HOW DO STUDENTS ACQUIRE AN UNDERSTANDING
OF LOGARITHMIC CONCEPTS?
A dissertation submitted to the
Kent State University Graduate School
of Education, Health, and Human Services
in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
By
Ellen Mulqueeny
August 2012
© Copyright, 2012 by Ellen Mulqueeny
All Rights Reserved
ii
A dissertation written by
Ellen Mulqueeny
B.C.E., Cleveland State University, 1979
M.S., Cleveland State University, 1993
Ph.D., Kent State University, 2012
Approved by
_______________________, Co-director, Doctoral Dissertation Committee
Michael Mikusa
_______________________, Co-director, Doctoral Dissertation Committee
Joanne Caniglia
_______________________, Member, Doctoral Dissertation Committee
Donald White
Accepted by
_______________________, Director, School of Teaching, Learning,
Alexa Sandmann
and Curriculum Studies
_______________________, Dean, College and Graduate School of
Daniel F. Mahony
Education, Health and Human Services
iii
MULQUEENY, ELLEN, Ph.D., August 2012
Teaching, Leadership and
Curriculum Studies
HOW DO STUDENTS ACQUIRE AN UNDERSTANDING OF LOGARITHMIC
CONCEPTS? (332 pp.)
Co-Directors of Dissertation: Michael Mikusa, Ph.D.
Joanne Caniglia, Ph.D.
The use of logarithms, an important tool for calculus and beyond, has been
reduced to symbol manipulation without understanding in most entry-level college
algebra courses. The primary aim of this research, therefore, was to investigate college
students’ understanding of logarithmic concepts through the use of a series of
instructional tasks designed to observe what students do as they construct meaning.
APOS Theory was used as a framework for analysis of growth.
APOS theory is a useful theoretical framework for studying and explaining
conceptual development. Closely linked to Piaget’s notions of reflective abstraction, it
begins with the hypothesis that mathematical activity develops as students perform
actions that become interiorized to form a process understanding of the concept, which
eventually leads students to a heightened awareness or object understanding of the
concept. Prior to any investigation, the researcher must provide an analysis of the
concept development in terms of the essential components of this theory: actions,
process, objects, and schemas. This is referred to as the genetic decomposition.
The results of this study suggest a framework that a learner may use to construct
meaning for logarithmic concepts. Using tasks aligned with the initial genetic
decomposition, the researcher made revisions to the proposed genetic decomposition in
the process of analyzing the data. The results indicated that historical accounts of the
development of this concept might be useful to promote insightful learning. Based on
this new set of data, iterations should continue to produce a better understanding of the
student’s constructions.
ACKNOWLEDGMENTS
Many people have supported my efforts during this long process. I would like to
acknowledge them and thank them for the many hours of their valuable time that they
have given to me. My advisor and co-director, Dr. Michael Mikusa, has provided support
since my arrival at Kent State University. He has offered his guidance and expertise,
encouraging me to complete this journey. He challenged me to think as a researcher and
not solely as a classroom instructor. This caused me to pause and forced me to
reconceptualize my research interests several times in the course of this journey.
Thanks also to Dr. Joanne Caniglia, my co-director, for her support and
encouraging words along the way. I greatly appreciated both of you offering ideas for
improving my study as well as your reading and guidance through the numerous drafts of
this dissertation as it took its final form. Both of you have been a source of scholarly
comfort for the past three years and I believe I am a better mathematics educator for this.
I would also like to recognize Dr. Donald White for his mathematical insights and
his critical analysis of this study. I do appreciate the opportunity you have given me to
accomplish my goal of earning a doctorate.
Such a long journey would not have been possible without the support of friends
and colleagues along the way, Teresa Graham, Carol Phillips-Bey, Peggy Slavik, Jason
Stone, Ieda Rodrigues, Kingsley Magpoc, and Wade Zwingler: thanks for just being there
when I needed your support.
iv
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................................................................................................. iv
LIST OF FIGURES ......................................................................................................... viii
LIST OF TABLES ............................................................................................................. ix
CHAPTER
I.
INTRODUCTION TO THE STUDY .........................................................................1
Background Information ............................................................................................4
Problem Statement ....................................................................................................10
Purpose of Research ..................................................................................................15
Research Questions ...................................................................................................16
Theoretical Basis for the Study .................................................................................17
A Constructivist Perspective ............................................................................17
Advanced Mathematical Thinking ...................................................................18
The action construct ................................................................................22
The process construct ..............................................................................23
The object construct ................................................................................24
The schema construct. .............................................................................26
Instructional Implications ........................................................................................28
Goals of the Study.....................................................................................................30
Methodological Considerations ...............................................................................30
Summary of Chapter 1 ..............................................................................................32
Definition of Terms ..................................................................................................33
II.
REVIEW OF RELATED LITERATURE ................................................................36
Introduction ..............................................................................................................36
Influence of Symbols in Algebraic Thinking ...........................................................37
Functions and Associated Learning Difficulties .......................................................46
Students’ Understanding of Exponents and Exponential Functions .........................51
Inverse Functions .....................................................................................................56
Historical Development of Logarithmic Concepts ...................................................59
Students’ Understanding of Logarithms ..........................................................62
Advanced Mathematical Thinking and APOS Theory .............................................69
Summary of Related Work ......................................................................................71
v
III.
QUALITATIVE METHODOLOGY........................................................................75
Introduction ...............................................................................................................75
Qualitative Versus Qualitative Research Design ......................................................76
Qualitative Methodology ..........................................................................................81
Qualitative Research in Mathematics Education .............................................82
Teaching Experiments .....................................................................................83
Research Design........................................................................................................87
Research Site ............................................................................................................88
Sample.......................................................................................................................90
Procedure ..................................................................................................................92
The Role of APOS Theory ...............................................................................93
The genetic decomposition .....................................................................95
Defining the genetic decomposition of logarithms ........................98
Overview of instructional design .................................................103
Protocols ........................................................................................................106
Triangulation ..................................................................................................107
Inter-Rater Reliability .............................................................................................107
Data Analysis ..........................................................................................................109
Instructional Tasks ..................................................................................................111
Initial Assessment/Pretest ..............................................................................111
Task 1 .............................................................................................................112
Task 2 .............................................................................................................113
Task 3 .............................................................................................................114
Task 4 .............................................................................................................116
Task 5 .............................................................................................................117
Task 6 .............................................................................................................119
Summary of Chapter 3 ............................................................................................119
IV. DATA COLLECTION IN THE TEACHING EXPERIMENT..............................121
Introduction .............................................................................................................121
General Knowledge of Exponents and Functions: The Pretest .............................123
Growth of Student Understanding of Exponential Functions: Task 1 ....................131
Developing a Relationship between Exponential and Logarithmic
Functions: Task 2 .................................................................................................138
Strengthening the Understanding of Logarithmic Concepts: Task 3 .....................149
How to Construct a Table of Logarithms ......................................................159
Deepening the Exponential-Logarithmic Connection: Task 4...............................169
Exploring Properties of Logarithms: Task 5 ..........................................................176
Summary of Chapter 4 ............................................................................................185
Impact of the Teaching Experiment on Students Knowledge of
Logarithmic Concepts ...................................................................................185
Summary of Tom’s Performance ..........................................................185
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Summary of Doug’s Performance ........................................................192
Summary of Jim’s Performance ............................................................199
Summary of Earl’s Performance ...........................................................205
Growth in Knowledge of Logarithmic Concepts ...........................................212
Conclusion .............................................................................................................215
V.
DISCUSSION AND IMPLICATIONS ..................................................................218
Introduction .............................................................................................................218
How Do Students Acquire an Understanding of Logarithmic Concepts? ..............219
Question 1a: How Do Students Assign Meaning to the Symbolic
Notation Associated with Logarithms? .......................................................219
Understanding of exponents and exponential expressions ...................220
Development of the inverse exponential function. ...............................224
How do students assign meaning to log b x?.........................................227
The influence of the laws of exponents on multiple
logarithmic terms ................................................................................238
Question 1b: What Are the Critical Events That Contribute to the
Total Cognitive Understanding of Logarithmic Concepts? .........................243
Historical development of logarithms ...................................................243
Revised genetic decomposition ............................................................252
Cautions with Interpretations ..................................................................................255
Implications for Instruction.....................................................................................258
Implications and Recommendations for Future Research ......................................260
Concluding Remarks and Chapter Summary .........................................................263
APPENDICES… .............................................................................................................266
APPENDIX A. INSTRUCTIONAL TASKS .........................................................267
APPENDIX B. MATHEMATICAL BELIEFS AND ATTITUDES SURVEY ...291
APPENDIX C. CLASS NOTES: USING SEQUENCES TO DEVELOP
LOGARITHMIC CONCEPTS ......................................................................302
APPENDIX D. HOW DID BRIGGS CONSTRUCT HIS TABLE OF
COMMON LOGS ..........................................................................................304
APPENDIX E. CLASS NOTES: NON-INTEGER FACTORING .......................308
REFERENCES ................................................................................................................311
vii
LIST OF FIGURES
Figure
Page
1
Illustration of Napier’s geometric model ..................................................................60
2
Schemas and their constructions ...............................................................................96
3
Initial genetic decomposition .................................................................................102
4
Using sequences to develop logarithmic concepts..................................................163
5
Table of common logarithms ..................................................................................167
6
Revised genetic decomposition ..............................................................................256
viii
LIST OF TABLES
Table
Page
1
Overview of Instructional Tasks .............................................................................104
2
Results of Initial Calibration ..................................................................................109
3
Inter-rater Reliability Summary .............................................................................110
4
Mathematical Beliefs and Attitudes .......................................................................127
5
Summary of Tom’s Performance ............................................................................187
6
Summary of Doug’s Performance ..........................................................................194
7
Summary of Jim’s Performance..............................................................................201
8
Summary of Earl’s Performance .............................................................................207
9
Summary of Growth in Understanding ..................................................................216
10
APOS Conception of Exponential Expressions and Functions after Task I ...........224
11
Understanding of the Notation log b x .....................................................................235
12
Understanding of a Single Logarithmic Term and Exponential Concepts
At the Completion of Task 4 ...................................................................................238
13
Understanding of Logarithmic Concepts at the Completion of Task 5 ..................242
ix
CHAPTER I
INTRODUCTION TO THE STUDY
Karl Marx (1851/1963) stated: “The traditions of dead generations weigh like a
nightmare on the brain of the living” (p. 15). His thinking is applicable to the practices in
today’s classrooms. Debates surrounding issues of standards, methods, and curricula
have pervaded American education throughout the twentieth century. Research in
mathematics education has consistently advocated a constructivist approach to the
teaching of mathematics but is “struggling mightily to escape the stranglehold of the
outdated behaviorist learning theory that has dictated the course of mathematics teaching
for more than 40 years” (Battista, 1999a, p. 11). Scientific research on how students
learn mathematics has consistently found that students learn better when they are viewed
as constructors of their own knowledge (Battista, 1999a); however, political watch
groups have mandated that our schools operate as efficient agents for the dispensing of
knowledge, which effectively has eliminated creative thinking and honest study.
“Walk into any classroom in this country and ask a teacher what seems to
‘govern’ the mathematics taught. . . . [T]hey are sure to point to their state or local school
district’s curriculum frameworks” (Fennell, 2006, p. 3). According to Henderson and
Gornik (2007), virtually every state in the nation has a set of accountability standards
similar to Ohio’s strands, benchmarks, and grade-level indicators, which assess student
achievement based on these curricular goals and objectives. Teachers are using these
frameworks to guide their units of study in order to achieve successful test
1
2
results that rank their school systems. Policy makers at the state level feel that successful
completion of these exams will ensure that students are armed with the knowledge they
need to be successful in higher educational pursuits as well as the jobs and careers of the
future (American College Testing Board [ACT], 2007a). While the standards do provide
structure for the written curriculum, they do not ensure that instruction is carried out in
the spirit intended. Frequently teachers end up teaching for the test, which in principle is
incompatible with constructivist learning theories that allow students opportunities to
discover their own mathematical understandings.
With state testing programs well under way nationwide, postsecondary instructors
of mathematics face increasing numbers of underprepared students in their classrooms
(ACT, 2007a). In spite of recent efforts to improve secondary mathematics education,
the lasting effects are not realized in our nation’s universities, colleges, and two-year
institutions. Sources from the American College Testing Board (ACT) report that results
from the spring 2007 college entrance exam in mathematics showed the gap is widening
between what high school seniors know and what colleges want incoming students to
know. Additionally, results of the National Curriculum Survey released by ACT, spring
2007, report high school teachers’ views of preparation for college mathematics are not at
all similar to the views held by college instructors. High school teachers who are being
held accountable to teach students the content and skills listed in state standards tend to
place greater importance on covering more content and practicing skill-based activities
than their postsecondary counterparts. Furthermore, the report indicates that increased
3
numbers of high school teachers feel that their state standards and assessment practices
are better preparing students for postsecondary mathematics, while the majority of
college mathematics professors “responded that their state’s standards did a poor or very
poor job of preparing students for postsecondary work” (ACT, 2007a, p. 32). To
understand the depth of the growing differences in expectations between high school and
college mathematics, participants were asked, “How prepared for college-level work are
today's graduating seniors compared with graduating seniors in the past 5–10 years?”
The study showed 35% of high school teachers felt students were better prepared while
only 9% of postsecondary instructors felt students were better prepared to succeed at
college level mathematics (ACT, 2007a, p. 64).
The professoriate in many mathematics departments nationwide want their
students to have an in-depth knowledge of fundamental skills, but, according to data
collected from postsecondary instructors, high school seniors do not have the necessary
prerequisite skills to be successful at college-level work in mathematics (ACT, 2007b).
With the current push for mandated state proficiency tests, high school curriculums tend
to provide instruction on a broad range of topics; unfortunately most are covered
superficially, with students retaining little of what they supposedly learned (ACT,
2007b). Secondary mathematics programs in the United States tend to be topic-driven.
According to Cuoco (2004),
By the time one reaches high school, we end up with an 18-chapter compendium
of topics that range from graphing equations to triangle trigonometry to data
4
analysis to complex numbers. These “monster” textbooks have become de
facto definitions for the American high school curriculum. (p. 1)
With too much information and not enough time in the school year to adequately
cover the material, students pass through most of their mathematical coursework in
classrooms that emphasize rote and procedural knowledge, leaving students unable to
construct any meaningful mathematical knowledge from their experiences (Glass, 2002).
Background Information
The subject of logarithms, like the notorious “asses’ bridge” in Euclid (Elements
I,5) for earlier generations, seems to mark an intellectual rite of passage: before
going over there is a sense of unfathomable mystery, even danger, ahead;
afterwards there is still some wonder and perplexity at just what one has learned.
Some stumble and feel forever excluded, like the lame boy of Hamelin; others
press on and on and still do not come to the end of what is undeniably a paradigm
of the rich complexity of mathematical concerns. (Fauvel, 1995, p. 39)
With the widespread use of calculators in today’s mathematics classroom, the role
of the logarithm in the mathematics curriculum has forever changed. No longer required
as a computational tool, its role is now seen as the inverse of the exponential function or
as a meaningful application in mathematical sciences; however, student understanding of
this mathematical concept is restricted. Students regularly report “seeing” the material in
earlier coursework but report they have forgotten the “rules.” However, once presented
the right rule they are able to perform the needed calculations, but are unlikely to explain
5
why the answer is correct. Why are students unable to remember information presented
at an earlier time? Did they ever understand logarithmic concepts? Research documents
that student understanding of this topic is limited (Berezovski, 2004; Chesler, 2006; Gol
Tabaghi, 2007; Kenney, 2005; Weber, 2002a). APOS (Action-Process-Object-Schema)
Theory defines this type of limited understanding as action-oriented. A procedure or
action will have a fairly rigid structure, which often is committed to memory; however,
the learner can only understand the procedure when presented with an external cue
detailing the next steps to make (Asiala et al., 1996; Dubinsky, 1994). Skemp (1977)
might refer to this as instrumental understanding. Students are aware of the rules used to
manipulate and solve logarithmic or exponential equations and can readily use the rules
to obtain correct solutions; however, they do so without any sound mathematical
reasoning. In both cases, students believe they understand the concepts; after all, they got
the right answer. For example, students will connect to a rule, which if followed
carefully will produce correct solutions. However, when asked a question that does not
quite fit the rule, they are powerless to begin; they have failed to develop any cognitive
structures that will allow them to connect new ideas to what they already know.
Understanding is localized; the learner may know what to do but is unable to reason
about it. There is no overall cognitive image of the concept (Battista, 1999a, 1999b;
Sfard, 1994; Skemp, 1977; Wilson, 2003).
Mathematics educators will agree that educational issues in the teaching and
learning of mathematics have not substantially changed over the years. What has
6
changed, however, is mathematics education as an independent field of research; and it
has gained momentum. Davis (1990) points out that research in mathematics education
has provided new conceptualizations and metaphors for thinking about and observing
mathematical behavior. Researchers are concerned not only with tools and methods that
facilitate learning, but also with learning theories that support students’ understandings as
well as teachers’ conceptions. Scientific research on how students learn mathematics has
consistently found that students learn better when they are viewed as constructors of their
own knowledge (Battista, 1999b); however, the predominant form of instruction in U.S.
classrooms has focused on rote learning and memorization. Using behaviorist-learning
techniques, students are unable to make any meaningful connections or constructions
needed to interiorize, encapsulate, and coordinate their experiences into new schemas.
Competing views of school algebra make it difficult for mathematics educators to
determine essential qualities desirable for upper-level algebra curriculum and how best to
achieve these goals. Leaders in the field concur that most students easily learn numbers.
Many even develop a large vocabulary of mathematical words, such as add, subtract,
multiply, divide, exponent, variable, and fraction; however, few have mastered the
structures of mathematics. Concepts like inequality, expression, equation, the
relationship between a polynomial and its roots, logical argument, exponentials and
logarithms, and the notions of function and limit— all are simple on the calculation level
but difficult conceptually (Hauk, 2002). Mathematicians who support the traditional
view of algebra place importance on students’ ability to manipulate algebraic
7
expressions. According to Saul (1998, as cited in Kieran, 2007), “This emphasis on
polynomial and rational expressions is oriented toward recognizing form, which is
considered one of the most important aspects of school algebra within this perspective”
(p. 709).
Teachers of mathematics who identify with this perspective often feel instruction
is best accomplished using drill and practice. It is felt that additional time spent
practicing will eventually lead to mastery. However, research indicates that techniques
required for proficiency are not difficult to master, but serious conceptual dilemmas arise
as students attempt to make sense of mathematical concepts learned in isolation (Sfard,
2000). This is because they are unable to retain the material long enough to form any
meaningful connections. In direct instruction, students are treated as empty vessels
waiting to be filled with knowledge. The learner is presented with a rule, which
describes how to perform certain operations on figures; when such steps are completed,
the proper result is obtained. However, no reasons were given for taking a single step;
and, when the learner had worked through the steps and obtained the result, he would
understand neither what the result was nor the use or application of the result. With
traditional instruction, the learner became fixated and wholly dependent on the book or
more frequently the teacher for their mathematical knowledge (Confrey, 1994).
“Traditional methods ignore recommendations by professional organizations in
mathematics education, and they ignore modern scientific research on how children learn
mathematics” (Battista, 1999b, p. 424). Yet traditional teaching pervades secondary
8
mathematics programs nationwide, despite evidence that it does not support development
of higher cognitive skills (Confrey, 1994).
With its roots firmly anchored in Edward L. Thorndike’s traditional bond theory,
traditional instruction can be seen as a relatively familiar sequence of events: “an
introductory review, a development portion, a controlled transition to seatwork, and a
period of individual seatwork” (Confrey, 2004, p. 107). Many reform efforts in
mathematics education conducted during the 1980s captured “educators’ attention for
short periods of time but failed to address critical issues that are at the root of students’
difficulties with mathematics” (Ellis & Berry, 2005, p. 7). These efforts, while
invaluable to later reform movements, failed to “substantially alter deeply held beliefs
about the nature of mathematics, how it is to be taught, the sort of learning that is valued,
and how success is determined” (Ellis & Berry, 2005, p. 8). Reform movements,
however, can lead to or directly influence transformative changes which redefine
epistemological positions in the learning and teaching of mathematics.
Education leaders acknowledge the importance of preparing our students to meet
the challenges of the technological forces at work in society yet acknowledge that this is
a daunting task. The National Research Council (1989), in its publication Everybody
Counts, characterized undergraduate mathematics as the linchpin for revitalization of
mathematics education and reminded us that critical curricular review and revitalization
take time, energy, and commitment. The Council asserted:
9
Research on learning shows that most students cannot learn mathematics
effectively by only listening and imitating yet, most teachers teach mathematics
just this way. Research on learning shows that students actually construct their
own understandings based on new experiences that enlarge the intellectual
framework in which ideas can be created. Much of the failure in school
mathematics is due to a tradition of teaching that is inappropriate to the way most
students learn. (p. 6)
The division between procedural skills and conceptual understandings, how much
of each should be taught, how they should be taught, and how and when to use
technology as an aid to assist with procedural skills and conceptual understandings is
the cornerstone of the paradigm shift in mathematics education (Shore, 1999). This
understanding seems to echo contemporary views of the National Council of Teachers of
Mathematics (NCTM) as well as countless others in mathematics education. NCTM in
its Principles and Standards for School Mathematics (2000) builds a case for learning
mathematics that requires construction of knowledge from prior experiences and
understandings, not passive reception of facts. It claims students need to be actively
engaged in the creation of their mathematical knowledge to take ownership of this new
information.
