e −1 ), (1, 0), (2, e 3 ), (3, e 8 )
10. Convert 135° to radians.
4.
Solve, then simplify the radical: x 2 − 2 = −4 x
5.
Find an equation for the line through the points (–1, –2) and
(1, 4). Give the slope, m, and the y-intercept, b.
Rueda & Sokolowski
(0,
11. Simplify in terms of sin θ : 1 − cos 2 θ = ?
12. Which of the following is greatest? sin 30° , sin 45° , sin 90° ,
sin 180°
33
The Mathematics Educator
2004, Vol. 14, No. 2, 34–37
In Focus…
Can the Ideal of the Development of Democratic Competence
Be Realized Within Realistic Mathematics Education? The Case
of South Africa
Cyril Julie
As is the case in any country there is a constant
search to improve the mathematics offerings presented
to school-goers in South Africa. The activity
surrounding this search was intensified after the
attainment of democracy. The primary aim of this
search was to establish a mathematics curriculum that
would result in productive learning and the mastery of
the goals set for the curriculum. These goals are
predetermined and are embedded within the country’s
ideological intent of its school educational endeavors.
Explicitly it is stated that school education should be a
mechanism to contribute towards the development of
“a participating citizen in a developing democracy
[who has] a critical stance with regard to mathematical
arguments presented in the media and other platforms”
(South African Department of Education, 2003, p. 9).
This goal was already proffered during the struggle for
liberation against apartheid and encapsulated in the
alternative school mathematics program during the
latter periods of that struggle. The alternative
mathematics curricular movement found expression in
People’s Mathematics (PM). People’s Mathematics
was an independent development in South Africa
during a particular historical moment but shared
commonalities with the varieties of Critical
Mathematics Education (Skovsmose, 1994;
Frankenstein, 1989). It differed from other varieties in
that it adopted the stance of critique but also
emphasized action against those practices which inhibit
human possibility.
The broad umbrella goals of People’s Mathematics
were “political, intellectual and mathematical
Cyril Julie obtained his doctorate in Mathematics Education
and Computer-Based Education from the University of Illinois
at Urbana-Champaign. He teaches courses at undergraduate
and graduate level in Mathematics Education at the University
of the Western Cape. He is currently involved in a major
project related to the development of research capacity
building in Mathematics Education in Sub-Saharan Africa. His
major research centers around the application and modeling of
school mathematics and the relevance of school mathematics
for students in grades 8 to 10.
34
empowerment” (Julie, 1993, p. 31). It is with these
goals in mind that Realistic Mathematics Education
(RME) was found a viable approach to school
mathematics with which the People’s Mathematics
movement could “live.” The particular characteristics
of RME that PM found useful were:
(a) It has a Lakatosian research program nature
(Gravemeijer, 1988). Being such a research
program there was some certainty of sustainability
due to Lakatos’s notion of strong research
programs fulfilling their predictions.
(b) The retention of the integrity of mathematics
through RME’s vertical and horizontal
mathematization (Streefland, 1990).
(c) The centrality of applications and modeling in
RME (De Lange, 1987).
(d) The seamless integration of the history of
mathematics and educational contexts from extramathematical domains (De Lange, 1987).
(e) Mathematics curriculum development that is
continuous and not of a once-off tinkering nature.
(f) A curriculum development research methodology
that is classroom-based and action-oriented
(Gravemeijer, 1994) with an accompanying
reporting strategy of research findings that is
understandable to practitioners.
In terms the umbrella goals enunciated above in
RME fulfilled the mathematical and intellectual ideals
((b) and (d)) of PM but not the political. This does not
imply that contexts of an overt political nature are not
included in the RME program. For instance, in the
following activity (De Lange & Verhage, 1992) around
national budgets and military expenditure, the overt
political dimensions are clearly discernable.
In a certain country, the national defense budget is
$30 million for 1980. The total budget for that year
was $500 million. The following year the defense
budget is $35 million, while the total budget is
$605 million. Inflation during the period covered
by the two budgets amounted to 10 per cent.
Democratic Competence
A. You are invited to give a lecture for a pacifist
society. You intend to explain that the defense
budget decreased over this period. Explain how
you would do this.
B. You are invited to lecture to a military academy.
You intend to explain that the defense budget
increased over this period. Explain how you would
do this.
What is not clear from these and other similar
activities is how these activities are to be used in
classrooms or how and whether there are follow-up
activities that take the intent of the activities beyond
the purely mathematical. The political empowerment
ideal within People’s Mathematics implied a
movement beyond this purely mathematical treatment
of issues of political import. This begs the question of
where does the political reside in mathematical
activity?