Unwilling or unable to take responsibility for their own learning once enrolled in
college, some students still expect and even demand that instruction continue in this
familiar format in their first college mathematics course; however, the expectations of
10
many college professors differ significantly. Students are expected to be able to apply
what they should have learned in high school. Skemp (1977) views this as relational
understanding: not only do students know how to use the rules fluently, they know why
they work and can use them as a framework to support further learning. Mathematics
educators who align themselves with the constructivist view of learning algebra agree
that mathematics is more than an accumulation of facts and rules. It is a language of its
own that needs to be learned. It needs to be cultivated and to have its structure developed
by an experienced instructor. It is about problem-solving and number sense, a search for
order, and an exercise in abstract thinking. It provides a succinct framework to evaluate
seemingly complex ideas with precise notation.
Problem Statement
Seeing there is nothing . . . that is so troublesome to mathematical practice, nor
that doth more molest and hinder calculators, than the multiplication, division,
square and cubical extraction of great numbers, which besides the tedious
expense of time are for the most part subject to many slippery errors, I began
therefore to consider in mind by what certain and ready art I might remove those
hindrances.
John Napier, A description of the admirable Table of Logarithms, 1616.
How did a topic that historically represented a major contribution to
computational mathematics become so meaningless to secondary mathematics students?
The discovery of logarithms supported the massive calculations needed for astronomy
11
and navigation; however, mathematicians took notice and logarithms that were once used
only as computational tools took on a life of their own. A method that began as strictly a
computational device later was shown to have a significant impact on understanding
inverse-function relationships, and it became clear that logarithms held the “vital key in
the new mathematics of calculus” (Smith, 2000, p. 773). An important tool for calculus
and beyond, the use of logarithms has been reduced to symbol manipulation without
understanding in most entry-level college algebra courses. In other words, what we end
up with is an “easily reproducible mental experience of a mark or character strings with
no other mental activity or structure beyond this primitive experience” (Harel & Kaput,
1991, p. 89).
Euler (1810), in one of his more notable publications, Elements of Algebra,
presents his definition of logarithmic functions:
Resuming the equation
, we shall begin by remarking that, in the doctrine
of Logarithms, we assume for the root a, a certain number taken at pleasure, and
suppose this root to preserve invariably its assumed value. This being laid down,
we take the exponent b such that the power
becomes equal to a given number
c; in which case this exponent is said to be the logarithm of the number c . . . . We
see, then, that the value of the root a being once established, the logarithm of any
number, c, is nothing more than the exponent of that power of a, which is equal to
c; so that c being equal
91)
, b is the logarithm of the power
. (Euler, 1810, p.
12
Most textbooks begin their discussion of logarithmic notation after exponential
and inverse functions have been introduced. Typically, a definition similar to Euler’s,
stating that every exponential function of the form ( )
is a one-to-one function
and therefore has an inverse function, is given. After the variables x and y have been
interchanged, the student has successfully created the implicit form of the inverse
function; however, the inverse function needs to be defined explicitly. Discussion then
continues along the line of something like this: “Noting that previous algebraic skills are
no longer adequate to solve the equation
for the exponent y, a “new” procedure
must be developed” (Aufmann, Barker, & Nation, 2005, p. 383). Students are next told
that a compact notation is needed to represent this procedure; hence, the rule
if and only if
is given. Next, the logarithmic function is defined as ( )
followed with a statement telling the student this is the inverse function of
( )
Prior instruction has fostered the notion that procedures are computational rules to
follow; however; students are now being asked to develop an entirely new type of mental
image. Logarithmic functions belong to a class of functions referred to as
“transcendental.” Logarithmic functions along with exponential and trigonometric
functions transcend algebra in the sense that these functions cannot be expressed in terms
of a finite sequence of algebraic operations. In other words, it is generally not possible to
relate the value of ( ) to its input x by a finite number of algebraic operations. Without
a clearly defined set of algebraic rules to follow, students struggle to make sense of the
13
concepts. What exactly is the nature of the concept image that students develop in regard
to the symbolism after instruction? Is it in conflict with existing knowledge structures?
Development of a concept image permits the researcher to examine how the
“interpretation of a concept may be accommodated in the mind and how the students may
fail to understand or misunderstand some aspect of the concept” (Mamona-Downs &
Downs, 2008, p. 159). Is the difficulty students encounter in concept acquisition
intertwined with the notation that “log” is the procedure? Do they see this as a word, a
variable name, or a procedure? Has their “loose” attention to the definition of the
symbols involved in the algebraic notion compromised mathematical meaning? If
students’ conceptual structures do not reflect conceptual understanding, how can teaching
of logarithmic concepts be improved?
Although the original motivation for the teaching and learning of logarithms has
all but disappeared from today’s mathematics curriculum, students and teachers are left
wondering: what are logarithms used for, and why are they still on the syllabus?
Calculations that once proved tedious for mathematicians are no longer problematic. The
development of tools to make computation easier, more accurate, and faster has
predicated a change in the approach to teaching this topic; however, for most students,
“log” is a mysterious button on their calculator.
The National Council of Teachers of Mathematics (NCTM) released An Agenda
for Action in 1980 recommending that technology be made available to all students so
that difficulties encountered with pencil-and-paper activities would not interfere with the
14
learning of problem-solving strategies (Klein, 2003). What implications does this have
for the learning and teaching of logarithms? Furthermore, the 1980 document warned, “It
is dangerous to assume that skills from one era will suffice for another” (NCTM, 1980,
p. 6). Before calculators were introduced into the classroom, pencil-and-paper
computation was the only accepted procedure available for use by the student and the
teacher. The de-emphasis of pencil-and-paper calculations signaled important changes in
classroom behavior and structure. Calculators, properly used, can act as a scaffolding
agent to enable the learner to bridge minor gaps in background knowledge to reach higher
levels of mathematical understandings. But how do we use this tool to bridge the
ever-widening gap between students’ procedural knowledge and conceptual
understanding of a logarithm? Because of the technology introduced in the late 1970s,
understanding of logarithmic concepts and their associated properties has plummeted
(Steele, 2007). In light of its changed role in the curriculum, what does it mean to
understand logarithms? Some may speculate on the usefulness of this topic in the
secondary curriculum. Still others will question its role at the postsecondary level. If it is
not used as a computational tool, is it necessary that we continue to teach this topic for
non-calculus bound students? In light of the push for quantitative literacy, it seems
unlikely that logarithms will disappear from the curriculum as they have many useful
real-world applications.
\
15
Purpose of Research
Napier was the first to develop a notational system to express how multiplicative
structures could be related to additive structures. Napier’s term logarithmus, or
“reckoning number,” does not offer any intuitive notion of the role that the symbol “log”
plays in the overall development of logarithmic concepts. Dreyfus (1991) suggests that
there must be some perception of the concept before a symbol for that notion can be
introduced. This implies that students need opportunities to engage in activities to
generate mental images that require a symbolic notation to make the conception more
transparent. The primary aim of this research is to investigate college students’
understanding of logarithmic concepts by using a series of instructional tasks designed to
observe what students do as they construct meaning using APOS Theory as a framework
for analysis of growth.
It is my belief that logarithmic concepts should initially be embedded in the
processes involved in constructing exponential functions. In this way, participants can
develop flexible ways to deal with the dual nature of the symbolism associated with
logarithms. The symbol itself
represents both the process for calculating
what the exponent “b” must be raised to in order to produce “x” as well as its own
conceptual identity. As mathematical concepts become more complex, the cognitive
needs for coping with mathematical content change. For change to occur, it is critical
that discrete topics lose their “intrinsic identity to categories of (non-specific) objects that
share a set of conditions” (Mamona-Downs & Downs, 2008, p. 156). This means that the
16
learner must reconstruct and reorganize actions and processes associated with learning
both exponential and logarithmic concepts into conceptual entities or objects. Harel and
Kaput (1991) suggested that the ability to form conceptual entities is directly related to
mathematical notation; the physical notational name must somehow be integrated into a
cognitive permanence if students are to engage in advanced mathematical thinking. Gray
and Tall (1994) speculated, “Interpreting symbolism in this flexible way is at the root of
any successful mathematical thinking . . . . [Its] absence leads to stultifying uses of
procedures that need to be remembered as separate devices in their own context” (p. 120).
Furthermore, they conjectured that “the good mathematician thinks ambiguously about
the symbolism for product and process . . . by replacing the cognitive complexity of the
process-concept duality by the notational convenience of the process-product ambiguity”
(p. 121). We can investigate this conjecture only by first examining how students acquire
knowledge of logarithmic expressions.
Research Questions
This study will investigate the following questions:

How do students acquire an understanding of logarithmic concepts?

How do they assign meaning to the symbolic notation associated with
logarithms?
 What are the critical events that contribute to the total cognitive
understanding of logarithmic concepts?
17
Theoretical Basis for the Study
A Constructivist Perspective
The Mathematical Association of America (MAA), a leading force in the
development and evaluation of undergraduate mathematics programs, has been active in
reconceptualizing the curriculum for the first two years of college mathematics. Reports
generated by this organization raise important issues regarding goals, content, and
pedagogy of college preparatory mathematics. The MAA advocates for a curriculum that
places an emphasis on depth over breadth, data collection and statistical inferences, and
mathematical modeling and communication (Gaunter & Barker, 2004). Its
recommendations resonate with the material contained in the NCTM Standards. They
would agree that the mindless symbol manipulation that dominates traditional school
algebra does not prepare students to meet the challenges of collegiate mathematics.
While many still disagree on the content of school algebra, research supports the idea that
reform-oriented material fosters critical thinking. Marcus, Connelly, Conklin, and Fey
(2007), reporting on the work of others, maintained:
One consistent and striking finding in evaluations of Standards-based middle and
high school programs is that students with extended experience with independent
and collaborative work on complex and open-ended mathematical tasks become
more capable, confident, and persistent problem solvers that those who have
always worked in a highly structured and guided learning environment. (p. 356)
18
Leaders in research in undergraduate mathematics education advocate for a
unified theoretical perspective in an attempt to understand how students construct their
understandings of mathematical concepts. Constructivism as a unifying philosophical
stance seems to offer promise because it offers alternative pedagogical strategies for the
classroom. According to Noddings (2004), “An acceptance of constructivist premises
about knowledge and knowers implies a way of teaching that acknowledges learners as
active knowers” (p. 10).
“Constructivists believe that knowledge is built up by students as they abstract
from and reflect on the mental and physical actions of their experiences and their
environment” (Mellilo, 1999, p. 2). Reflective abstraction, first introduced by Piaget,
describes the mechanisms by which mathematical structures are formed. Piaget believed
it was “the mechanism by which all logico-mathematical structures are constructed, and
he felt that it alone supports and animates the immense edifice of logico-mathematical
structure” (as cited in Dubinsky, 1991a, p. 160). In response to Piaget’s findings,
Dubinsky (1991a) has been developing a framework for “observing students in the
process of trying to learn mathematics concepts at the late secondary and early
postsecondary levels” (p. 165).
Advanced Mathematical Thinking
As students advance into postsecondary mathematics, instructors expect that students can
“understand given sets of families of objects with a certain coherent set of relationships
between them” (Mamona-Downs & Downs, 2008, p. 166) as collective wholes. This
19
means the student should be able to link large portions of mathematical knowledge in
logically structured ways. Piaget and others use abstraction, reflection, and
accommodation to describe this constructive process (Battista, 1999a, 1999b; Tall, 1991).
Tall (1991) claimed that as students recursively cycle through these stages of learning,
they develop more sophisticated mental images which enable them to connect previously
unconnected concepts.
At the tertiary level, we are asking students to consider the importance of
processes and objects in the development of mathematical concepts, whereas traditional
secondary mathematics education seems to focus on the learning of syntactic rules for
manipulation. At the advanced level, we are asking students to move from “describing to
defining, from convincing to proving in a logical manner based on definitions” (Tall,
1991, p. 20). This transition is the heart of the advanced mathematical thinking students
are required to develop in college-level mathematics coursework.
Advanced mathematical thinking (AMT) is best defined as the cognitive tools
needed to succeed with the mathematical content typically treated at the postsecondary
level of education. Is this different from mathematical thinking at the secondary level?
Arguably, the topics covered in a first-year college mathematics course are similar to
topics found in either Algebra II or beyond at the high school level, but what is different
is student acuity. Schoenfeld, a leading figure in research on problem solving, cited
monitoring and control as critical for AMT (Mamona-Downs & Downs, 2002, 2008). At
the AMT level, students are able to evaluate their work and switch their solution path
20
based on structural factors. While secondary students can effect change in solution
strategies, their rationale is not prompted by structural factors (Mamona-Downs &
Downs, 2002, 2008). Furthermore, secondary students perceive mathematics as changing
character once they enter a postsecondary program.
At the collegiate level, the beliefs of students taking first-year mathematics
courses have been problematic. Studies indicate that students’ perceptions of
mathematics as a useful subject are narrow. Combined with the fast-paced presentation
of content and a right-answer mentality, success is dependent on one’s ability to
memorize and repeat routine tasks, making it difficult for an increasingly diverse student
body entering college to manage the transition between school mathematics and tertiary
mathematics (Linn & Kessel, 1996; Ricks-Leitze; 1996; Yosof & Tall, 1999). At the
postsecondary level of instruction, instructors do not face problems of rigor, but of
students’ inability to develop meaning. Students are unable to make an internal mental
construction upon which they can build mathematical objects, and their mathematical
thinking is fragile at best. Concept acquisition is weak because of their inability to
develop a coherent concept image. Tall and Vinner (1981) describe concept image as the
“total cognitive structure that is associated with the concept” (p. 99). In other words, the
learner has been able to make connections between existing knowledge structures. The
learner has built an image through experiences in the classroom, constantly changing as
the individual meets new stimuli (Tall & Vinner, 1981). Development of a concept
image enables the researcher to determine two things about the learner: “how
21
interpretation of a concept may be accommodated in the mind and how the practitioner
may fail to understand or misunderstand some aspect of the concept” (Mamona-Downs &
Downs, 2008, p. 159). In the attempt to understand how logarithms are learned and what
can be done to facilitate this learning, a sound learning theory grounded in rich
qualitative data must be applied. APOS Theory is thought to hold promise in
undergraduate mathematics educational research.
APOS Theory
APOS theory is a useful theoretical framework for studying and explaining
conceptual development. Closely linked to Piaget’s notions of reflective abstraction, it
begins with a hypothesis that mathematical activity develops as students perform actions
that become interiorized to form a process understanding of the concept, which
eventually leads students to a heightened awareness or object understanding of the
concept. Furthermore, the student is then able to organize the totality of these mental
images as a schema to make sense of mathematical problem situations. In this respect,
the theory includes implications for both methodology and pedagogy. The essential
components of this hierarchical theory are actions, processes, objects, and schemas. Prior
to any detailed investigation, the researcher must provide an analysis of the concept
development in terms of these specific constructs. “The description resulting from this
analysis is called a genetic decomposition of the concept” (Dubinsky, 2001, p. 11).
According to Dubinsky (1991a), a genetic decomposition “for a topic is a detailed
description of what a schema for this topic may look like and how the . . . construction of
22
processes and objects could be used to construct a schema” (p. 182). The purpose of this
theoretical analysis is to suggest a framework a learner may use to construct the meaning
of a mathematical topic, with the researcher then designing mathematical tasks to move
the student forward in their mathematical understandings of the concept. Initially, this
analysis is grounded in the researcher’s own beliefs about concept development, but as
the researcher cycles through different learning episodes with study participants, the
researcher evaluates the data to make informed instructional decisions in an attempt to
move students’ thinking forward.
The action construct. According to Asiala et al. (1996), “An individual whose
understanding of a transformation is limited to an action conception can carry out the
transformation only by reacting to external cues that give precise details on what steps to
take” (p. 10). Students see algebraic expressions as commands to calculate. For
example, a student who is unable to evaluate a logarithm to a base other than 10 or e
unless given the change-of-base rule is restricted to an action concept of logarithm. The
student is unable to think of this expression as the exponent of the base b that produces x,
and is unable to reason about or explain the meaning beyond what the rule “tells” them to
do. Furthermore, the student has no cognizant awareness of its usefulness. Given the
intrinsic nature of logarithms, it seems clear that attempts to alleviate students’
misconceptions at the action level have to address the symbols. Working with symbols
that are not understood leads to incorrect solution methods. Students will invent their
own mathematics to deal with symbols they do not fully understand.
23
An expression such as log3 7 is composed of three distinct pieces, each of which
needs to be addressed (Woods, 2005). This poses a significant cognitive obstacle for the
learner and a challenge for the educator. Students have been exposed to a new symbol
system unlike any they have seen previously and are unable to make sense of its semantic
structure. The symbol “log” does not offer any intuitive notion of the role that a
logarithm performs in the way that the symbol
does, nor does the subscripted notation
for the parameter b offer any indication of the role it will play when one examines the
family of functions called logarithms. Students struggle to see how this new information
can “fit” into their existing cognition.
The process construct. As the action is repeated and the individual has the
opportunity to reflect on the procedure, it becomes interiorized; that is, the process takes
place entirely in the individual’s mind. The learner “performs the same action, but now,
not necessarily directed by external stimuli” (Asiala et al., 1996, p. 11). Dubinsky
(1991a) posits, “The student has constructed an internal process as his or her response”
(p. 170) in an attempt to organize a coherent concept image. In other words, the actions
have become part of a generalized process. Actions and processes then operate in tandem
or as complementary tasks. Sfard (1991) refers to this as operational conceptions. She
implies that interpreting an action as a process is “static, instantaneous, and integrative”
(Sfard, 1991, p. 4), which enables the learner to regard the mathematical content for its
potential rather than as an entity. This is in contrast to “something one does in response
to external cues” (Asiala et al., 1996, p. 11). In other words, the learner can reflect on,
24
describe, or reverse the steps of the procedures without actually performing those steps.
If a learner has moved to this next level of understanding, they should be able to see that
is the same as
without an external cue. The learner is able to reverse
the process invoked by the logarithmic notation to obtain the inverse exponential
function. The learner is able to reflect on a set of possible inputs and outputs and explain
how they are interconnected for logarithmic and exponential functions. The learner with
a process conception is able to see what the expression or the function stands for without
evaluating it. Classroom instruction supplies students with notation used to represent an
inverse relationship between exponentials and logarithmic functions, but typically do not
attend to developing appropriate mental referents (Kenzel, 1999; Ursisi & Trigueros,
1997, 2004). Explicitly attending to this shift of attention has the potential to support the
development of symbol sense (Bills, 2001; Kenzel, 1999) and move the student to a
process level of understanding.
The object construct. As the learner reflects and revises his or her concept
image as it pertains to a particular concept he or she is encapsulating the processes,
giving structure to his or her mathematical understandings. As the process is
encapsulated into a cognitive object, the learner is able to reflect on the many different
representations of the concept. Dubinsky (1991a) speculated that encapsulation is
difficult to see and researchers can only infer that this level of understanding has been
achieved from statements made by the subject. Asiala and others (1996) described this
phenomenon as the ability of an individual to “reflect on operations applied to a
25
particular process, become aware of the process as a totality, realize that transformations
(whether they be actions or processes) can act on it, and [can] actually construct such
transformations” (p. 11). At this stage, the learner sees the expression log
log
as
an entity that can be transformed itself. They would be able to reconstruct this expression
as
log
, evaluate it, and justify how this differs from
log
Sfard (1991) describes this ability as structural thinking. Seeing a mathematical
concept as an entity enables the learner to “recognize the idea at a glance and manipulate
it as a whole, without going into detail” (Sfard, 1991, p. 4). At this developmental stage,
thinking is detailed and dynamic. The learner is able to move freely from object to
process. This type of flexible thinking has been described as de-encapsulating (Asiala et
al., 1996). Often this occurs as the learner attempts to perform a process on an object.
Before they can make sense of the mathematical object they must “de-encapsulate the
object back to the process from which it came in order to use its properties in
manipulating it” (Asiala et al., 1996, p. 11). Working with logarithmic functional
notation often requires the learner to use the process from which it came,
and use
prior knowledge structures to manipulate the object.
In studying a graphical representation of a logarithmic function, the learner can
grasp the many different components of the function itself simultaneously. For example,
connections between the domain and range of the logarithmic function to its inverse
exponential function are clear if the learner has been able to coordinate his or her concept
image with the corresponding formal concept definition. According to Tall and Vinner
26
(1981), a student’s concept image and concept definition frequently come into conflict
with each other when the student’s own concept definition is not coherently related to
other parts of the individual’s concept image. This conflict may occur as a result of a
student’s concept image of function. Students may see function as a concept that has an
input and a rule to generate the output; however, in order to understand fully the concept
definition of a logarithmic function the individual needs a broader, more flexible
understanding of the role of inputs, outputs, and inverse operations.
To develop a rich concept image that supports an individual’s concept definition
image an abstract understanding must be achieved independent of the numerical context
(Kieran, 2007, p. 729). This requires that the instruction focus on the general laws and
forms at length and not on detailed calculations of restrictive activities. This strategy
supports students as they move from a process orientation to an object orientation by
“reifying” their process views. According to Sfard and Linchevski (1994), “The
mathematical objects are outcomes of reification, of our mind’s eye’s ability to envision
the results of processes as permanent entities in their own right” (p. 194).
The schema construct. The highest level of abstraction is referred to as the
schema construct. Once a student’s mathematical understanding is at the object level, the
actions, processes, and objects associated with a specific mathematical concept “can be
organized in a structured manner to form a schema” (Asiala et al., 1996, p. 12). At the
core of this constructive process to form a coherent entity are interiorization,
encapsulation, coordination, reversal, and generalization (Dubinsky, 1991a, 1991b).
27
Schemas are dynamic; they are attempts to make sense of mathematical experiences
encountered during learning. Asiala et al. (1996) posited, “Schemas themselves can be
treated as objects and included in the organization of higher level schemas” (p. 12).
Being able to see these invisible objects as a whole seems to be an essential component of
advanced mathematical thinking. Dubinsky (1991a, 1991b) summarized the construction
of schemas as a cyclic process consisting of a collection of cognitive objects and internal
processes where the learner is able to perform actions on these objects by coordinating
two or more processes to obtain a deeper understanding. Sfard (1994) described this as
reification, the treatment of something abstract as a material or concrete thing. Dubinsky
and others have indicated that the idea of a schema is not as detailed as other aspects of
APOS Theory, in part due to the difficulty of tapping into a learner’s subconscious mind.
The schema level of understanding logarithms might consist of being able to
conceptualize the many relationships between logarithmic and exponential functions.
When solving an exponential equation for the exponent, a student must first reconstruct
or coordinate his or her mental image with that of a logarithmic expression. In effect, the
learner needs to coordinate an existing schema for exponential functions with existing
knowledge about logarithms. On another level, the objects associated with logarithmic
properties may be used to construct formal proofs of subsequent properties. The ability
to retrieve appropriate schemas to construct new understandings and insights is
characteristic of thinking at this level.
28
Instructional Implications
APOS theorists claim that students do not generally learn material in a logical
sequential order as presented in most traditional textbooks. Instead, they claim students
gain partial knowledge and repeatedly return to this knowledge in an attempt to organize
their knowledge structures (Asiala et al., 1996). The instructional approach developed
from APOS Theory begins with a genetic decomposition of the topic as described from
the experiences of the researcher. The researcher proposes a set of mental constructions
that a learner might form as they begin to explore the concepts. This provides an initial
theoretical perspective used to guide instruction. The analysis proposes the student
begins instruction with explicit directions, enabling the student to carry out routine
procedures. Repeating these actions, coupled with instructor-guided questioning
strategies to encourage reflection, provides a framework for the development of an action
conception of the concept. At this level, we are in fact giving students tools to think with.
When the student no longer requires an external cue to evaluate a logarithmic expression,
they begin to see how symbolic notation is related to the exponential function, its inverse,
then to interiorize these actions to form processes the learner needs in order to develop a
more sophisticated concept image. With a process conception, the student can reverse
the process of obtaining a logarithm by imagining the process of associating the value of
the logarithm with an exponent. In other words, the student begins to understand the
symbolism as a concept to know and a procedure to do.
29
The student can achieve this level of understanding through the use of
visualization activities in which the student constructs a graphical representation of an
exponential function and is asked to solve for the exponent that produces a specific
output value. This type of problem requires that the student have a flexible
understanding of variables. Once these new connections are formed at the next
developmental stage, the learner no longer requires a visual cue to connect logarithmic
and exponential concepts. Because of applying actions and processes to examples of
logarithms and exponentials, the learner is able to encapsulate these collective images
into objects. It has been pointed out that in the course of performing actions or processes
on objects, the learner may need to “de-encapsulate the object back to the process from
which it came in order to use its properties in manipulating it” (Asiala et al., 1996, p. 11).
As processes and objects become linked in the mind of the learner, the learner is able to
conceptualize the relationships between exponential and logarithmic functions. As the
learner begins to organize the mathematical concepts in a structured manner, a schema
begins to develop.
According to Dubinsky and McDonald (2001), after students have been through a
cycle of learning activities, data analysis provides feedback on the effectiveness of the
instructional program and suggests revisions to the genetic decomposition originally
framed by the researcher. “This way of doing research and curriculum development
simultaneously emphasizes both theory and applications to teaching practice” (Dubinsky
& McDonald, 2001, p. 279).
30
Goals of the Study
The purpose of this study is to investigate how students at the collegiate level
acquire an understanding of logarithmic concepts and how the symbolic notation
contributes to this cognitive understanding by attempting to find pedagogical strategies
that help students move from one level of abstraction to the next as outlined by APOS
Theory. Using a sequence of instructional tasks designed intentionally to evoke
disequilibrium, the researcher hopes to observe the development of an object conception
of logarithmic notation and to investigate how this affects students’ subsequent
understanding of logarithms as a mathematical concept itself. Specifically, this research
will elaborate on the genetic decomposition of the logarithmic concept using APOS
Theory as a framework in an attempt to understand and explain the difficulties students
encounter and to suggest possible strategies to help students learn these concepts.
Methodological Considerations
Because this research is grounded in the philosophical beliefs of constructivism
and focuses on the mental constructions made by students as they attempt to make sense
of logarithmic concepts, a qualitative approach was used to collect and analyze data.
While the scientific community does not always accept qualitative research as a
legitimate form of inquiry, this view is not the consensus in the mathematics education
community.
In mathematics education research, some believe that “excessive reliance on
statistical measures strips away context and hence meaning” (Rubin & Rubin, 2005, p.
31
30); therefore, alternative worldviews suggest that qualitative methods employed
rigorously are effective. Furthermore, others suggest that part of the scientific method “is
to be open-minded about method and evidence” (Bogdan & Biklen, 2003, p. 37).
Qualitative research does not begin with fixed methodology, but rather from a set of
broad objectives. It is an attempt to understand how humans construct meaning in their
environment. Gertz (as cited in Maxwell, 1996), believed that qualitative research is an
iterative process requiring the researcher to frequently refocus and adjust original
interests. This iterative process is consistent with APOS Theory.
According to Asiala et al. (1996), the cognitive growth experienced by individuals
trying to learn a particular mathematical concept begins with a conjecture based primarily
on the researchers’ understandings and experience with the content. In studying how
students might learn a particular concept, the researcher provides the genetic
decomposition that guides instructional design and theoretical analysis. The researcher
postulates certain mental constructions in terms of actions, processes, objects, and
schemas that a learner may construct as they attempt to learn the material. As the
researcher tries to explain the performance of individual students on researcherdeveloped tasks in terms of whether or not they constructed the proposed actions,
processes, and objects, the researcher may wish to either add or drop some proposed
constructions from the original genetic decomposition. This, in turn, provides the
researcher an opportunity to revise the proposed theoretical analysis, laying the
foundation for the next cycle of learning activities.
32
Similar to the genetic decomposition that suggests specific mental constructions a
student might make, the conjecture-driven teaching experiment “is a means to
reconceptualize the ways in which to approach both the content and pedagogy of a set of
mathematical topics” (Confrey & Lachance, 2000, p. 235). This theoretical analysis will
provide a framework to structure learning activities much like that of the transformative
and conjecture-driven teaching experiments described by Confrey and Lachance (2000).
The teaching experiment will involve a dialectical relationship between the genetic
decomposition and the instructional tasks in which the researcher is compelled to
examine his or her own understanding of how students learn this topic as he or she tries
to make sense of an individual’s response or reactions to a certain task.
Summary of Chapter 1
It has been acknowledged that thinking at the collegiate level is different from
what is encountered in school mathematics. In an attempt to understand the nature of
learning at the collegiate level, APOS Theory was developed as an extension of Piaget’s
work. Grounded resolutely in the tenets of constructivism, it contends that learning is not
passively received, but constructed by an active learner. APOS Theory provides the
researcher with both theoretical and methodological considerations.
Research documents that student understanding of this topic is limited (Kastberg,
2002; Kenney, 2005; Weber 2002a, 2002b). Students typically exhibit a disposition
toward a procedural approach characterized by overreliance on memorized rules. This
study will explore how students develop a wider mental schema for logarithmic concepts.
33
Definition of Terms
Abstraction: “The process by which the mind selects, coordinates, and registers
in memory a collection of mental items or acts that appear in the attentional field”
(Battista, 1999a, p. 5).
Action: “An action is a transformation of objects, which is perceived by the
individual as being at least somewhat external. It is a repeatable mental or physical
manipulation of objects” (Asiala et al. 1996, p. 10).
Assimilation: When confronted with new information, an individual will first
access their existing knowledge structures in an attempt to make sense of this new
information. In other words, they are trying to make sense of new ideas with existing
knowledge structures.
Coordination: The ability to take two or processes and use them to construct a
new process.
Emergent Process: This term implies that an individual understanding of
logarithms is in transition from action to process level conception.
Emergent Object: Because cognitive transitions in states of understanding are
difficult to determine with certainty, an emergent object conception implies that the
individual is in transition from process to object level understandings.
Encapsulation: According to Dubinsky (1991a), the student has been able to
construct an internal process, but in addition to just thinking about the process, the learner
seems to be considering the process as an object.
34
Epistemological obstacles: “This term may be described in terms of the old and
trusted knowledge suddenly becoming inadequate in face of new problems, or as
discontinuities occur between common thinking and scientific thinking” (MamonaDowns & Downs, 2008, p. 159).
Genetic Decomposition: (Dubinsky, 2000). A theoretical analysis of the specific
mental constructions of actions, process, objects, and schemas one may use to learn a
particular mathematical concept.
Generalization: According to Dubinsky (1991a), this term means the ability to
use an existing schema in a new situation that is different from its previous use.
Instrumental Understanding: Skemp (1977) describes this term as the ability to
use rules without understanding the reasoning behind the rules that make them work.
Interiorization: A means to reorganize one’s knowledge. It is the most general
form of abstraction. Actions on objects have been organized with an awareness of a
coherent totality.
Logarithm: From the Greek word logarithmus, is composed of two words
meaning ratio and number, or the symbol of the number of times a number must be
multiplied by itself to equal a given number.
Object: When an individual reflects on operations applied to a particular process,
becomes aware of the process as a totality, realizes that transformations can act on it, and
is able to construct such transformations, the individual is said to possess an object
conception” (Asiala et al. 1996, p. 11).
35
Process: “A process is perceived by the individual as being internal, and under
one’s control, rather than as something one does in response to external cues” (Asiala et
al., 1996, p. 11).
Relational Understanding: According to Skemp (1997), relational understanding
involves knowing not only what method worked, but also why it worked in this particular
situation.
Reversal: The ability to take a mathematical procedure or process and decompose
or mentally reverse this process.
Schema: (Dubinsky, 2000). A collection of actions, processes, objects, and other
schemas organized in a structured manner to form a complete understanding of a
mathematical concept.
Understanding: (Sfard, 1994). Understanding is conceived of as grasped
meaning. It is a process that mediates between the individual mind and the universally
experienced. It consists of building links between symbols and certain mind-dependent
realities.
CHAPTER II
REVIEW OF RELATED LITERATURE
Introduction
What exactly does current research say about algebraic meaning, where does it
come from, and how does it influence the learning and teaching of algebra as it relates to
the concept of logarithms? This chapter summarizes topics that the author deems to be
critical to understanding how students build meaning for mathematical concepts. The
chapter is divided into six parts: influence of symbols in algebraic thinking, functions and
associated learning difficulties, student understanding of exponents and exponential
functions, inverse functions, history of logarithms and student understanding of
logarithms, and advanced mathematical thinking combined with APOS Theory. Each
topic highlights an important theme in the development of logarithmic concepts.
The first part describes the influence of symbols in algebraic thinking. If students
are unable to see abstract ideas beneath the symbols, they develop an impoverished
understanding of algebra. As students matriculate into secondary and postsecondary
mathematics programs, most mathematicians and mathematics educators agree that
functions play an indispensable role in their mathematical education. The second section
of this chapter discusses research on students’ difficulty with functions as they study
algebraic concepts, another symbolic format that emphasizes the relationship between
variable quantities. Laws of growth characterize two important families of functions,
36
37
exponential and logarithmic. These functions model a wide array of applications, making
the mathematics relevant and accessible; however, research on students’ understanding of
exponents and exponential functions indicates that if students are to integrate their
understanding of functions with their operational understanding of exponential growth, a
relationship between multiplicative growth and additive structures must develop.
Exponential relationships are the focus of the third section of this chapter. Inverse
function constitutes the fourth section of this literature review. To develop a process
understanding of exponential functions, the subject must be able to think about the
function in its original direction but also be able to reverse the process. Therefore, while
limited research has been conducted pertaining to inverse functions and how students
develop an understanding for them, inverse functions play a pivotal role in the
development of logarithmic concepts. The chapter finishes with first a brief overview of
the historical developmental of logarithmic concepts, followed with a review of studies
concerning students’ understanding of logarithmic concepts and concludes with a brief
overview of advanced mathematical thinking and APOS Theory.
Influence of Symbols in Algebraic Thinking
Learning how to teach more effectively depends upon more than just recognizing
typical errors students make when working with exponential and logarithmic functions;
one needs to develop a mathematically sound pedagogical framework. In essence,
effective teaching requires an “understanding of how people come to know mathematical
ideas, or more specifically being able to specify the operations involved in constructing a
38
mathematical reality” (Smith & Confrey, 1994, p. 331). Observations from extant
research literature indicate that to further our understanding of how students develop
meaning from algebraic activity we need to understand the source of algebraic meaning,
and how it affects student performance.
Kieran and Wagner (1989) posited that the coalescence of mathematics education
research has led to a reconceptualization of what students do as they attempt to learn
school algebra. Questions focus on the processes of learning rather than its outcomes.
Instructional activities therefore need to support the development of algebra as a sensemaking activity. “Students should see algebra as an aid for thinking rather than a bag of
tricks” (Thorpe, 1989, p. 12). If school algebra is a procedural tool and not viewed as
representational, how does advanced mathematical thinking develop in a program that
places a great deal of emphasis on functions? Where does the meaning come from?
Kieran (2007) explored four main sources of meaning: meaning from the letter-symbolic
form, meaning from multiple modes of representation, meaning derived from content of
the problem, and meaning derived from external factors.
Algebra is typically thought of as the part of mathematics used to express
generalities about number relationships where the use of symbolism is indispensable.
Stacey and MacGregor (1997) contended, “Algebra is a special language with its own
conventions where oftentimes mathematical ideas need to be reformulated before they
can be represented as algebraic statements” (p. 308). Without this framework, algebra is
disconnected for many, which renders their mathematical experiences meaningless. A
39
topic in algebra that students struggle with at both the secondary and college level is
logarithmic functions. Students typically have little if any intuition about logarithms and
their connections to exponential functions. Is this a result of their lack of understanding
of the algebraic structure itself? What is it about this algebraic representation that is
difficult for students to process? According to Sfard and Linchevski (1994), algebraic
symbols do not speak for themselves. They depend on what a student is prepared to
notice and able to perceive. In other words, meaningfulness comes from the ability to see
abstract ideas beneath the symbols. How does this happen? Is the structure of the
concept itself problematic? Have we asked students to abandon their cognitive structures
previously developed in favor of concepts developed by logical deduction and axioms?
Successful transition from arithmetic thinking to algebraic thinking involves the
ability to make sense of the symbolic notation. Herscovics and Linchevski (1994)
conducted a study to determine how to bridge the cognitive gap characterized by “the
students’ inability to operate spontaneously with or on the unknown” (p. 75) as students
move to algebraic thinking. These researchers reported that students must not only be
able to view literal symbols as generalized numbers but also be able to operate with the
symbols themselves. Without sufficient time to develop this type of meaning, many
students fail to move from arithmetic thinking to algebraic thinking. As a result, the
researchers claimed, many students “fail to construct meaning for the new symbolism and
are reduced to performing meaningless operations on symbols they do not understand”
(p. 60).
40
Kieran (2007) reported that several research studies indicate that with advanced
students there was a preference for the use of traditional symbolic manipulation, even
when symbolic manipulating tools such as hand-held calculating machines were
available. This indicates a need to understand the ways in which students at this level
construct their understandings of algebraic symbols and notations. Is this based on their
past success, or does it point to a weak understanding of the conceptual underpinnings
associated with the expressions?
In algebra, symbols can be interpreted in various ways. Linked to their purpose,
literal symbols have different uses. Students need to differentiate between the many
representations. For example, does the symbol represent the variable as an unknown
quantity, a generalized number, or a functional relationship? Students need to develop
fluency with the multiple uses of symbols; however, students often try to “fit the idea of
variable into a single conception” (Usiskin, 1988, p. 10), leading to an oversimplification
of the nature of algebra. Kieran (2007) reported on the work of others who posited that
the source of the meaning students derive from letter-symbolic forms provides
connections to “property-based manipulation activity” (p. 711). But is successful
manipulation of symbols the purpose of algebra? Usiskin (1988) claimed the purpose for
teaching and learning algebra is intrinsically linked to the various uses of variables.
Without a clear understanding of the purpose of algebra, the relative importance of
variables is skewed. In a typical high school course, students first encounter algebra as
generalized arithmetic, only to learn later on that it is the study of relationships among
41
quantities, or, perhaps, a study of procedures to solve certain types of problems. For
example, what is it that we are asking students to do when we ask the following: As x
gets infinitely large, what happens to the value of
1
? To the mathematics teacher it is
x
clear, but students struggle. “We have not asked for a value of x, so x is not an unknown.
. . . There is a pattern to generalize, but it is not arithmetic” (Usiskin, 1988, p. 13). To
complicate the issue further, students must do something different as their study moves to
the structure of algebra. We have previously asked students to use the underlying
numeric structure to think about the symbol system, yet here that mental image is not
necessary; we are merely asking students to manipulate the symbols. When we ask
students to solve the logarithmic equation
before the solution process can
begin, they must manipulate the equation into a different form “using properties that are
just as abstract as the identity we wish to derive” (Usiskin, 1988, p. 16). Usiskin further
writes:
In these kinds of problems, faith is placed in the properties of the variables . . . be
they addends, factors, bases, or exponents. The variable has become the arbitrary
object in a structure related by certain properties. Much criticism has been
leveled against the practice by which symbol pushing dominates early experiences
with algebra. We call it “blind” manipulation when we criticize, “automatic”
skills when we praise. Ultimately, everyone desires that students have enough
facility with algebraic symbols to deal with the appropriate skills abstractly.
What is ironic . . . is those who favor manipulation on one side (the traditional
42
approach) and those that favor theory on the other side (functional orientation)
both come from the same view of variable. (p. 16)
What do students see as the variable in the above equation? Just the letter x or the entire
expression,
? What would happen if we drop the base value or change it to an
unknown quantity?
Furinghetti and Paola (1994) claimed parameters are as ambiguous as the
concepts of variables and unknowns. They have found that “the difficulties encountered
by students are of a dual nature: syntactic-manipulation and semantic-conceptual” (p.
368). When considering logarithmic expressions and/or equations, are students aware of
the role of the variables and the parameters at work? Combined with this difficulty is the
word “log” itself. What type of concept image does the word “log” invoke? When
studying other algebraic functions such as quadratic or linear, the word line or quad does
not appear in the notation; now, suddenly the name of the function is also a symbol to
manipulate. “Log” represents a value of an exponent and the graph of a function. Is the
entire expression
the symbol? The knowledge of which letters represent
parameters and which represent variables is apparently not clear in the equation itself.
Bloedy-Vinner (1994) speculates:
Understanding algebraic language related to parameter means understanding from
the context, which letters are used as parameters, and understanding the role of
the parameter as opposed to the role of the unknowns or the variables. The
different roles are explained by the fact that the equations of functions with
43
parameters stand for a family of equations or functions, where specific instances
may be created by substituting numbers for the parameters while letters still
assume the roles of unknowns or variables . . . . The difference between the role
of parameters and the role of other letters is related to a dynamics of possible
solutions: first substitute for the parameter, get an equation, then substitute for the
unknowns or variables to check to see if the equality holds. (p. 90)
Bloedy-Vinner (2001) claims that part of the difficulty students have with
differentiating the role of parameters and variables may be the inability to see the secondorder nature of the function. She has found that students typically quantify the variables
x and y before the equation, which in turn will lead to a wrong order of substitution.
Students’ performance may seem skillful at first, but when they were asked to reveal their
thinking, conceptual understandings were deficient. “Of course you could always tell the
students which letters hold which role but if students do not understand the logical
distinction between the roles, specifying the names of the roles may be meaningless”
(Bloedy-Vinner, 2001, p. 184).