The political find expression in at least three areas
of mathematical activity. They are all within the arena
of the applications and modeling of mathematics. The
first of these is akin to the activity of De Lange and
Verhage above. What is added to this is opportunity for
overt reflection on those issues that relate to inequality
and discrimination on the basis of race, sex, social
class and economic developmental level of countries.
This is aptly illustrated by the following activity
(Frankenstein, 1989, p. 140):
Review the comparisons made in the following
three tables and write briefly about the connections
among the data in each table and any conclusions
and any questions you have about the given
information. (Note: Only one table is given here.)
Median incomes of full-time workers by occupation
(persons fourteen and above)
Major occupation group
Professional and technical workers
Non-farm managers & administrators
Sales workers
Clerical workers
Operatives (including transport)
Service workers (except private household)
1976 income ($)
Women Men
11,081
10,177
6,350
8,138
6,696
5,969
16,296
17,249
14,432
12,716
11,747
10,117
A second area where the political is overtly present
is during the model construction process. When a
mathematical model is constructed, interpretations and
translations take place. The given reality situation from
outside of mathematics is stripped down to make it
amenable for mathematical treatment, and the resulting
mathematical model is more a mathematical
representation of a stripped-down version of the
Cyril Julie
situation. In essence, there are three domains involved
in mathematical model making. These are the extramathematical reality, the consensus-generated reality
domain, and the intra-mathematical domains. The
characteristics of these domains are summarized in
Figure 1.
ExtraMathematical
Reality Domain
Issues of a
technical,
physical,
financial, social,
political,
environmental,
and so forth
nature are at
stake.
ConsensusGenerated
Reality Domain
IntraMathematical
Domain
Issues are
stripped of some
of the influencing
factors.
Mathematical
procedures and
ideas are
developed and
used.
Consensus is
reached based on
purposes and
interests.
Mathematical
conclusions are
reached.
Issues are
complex and
under a variety of
influencing
factors.
Figure 1. The translation of reality issues through different
domains.
It is during this process of translation that issues of
interests, ideological preferences and power are at
stake and contestations manifest themselves. These
contestations occur prior to the subjucation of the issue
for which a mathematical representation is to be
constructed to mathematical treatment. They occur
between the domain of the real and that of
mathematics. The resolution of conflicts,
interpretational variances, and interests render a
different reality. This reality is realized through
consensus and hence the postulation of a consensusgenerated reality domain as outcome of deliberations
on differences, interests, and intentions. It is within the
consensus-generated reality domain that ideological
intentions are explicitly revealed. A resulting
mathematical model is always a product of its
consensus-generated domain and thus different and
non-equivalent models might result for the same
phenomenon.
Lastly, the political rears its head during the final
phase of the modeling process when the adequacy-offit of the model to the reality situation is considered.
Here the issue is whether the derived mathematical
conclusions should be accepted or not. Consider, for
example, the mathematics of voting. In this instance a
35
dictatorship is defined as “[a member] in [a voting set]
A is called a dictator in A if and only if {x} is a
minimal coalition” with a minimal coalition “a subset
of K…if and only if K is a winning coalition and no
proper subset of K a winning coalition” (Steiner, 1968,
p. 189, 184). From this definition, if the results of a
general election in a country is such that a political
party has a majority such that they need to form no
coalition with any other party to carry an issue, then
that political party, and by implication, the government
of the day, is a dictatorship, at least mathematically.
For example, for the South African 2004 election, the
African National Congress (ANC) commanded that
69.75% of the parliamentary seats and a two-thirds
majority was needed to carry any decision. Thus,
according to the mathematical definition of a
dictatorship, South Africa is under the dictatorship of
the ANC. However, the lived experiences in South
Africa are such that this is not the case. Here then it is
clear that mathematical conclusions and lived
experiences might at times be in conflict and for all
intents and purposes it is sometimes wiser to be guided
by lived experiences rather than by the dictates of
mathematical conclusions.
One of the features that stand out in modern school
mathematics curricula in terms of the goals that are
offered is the development of democratic competence.
This competence is an individual’s (and a collective’s)
ability to make sound judgments about those issues
which structure and steer the affairs and practices of
humankind. The judgments are about the
appropriateness or not of the development and
implementation of the mechanisms that guide these
affairs and practices. These mechanisms are
profoundly driven by worldviews on issues such as
race, gender, and class differentials and generally the
kind of world that is envisioned. During the model
construction process issues of this political nature
come into play and hence the need for democratic
competence as stated. Further, as enunciated above,
part of this competence is a considered skepticism
towards being convinced through mathematical
argumentation. Democratic competence is normally
captured in the definition of Mathematical Literacy as,
for example, given below for the Programme for
International Student Assessment (Organization for
Economic Co-Operation and Development, 2003, p.
24):
Mathematical literacy is the individual’s capacity
to identify and understand the role that
mathematics plays in the world, to make wellfounded judgments and to use and engage with
36
mathematics in ways that meet the needs of that
individual’s life as a constructive, concerned, and
reflective citizen.