In the case of logarithms, the inability to distinguish the role of each letter may
explain why students are unsure of the meaning of the expression
. To complicate
matters, frequently the parameter is implied in the written format. For example, after the
meaning for logarithmic notation is developed, we suddenly tell students that if the
subscripted notation is no longer present we are working with common logs. The
implication is that the parameter is still present, but we no longer require the written
44
symbol to represent its value. To exacerbate an already difficult concept, we suddenly
replace the entire notation when we refer to a logarithm that has a base value of e.
Students are given yet another instance of the increasing difficulty of the concept when
they are told that
.
When asked to evaluate the meaning of this expression, the less able students are
powerless to begin. They have difficulty comprehending this duality and rely on their
piecemeal rearrangement of memorized manipulations (Kenney, 2005). Students need to
be able to see the dual role of this expression, a numerical value dependent on the value
of a parameter, which is in fact an exponent. Before students can work successfully with
parameters, they need to realize that a parameter can have multiple roles. In the case of
logarithms, the perception of the parameter as a general number should enable the student
to relate it to a family of functions with this particular characteristic. Data gathered by
Ursini and Trigueros (2004) indicated that students view parameters as general numbers
and experience difficulty interpreting them as other variables. They posited that in order
to understand students’ difficulties with parameters, “It is necessary to focus on their
capability to interpret them, to symbolize them, and to manipulate them in different
contexts” (p. 362). This ability requires flexible thinking on the part of the learner; he or
she must be able to reflect on the actions required to operate on the symbols and
internalize the process to manipulate mentally the cognitive objects.
Bills (2001) contended that the role of a literal symbol within an equation
involves a subtle shift in meaning, which in turn allows students to perform standard
45
algorithmic procedures flexibly and not by rote memorization. The kind of shift she
envisioned is in the mind of the individual, where the focus of attention is on the role of
the symbol within the context of the problem. She identified what she believes to be four
different kinds of shifts dependent upon the routine procedures of the algebra activity:
variable to unknown-to-be-found, placeholder-in-a-form to unknown-to-be-found,
unknown-to-be-taken-as-given to unknown-to-be-found, and unknown-to-be-taken-asgiven to variable. Of interest to the topic of logarithms could be her notion of the shift
that takes place “when a quantity which is originally conceived of as a constant is
allowed to vary, that is, shift from unknown-to-be-taken-as-given to variable” (Bills,
2002, p. 166). Students may view the point (x, y) as a point on the graph of a logarithmic
function but are unable to see how the role of the parameter b influences the relationship
between the variables. Understanding of the routine used to evaluate logarithms does not
necessarily imply an appreciation of the shift in meanings as parameters change, affecting
the structure of logarithmic equations.
Competing views of the purpose of algebra will continue. What is certain,
however, is that symbolic manipulation should not be the major criterion by which
algebra content is determined. The content of algebra should be based on a framework
that uses variables to describe the underlying mathematical structure of a society that has
increased in its mathematization.
46
Functions and Associated Learning Difficulties
In most secondary and entry-level college mathematics courses, functions play an
important role in the curriculum. The heavy reliance on algebraic manipulations in most
textbooks commonly used for these courses tends to present a correspondence definition
for functions. For example, Lial, Hungerford, and Holcomb (2011) provided the
following definition: “a function consists of a set of inputs called the domain, a set of
outputs called the range and a rule by which each input determines exactly one output”
(p. 131), while Sullivan (2007) states, “a function from x into y is a relation that
associates with each element of x exactly one element of y” (p. 298). Introducing
functions using definitions based on the Dirichlet-Bourbaki definition plays a major role
in the difficulties students have in learning functions:
[It] requires students to learn a definition, which is separated from the functional
thinking they do outside of mathematics class. It does not build on experiences
they have had with functional relationships in their world in which one quantity
varies with or depends upon another. (Rizzuti, 1991, p. 25)
In other words, applications are not used to connect students’ natural tendencies
to the definitions and rules typically encountered in the textbook. Presumably, the
textbook with its formal treatment of the function concept should not represent the
complete instructional program. Studies have found that “it becomes obvious that a
formal treatment of a mathematical idea is simply inadequate to promote insightful
learning” (Confrey, 1991, p. 127).
47
The literature reflects that many studies have been conducted to explore students’
understandings of the function concept. Because there are various ways to represent
functions, difficulties arise as students attempt to make connections between these
representations. Findings indicate that many students think a function is a rule that can
be operated on only when given a specific numeric input (Breidenbach, Dubinsky,
Hawks, & Nichols, 1992; Even, 1998). Generally, students who think of functions in
these limited terms have difficulty modeling function relationships (Carlson & Oehrtman,
2005). Thompson (1994) cited students’ lack of a fully developed number sense as a
starting point for misconceptions regarding the function concept. He felt this
“contributes substantially to their developing an orientation towards memorizing
meaningless symbol manipulation . . . as a mechanism for coping with an . . . intolerable
situation” (Thompson, 1994, p. 24) that later manifests itself in our college classrooms.
Carlson and Oehrtman (2005) in their report noted misconceptions and common
conceptual obstacles observed by students as they attempt to construct meaning for the
function concept. In an earlier study conducted by Carlson (1998), she reported:
Forty-three percent of “A” students at the completion of college algebra attempted
to find (
) by adding a onto the end of the expression for f rather than
substituting x+a into the function rule . . . . [Furthermore], only 7% of these same
students could produce a correct example of a function all of whose output values
are equal to each other. (p. 2)
48
Carlson and Oehrtman (2005) claimed that instruction needs to focus not only on
repeatable actions but also on a process view of function to enable students to think
flexibly about the concept. Without the ability to interpret functions more broadly, many
students fail to acquire functional reasoning.
Without a generalized view . . . students cannot think of a function as a process
that may be reversed but are limited to understanding the related procedural
tasks. . . .This procedural approach to determining an answer has little or no real
meaning for the student unless he or she also possesses an understanding as to
why the procedure works. (Carlson & Oehrtman, 2005, p. 6)
Specifically, the authors were interested in describing the mechanisms that enable
students to move from an action view of functions to a more robust process
understanding.
Breidenbach et al. (1992) posited that if understanding of the function concept is
to extend beyond rote memorization and manipulation of symbols, the student must attain
a process conception of function as defined by APOS Theory. Working with pre-service
mathematics teachers, the researchers compared pre-service mathematics teachers’
responses to the question “What is a function?” before and after instructional treatment.
In their study, they identified three ways students typically think about functions: prefunction, action, and process. At the pre-function level, students have little or no
understanding of the function concept. When asked to respond in writing to the question
posed by researchers “What is a function?” those operating at the lowest level would
49
respond with comments such as “I don’t know” or “A mathematical statement that
describes something” (p. 252). If a student has an action understanding of the function
concept, their response would contain images of repeatable procedures. For example, a
student might respond to the question in this way: “A function is an equation in which a
variable is manipulated so that an answer is calculated using numbers in place of that
variable” (p. 252). The ability to interpret functions more broadly would require the
ability to disregard specific computational procedures and think more globally in order to
reason about the function concept as it behaves across its domain (Carlson & Oehrtman,
2005). Breidenbach et al. (1992) described this as a “dynamic transformation of objects
according to some repeatable means . . . that will always produce the same transformed
object” (p. 251). Subjects operating at this level of sophistication were able to
demonstrate this by integrating notions of input, output, and transformation in their
responses to this question. An example in this category might include descriptions such
as “A function is an algorithm that maps an input into a designated output” (p. 252).
Having identified student difficulties and levels of understanding, the teacher designs
instructional units that attempt to perturbate student thinking using the theoretical
framework provided by APOS Theory.
Dubinsky and Harel (1992), in their attempt to understand how students achieve a
process understanding of the function concept, conducted a teaching experiment with 22
undergraduate mathematics students enrolled in a discrete mathematics course. Using the
programming language ISTEL, which according to the researchers is grounded in a
50
constructivist framework, students responded in writing to 24 different situations. They
contended that the function concept is “very complex and consists of many notions, all
depending, to some extent, on the student’s prior experiences with . . . functions” (p. 90).
The researchers also claimed that acquisition of a process conception cannot be measured
in terms of a linear progress scale, but according to four factors that emerged from their
data (p. 86):
1. Restrictions students possess about what a function is. The three main
restrictions observed were (a) manipulation restriction (one must be able to
perform explicit manipulations or a function does not exist; (b) the quantity
restriction (inputs and outputs must be numbers); (c) the continuity restriction
(a graph representing a function must be continuous).
2. Severity of the observed restriction as described above.
3. Ability to construct a process when none is explicit in the situation, and
students’ autonomy in such a construction.
4. Uniqueness to the right condition, confusion with one-to-one.
What is clear is that a curriculum that fosters only procedural fluency is not
effective for building foundational function concepts. Students need to have a
generalized view of inputs and outputs before they can effectively build models to
reverse this process. This is critical in the development of logarithmic concepts;
however, before this process can begin we need an understanding of how students
construct meaning for exponential functions and related constructs.
51
Students’ Understanding of Exponents and Exponential Functions
Exponential and logarithmic functions model a wide array of environmental
concerns, creating a strong case for their inclusion in the secondary and postsecondary
curriculum. “Furthermore, conceptually and empirically, exponential functions offer a
unique opportunity to explore the relationship between mathematics and nature and, in
doing so, can make mathematics relevant and accessible” (Confrey, 1994, p. 294).
Educators and researchers will agree that the body of literature that reflects how students
come to know and understand exponential and logarithmic equations is sparse. Jere
Confrey (1994) speculated, “Models of multiplication based on counting or repeated
additions do not . . . explain the contextual situation typically modeled with exponential
and logarithmic functions” (p. 292). It is assumed that once students can operate with
exponential expressions, exponential functions can be defined and the student can move
operationally between exponential functions and their inverse functions, logarithms.
However, teachers of secondary mathematics continually find that students are unable to
form any meaningful connections between exponential and logarithmic functions.
Confrey (1991) described how students construct understanding of exponents and
related functional concepts, first by examining the traditional approach taken by most
mathematical textbooks. Next, she explored student thinking by conducting six case
studies to determine whether the student’s developmental route resonates with or deviates
from the mathematical conventions expressed in the text.
52
A typical treatment of simple exponential expressions and functions found in
precalculus is relatively uniform. Students must extend their thinking using what
Confrey refers to as the “plausibility” argument, beginning with exponents used as
counters to represent repeated multiplication applications. While most textbooks will
present a plausible reason for the inclusion of all rational numbers, students are unwilling
to make this jump. Is this due to their lack of understanding of the multiplicative
structure itself? She explains that the “extension of the domain to all real numbers is
achieved by sacrificing the intuitive meaning and gaining the elegance of a broad
isomorphic relationship between exponents and exponential expressions” (p. 125);
however, students are not given opportunities to explore this relationship. The student
then is asked to perform flawlessly in the application of these rules on complex
expressions. Next, a cursory introduction to another class of functions follows. The
exponential function ( )
is introduced, with emphasis placed on graphing this
relationship and possible domain and range restrictions. Logarithmic functions are then
developed as inverse exponential functions. The formal definition given,
and only if
if
follows with exercises that stress ways to switch from logarithmic
form to exponential form or from exponential form to logarithmic form. Some reference
to logarithmic applications may be introduced, but only as formulas to manipulate. A
cursory discussion of alternate bases will likely expose students to the number e and its
equivalent logarithmic notation,
.
53
Based on her own constructivist teaching theories, Confrey outlines shortcomings
of the traditional approach presented in the many texts she reviewed:
1. The material is offered in a logical sequence that covers the content but pays
no attention to psychological issues that may arise.
2. Applications are cast as circumstances for demonstrating usefulness, but do
not encourage students to consider why this is an appropriate model.
3. Students are required to demonstrate facility in manipulate symbols
efficiently.
4. The argument structure in the most textbooks is as follows; definitions, rules,
plausibility by extension, more definitions, proofs, and elaborated exercises,
concealing what is actually a complex and difficult set of issues concerning
the isomorphism between structures involving exponential expressions.
5. The traditional presentation minimizes the underlying multiplicative
operation. This failure to emphasize the operational character of functions
increases the likelihood that students will fail to recognize the call for a
particular function in a contextual situation.
Results indicated that interview transcripts “are not only a psychological portrayal
of an individual student, they are part of an epistemological attempt to provide a portrayal
of the meaning of the concept of exponential function as it is constructed by humans” (p.
130). Confrey identified five overlapping interpretative frameworks to model student
thinking:
54
1. exponents and exponential expressions as numbers;
2. exponential expressions and local operational meaning;
3. exponents as systematically operational;
4. exponents as counters;
5. exponents as functions.
Of note regarding this last framework, Confrey found that the students’ natural tendency
was to work in a recursive format rather than to develop a model to fit the ( )
format. Confrey found it was difficult for the participants to go from the initial point to
the final point using a symbolic representation; rather, they had to find all the
intermediate values using a recursive technique. She refers to this strategy as an example
of a “critical barrier to understanding” (p. 153). She claims, “To integrate understanding
of functions with their operational insights into repeated multiplication requires a
powerful act of reflective abstraction” (p. 155).
In previous work, Confrey identified an epistemological obstacle students
encounter when trying to develop understanding of exponential functions: how to
reconcile the use of repeated multiplication and rate of change. Confrey and Smith
(1995) extended this research by identifying covariation as an alternate approach to
understanding exponential functions. As cited earlier (Carlson & Oehrtman, 2005;
Confrey, 1991; Rizzuti, 1991), most curriculums tend to emphasize the correspondence
approach to the function concept; however, research indicates that students find the
covariation approach more intuitive (Confrey, 1991). A covariation approach “entails
55
being able to move operationally from ym to ym+1, coordinating movement from xm to
xm+1” (Confrey & Smith, 1994, p. 137). Using this approach students, can consider how
one variable changes with respect to another. As students coordinate such changes, they
must be able to determine important features in the shape of the related graph of the
exponential function. Building on students’ intuitive understandings of functional
relationships, instead of an approach based on formal definitions, allows a more
sophisticated understanding of the function concept to develop (Rizzuti, 1991).
Confrey (1991, 1994) documented the notion that multiplication as repeated
addition does explain the contextual situations modeled by exponential and logarithmic
functions. Confrey has used the label splitting to describe the cognitive schema students
use to make sense of concepts such as scaling, magnitude, and growth. In a splitting
world, where the focus is on recording the number of splits, one is the unit of origin,
whereas zero is the unit of origin in a counting world.
The independence of splitting from counting can be demonstrated by showing that
the requirement of equal-sized partitions can be obtained by arguments of
symmetry and congruence by folding continuous planar objects or, in the case of
discrete objects, by testing for one-to-one correspondence. (p. 300)
As students engage in splitting activities, they use counting numbers to keep track of the
numbers of splits. This in turn leads to the development of a unique number system.
This action is equivalent to the mapping of positive counting numbers onto positive
geometric sequences.
56
Confrey and Smith (1995) claim that “the construction of a counting and a
splitting world and their juxtaposition through covariation provide the basis for the
construction of an exponential function” (p. 80). In an attempt to understand how
students coordinate the “the isomorphic mapping between an addition of exponents and
multiplication of exponential expression” (Confrey, 1991, p. 125), Confrey and Smith
suggest giving students opportunities to construct meaning by having them explore
functional relationships in contextual situations first. This strategy allows students to
build both counting structures and splitting structures separately. Once students
understand the operational equivalence between arithmetic and geometric sets, they
should be able to “create an isomorphism between counting (additive) and splitting
(multiplicative) worlds” (p. 80). They posit this “can take the mystique out of the
eventual formal introduction of the rules of logarithms” (p. 81).
Inverse Functions
Logarithmic functions typically are introduced as follows:
if
if and only
. To begin to make sense of this definition, students need an understanding of
inverse functions. Do we trivialize the understanding of inverse functions? Some
textbooks leave this discussion out of the textbook completely, preferring to move
directly from exponential functions to logarithmic functions. How does this omission
influence student understanding of logarithmic functions? Even if students receive
formal instruction on inverse functions prior to the introduction of logarithms, is their
understanding procedural? When given the implicit form of the inverse exponential
57
function,
students hear that there is not an algebraic procedure to solve for y.
Next, they learn the definition,
which represents the explicit form of the
inverse exponential function, or the solution to
. What does this do to their
understanding of inverse functions?
In a study conducted by Bayazit and Gray (2004) to determine if teachers’
instructional practice affected student learning of the inverse function concept, they
reported that when instructional practices focused on making connections between
multiple representations, students were more likely to articulate an idea that correctly
illustrated the concept of an inverse function. In summary, instruction “aligned to the
logic of inverse operation to the procedural knowledge of doing but not to the conceptual
knowledge of undoing” (p. 105) does not facilitate student understanding that an inverse
function undoes what a function does. Do secondary mathematics teachers consider the
acquisition of procedural rules for finding an inverse function essential for understanding
inverse relationships? If so, how do they justify the statement
? Since no procedure exits for solving
if and only if
for y, is this just another fact
that students need to memorize?
Vidakovic (1996) conducted a study to examine how college students enrolled in
an introductory calculus course acquire the concept of inverse function. In order to
achieve this goal he first proposed a genetic decomposition for the inverse function
concept. Using Dubinsky’s ideas espoused in APOS Theory, he gives a possible
description of the construction methods that a student might use for developing schemas
58
for the inverse function concept. Initially he describes a preliminary version of the
genetic decomposition for the inverse function (p. 305):
1. Student has developed a process or object concept of function.
2. Student is able to coordinate two or more function processes to define the
composition of two functions.
3. Student then uses a previously constructed schema of a function and the
composition of functions to define an inverse function.
4. Student understands and applies the inverse process to specific situations.
Vidakovic collected data from his participants via clinical interviews. Subsequent
analysis of the data revealed that students acquire meaning for inverse functions in a
slightly different format than originally envisioned by the researcher. The study showed
that students placed an emphasis on the de-encapsulation of the function concept into
associated processes in an attempt to make sense of the concept operationally. In other
words, students obtained the inverse function through the action of switching the
dependent and independent variables and then solving for the dependent variable. The
researcher designed instructional programs to help facilitate students’ development of the
inverse function concept “as they go through the steps of reflective abstractions which
appeared in their genetic decomposition of the inverse function” (p. 310).
Snapper (1990) suggests that the concept of inverse “should first be explained on
the set-theoretic level” (p. 145). This means that domain and range of the function
should not be restricted to real numbers, but to sets of arbitrary values, to encourage
59
students to connect interchanging sets of inputs with sets of outputs to form the inverse
function. Of course, this does not lead to an explicit expression for the inverse function,
but in the case of the exponential function and its associated inverse logarithmic function
it can lead to a strong visual representation. Once the exponential function has been
graphed, by interchanging the x and y-axes one can view its inverse. Snapper goes on to
say that while the position of the axes is unnatural, an efficient way to view the inverse in
its natural position is to simply look at the graph of the inverse function through the back
of the paper, with the axes in their natural position.
Historical Development of Logarithmic Concepts
Smith and Confrey (1994) present a historical analysis of the development of
logarithmic concepts. While not advocating for instruction to follow the historical
development of this concept, they claim much can be gained from the knowledge of how
mathematicians invented logarithmic notation to reconcile complex multiplications that
were becoming commonplace as mercantilism and astronomy flourished in the fifteenth
and sixteenth century. Understanding the historical genesis of logarithms provides the
instructor a different perspective for evaluating student thinking.
According to historians, Napier coined the term “logarithm.” He defined it as
“reckoning number,” which signified the number of ratios used (Burton, 2007). Since
Napier was familiar with the idea of the juxtaposition of arithmetic and geometric
sequences, his challenge was to develop a method that would substitute the operations of
addition and subtraction for those of multiplication and division that would hold for all
60
real numbers. The process consisted of having a line segment and a ray where a particle
was made to move on each, both starting at the same time and moving to the right as
shown in Figure 1 (Oliver, 2000). The particle starting at A' moves at a constant speed
and the other, starting at A, moves according to the following rule:
When particle p' has reached P' particle p has reached P such that the speed of p is
proportional to the distance remaining to Z and the initial speed of p is the same as
the initial speed of p', then y = naplog (x) where x = PZ, y = A'P', and “naplog”
refers to the logarithm of x as defined by Napier. (Oliver, 2000, p. 10)
Figure 1. Illustration of Napier’s geometric model.
In other words, Napier had defined the distance traveled by the arithmetically
moving point as the logarithm of the distance remaining to be traveled by the
geometrically moving point. This cogeneration of additive and multiplicative structures
was essential to the development of Napier’s logarithm (Confrey & Smith, 1995; Smith
& Confrey, 1994). “Napier constructed two independent worlds, a particle moving
arithmetically and one moving geometrically, and by using time as a basis to visualize
their cogeneration, created a relationship that we now call a log function” (Smith &
Confrey, 1994, p. 338).
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The uniqueness of Napier’s mathematical ideas led to the development of
“increasingly dense tables of juxtaposed arithmetic and geometric series . . . by
embedding what had previously been seen only as discrete sequences into these two
continuous worlds” (Confrey & Smith, 1995, p. 80). Exploring these types of
progressions encourages covariational thinking, which involves being able to move
“operationally from ym to ym+1 coordinating with movement from xm to xm+1” (Confrey &
Smith, 1994, p. 137). A covariational approach to understanding logarithmic function
concepts makes the rate of change more visible, allowing students greater access to
logarithmic concepts and notation. Research indicates this idea is problematic for
students as they attempt to generalize relationships (Confrey & Smith, 1994, 1995).
Students who have explored some of these historical ideas are in a stronger
position to take in the Napierian approach to logarithms at a stage when it is appropriate
in their mathematical studies (Fauvel, 1995). The historical analysis may also serve as a
guide for how students make sense of its modern applications and may act as a lens
through which to view student actions during teaching episodes. Additionally, the
historical development highlights the importance that representation played in the
development and creation of the logarithmic concepts (Kastberg, 2002), and can alert the
researcher to difficulties students may encounter when introduced to logarithmic
notation.
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Students’ Understanding of Logarithms
Weber (2002a) conducted a study to describe instruction intended to facilitate
student learning of the concepts of exponents and logarithms. In his work, he
hypothesized that students first need a process understanding of exponentiation, or, in
other words, they need to learn to understand exponentiation as real valued quantity. He
posits,
The most plausible way that a student can learn to understand real-valued
functions is to understand first exponential functions with their domain restricted
to the natural numbers. The student must then generalize his or her understanding
of this process to make sense of what it means to be “the product of x factors of
a” when x is not a positive integer. (p. 3)
Without this initial understanding, students are unable to apply the concepts in
novel situations in meaningful ways. For example, can a student really grasp the notion
of what it means to measure the intensity of an earthquake without some basic
understanding of what logarithms represent? Can they use logarithms meaningfully to
compare the intensity of different types of sounds? Deeper understanding leads students
to think like experts, to make associations and transfer knowledge across situations, as
opposed to instruction that features “teaching to the test,” which will produce
“information-crammed but still ignorant adult[s]” (Gardner, 2000, p. 123).
Using APOS methodology to describe student constructions at the action and
process levels of understanding, Weber (2000a) developed instructional units to
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encourage students to make constructions consistent with a process view of
exponentiation. In an attempt to further clarify stages of students’ understandings, he
referenced the work of Sfard to differentiate between “an operational understanding of a
concept—which focuses on its algorithmic nature—and the structural understanding of a
concept—which treats the result of a process as an object in its own right” (p. 3). Weber
claims that at the action level, the student will not be able to do much more than evaluate
situations where the exponents are given as positive integers. Evaluating exponential
expressions where exponents are given as positive integers closely resembles the first
level of interiorization described by Melillo (1999) in her dissertation. She claimed that
students “have abstracted only the sequence of actions in the procedure” (p. 8). The
student will always know what to do next at this level of sophistication, but is unable to
reason about why this process works.
At the process level, the student is able to reflect on the action and begin to
interiorize the action as a process. They can decompose a particular procedure into its
components and reassemble them in novel ways. Weber (2002b) posited that students at
this level of sophistication “can view exponentiation as a function and reason about
properties of this function. They can also imagine the process obtained by reversing the
steps of exponentiation to form the process of taking logarithms” (p. 4). Research also
indicated that students needed to develop a structural understanding of a concept, which
according to Sfard (1991) treats the result of the process as an object in its own right,
which appears to be quite difficult for the student to obtain. She refers to the “ability to
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envision the result of processes as permanent entities” (p. 194) as reification.
Specifically, reification implies that the student “understands
product of x factors of b and
as the number that is the
as the number of factors of b that are in m” (Weber,
2000a. p. 3). To report on the effectiveness of the proposed pedagogy, Weber conducted
a pilot study with students enrolled in a college algebra course in a regional university in
the southern United States. Results indicated that when compared to students who
received traditional instruction, students who completed the newly designed instructional
activities were able to reconstruct forgotten symbolic knowledge in more meaningful
ways.
Weber (2000b) provided a detailed analysis of student understanding within the
context of the theories proposed by Dubinsky (APOS) and Sfard (operational-structural
thinking), reported in Weber (2000a), and reiterated in this document. Participants in this
study were 15 students enrolled in a traditionally taught precalculus course at a university
in the southern United States. Three weeks after the students had learned about
exponential and logarithmic functions, researchers interviewed them and asked them a
wide variety of questions. The main finding was that students traditionally taught could
understand exponentiation only as an action. When asked to evaluate
, eight
students were unable to propose a way to compute this, three believed the answer would
be the fifth root of 78125, and four students knew they must find an x such that
but were unable to find a way to determine what this x was. Additionally, Weber
found that when students in his study “were confronted with unfamiliar problems, they
65
could only resort to crude symbolic techniques such as looking at specific cases and trial
and error” (p. 7).
Kenney (2005) investigated how college students interpret logarithmic notation
and how they use these understandings to solve problems that involve logarithms. Using
the Procept Theory formulated by Gray and Tall (1994), which considers the duality of
using the same symbolism to represent both a process and a concept, Kenney (2005)
collected data from 59 college students enrolled in two different precalculus courses
taught by her. Proceptual thinking is characterized by the ability to “compress stages in
symbol manipulation to the point where symbols are viewed as objects that can be
decomposed and recomposed in flexible ways” (Gray & Tall, 1994, p. 132), whereas in
procedural thinking the focus is on the algorithm. This overdependence on procedural
thinking adversely affects the learners’ ability to see the relationship between input and
output. Kenney explains that students who can utilize this framework think
unambiguously about the dual role of the symbolism, while the less able rely on
memorized procedures evoked by the symbolism encountered. When the logarithmic
function is introduced with the definition
if and only if
students are,
according to Hurwitz (1999), “bereft of a succinct way to verbalize the operation
performed on the input” (p. 344). The notation is not at all similar to the familiar ( )
notation and its subsequent connection to the inverse function concept; and when asked
to evaluate a logarithmic function, students are powerless to begin. Kenney’s results
indicated that, in general, students did not have a proceptual understanding of logarithms.
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Instead, she found that most students “invented their own solution methods for getting rid
of the logarithmic notation” (p. 7) when the logarithmic form involved more than one
term.
Kastberg (2002) conducted a study to “develop descriptions of students’
understanding of the logarithmic function, of changes in their understanding of the
function, and ways of knowing that they use to investigate problems involving the
logarithmic function” (p. 3). According to Kastberg, a student understands a concept
when his or her beliefs are consistent with those held by the mathematical community.
Using various instruments to collect data from four students, the researcher developed
case studies based on the evidence gathered. The researcher made inferences using four
categories of evidence: (a) conception, a students communicated feelings about the
concept; (b) representation, the symbolism a student uses to communicate the concept;
(c) connection, the relationships between the representations; and (d) application, the use
of the concept to solve problems. For example, a student’s conception of the logarithmic
function might be confusing or just a collection of letters with no meaning. A student’s
view of representation may have multiple formats. Kastberg reported that representation
of a concept requires the use of symbols to communicate thinking. This presentation
could be written form, graphic image, tabular compilation of data, or oral representation.
She speculated, “Students’ uses of representations are indications of their understanding
of a mathematical concept” (p. 8); however, she contended that representations adopted
from instruction are not at all similar to representations they develop. A student is said to
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have made a connection if he or she can translate a representation from one mode to
another or form some type of link between his or her various ways of knowing. For
example, if a student rewrites the written representation
, he or she has
translated the representation. If a student can apply a concept to a novel problem
situation, he or she must understand something about the nature of the problem. For
example, if a student is given the value of
value of
and uses this information to find the
this is evidence a student understands at the application level. The
researcher gauged changes in students’ understanding based on their initial beliefs or
understanding of the topic. In attempting to describe students’ understandings and
changes to their understanding, Kastberg also documented the operations and strategies
used to investigate problems. She claimed, “A student’s ways of knowing provides
insight into how a student’s understanding of the logarithmic function can grow” (p. 11).
Kastberg designed interview protocols that included a variety of tasks and
activities administered over the course of a college semester. The researcher collected
data and broke it into three phases: preinstructional, instructional, and postinstructional,
in an attempt to gather evidence of student beliefs about logarithms and how they may
have changed. Results indicated that during each phase of the study, students viewed and
described their understanding of a logarithmic function as being able to do a collection of
problems. Kastberg identified four categories of beliefs associated with students’
understanding: level of difficulty, type of problem, tools to solve the problem, and
characteristic of the logarithmic function. All remained similar during each instructional
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phase except for category of the tools needed to solve problems. For example, in the
preinstructional phase all participants claimed that if they knew how to use the log key on
their calculators, they could solve the problem. During the instructional phase, the
students acquired more tools to use in addition to the calculator, such as facts and
formulas. Linked together, the facts and formulas became procedures to solve problems.
For example, they knew such things as the fact that when you add you multiply; however,
they did not know why any of their tools worked—just that they did. During the
postinstructional phase, their beliefs were fewer in number; however, distortions of
beliefs they formed during instruction became evident.
Kastberg’s study also suggested that students believed that performance is
understanding. In particular, during the postinstructional phase they aligned their
performance to their ability to reconstruct the “tools” used during the instructional phase.
For example, when asked to simplify
a student responded:
(
).
The student suspected the answer was incorrect since it violated one of her facts: you
cannot take the log of a negative number; but her lack of any logical basis for the
formulas and facts she had adopted for use as tools made it difficult for her to simplify
this expression. While all four participants performed above average on classroom
assessments during the instructional phase, postinstructional information revealed
students could not use their tools to solve problems correctly. Researchers made this
assumption because, during the instructional phase, students remembered the tools for
one purpose: performance on an exam.
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Kastberg noted that the focus in the classes she observed was on doing problems.
Teachers would demonstrate flawlessly how to do problems and then students practiced
doing them. Without any opportunities to develop higher-order thinking skills, such as
those advocated by APOS Theory, students are left with an action understanding of the
concept of logarithmic function, which may be quickly forgotten. If students continue to
experience college mathematics as a collection of rules to be memorized and applied
correctly during examinations, it is likely that their beliefs about mathematics as a rigid
set of rules that need not make sense will not change. However, if we want students to
make sense of their mathematical experiences, then as educators we need to promote
ways to think mathematically.
Advanced Mathematical Thinking and APOS Theory
Current approaches to mathematical teaching at the undergraduate level tend to
give students the products of mathematical thought rather than the process of
mathematical thinking (Tall, 1991). Skemp (1977) calls this instrumental learning.
While instrumental learning does produce immediate rewards, it is not easy to apply to
new problems and students have a hard time remembering the procedures. At the
undergraduate level, students need to move from elementary to advanced mathematical
thinking. Tall (1991) posited that this movement will involve a significant transition
from describing to defining and from convincing to logical justification based on those
definitions. “This transition requires cognitive reconstruction which is . . . the transition
from coherence of elementary mathematics to the consequence of advanced mathematics,
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based on abstract entities which the individual must construct through deductions from
formal definitions” (p. 20). In other words, it is the learner’s ability to coordinate
previous knowledge structures with new experiences into a coherent collection of
cognitive objects. The conscious effort to coordinate these actions is commonly referred
to as reflective abstraction, which according to Piaget plays a pivotal role in development
of mathematical thought (Dubinsky, 1991a, 1991b). As the learner attempts to transition
from elementary mathematical thinking, how the learner accommodates the complexity
of increasingly difficult subjects will influence his or her success with topics typically
encountered in undergraduate mathematics.
APOS Theory provides a vehicle to explore the mechanisms of reflective
abstraction as students attempt to learn collegiate mathematics. The ideas of APOS
Theory extend the work of Piaget and provide a framework to cultivate an understanding
of the constructions the learner must develop before moving to the next level of
sophistication. This leads to the design of instructional material aimed at getting the
student to construct the necessary images to move to the next level of abstraction.
APOS Theory begins with the hypothesis that “mathematical knowledge consists
in an individual’s tendency to deal with perceived mathematical problem situations by
constructing mental actions, processes, and objects and organizing them into schemas to
make sense of the situations and solve problems” (Dubinsky & McDonald, 2001, p. 274).
Once the learner is able to reflect on their actions and interiorize this action, they may
then begin to view the action as a process. Upon completion of instructional activities at
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this level, the learner would then explain and justify their actions on slightly more
abstract concepts, after which they must reflect on the mathematical process used and
evaluates the mental images evoked by the context. As the learner moves from thinking
about and performing an action on a series of tasks they begin developing an object
conception of the related ideas. The learner at this level is capable of thinking abstractly
about the topic without actually performing an action. The final stage of APOS Theory
speculates that the collective whole of a learner’s action, process, and object
understandings makes up an individual’s schema. While the four components of this
theory have been presented in a hierarchal ordered format, in reality, “when an individual
is developing his or her understanding of a concept, the constructions are not actually
made in such a linear manner” (Dubinsky & McDonald, 2001, p. 277).
Summary of Related Work
Research clearly indicates that successful transition to algebraic thinking involves
the ability to make sense of symbolic notation. Developing appropriate mental referents
for the many uses of symbols and explicitly attending to this shift of attention has the
potential to support the development of symbol sense (Bills, 2001; Kenzel, 1999);
however, there is no body of research that investigates how this symbol sense is
developed as it relates to logarithmic functions. Students typically can give a literal
interpretation of linear, quadratic, or maybe even exponential functions because the
symbols themselves tell students what to do, but the notation used for logarithmic
expressions is more ambiguous.
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Research on the development of the function concept suggests that students often
view a function as a rule with symbols to manipulate or as a correspondence between two
sets of numbers. Breidenbach et al. (1992) posited that if understanding of the concept
function is to extend beyond rote memorization and manipulation of symbols, students
must attain a process conception of function as defined by APOS Theory. While many
(Breidenbach et al., 1992; Carlson & Oehrtman, 2005; Dubinsky & Harel, 1992) have
detailed how students obtain a process understanding for the function concept, research
on students’ understanding of a logarithmic function as a process is sparse.
A growing body of research led by the efforts of Jere Confrey documents how
students build meaning for exponential expressions and functions, yet there is still not a
clear link between how students build meaning for an exponential function and its inverse
logarithmic function. As a first step toward understanding this connection, students need
to understand the role “splitting” and covariation play in the development of exponential
functions. The literature has revealed that multiplication as repeated addition does not
adequately explain the contextual situation modeled by exponential and logarithmic
functions. Research has also documented the need for instructional plans to include
activities in which students can develop generalized rules for constructing exponential
functions using a covariance approach, as opposed to the traditional rule-based
correspondence approach. Using these strategies, a student may develop both a counting
structure and a scaling or “splitting” structure to account for exponential growth, enabling
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them to coordinate the isomorphism of the arithmetic set to the geometric set, a necessary
condition to understand logarithmic properties.
Since it is no longer viewed as a computational tool in the modern mathematics
curriculum, the role of logarithmic functions has changed. Because an exponential
function is either increasing or decreasing across its entire domain, it is a one-to-one
function. Therefore, it has an inverse, which is called the logarithmic function. This is a
typical introduction to logarithmic concepts. Vidakovic (1996) suggested that students
need to have a process-object understanding of the concept of function first, before they
can begin to develop an inverse function schema. He found that even if students’ concept
image of function was at the process-object level, the inverse function concept lacked any
structural meaning. If students view the inverse function concept as an action of
switching the independent and dependent variable, how does this help to explain the
symbolic notation used for logarithms?
While understanding of logarithmic concepts does begin with the symbolic
notation, can understanding of logarithms develop in isolation? Should it be linked to its
historical development? Research on logarithmic expressions and functions has
concentrated on student understanding. Traditionally taught classrooms have been the
fodder for research data, which suggests an orientation toward procedural understanding.
For the most part students believed they understood the concept if they were able to get a
right answer, but they were unable to give meaning to their actions. In other words, the
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students only understood the procedure locally; they were unable to link different aspects
of the concept.
The collective summary of literature reviewed provides much of the structure to
the current study. In particular, APOS Theory provided the theoretical framework that
allowed the researcher to develop a genetic decomposition for the topic leading to the
development of instructional activities; however, the researcher also needed to
understand the nature of how students develop meaning for algebraic notation before a
sound pedagogical plan could be developed.
CHAPTER III
QUALITATIVE METHODOLOGY
Introduction
The purpose of this study was to investigate how students at the collegiate level
acquire an understanding of the concept of logarithms by using pedagogical strategies
that help them move from one level of abstraction to the next, as outlined by APOS
Theory. The theoretical framework for the study suggests a qualitative research design.
“Qualitative research involves a particular chain of reasoning that is coherent, shareable,
and auditable and that should be persuasive to a well-intentioned skeptic” (Lesh, Lovitts,
& Kelly, 2000, p. 20), and does not lend itself to the long-standing practice of empirical
methods. To further support this position, Selden and Selden (1992) report that statistical
data rarely provides information about students’ thinking.
In an effort to understand student thinking and the interaction between teacher and
students, the use of the teaching experiment as an accepted form of research in
mathematics education has gained acceptance. The teaching experiment illuminates the
“distinctive characteristics of research in mathematics education” (Kelly & Lesh, 2000, p.
191). This emphasis on understanding student thinking makes the teaching experiment a
particularly good methodology for investigating the question of how students acquire an
understanding of logarithmic concepts by addressing these two main issues:
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76

How do they assign meaning to the symbolic notation associated with
logarithms?

What are the critical events that contribute to the total cognitive understanding
of logarithmic concepts?
Multiple data sources were used to triangulate results, allowing the researcher to
develop a plausible account of the way in which students acquire understanding of
logarithmic concepts and how the symbolic notation influences this acquisition. The
intent of this research was to provide a model that in principle can be applied to settings
beyond the one that gave rise to this genetic decomposition. “It is how students
conceptualize material that determines their degree of success in mastering it.
Consequently, it behooves us, as teachers, to incorporate into our pedagogy an approach
that facilitates the construction of the concept” (Dubinsky & Lewin, 1986, p. 91).
Quantitative Versus Qualitative Research Design
Some believe that “excessive reliance on statistical measures strips away context
and hence meaning” (Rubin & Rubin, 2005, p. 30); whereas, alternative worldviews
suggest that part of the scientific method “is to be open-minded about method and
evidence” (Bogdan & Biklen, 2003, p. 37). This suggests a necessary shift in how
educational research is conducted. As societal values change, computational skills once
highly prized as the only vehicle for measuring success in an industrialized society are no
longer the sole measure of mathematical ability. Critical thinking, problem solving, and
quantitative literacy have replaced traditional computational skills as the measure of
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success of an informed citizenry, thus altering the research agenda in mathematics
education. These qualities cannot be identified through the use of research measures that
quantify students’ ability to memorize facts and use blind procedures to “come up with an
answer.” Reform-based instructional practices guide students instead to uncover valuable
meaning-making, utilizing the required subject matter skills to accomplish this (Confrey,
1991; Kieran, 2007; Sfard, 2000; Shore, 1999; von Glaserfeld, 1987).
Although steeped in years of tradition, the belief system characteristic of the
conventional or scientific paradigm is now being challenged (Guba & Lincoln, 1989).
Since the goal of quantitative research is to predict, control, confirm, or test research
hypotheses using the long-standing practice of empirical methods, researchers who
subscribe to this tradition maintain the existence of an objective reality. They claim that
by asserting tight control on all variables it is possible to strip the context of confounding
variables and hence discover an objective reality (Guba & Lincoln, 1989; Johnson &
Onwuegbuzie, 2004; Kohlbacher, 2006). Opponents of this position maintain that the
existence of a single reality, which operates according to irrefutable established laws, is
unlikely. “Clearly the issue of what reality is, is very much up for grabs even in areas
like physics due to recent advances in cognitive learning theories” (Guba & Lincoln,
1989, p. 92). However, due to long-standing traditions in Western cultures, quantitative
purists maintain that hard generalizable data can be determined reliably and validly “in
order to gain an ontologically objective understanding of the events and objects we
study” (Eisner, 1992, p. 4). “According to this school of thought, educational researchers
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should eliminate their bias, remain emotionally detached and uninvolved with the
subjects, and test or justify empirically their stated hypotheses” (Johnson &
Onwuegbuzie, 2004, p. 14). When investigating the mechanisms that students use to
construct meaning, can researchers maintain objectivity and control for all variables
encountered in the teaching and learning of mathematics?
Research questions, which serve as a guideline for conducting research, are
typically framed from two different perspectives (Maxwell, 2005). Qualitative research
tends to focus on questions that deal with process theory (Maxwell, 2005) rather than
accounting for experiences through quantitative methods. Process Theory attempts to
explain the mechanism by which human needs change, whereas Variance Theory that
typifies quantitative work is concerned with finding differences and correlations between
variables of interest (Maxwell, 2005). Qualitative research strives to offer theories or to
synthesize commonalities shared by groups in an attempt to understand how humans
construct meaning. Its greatest strength according to Hull (1997) is its naturalistic
approach, studying individuals in their “natural” environment, allowing participants
opportunities to construct their own realities.
Certain issues are pivotal in the social sciences and cannot be adequately
addressed using empirical methods. Quantitative data in mathematics education research
rarely provide insight into students’ cognitive processes (Selden & Selden, 1993);
however, they can illuminate certain trends in academic achievement for large
populations and guide educational policies. According to Guba and Lincoln (1989),
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As soon as inquiry is extended to include human behavior, that phenomenon can
no longer be disregarded. Human respondents are not inert passive objects. They
are capable of a variety of meaning-ascribing and interpretative actions, and those
possibilities are certainly not held in abeyance simply because the people are
labeled subjects in an inquiry. (p. 99)
Qualitative research provides a platform to explore students’ cognitive processes.