A question that can be asked is whether the goal of
the development of democratic competence can be
realized within the Realistic Mathematics Education
framework. What was indicated above is that RME
falls short as a paradigm in this regard. It does allow
for reflection on issues of a political nature but remains
at an awareness and conscientization level. It does not
allow for a consensus generation phase in modelconstruction. Neither does it explicitly allow for the
questioning of mathematical conclusions in relation to
lived and other experiences. It is suggested that RME
needs to be broadened to incorporate at least these
three issues in order to contribute more to the goal of
development of democratic competence. In doing so
there is a need to move beyond awareness and
conscientization. This “moving beyond” is what Ellis,
a leading South African and internationally recognized
cosmologist, suggested about 20 years ago. He stated:
…the only true basis of freedom is a realistic vision
of the alternative possibilities for change before us.
Mathematical studies can sometimes help us in
understanding what these alternative possibilities
are. But such an understanding is quite valueless
unless it affects our actions. An understanding of
the causes of any social wrong, which does not
lead to some corrective action to right that wrong,
is meaningless. (Ellis, 1974, p. 17)
Democratic competence is thus about an
individual’s (or collective’s) capacity to interact with
mechanisms which affect their lives and those of
society and to act where such mechanisms are to the
detriment of humankind. Where these mechanisms
have a mathematical base, or where they can be
explained and understood through mathematical
means, necessitates that schooling in mathematics be
called upon to provide the spaces for such interactions
and actions.
REFERENCES
de Lange, J. (1987). Mathematics, insight and meaning. Utrecht
University: Utrecht, Netherlands: OW & OC.
de Lange, J., & Verhage, H. (1992). Data visualization.
Pleasantville, New York: Sunburst.
Ellis, G. (1974). On understanding the world and the universe.
Professorial Inaugural Lecture, University of Cape Town,
Cape Town, South Africa.
Frankenstein, M. (1989). Relearning mathematics: A different third
R – Radical maths. London: Free Association Books.
Gravemeijer, K. (1988). Een Realistisch research programma. In
K. Gravemeijer & K. Koster (Eds.), Onderzoek, ontwikkeling
Democratic Competence
en ontwikkelingsonderzoek (pp. 106–117). Utrecht University,
Utrecht, Netherlands: OW & OC.
Gravemeijer, K. (1994). Educational development and development
research in mathematics education, Journal for Research in
Mathematics Education. 25(5), 443–471.
Julie, C. (1993). People’s mathematics and the applications of
mathematics. In J. de Lange, C. Keitel, I. Huntley, & M. Niss
(Eds.), Innovation in maths education by modelling and
applications (pp. 31–40). Chichester: Ellis Horwood.
Organisation for Economic Co-operation and Development
(OECD). (2003). The PISA 2003 assessment
framework—Mathematics, reading, science and problem
solving knowledge and skills. Retrieved November 23, 2004
from http://www.pisa.oecd.org/
Cyril Julie
Skovsmose, O. (1994). Towards a philosophy of critical
mathematics education. Dordrecht, Netherlands: Kluwer
Academic Publishers.
Steiner, H. G. (1968). Examples of exercises in mathematization in
secondary school level. Educational Studies in Mathematics,
1, 181–201
South African Department of Education. (2003). The national
curriculum statement: Mathematical literacy. Pretoria, South
Africa: Government Printers.
Streefland, L. (1990). Fractions in realistic mathematics education:
A paradigm of developmental research. Dordrecht,
Netherlands: Kluwer Academic Publishers.
37
CONFERENCES 2005…
MAA-AMS
Joint Meeting of the Mathematical Association of America
and the American Mathematical Society
http://www.ams.org
Atlanta, GA
January 5 – 8
AAMT 2005
Australian Association of Mathematics Teachers
http://www.aamt.edu.au/mmv
Sydney, Australia
January 17 – 20
AMTE
Association of Mathematics Teacher Educators
http://amte.sdsu.edu/conf_info_2005.shtml
Dallas, TX
January 27 – 29
RCML
Research Council on Mathematics Learning
http://www.unlv.edu/RCML
Little Rock, AK
February 24 – 26
Mα
The Mathematical Association
http://m-a.org.uk/resources/conferences/
Coventry, UK
March 30 – April 2
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org/meetings/index.htm#annual
Anaheim, CA
April 6 – 9
AERA
American Educational Research Association
http://www.aera.net/meeting/
Montreal, Canada
April 11 – 15
AMESA
Eleventh Annual National Congress
http://academic.sun.ac.za/mathed/AMESA/Index.htm
Kimberley,
South Africa
June 27 – 30
PME-29
International Group for the Psychology of Mathematics Education
http://staff.edfac.unimelb.edu.au/~chick/PME29/
Melbourne, Australia
July 10 – 15
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings/jsm/2005/
Minneapolis, MN
August 7 – 11
GCTM
GCTM Annual Conference
http://www.gctm.org/georgia_mathematics_conference.htm
Rock Eagle, GA
TBA
38
The Mathematics Educator (ISSN 1062-9017) is a student-produced journal published semiannually by
the Mathematics Education Student Association (MESA) at The University of Georgia. The journal promotes the
interchange of ideas among the mathematics education community locally, nationally, and internationally, and
presents a variety of viewpoints on a broad spectrum of issues related to mathematics education. TME also provides a
venue for the encouragement and development of leaders and editors in mathematics education. The Mathematics
Educator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical
Education).