It supports constructivism as an orienting idea in research endeavors. It does not begin
with fixed methodology, but rather starts from a set of broad objectives requiring the
researcher to frequently refocus and adjust original interests (Guba & Lincoln, 1989;
Maxwell, 2005). Proponents of this methodological position assert, “Inquiry must be
carried out in a way that will expose the constructions . . . and provide the opportunity for
revised or entirely new constructions to emerge” (Guba & Lincoln, 1989, p. 89).
Emerging from what Gage termed the paradigm wars of educational research in the
1980s, educational researchers now find “themselves in the peculiar position of having
achieved orthodoxy and have become part of the dominant methodological
establishment” (LeCompt, Millroy, & Preissle, 1992, as cited in Teppo, 1998, p. 2).
Citing its myriad genres, each with its own perspective, skeptics question the
validity of qualitative research because of the lack of a single common defining thread or
generic model. However, those who are committed to qualitative research find coherence
in the fact that “several key characteristics cut across the various interpretative qualitative
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research designs” (Merriman, 2002, p. 4). Schram summarizes these characteristics as
follows:
A commitment to direct experiences with people, situations, and ideas as they
naturally occur; an acknowledgment of the interactive and intersubjective nature
of constructing knowledge; the need to be sensitive to context as a means to
understand the complexity of phenomena; the value of attending to the particular,
unpredictable, and complex nature of specific cases; the logic and necessity of an
interpretative frame of references; and the selective nature of qualitative research.
(Schram, 2003, p. 15)
While the scientific community does not always accept qualitative research and
the methods employed used to evaluate researchers interests as a legitimate form of
inquiry, this narrow view is not the consensus. In 2006, President George W. Bush
commissioned the National Mathematics Panel to investigate the best use of scientifically
based research, as defined by the evidence standards provided by the What Works
Clearinghouse of the Department of Education, to advance the teaching and learning of
mathematics (Boaler, 2008). In its Final Report, issued March 2008, the Panel found
only eight studies that adhered to this rigid definition of research criteria; this strongly
suggested that mathematics education research using the kind of randomized controlled
experiments that typify quantitative research methodologies is not common (Boaler,
2008). Therefore, important research in mathematics education that studied children in
their natural settings was not included in this report to the nation. “Researchers in
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mathematics education, like researchers in other fields, choose different methods to
answer different questions” (Boaler, 2008, p. 592), thus necessitating research
methodologies that highlight the different disciplinary perspectives.
Qualitative Methodology
The research philosophy employed by those engaged in qualitative work derives
from an interpretive constructionist approach, which guides observational and in-depth
interviewing projects (Ruben & Ruben, 2005). Interpretive constructionist theory is
situated within the reality of each individual, examining the knowledge and
understanding they have constructed. Rubin and Rubin (2005) describe interpretivism as
an attempt to analyze human behavior. “Interpretivist researchers try to sort through the
experiences of different people as interpreted through the interviewees’ own cultural lens
. . . to put together a single explanation” (Rubin & Rubin, 2002, p. 30).
In the quest to understand the participants’ perspective, Rubin and Rubin (2005)
posit that interpretive constructivism effectively provides a framework for qualitative
research. It enables us to obtain a “thorough and credible” (Maxwell, 2005, p. 20)
explanation for how something happens by observing, listening, and talking to individual
participants. As Merriam (2002) stated:
The product of qualitative inquiry is richly descriptive. Words and pictures rather
than numbers convey what the researcher has learned about a phenomenon.
There are likely to be descriptions of the context, the participants involved, and
the activities of interest. In addition, data in the form of quotes from documents,
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field notes and participant interviews, excerpts from videotapes, electronic
communications, or a combination thereof are always included in support of the
findings of the study. (p. 5)
Qualitative researchers, according to Maxwell (2005), typically structure research
questions in one of three ways:
(a) questions about the meaning of events and activities to the people involved in
these, (b) questions about the influence of the physical and social context on these
activities, and (c) questions about the process by which these events and activities
and their outcomes occurred. (p. 75)
Qualitative Research in Mathematics Education
“Traditionally, mathematics educational research in the U.S. has focused on
largely isolating variables in a student’s environment that play an appreciable role in the
way they learn” (Thompson, 1979, p. 2). In a tradition grounded in quantitative
methodology, mathematics education researchers focused on group effects, but not on
how individual students come to know mathematics based on their experiences. When
the nature of the research questions focuses on the development of student thinking
regarding a topic such as logarithms, emergent perspectives in constructivist research in
mathematics education allow the controlled experiment, with its emphasis on statistical
tests, to be replaced with qualitative studies that involve the experience of the researcher
and socially situated learning episodes viewed from multiple perspectives (Steffe &
Kieran, 1994; Steffe & Thompson, 2000). Tracing its origins to Piagetian-style inquiry
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methods, qualitative research methods in mathematics education support field-based
research designs where individuals can be observed in their natural settings.
The current view held by leading researchers in mathematics education is that
students are constructors of their own knowledge based on their experiences; this implies
that epistemological problems encountered as students attempt to make sense of
mathematical ideas are ideally suited for investigation using qualitative methods. Annie
and John Selden (1993), in an attempt to outline what qualitative research in collegiate
mathematics education would look like, posited:
Research in mathematics education is not mathematics; however, on the
university level, it is used by and can be produced by mathematicians. It is not
curriculum development; however, research results can speak to curriculum
development. Making changes in curriculum or teaching methods with
inadequate knowledge of how students learn is like designing flying machines
with little knowledge of aerodynamics. It is possible, but requires a lot of time,
patience, and test pilots. Conversely, expecting extensive how-to-do-it teaching
information from a research project in education is like expecting the typical
mathematics paper to affect engineering practice directly. (p. 432)
Teaching Experiments
In mathematics educational research, the term “teaching experiment” rapidly took
hold as researchers tried to build accounts of how students learn mathematics.
Researchers were interested in not only understanding how students view particular
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concepts, but also measuring the progress that students make over time. Unlike classical
methodologies which “can make sense only if the researcher posits tacitly or explicitly;
(1) the predominance of the students’ environment as a determiner of their behavior and
(2) that the students’ behavior is structurally determined by the structure of their
environment” (Thompson, 1979, p. 1), the teaching experiment attempts to address
epistemological difficulties that individual students encounter during instruction. The
researcher is no longer comparing students against some prefabricated ideal, but is
learning how to use their own mathematical knowledge to understand student thinking
(Kelly & Lesh, 2000; Steffe & Thompson, 2000).
Determined to capture what children do when they construct mathematical
meaning, Robert Davis was one of the first researchers to adopt a methodological
position that documented children doing mathematics in hopes that others could learn
from these experiences (Davis, 1964). He envisioned a school environment where
“mathematics is more natural, fitting better into the context of children’s lives” (Davis,
Maher, and Noddings, 1990, p. 1). Teachers, according to Davis’s perspective, must
focus on methodological issues to develop appropriate experiences that will lead the
student “into a direct face-to-face confrontation with mathematics itself” (Davis, 1964, p.
147). His forward thinking played a supporting role in both the pre-constructivist
revolution of the 1970s and later as constructivist ideas permeated research in
mathematics education in the 1980s and beyond (Steffe & Kieran, 1994). In the early
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1980s, mathematics education researchers began in earnest to implement teaching
experiment methodology.
Teaching experiments enable the experimenter/researcher to observe firsthand the
evidence of students’ reflective thought about their operative knowledge (Steffe &
Thompson, 2000). As in Piaget’s use of the clinical interview, the basic goal of the
researcher in a teaching experiment “is to construct models of students’ mathematics . . .
by looking behind what students say and do in an attempt to understand their
mathematical realities” (Steffe & Thompson, 2000, p. 269). However, in ways that differ
from the methods of the clinical interview the “experimenter hypothesizes pathways to
guide the child’s conceptualizations towards adult competence” (von Glaserfeld, 1987, p.
13). It is a dynamic process that enables the teacher/researcher to think about the
dynamic aspects of other students’ constructions outside of the teaching experiment in a
meaningful way. In other words, the results are useful for organizing and guiding
subsequent experiences of students doing mathematics (Steffe & Thompson, 2000).
In a teaching experiment, the researcher must always remain aware of the
participants’ current thinking. Using a carefully designed sequence of teaching episodes,
the researcher hopes to optimize the chances that relevant developments will occur in a
way that reveals students’ current thinking. A teaching episode, according to Steffe and
Thompson (2002), should “include a teaching agent, one or more students, a witness of
the teaching episode, and a method of recording what transpires during the episode” (p.
274). The teaching agent must use his or her knowledge about the subject as an orienting
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idea, not a blueprint for what students should learn, enabling the researcher to explore
students’ different ways of knowing. The experimental part of this methodology is the
guidance that takes place during the teaching episode. The guidance provided by the
researcher “must take the form either of questions or of changes in the experiential field
that lead the child into situations where her present way of operating runs into obstacles
and contradiction” (von Glaserfeld, 1987, p. 14). Subsequent analysis of each teaching
episode should then follow a format that allows the researcher to account for individual
differences in how students learn mathematics and enables the researcher to engage in
“responsive and intuitive interactions with students when in fact they are puzzled about
where the interactions may lead” (Steffe & Thompson, 2000, p. 278). The end product
should provide sound content-specific learning theories that in turn can be used to make
predictions, provide a platform for generalizations, and make recommendations for
implementation of further instructional strategies. This methodology “clearly illustrates
the distinctive characteristics of research in mathematics and science education” (Kelly &
Lesh, 2000, p. 192).
In this study, selection of participants was guided by an initial hypothesis
pertaining to prerequisite knowledge: specifically, that students perform better on tasks
designed to explore how they come to understand logarithmic concepts and the associated
symbolic representations if their understanding of exponents has been at a process level
conception as defined by APOS Theory. A pretest/initial assessment was administered to
students enrolled in Precalculus I and suitability for participation was determined. The
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teaching episodes delivered the potential perturbations needed to move students forward
in their thinking with regard to logarithmic concepts. APOS Theory provided the
framework for evaluating student growth.
Research Design
Because the emphasis of this study was on investigating how students at the
collegiate level acquire an understanding of logarithmic concepts and the role that
symbolic notation plays in this understanding, teaching experiment methodology was
used. According to Battista (1999b), teaching experiment methodology consists of four
components: (a) preliminary work, (b) teaching, model building, and hypothesis testing,
(c) retrospective analysis, and (d) scientific model building.
In this study, the preliminary work consisted of a pretest/initial assessment to
determine the cognitive structures of exponential concepts of the potential participants.
The proposed genetic decomposition for logarithms was used for the development of
learning activities to encourage the participants to make the required reflective
abstractions. In other words, the experiences provided by the instructional tasks
encouraged participants to “press for adaptation by facilitating the construction and
testing of basic constructs so that some will be ruled in and others ruled out” (Lesh &
Kelly, 2000, p. 202). By creating conditions that helped students move from one level of
abstraction to the next, without dictating the direction that this development must take,
the researcher observed and documented the process of learning the concept of
logarithms. Using the processes and mechanisms as defined by APOS Theory, the study
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provided rich descriptions of this learning process and investigated how this might affect
subsequent understanding of logarithms as a mathematical concept itself.
As the teaching experiment unfolded, the researcher gathered and analyzed the
responses of the study participants, refining both the initial analysis and the instructional
treatment as needed. This gave insight to the researcher as to what it “means to
understand a concept and how that understanding might be constructed or arrived at by
the learner” (Selden & Selden, 2001, p. 242), allowing the researcher to categorize her
inferences about students’ beliefs and understandings as they interacted with the
mathematical concepts.
According to Dubinsky and McDonald (2001), “This cycle is repeated as often as
necessary to understand the epistemology of the concept and to obtain effective
pedagogical strategies for helping students learn it” (p. 279). This retrospective analysis
of the entire data set can provide a broader theoretical context of students’ mathematical
understandings of logarithmic concepts. Consequently, teachers can then design
activities that facilitate the construction of logarithmic concepts by specifying the
cognitive acts students actually perform during concept acquisition (Dubinsky & Lewin,
1986).
Research Site
A moderately large urban university located in the Midwestern United States was
the site of this research. Enrollment figures for this university indicate that 80% of the
student population is from the surrounding county, 18% from other areas of the state, and
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the remaining 2% from areas outside the state. Women make up the majority of students
attending the university, at 57%, versus 43% for men. Minority enrollments constitute
27% of the student body, with African-Americans making up the largest group within this
population, 18% of that total. All degree-seeking students at this university must earn at
least six credit hours in credit-bearing mathematics courses.
To focus this study, the researcher investigated the population enrolled in day
sections of Precalculus I Mathematics. This course is offered by the mathematics
department, which is housed within the College of Science, and is a requirement for all
students seeking a degree in the sciences or engineering. The College of Science is the
second largest of the university’s seven colleges. It enrolls more than 2,000 students and
has almost 100 full-time faculty. One of the many goals of the programs offered within
the college is a strong commitment to the individual learner. Faculty recognize that
learners at different stages in the educational process have different needs and interests.
In 2007, the Mathematics Department, in an attempt to improve student retention,
appointed a course coordinator for the Precalculus sequence. The goal of this coordinator
was to ensure uniformity in the course at the assessment level as well as in course
content. Additionally, across the college, initiatives were developed to support the
engaged-learning theme that emerged in 2008 as part of the university’s self-evaluation
process. Programs were developed to create new structures to support this mission. In
fall 2008, the Supplemental Instruction program (SI) and Structured Learning Assistance
program (SLA) were introduced campuswide. SI courses provide additional academic
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support in a variety of classes at this university. SI courses incorporate an SI leader,
usually a student peer or graduate student, who attends all of the course lectures. The SI
leader then conducts two to three scheduled voluntary review sessions each week based
on the course lectures. SLA courses provide additional support much in the same way as
SI supported courses, with two main differences. First, the SLA course review sessions
are built into the course, in much the same way as a lab session is built into a science
course. When a student enrolls in an SLA course, he or she also enrolls in the SLA
review session. Second, the SLA courses have an attendance policy for their review
sessions. All students attend the SLA sessions at the beginning of the term. Continued
attendance becomes mandatory for students who are below the expected performance
goal as set by the course instructor.
Sample
Research participants were selected from a pool of students enrolled in math 167,
Precalculus I. Students enrolled in this course are typically traditional first-year college
students between the ages of 18 and 22 who either are placed into this course by a
qualified ACT/SAT score, a placement exam score commensurate with the student’s
level of competency, or matriculate into the course through the prerequisite courses Basic
Mathematics or Algebra for Business and Science Majors.
Students enrolled in a summer 2010 Precalculus I course were all administered a
pretest/initial assessment. The assessment was given approximately one week before the
completion of the summer semester. After this initial assessment, suitable participants
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were selected based on predetermined criteria: an action-level understanding of function,
a process-level understanding of exponents as defined by APOS Theory.
By the time they complete Precalculus I, students have been exposed to properties
of integer value exponents, functions, and inverse functions. If students are to make
sense of the notation
if and only if
, they need to understand
exponentiation as more than just repeated multiplication of a, y times. According to
Weber (2002b), they need to see this as both a process and an object. In other words,
they need to understand the dual nature of the symbolic notation. If a student has moved
on to the next level of understanding outlined in APOS Theory, they have begun to
interiorize the action of exponentiation as a process. What this implies is that the student
can think about properties of the expression
without being given specific details on the
operations to perform. In other words, a student will be able to transform exponential
expressions by performing repeatable mental actions that are organized first as a
procedure and then internalized as a dynamic process that can be operated on. For
example, students would be able to tell that larger values of x lead to larger results
without calculation, or that negative values imply a different representation. It is then
possible to use this process conception to form other processes.
This researcher believed that in order to investigate the learning process one goes
through to make sense of logarithmic concepts, participants had to be able to consider
exponential expressions as objects. Weber (2002b) supports this conjecture, finding that
most students who participated in his project were unable to view exponentiation as a
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process, thereby hindering any progress they might have made in understanding more
advanced mathematical concepts such as logarithms. While successful completion of
Precalculus I did not guarantee that students had attained this level of understanding, by
setting criteria for participation the researcher was able to select a suitable sample of
students.
Procedure
After the preliminary screening process, four subjects agreed to participate in a
teaching experiment. For all participants, this was the first time they had been asked not
only to think about their thinking, but also to verbally express their understandings. Prior
to beginning the instructional sessions, the participants completed a mathematical beliefs
survey. The Beliefs Survey uses a five-point value scale; Yackel (1984) developed it in
an attempt to measure college students’ perceptions of mathematics. Use of this survey
provided the researcher with valuable background information. Knowing how the
participants viewed mathematics could help to explain their individual predispositions for
responding to the planned activities. Additionally, it could assist the researcher with
interpretations of students’ responses as they completed the tasks collectively as a group.
Six teaching/learning sessions were conducted with the four participants selected.
Teaching sessions were held on Monday, Wednesday, and Friday for two weeks, each
lasting approximately 60 minutes. The sessions were designed to get students to make
the desired mental constructions of actions, processes, objects, and schemas in an attempt
to understand, from their perspective, the learning that occurred. Their responses to the
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activities were subsequently used to construct an understanding of how students might
acquire this knowledge.
As the subjects completed the instructional tasks, the teacher-researcher
continually questioned possible meanings lying beneath the students’ language and
actions. Because of the unanticipated ways students engaged with the material during a
teaching episode, it was necessary at times to abandon or revise portions of the
pedagogical strategies developed in order to help the students make the desired mental
constructions. Dubinsky and McDonald (2001) claim, “This way of doing research and
curriculum development simultaneously emphasizes both theory and applications to
teaching practice” (p. 279).
The Role of APOS Theory
APOS Theory arose out of an attempt to extend the mechanisms of reflective
abstraction to advanced mathematical topics (Dubinsky, 1991a). The framework for
APOS Theory has three individual components: a theoretical analysis of the
mathematical concept, design of instructional activities based on the theoretical analysis,
and implementation of the instructional plan (Asiala et al., 1996). APOS Theory offers
researchers an opportunity to coordinate a theoretical approach and instructional
treatment based on this theory in order to propose a pedagogy that can assist the process
of learning mathematical concepts. Dubinsky and MacDonald (2001) describe the theory
as follows:
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The theory we present begins with the hypothesis that mathematical knowledge
consists in an individual’s tendency to deal with perceived mathematical problem
situations by constructing mental actions, processes, and objects and organizing
them into schemas to make sense of the situation and solve the problems. We
refer to these mental constructions as APOS Theory. (p. 276)
At the collegiate level, APOS Theory, then, can be used to describe “how actions
become interiorized into processes and then encapsulated as mental objects, which then
take their place in more a sophisticated cognitive schema” (Tall, 1999, p. 1). Actions are
thought of as lower-level computational procedures, where, given a rule, the student can
operate on it, either from memory or by using step-by-step instructions. If the student is
functioning only at this level, they will not be able to visualize what must be done to
complete the procedure. In other words, they are not able to think about the
mathematical construct as a process; they can only perform an action. “An object is
constructed from a process when an individual becomes aware of the process as a totality
and realizes that transformations can act on it” (Dubinsky & McDonald, 2001, p. 276).
Finally a schema for a certain mathematical concept is formed when an individual’s
collection of actions, processes, and objects “are linked by some general principles to
form a framework in the individual’s mind” (Dubinsky & McDonald, 2001, p. 277).
Based on prior research, it has been noted that most students will be observed performing
at the lower two levels of operation (Kenney, 2005; Weber, 2002a, 2002b).
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The learning process begins with a theoretical analysis based on general APOS
Theory and the researcher’s understanding of the mathematical concept (Dubinsky, 2000;
Dubinsky & McDonald, 2001). Asiala et al. (1996) refers to this as the genetic
decomposition of the concept. In other words, the researcher first must propose a model
of cognition that a typical student might use to construct an understanding of logarithmic
concepts. This model will allow the researcher to design activities to help students make
these mental constructions and relate them to logarithmic concepts.
The genetic decomposition. The genetic decomposition is a conjecture, based on
this researcher’s experiences and relevant literature pertaining to how certain
mathematical concepts could be constructed, that provides the foundational description of
the construction of logarithmic concepts. According to Dubinsky (1994), APOS Theory
is an elaboration of the mental constructions of actions, process, objects, and schemas a
student may make as they attempt to learn a particular mathematical concept. However,
to evaluate the strengths or weaknesses of the constructions of the participants, the
researcher must provide a genetic decomposition of the concept in terms of these four
specific mental constructs: actions, processes, objects, and schemas. Figure 2 provides an
overview of this cyclic process of the construction of schemas in advanced mathematics.
Actions are at the lowest level of understanding, perceived as external to the
object requiring either step-by-step or from-memory instructions. For example, at this
level students will be able to graph, using appropriate hand-held technology, an
exponential and logarithmic function on the same set of axes and complete a table of
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Interiorization
Action
PROCESSES
OBJECTS
Coordination
Reversal
Encapsulation
OBJECTS
Generalization
Figure 2. Schemas and their construction. Adapted from “Reflective Abstraction in
Advanced Mathematical Thinking,” by Ed Dubinsky, 1991b, in D. Tall (Ed.), Advanced
OBJECTS
Mathematical Thinking, p. 107.
values for specified inputs for both functions. Calculations could be made with basic
exponential or logarithmic expressions. Students could rewrite exponential equations as
Generalization
logarithmic equations, and logarithmic equations as exponential equations, given the
formal definition of a logarithm; however, at this level of conception it would be hard for
students to verbalize an understanding of the relationship between the two functions.
In order to advance their mathematical understanding of the concept, the learner
interiorizes actions. “Interiorization permits one to be conscious of an action, to reflect
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on it, and to combine it with other actions” (Dubinsky, 1991b, p. 107). Dubinsky and
others refer to actions that have been interiorized as processes. As students complete
several tasks at the action level, they should begin to form some type of connection
between exponents and logarithms as they graph both exponential and logarithmic
functions of various bases on the same set of axes. They begin to see how the role of the
base value, which is considered a parameter, influences the output, or how its
corresponding exponential form influences the symbolic notation associated with
logarithms. They know that somehow the “log” button on their calculator relates to the
exponential notation; however, they no longer need a visual representation of the two
graphs to see the connection or the formal definition to perform the required
transformation; they have internalized these processes. Once a process understanding has
been achieved, the student can continue to work with existing processes to form new ones
by reversal and/or coordination.
The ability to reverse the process of exponentiation is critical to achieving a
process understanding of logarithmic concepts. By asking students to verbalize how they
might find the value of an exponent if the output is given, the instructor/researcher helps
the students begin to see that this is a reversible process that requires them to coordinate
what they know about exponential functions with the newly acquired information on
logarithmic forms. Once the student is able to condense or encapsulate the meaning of
both exponential and logarithmic processes, it is possible then to convert this knowledge
into an object understanding. However, this transformation is difficult for the researcher
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to see directly and typically, it must be inferred from students’ responses as they attempt
to make sense of the material.
In isolating small portions of the structure of logarithms and giving explicit
descriptions on possible relations between existing cognitive structures commonly
encountered during instruction the researcher offers the students one possible way to
construct a schema for logarithms. This detailed description provides the genetic
decomposition for the concept, which in turn provides theoretical constructs for
evaluating student learning. Using this framework to guide instruction, the
researcher/teacher is able to recognize the representations that a student may be using to
make sense of the material; this in turn enables the researcher to make contact with these
representations in meaningful ways.
Defining the genetic decomposition for logarithms. Understanding of
logarithmic concepts begins with exponential expressions with whole number exponents
as the objects. Initially the student is able to perform actions on these exponential
expressions by successfully completing routine simplifications using multiplication and
division. Furthermore, he or she can explain why the rules for operating with wholenumber exponents are valid. Successful demonstration of these skills is essential before a
student is able to move from a process understanding to an object understanding of
exponents. The next step is to provide opportunities for students to generalize the
exponential expression and function schemas to include all real numbers as exponents.
To preserve consistency in the rules students are first offered plausible arguments that
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allow them to extend the rules of exponents to include zero and integer values; however,
additional instructional activities are needed to promote insightful learning. By asking
students to consider how the graph of the exponential function could be used to evaluate
such expressions as
√
, the researcher/teacher will help students will develop a broader
sense of the multiplicative structure of the function and abandon the view of the exponent
as nothing more than a counter. This generalization allows the participants to extend
their understanding of exponential concepts to form a new cohesive set of objects. This
implies that students must abandon the intuitive appeal of the meaning of an exponent; in
essence, a fundamental shift must occur in the meaning of exponents. “This isomorphism
becomes the basis for meaning rather than the view of the exponent as a counter”
(Confrey, 1991, p. 127).
What follows next is the construction of a process conception. According to
Dubinsky (1991a, 1991b), a process conception begins with the interiorization of actions
on this set of objects labeled exponentials. Instruction would begin with a task that
would help participants reverse the exponential process by asking them to look at a graph
of a corresponding exponential function and predict the input based on a given output.
Once subjects see a need to reverse the process of exponentiation and realize that a
mathematical procedure could be developed to assist with this task, they are ready for the
concept of logarithm to be introduced. To develop a need for a “new” mathematical
procedure, students complete a series of tables designed to get them to see the connection
between exponents and the “log” button on their calculators. Using TI-84 calculators
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equipped with the latest operating system, students are able to evaluate logarithms to
bases other than 10 by using the “logBASE” function. Using calculators equipped with
this technology eliminates the need, at this point, to introduce the change-of-base rule.
Once students begin to interiorize the actions, the formal definition of a logarithm can
then be introduced. Students then practice rewriting exponential equations as log
equations and log equations as exponential equations using the definition log
and only if
if
As these actions become interiorized, the students are able to
mentally construct a meaning for a logarithmic expression without explicit instructions.
They can compare the relative sizes of logarithmic expressions and explain why the
results make sense. Operating at this level of understanding, students are coordinating
the process of exponentiation with the process of finding the logarithm of a given
number. Students begin to understand the role of the parameter and how it influences the
output in a logarithmic expression, and they may be able to think about the process
involved in evaluating a logarithmic expression without actually performing the
manipulation. For example, since no explicit instructions are given for obtaining an
output of
( )
for a given input, a process conception of logarithms is
necessary. One must be able to form a mental image of the process of associating a real
number with its logarithm for a given parameter.
“An object is constructed from a process when the individual becomes aware of
the process as a totality and realizes that transformations can act on it” (Dubinsky &
McDonald, 2001, p. 276). For example, an individual understands logarithms as objects
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if he or she can compute log
log
and compare it to log
and explain why the two
expressions are not equivalent. Asiala et al. (1996) refer to this as encapsulation.
According to Sfard (1991),
Seeing a mathematical entity as an object means being capable of referring to it as
if it was a real thing, a static structure, existing somewhere in space and time. It
also means being able to recognize the idea “at a glance” and to manipulate it as a
whole, without going into details. (p. 4)
Finally, a schema conception is achieved when an individual can think of a
logarithm as an object that can be operated on and can link this understanding to
exponential functions. This understanding forms a coherent framework in the
individual’s mind. For example, when solving an exponential equation, the student
knows when it is appropriate to apply either the natural logarithm or the common
logarithm to extract the required solution. According to Dubinsky and McDonald (2001),
using the processes and mechanisms as outlined by APOS Theory, “This framework must
be coherent in the sense that it gives, explicitly or implicitly, means of determining which
phenomena are in the scope of the schema and which are not” (p.277). However, this
understanding is difficult to measure since it exists in the mind of the individual. What
can be determined is the distinction between the proposed genetic decomposition and the
subject’s progress along the genetic decomposition pathway. Figure 3 provides a visual
representation of the genetic decomposition.
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Figure 3. Initial genetic decomposition.
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Overview of instructional design. Advanced mathematical concepts require
advanced mathematical thinking. What this means is that students’ first exposure to these
topics should not be presented in a neat, polished format, typical of those commonly
found in first-year university-level mathematics courses, but in a format that allows
students to experience a concept’s development (Dreyfus, 1991; Tall 1991). As the
material becomes more abstract, no evidence exists to suggest that students will discover
these ideas without considerable “orchestration of their learning activities” (Dubinsky &
Lewin, 1986, p.85). Even the best and most dynamic instruction will fail “if it does not
take into account the cognitive structures of the knower as well as the process by which
these constructions take place” (Dubinsky & Lewin, 1986, p. 85). This statement implies
that concept acquisition does not develop devoid of specific mental constructions, but
rather through a series of cognitive connections of previously learned component
concepts (Confrey, 1991; Dubinsky & Lewin, 1986; Dubinsky & Harrell, 1992).
The instructional treatment, then, an essential component of the teaching
experiment, must be carefully structured to provoke students to perform the reflections,
abstractions, and accommodations needed to promote cognitive growth (Battista, 1999a,
1999b; Dubinsky & Lewin, 1986). It was the researcher’s hope that by completing tasks
outlined in Table 1 (see Appendix A for a complete description of instructional tasks),
which model the genetic decomposition proposed by the researcher, the participants
would develop a more sophisticated repertoire of cognitive building blocks which would
interact much like the “working methodology of a mathematician” (Dreyfus, 1991, p.28).
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Table 1
Overview of Instructional Tasks
Observable Skills/
Evidence of
Understanding
Explain why their
solutions make sense
Students’ Understandings as
Outlined in APOS Theory
Extend this to include all real numbers as
exponents
Pretest, Part C, D
Explain why the rules
work
Still at an action level, but students
are beginning to generalize to a
larger set of inputs for the exponent
Using the graph of
we will fill in
the “holes” by including integer values,
rational and irrational numbers in the
domain, move to generalizations about
the function ( )
Task 1, Part A
Correctly complete
worksheet and make
the table, draw the
graph for each
function on the same
set of axes
Process level understanding begins to
develop when the student can
generalize about overall
characteristics of the function
without actual values for the base and
the exponent
Understand the limitations of the set of
inputs/outputs for the function,
differentiate between parameters and
variables
Task 1, Part B
Response to
questions posed by
researcher
Still at a process level of
understanding; however, when asked
to explore further, they begin to
move to next level of understanding
Explain what it means to reverse
exponentiation, solicit conversations
about how to accomplish this task
Task 1, Part B
They can use a graph
to find the value of
the exponent
Object level understanding of
exponents and exponential functions
Ask students to describe the role of the
parameter and the variables, do they
even recognize the difference?
Task 2, Part B
Looking for
something like:
means x is the
product of how many
factors of a
Trying to move to a process level
understanding of the notation
associated with logs
Subjects will be asked to complete
several tables using exponents and logs
Task 2, Part B
Complete a table of
values, verbalize any
connections between
exponents and logs
Action level; however, subjects
should be attempting to make
connections between exponents and
current work with logs
Introduce the formal definition of the
logarithmic notation, ask subjects to
perform several transformations from
logs to exponents and the reverse
Task 3, Part A
Correctly complete
the exercises
Action level, subjects have a rule to
follow to complete the activity
Instructions/Activities/Tasks
Basic operations with whole number
exponents
Pretest, Part A, B
Action level of understanding of
exponentiation
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Table 1 Continued
Subjects should evaluate the
expression without the use of a
calculator. Students are being
asked to coordinate what they
know about logarithmic
notation and exponents
Task 3, Part B,C
Correctly complete
the exercises
Process level understanding: the
format has changed and technology is
prohibited; subjects are no longer
looking at an equation and just figuring
out how to rearrange the “pieces.”
They need to understand what each
“symbol” in the expression represents
Students are asked to solve
exponential equations
Task 4, Part A
Correctly complete
the exercises
By guessing answers, students
demonstrate they have not interiorized
logarithmic rules, operating at action
level. If they verify responses using
logarithmic notation, classify as
process level understanding
Students are being asked to use
the two ideas in combination
with each other, flexibly
moving from one form to the
other to make sense of notation
Task 4, Part B
I would consider this to be process
level understanding if explanations
contain ideas that demonstrate ability
to move freely between representations
Development of logarithmic
rules
Task 5, Part A
Complete the tables
and offer some type
of generalizations
Action level if only able to complete
table; process level if a proof is
formulated
Have students complete an
activity demonstrating
understanding of how to apply
logarithmic rules
Task 5, Part B
Can answer a series
of true-or-false
questions and justify
responses
Process level if justification includes
comments like “You add exponents
when you multiply with the same base
value”
Complete the table without a
calculator, answer questions
using their table
Task 6, Part A
Can complete
without the
assistance of
technology or a rule
Process level understanding if they can
apply previously learned material to
complete exercises
Complete a similar table, less
information given, looking for
more conceptual understanding
of when these laws of logs will
not apply
Task 6, Part B
Can answer the
Process level understanding moving
questions posed
toward object level—students see that
about the given table when asked to evaluate a log of a
prime number they can only give an
approximate answer without
technology
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At the completion of each task, students were evaluated as successful or unsuccessful. If
successful, the subject was said to have assimilated the new knowledge into an existing
cognitive structure. If the student had not succeeded, a perturbation would have to occur
before an accommodation could be made to handle the new information (Battista, 1999a;
Davis, 1984; Dubinsky, 1994). This perturbation would be provided either by the task
itself or by questioning the student about their current level of understanding; however,
the most important part of the teaching experiment was to focus on the nature of the
developing ideas and model the students’ responses into a coherent picture, and then, use
them to help students construct an understanding of the concept.
Protocols
All participants in this teaching experiment worked collectively as a group during
the teaching episodes. Building on their understandings of exponents, participants
explained their thinking as they completed a series of tasks designed to help them build
or revise existing mental representations of exponential functions, which later would be
called upon to build a coherent image of logarithms. However, before the start of Task 1,
a discussion was held to clarify misconceptions students had experienced in completing
the pretest/initial assessment.
After each teaching episode, the participants completed an additional informal
assessment. The informal assessment was designed to encourage the participants to think
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about their thinking and identify the interconnection between logarithms and exponents
rather than as a series of isolated tasks. Furthermore, the hope was that by reflecting on
the mathematical concepts, participants would come to understand this mathematical
concept as more than an activity involving only symbolic manipulation.
Triangulation
To establish the reliability and internal validity of the findings the researcher used
multiple data sources to corroborate the convergence of data. Video and audio
recordings, along with written work, interviews, and observer/research notes, were the
data sources. All teaching episodes were both audio and videotaped for subsequent
analysis. All audio recordings obtained via classroom sessions or individual interview
were transcribed and compared to the video recordings. The videotaped data gathered
during each session was supplemented by the researcher’s notes as well as by written
student work. This provided additional insight in a way not captured in the video or
audio recordings. Collectively, these data sources provided a glimpse into the reality as
understood by the participants.
Interrater Reliability
There is common belief that reliability is an important property in educational
measurement. A verification tool known as inter-rater reliability addresses the
consistency of data analysis. According to Marques and McCall (2005), interrater
reliability is used not only as a tool to verify coherence of understanding, but also as a
method to strengthen the overall findings of a qualitative study. APOS Theory, which
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uses a qualitative framework to assess whether or not students have made the mental
constructions of actions, processes, objects, and schemas proposed by the theory, requires
subjective analysis of the data gathered from each participant in terms of these four
constructs.
Because the researcher served as the primary instrument in rating the subjects’
level of understanding, interrater reliability was used to establish solidification of
understanding. Interrater reliability depends upon the ability of two or more individuals
to be consistent in their observations of the same phenomena.
By using interrater reliability as a solidification tool, the interrater could become a
true validator of the findings in a qualitative study, thereby elevating the level of
believability and generalizability of the outcomes of this study. . . . Hence, the act
of involving independent interraters, who have no prior connection with the study,
in the analysis of the obtained data will provide substantiation of the “instrument”
and significantly reduce the chance of bias influencing the outcome. (Marques &
McCall, 2005, p. 440)
Clear guidelines were established with the outside observer prior to the rating of
individual observations cited by the researcher. The observer was given a seminal article
that presents a detailed description of the components of APOS Theory. The researcher
and the outside observer held a calibration session in order to ensure that the
interpretation of data was consistent with the tenets of APOS Theory.
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Satisfied that the outside observer’s understanding of APOS Theory was
consistent with her own beliefs, the observer received a table that summarized each
participant’s performance on the instructional activities, citing incidents that were
insightful. She was then directed to describe the level of understanding the student
appeared to have obtained in accordance with APOS Theory. Upon completion of the
first table, another meeting was held to compare and calibrate consistency between the
researcher’s observations and the outside observer’s. Table 2 summarizes these results.
Table 2
Results of Initial Calibration
Participant
Consensus
Agreement ±
Tom
6 of 9 items
1 of 9 items
Agreement ± 1
2 of 9 items
After a short discussion, it was agreed that understandings were similar. Three
additional data tables one for each of the other students were presented to the observer,
who independently completed her review. In addition, she completed an analysis of the
final two items in the table regarding Tom’s performance used for the initial calibration.
The results of this analysis are presented in Table 3.
Data Analysis
The theoretical framework for this study was APOS Theory, as described in
Chapter 1. Using the mechanisms offered by APOS Theory to analyze the data provides
considerable explanatory power. Dubinsky (2001) writes:
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Our method of analyzing data . . . looks at interview transcripts in very fine detail.
We try to find mathematical points as narrow as possible on which there is a
range of student performance. Then we try to find explanations for the
differences in terms of constructing or not constructing specific actions,
processes, objects, and/or schemas. In the totality of these local explanations,
APOS Theory offers explanations of student success or failures. (p. 14)
Table 3
Interrater Reliability Summary
Participant
Consensus
Agreement ±
Agreement ± 1
Tom
8 of 11 items
1 of 11 items
2 of 11 items
Doug
10 of 11 items
Jim
6 of 11 items
3 of 11 items
Earl
7 of 11 items
3 of 11 items
1 of 11 items
2 of 11 items
As the students’ perspectives on the effectiveness of this experience were
captured, various themes and categories developed. APOS Theory, with its description
of an individual’s journey from action to process to object to schema reification, fit nicely
with the strategies recommended for working out ways of recognizing and labeling a
concept when coding data (Rubin & Rubin, 2005). At the same time, it was imperative
that the data document the experiences of the participants through their own cultural lens.
“To complete the analysis, you still have to put these concepts and themes
together, show how they answer your research question, and pull out broader
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implications” (Rubin & Rubin, 2005, p. 223). A retrospective analysis of the videotapes,
interviews, student worksheets, written reflections, and other informal assessments was
conducted to gather evidence of students’ understandings. This evidence of students’
cognitive acts of construction as induced during the teaching episodes was summarized to
form a genetic decomposition from the students’ perspective, documenting how they
acquired their understanding of logarithmic concepts and the role symbolic notation
played in its development. In this way, the summary forms a basis for future pedagogical
approaches to the teaching and learning of logarithmic concepts.
Instructional Tasks
Designed according to the researcher-proposed genetic decomposition, the
instructional tasks focused on helping students attain at least a process or object
conception of logarithms. There were six tasks; however, time restrictions did not permit
the completion of Task 6. A rationale for each is briefly described below. Appendix A
contains a complete description of each task.
Initial Assessment/Pretest
Proficiency in problems in parts A, B, and C using the given instructions
demonstrated participants possessed an action understanding of exponentiation. As
students worked through each section, they realized they no longer needed to write out
what the expression represented. This ability was an indication that they were beginning
to internalize the process, moving to a more compact notation to explain what was
happening as they simplified. If students were using rules from memory, they explained
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why the rules worked, if possible. When students explained why the rules worked,
depending on the sophistication of their responses, their understandings were classified
using APOS constructs. Moving to Part D required students to reorganize their
knowledge. The participants were being asked to extend their knowledge of whole
number exponents to include integer values. If students offered plausible explanations of
why exponents can be extended to include integer values, this was classified as a preobject conception of exponentiation. The last exercise required students to understand
how to evaluate a function when given a series of inputs. Successful completion
demonstrated an action level understanding of functions: given a rule, the student
replaced the variable x and computed the corresponding output ( )
Task 1
Task 1 (see Appendix A) attempted to extend the participants’ understanding of
exponents to include all real numbers. Participants needed this understanding in order to
explore exponential functions. The idea of domain and range for functions became
critical. We talked about the reversal process, and the researcher informally presented it
with notation that was intuitive to students’ understanding of functions. At the
completion of the graphing portion of this task, the students responded in writing to the
question, “How do you identify characteristics of the graph of an exponential function?”
Evidence of a process or preprocess understanding was assumed if a verbal or written
response involved some type of dynamic transformation of objects according to some
rule. For example, if participants’ responses made mention of some sort of input, which
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is the exponent, and is processed according to a rule that produces some sort of output,
they were considered to be at this level of understanding. Their responses might also
include a reference to domain and range; however, the participants might be unable to do
anything more.
Part B of this task attempted to get participants to think about the reversal of this
process. As students constructed this type of understanding, they began to encapsulate
this knowledge, which allowed them to form an object conception of an exponential
function. For example, when asked to find how long an investment of $1,000 (from the
pretest) would take to double in value, a student might try algebraic methods to solve the
equation but be unable to find “something that works.” The variable is now the
exponent, and division by 1.07 is not valid mathematically since exponents are the higher
order of operation, so the student begins to think of ways to “undo” the power of n.
While the notation may have been problematic, the researcher hoped that the participants
would reorganize their cognitive structures to accommodate the reversal process of
exponentiation as being more than just another algebraic manipulation.
Task 2
Task 2 required the students to cycle back to an action level understanding. Using
instructor-defined tables, the participants were to graph an exponential function and its
corresponding inverse logarithmic function on the same set of axes for several different
pairs of functions, without being told the log function was the inverse. The desired
outcome was that students would see a connection between the pairs of functions. Part B
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asked students to verbalize this connection by exploring the “log” button on their
calculator; however, the graphical representation and/or the tabular representation
provided by the graphing activity was not available. The hope was that participants
would see how the role of the base, which is considered a parameter, influenced the
output. Participants came to realize that the “log” button on their calculator was more
than just a numerical value for a particular input; it was somehow related to exponential
notation. By the time of the completion of this activity, it was intended that the tasks
would have provided the participants with enough cognitive dissonance to force them to
expect some change in the way they thought about the relationships between logarithms
and exponentials. If successful, a participant would be able to verbalize these
connections, treating logarithmic functions as a separate concept with distinct
characteristics that were directly related to exponential functions. The successful
students were able to understand the implied question embedded in the notation and to
conjecture about the value of
without the aid of technology.