The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other
professionals in mathematics education including:
•
•
•
•
•
•
•
•
reports of research (including experiments, case studies, surveys, philosophical studies, and historical studies),
curriculum projects, or classroom experiences;
commentaries on issues pertaining to research, classroom experiences, or public policies in mathematics
education;
literature reviews;
theoretical analyses;
critiques of general articles, research reports, books, or software;
mathematical problems (framed in theories of teaching and learning; classroom activities);
translations of articles previously published in other languages;
abstracts of or entire articles that have been published in journals or proceedings that may not be easily
available.
The Mathematics Educator strives to provide a forum for collaboration of mathematics educators with varying levels
of professional experience. The work presented should be well conceptualized; should be theoretically grounded; and
should promote the interchange of stimulating, exploratory, and innovative ideas among learners, teachers, and
researchers.
Guidelines for Manuscripts:
•
Manuscripts should be double-spaced with one-inch margins and 12-point font, and be a maximum of 25 pages
(including references and footnotes). An abstract should be included and references should be listed at the end of
the manuscript. The manuscript, abstract, and references should conform to the Publication Manual of the
American Psychological Association, Fifth Edition (APA 5th).
•
An electronic copy is required. (A hard copy should be available upon request.) The electronic copy may be in
Word, Rich Text, or PDF format. The electronic copy should be submitted via an email attachment to
tme@coe.uga.edu. Author name, work address, telephone number, fax, and email address must appear on the
cover sheet. The editors of The Mathematics Educator use a blind review process therefore no author identification
should appear on the manuscript after the cover sheet. Also note on the cover sheet if the manuscript is based on
dissertation research, a funded project, or a paper presented at a professional meeting.
•
Pictures, tables, and figures should be camera ready and in a format compatible with Word 95 or later. Original
figures, tables, and graphs should appear embedded in the document and conform to APA 5th - both in electronic
and hard copy forms.
To Become a Reviewer:
Contact the Editor at the postal or email address below. Please indicate if you have special interests in reviewing
articles that address certain topics such as curriculum change, student learning, teacher education, or technology.
Postal Address:
The Mathematics Educator
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
Electronic address:
tme@coe.uga.edu
39
The Mathematics Education Student Association is an official affiliate of the
National Council of Teachers of Mathematics. MESA is an integral part of
The University of Georgia’s mathematics education community and is
dedicated to serving all students. Membership is open to all UGA students,
as well as other members of the mathematics education community.
Visit MESA online at http://www.coe.uga.edu/mesa
TME Subscriptions
TME is published both online and in print form. The current issue as well as back issues are available online at
http://www.coe.uga.edu/tme. A paid subscription is required to receive the printed version of The Mathematics
Educator. Subscribe now for Volume 15 Issues 1 & 2, to be published in the spring and fall of 2005.
If you would like to be notified by email when a new issue is available online, please send a request to
tme@coe.uga.edu
To subscribe, send a copy of this form, along with the requested information and the subscription fee to
The Mathematics Educator
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
___ I wish to subscribe to The Mathematics Educator for Volume 15 (Numbers 1 & 2).
___ I would like a previous issue of TME sent. Please indicate Volume and issue number(s):
Name
___________________
Amount Enclosed ________________
subscription: $6/individual; $10/institutional
each back issue: $3/individual; $5/institutional
Address
40
In this Issue,
Guest Editorial… Researching Classroom Learning and Learning Classroom Research
DAVID CLARKE
The Consequences of a Problem-Based Mathematics Curriculum
DAVID CLARKE, MARGARITA BREED, & SHERRY FRASER
Impact of Personalization of Mathematical Word Problems on Student Performance
ERIC T. BATES & LYNDA R. WIEST
Mathematics Placement Test: Helping Students Succeed
NORMA G. RUEDA & CAROLE SOKOLOWSKI
In Focus… Can the Ideal of the Development of Democratic Competence Be Realized
Within Realistic Mathematics Education? The Case of South Africa
CYRIL JULIE