Task 3
Using the information provided, students completed Part A of Task 3. Successful
completion of Part A indicated an action level of understanding of logarithms at a
computational level, whereas successful completion of Task 2 had indicated an action
level of understanding of logarithmic functions. Combining the actions of both Task 2
and Part A of Task 3, the students would begin to interiorize both actions to form a
cohesive view of logarithm as a numeric value and logarithms as function that ultimately
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represented the same thing: the value of the exponent in an equivalent exponential form.
It was hoped that the students would begin to develop a process understanding of the
basic structure of a logarithmic expression; if nothing more, they would be able to
verbalize the role of each symbol in both the exponential expression and its
corresponding inverse logarithmic notation.
In Part B, students were asked to evaluate a series of logarithmic expressions
without a calculator. Specifically, if students justified their responses for items 1 through
10 by including some mention of rewriting the expression in a compatible exponential
form, they were moving toward a process level understanding. In item 10, understanding
of the domain of a logarithm was critical. The ability to recognize the properties of the
function indicated that the subject was not just thinking about logarithms as
computations. To deal meaningfully with this situation, the student had to encapsulate all
possible values for the domain and range of this expression into a single conceptual
entity.
Part C encouraged students to think about the expression as its own entity in
order to compare the magnitude of each expression. This required the reversal of the
logarithmic notation in order to evaluate the expression correctly; however, if participants
completed these problems using generalizations about the meaning of the notation itself,
then it was possible to identify this as an object level understanding. Students were able
to think about logarithms not only as a computation to do, but as a function itself,
meaning that given an input, an output was generated. This ability required an
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understanding of the role the parameter plays in determining the desired output for the
given input. In other words, the participants should be able to see the symbol
as
two separate, yet connected ideas.
Task 4
If the subjects completed Task 4 using the graphs constructed earlier, they were
still at an action level of understanding. If, however, they “guessed” an answer and
checked the reasonableness of the solution by rewriting it in its corresponding
logarithmic form, the subjects had a process level understanding of logarithmic concepts.
By paying close attention to the thought process as verbalized by the participants the
researcher was able to determine their level of understanding. Guessing alone suggested
that a student was attempting to interiorize the action but was unable to coordinate his or
her knowledge of exponents and logarithms in a meaningful way. A process level
understanding was not obtained until the subjects could see the need to reverse the
process of exponentiation by writing an equivalent logarithmic expression and then use
the “log” button on their calculator to find an approximate value.
Part B of Task 4 consisted of conceptual-type questions in which students had to
apply the “rules” for logarithms and to understand why they worked. The subjects
needed to make sense of the questions using their existing knowledge structures.
Exponential functions were no longer an isolated concept, but part of what it means to
know and understand logarithmic concepts. An object level understanding was achieved
if the subjects were able to conceptualize the relationships between exponentials and
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logarithms. In other words, participants were able to use logarithms to answer questions
about exponentials, and to use exponentials to answer questions about logarithms, with
understanding. In essence, participants took large “chunks” of information and
compressed them into a single entity. Harel and Kaput (1991) posited that unless the
learner is able to consolidate knowledge into conceptual entities, the mind, with its
limited capacity for processing, especially as it pertains to working memory, would
experience considerable strain in dealing with complex mathematical topics. To this
point, the instructional tasks had dealt only with single terms. When students were asked
to solve equations where multiple logarithmic terms were involved, how flexible was
their thinking?
Task 5
Task 5 was structured to allow participants to explore the development of
logarithmic properties after participants had obtained a process level understanding of a
single logarithmic term. Participants were asked to identify the role of the parameter and
how it influenced a logarithmic function. Additionally, participants with a process level
of understanding coordinated their existing knowledge about exponents and exponential
functions to move flexibly between formats to solve either logarithmic or exponential
problems. Since logarithms represent exponents, it follows that “rules” associated with
exponents apply to logarithms.
Initially, students were to complete a table of values. Successful completion
demonstrated an action level conception, but if participants generalized a relationship
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between addition of logarithms and multiplication of exponential expressions, they had
internalized the action, indicating a possible shift to process level understanding. Moving
to a process understanding required students to use and construct some type of formalized
notation to represent the relationship. For example, the participants might be able to
verbalize the notion that when you add two logarithmic terms you must multiply the
inputs, but might be unable to express this understanding symbolically. It was not until
they developed an understanding for the symbolic notation used to communicate this idea
that growth in understanding was achieved. APOS Theory would describe this as
encapsulation: the learner had coordinated their knowledge about an individual
logarithmic term, allowing the individual to work with ideas that were more complex.
Participants were then asked to construct their own proof for the properties of
logarithms; however, this activity might require the use of numeric examples first, before
a more generalized format was developed. It was in Task 5 that the most profound
change in instruction occurred. The researcher introduced the historical genesis of
logarithms; and based on this new knowledge about the historical development of
logarithms, students were able to intuitively develop an understanding of logarithmic
properties. Participants used these ideas to evaluate the truthfulness of several
statements. The ability to do this demonstrated some type of process understanding.
Since the formalized properties were not given, they had to rely on knowledge they had
internalized.
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Task 6
Task 6 asked students to complete a table of values without the use of technology.
In order to complete this activity, participants needed at least a process level conception
of logarithmic concepts. A prompt for the first table, “Using what you know about
properties of logs, complete the table,” was included; however, the visual cue of the
printed rules was not available. If participants completed the table using memorized
rules, they had not developed an object level conception; they were more than likely
operating at a process level understanding. As they moved to Part B, the only prompt
was “Complete the table.” The table had values for
and
listed. Participants
discussed whether other representations were useful for completing the table; however,
this required coordination of existing knowledge. They needed to see that
Presumably, they knew
Therefore, if given
the participants should be able to estimate the value for
Participants would
complete the tables using these types of relationships. Time constraints and the
introduction to the use of logarithmic tables and how they were created prevented the
completion of this task.
Summary of Chapter 3
Using a teaching experiment methodology, this study described how students
built meaning for logarithmic concepts and how the symbolic notation contributed to this
cognitive understanding. Using APOS Theory as a framework to guide instruction and
subsequent analysis of the data collected through student worksheets, interviews, written
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reflections, and video analysis of teaching sessions, it was the intention of this research to
go beyond observation and classification of student difficulties with logarithmic
concepts. A mathematical model was formulated for a possible set of mental
constructions that an individual might make as they attempt to make sense of logarithmic
concepts and the associated symbolic notation. It was expected that some obstacles
would occur during the course of the teaching experiment. Revisions to the original
mathematical model were made, based on the cognitive conflicts encountered.
Educators at the university level are aware that most entering freshmen are not
prepared to do college-level mathematics. But what are educators willing to do in order
to meet the students at their level of development? Professionals in the field need to
utilize research-based findings to improve instructional programs. Understanding how
individuals approach certain topics may be the first step towards developing effective
programs of instruction.
CHAPTER IV
DATA COLLECTION IN THE TEACHING EXPERIMENT
Introduction
This chapter contains the analysis and interpretation of the data collected in the
course of the researcher’s teaching experiment, which was conducted during the summer
2010 semester. The data is organized by the tasks or instructional activities completed by
the participants in the teaching experiment. It answers the question: How do students
acquire an understanding of logarithmic concepts? Using APOS Theory as a framework
to study the cognitive development of logarithmic concepts, this research began with a
theoretical analysis modeling the epistemology of the concept in question. This analysis
is referred to as the genetic decomposition of the concept. Initially based on the
researcher’s understanding of the concept and general APOS Theory, the genetic
decomposition is a set of mental constructs that describes how a concept might develop in
an individual (Asiala et al., 1996; Dubinsky & McDonald, 2001). The researcher
designed instructional activities and collected data based on the genetic decomposition.
The researcher hoped that by completing the instructional tasks (see Appendix A for
complete description of instructional tasks) which modeled the genetic decomposition the
researcher proposed, the participants would develop a more sophisticated repertoire of
cognitive building blocks. Such building blocks would interact much like the “working
methodology of a mathematician” (Dreyfus, 1991, p.28), allowing participants to develop
an understanding of basic logarithmic concepts.
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Integral to the development of understanding for logarithmic concepts are
exponential functions. Weber (2002b) stated, “It is critical that students be capable of
understanding exponentiation as a mental process” (p. 3). Without this type of
understanding, it is doubtful that students will be able to view expressions such as bx as
both an operation to perform and a number that is the result of applying the operation of
exponentiation. Cognitive structures to support this type of thinking become increasingly
more difficult to develop when we expand the definition of an exponent to include all real
numbers. Typically, we ask students to sacrifice the intuitive appeal of exponentiation as
repeated multiplication without providing appropriate learning opportunities to flesh out
difficulties students experience “concerning the isomorphism between the structures
involving exponents and exponential expressions” (Confrey, 1991, p.127).
In his study, Weber (2000b) reported that when asked to recall properties of
exponents and logarithms after traditional instruction at the university level, students
could understand exponentiation only as an action. A “tenuous series of rules and
definitions followed by extensive practice in symbol manipulations masking the broader
systematic qualities of the relationships” (Confrey, 1991, p. 127) typically dominates
traditional instruction. Chesler (2006) conducted a similar study to assess the level of
understanding within the APOS framework that students possessed after completing a
unit on exponential and logarithmic functions. He found most students “had some sense
that a relationship exists between logarithms and exponents; however, [they] were
generally unable to communicate it precisely” (p. 5). He attributed their success, or lack
thereof, to their ability to construct models of inverse functions. Vidakovic (1996)
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hypothesized that students must have at least a process or object understanding of
functions, as detailed by Dubinsky, in order to construct models of inverse functions. He
felt this level of understanding would allow students to understand the action of
switching the independent and dependent variable. Without this process understanding of
exponential functions, it is doubtful that any meaningful learning of logarithmic concepts
will occur.
The following sections detail the results of this study’s teaching experiment, while
Chapter 5 will integrate the results of this study and of previous research with the genetic
decomposition that was hypothesized for the study.
General Knowledge of Exponents and Functions: The Pretest
This section categorizes the participants’ understanding of exponentiation and
functions. It is based on the results of the pretest administered used for selection of
participants and on subsequent conversations with the subjects during the first meeting.
This process allowed the researcher to categorize each participant’s initial level of
understanding using APOS terminology.
Logarithm, by definition, is the power to which a base must be raised to in order
to yield a given number. This implies that it would have been futile to select participants
who had not yet achieved an action conception of exponentiation; participants needed to
be proficient at simplifying exponential expressions with integer value exponents.
At the collegiate level, students typically encounter a cursory review of
exponential expressions followed by a list of rules to memorize, culminating in extensive
practice in symbol manipulation before exploring exponential and logarithmic functions.
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The instructional methods and textbook used by the instructor for this particular
Precalculus course from which participants were selected to participate in this study was
no exception to this model. Traditional instruction assumes that students can successfully
accommodate the extension of the role of the exponent as a counter to include all realnumber values as exponents through extensive practice. Instruction rarely pays attention
to the cognitive difficulties students encounter when the value of the exponent is
extended to include non-positive integers. Unaware of students’ difficulties, instructors
quickly move to include irrational exponents. Instructors rarely discuss the meaning of
irrational exponents openly in classroom discourse (Confrey, 1991); however, they
expect students to demonstrate flawless symbol manipulation. As instruction moves to
visual representations for this class of functions, the focus shifts to obtaining the correct
graph, with the question of continuity seldom explored.
Constructivist theory posits that students actively construct meaning for
mathematical concepts; APOS Theory establishes a metaphor for describing these
hierarchical levels of understanding. Using APOS Theory as a tool, this researcher
sought to explore how students enrolled in a first-year college Precalculus course
constructed meaning for logarithmic concepts by first examining their understanding of
exponential functions and the role of the inverse function.
All students enrolled in a summer-semester Precalculus mathematics course at a
large Midwestern urban university took a voluntary pretest/initial assessment to assess
their level of understanding of integer exponents and their ability to evaluate functions
prior to instruction on exponential and logarithmic functions. No points were offered
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toward the course grade for participation, but students were informed that it was possible
that they would be asked to continue with the study once the results of the assessment
were tallied.
The pretest consisted of five parts with an average of five questions each. Using
only the rule supplied, for any counting number n,
, n times,
students were asked to simplify a variety of routine exponential expressions. Building on
the basic definition, students simplified complex expressions, which should lead them to
a deeper understanding of why the rules for integer exponents were valid. After each
group of problems, students generalized about their work. This was done in an attempt to
recruit participants who were able to communicate their ideas both verbally and through
their written work. The last part of this assessment asked students to evaluate
exponential functions for various inputs.
The students were given time in class to complete the assessment (20 minutes);
however, all students required additional time after the scheduled end of the class period
in order to complete the assessment. Time constraints prohibited two of the 11 students
from completing the assessment in its entirety, and these students were not considered in
the selection process. The remaining nine students completed the assessment; however,
two these students did not attempt to generalize their understandings. In the remaining
pool of seven students, four were able to correctly answer most parts of the pretest and
offer some type of explanation; all four of these students agreed to participate in phase
two of the study.
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The researcher obtained background information from each participant to
determine their previous exposure, if any, to logarithms. The oldest participant had taken
a college algebra course in the late 1950s and was a non-degree seeking student. Two of
the three remaining participants were first-time degree-seeking students who had
matriculated into this course through two lower-level mathematics courses offered at this
university. The remaining participant was a post-baccalaureate non-degree-seeking
student who had previously completed this course unsuccessfully. Three out of the four
admitted they had heard of logarithms but did not know exactly what they were.
The researcher administered a mathematical beliefs survey during the first
meeting. Yackel (1984) developed the survey instrument at Purdue University to
determine college students’ beliefs about mathematics. “Yackel (1984) based the design
of her instrument on the long-time research of Skemp” as it pertained to beliefs held by
instrumental and relational learners (as cited in Quillen, 2004, p. 22). Three of the four
participants felt that mathematics consists mainly of using rules, memorizing procedures,
and manipulating formulas. The fourth participant (the oldest of the group) had a much
different view, strongly disagreeing with these statements. Results of this assessment are
summarized in Table 4. Complete results are included in Appendix B.
Prior to beginning Task 1, it was necessary to probe deeper into the participants’
initial understanding of exponential concepts. Because the first task asked students to
extend their understanding of exponents to include all real-number values as exponents, it
seemed reasonable to require that participants understand why the rules work. For
example, when asked how he simplified
, Doug responded, “I just knew it was 1”; but
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Table 4
Mathematical Beliefs and Attitudes
Guiding Theme
SD
D
U
A
SA
Mathematics by imitation
(Questions 1, 2, 5, 6, 10, 11, 14, 18)
5
8
7
10
2
Mathematics promotes deeper understandings
(Questions 3, 8, 9, 13)
0
0
4
4
8
Mathematics is about right answers
(Questions 4, 7, 12, 15, 16, 17, 19, 20)
7
15
2
7
1
Note. SD = strongly disagree; D = disagree; U = undecided; A = agree; SA = strongly agree.
Numbers in table represent frequency of responses for the grouped questions.
when asked to evaluate 20, he initially wrote, “This doesn’t fit into my mind.” However,
he was able to simplify
problem,
but added, “This is just a guess.” Tom explained for this
, “I see 3-3 which is zero, so 20 is 1.” Doug responded, “So you are saying
is equivalent to 23-3? Oh! Now I get it — it is not immediately intuitive, but if you do it
this way, then yes, zero powers are equal to one.” Doug’s written work and comments
suggest he was operating at an emerging process level; he knew most of the rules, but
could not immediately offer a plausible explanation for the extension of the rules to
create meaning for zero and negative exponents. However, he was attempting to
assimilate this new information. Breidenbach et al. (1992) refer to this understanding as
“sort of a pre-process conception” (p. 251). Furthermore, they suggest that as individuals
transition from action to process understanding the shift is never in a single direction,
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making it difficult to determine if a particular student’s understanding is limited to an
action or process.
When asked on the pretest to make conjectures or generalize their results, two of
the four participants used memorized rules to generalize their findings rather than writing
out each expression in expanded form and trying to extrapolate meaning from this
exercise. Tom wrote, “When I see
” When questioned
about the use of the notation he explained, “You know if the subtraction is positive it is
just the answer but if it is negative it goes in the bottom as a positive exponent.” Jim
responded, “Expanding the exponents makes little sense when compared to using the
laws of exponents.” Earl wrote out the expanded form as instructed, but then added,
“This process is the same as subtraction or addition,” indicating that he was perhaps
moving toward a more sophisticated understanding than either Jim or Tom but did not
share how he was legitimizing his definitions for negative and zero exponents. Doug, on
the other hand, just wrote out the expanded form, then added, “You could add them [the
exponents], but when the exponents are part of fractions you cancel, as long as the base is
the same.” He stated that he knew
but could not explain why, admitting it was
just memorization on his part. Tom, when asked about the expression 2-2, knew the result
was
but said, “Other than the rule I really don’t see it. I just know how to use the
rule.” Doug said, “Well let’s use the logic of zero exponents.” And he wrote, “
, so
let’s use the rule, so it is 22-4 and when you subtract you get 2-2 , so yes, I can see what
you are trying to get at!”
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When asked if by itself
made sense, all agreed that the exponent could no
longer be used as a “counter.” In an attempt to get all participants to create meanings for
zero and negative exponents, they were asked to write a problem that has
Earl responded by writing
responded with
as its result.
. Tom and Doug did something similar, while Jim
. This answer was not what was expected since he only rewrote the
expression without a negative exponent; however, he demonstrated an understanding of
how to apply the laws of exponents. With the exception of Jim, all others seemed to be
considering how the rules for positive value exponents could be extended to include zero
and negative values. This outlook was not at all surprising. On an attitudinal survey,
when asked if doing mathematics consists of mainly using rules, Jim answered strongly
agree. He also strongly agreed with the statement that learning mathematics mainly
involves the memorization of procedures and formulas.
When the group was asked to further clarify the meaning of fractional exponents,
Jim was quick to reply that the “bottom number indicates the root, the top number
indicates the power, so
would be the square root of 9 which is 3 then cube that.” Earl
replied that it indicated you were taking some kind of root while Tom and Doug admitted
they had seen fractional exponents but had forgotten their meaning. On the pretest, Doug
indicated
and Tom wrote
, indicating that neither student understood the
meaning of fractional exponents; they seemed to be relying on rules that they could not
remember. Confrey (1991) suggested that “Traditional rule-oriented approaches seem to
emphasize how to move expressions around without enough focus on the operation which
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underlies that movement” (p. 137). Jim and Earl seemed to possess solid algorithmic
computation skills, but they did not seem to be able to verbalize their respective
understandings. Tom and Doug, on the other hand, while they could move the expression
around if given the rule, seemed to experience cognitive difficulties when they attempted
to extend the domain of allowable exponents. Note the comments in the excerpt below,
where the letter I indicates the interviewer’s comments and the letters T, D, J, and E
represent the responses of the participants, Tom, Doug, Jim, and Earl respectively:
D: Well, does that mean any real number can be an exponent, or can any real
number be a base?
I: Any real number can be an exponent.
J: Any real number is just two to whatever it is [referring to the exponent].
D: Like 0.2395 can be the exponent?
T: Yes, I get it; it is similar to the first power but only smaller.
All agreed that trying to put into words what they were thinking was somewhat difficult,
claiming they knew what operation or procedure to do when, but were unsure of what
they needed to say.
Each participant was able to evaluate a function correctly. Because students
would be working with both exponential and logarithmic expressions during the course of
the teaching experiment, it was imperative they possess an action conception for function.
An action conception for function has been described as the ability to perform repeated
mental or physical manipulation such as substituting numbers into an algebraic
expression and evaluating it (Breidenbach et al., 1992). For example, when asked to
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evaluate ( )
and ( )
( ) for integer values of x, all participants correctly
completed each exercise. This demonstrated their ability to carry out the calculation by
reacting to the formulas
and ( ) that give precise details on what steps to take to
manipulate the formula.
Growth of Student Understanding of Exponential Functions: Task 1
The second section of this chapter characterizes the growth of student
understanding of exponential functions by using each student’s comments and written
work that characterize the APOS levels of understanding. Specifically, Task 1 explored
the initial conceptions held by the participants about graphs of exponential functions.
Students typically graph functions using discrete values of x; however, when graphed, the
rendering is a continuous function. Does this imply that students intuitively understand
that the domain for an exponential function is all real numbers? This task attempts to get
students to consider the graphical representation and its implications about the domain,
range, and limitations.
When the researcher was satisfied that all students were moving toward a process
understanding of exponentiation and could comfortably work with functional notation,
Task 1 was introduced. The researcher explained that participants would be working
with a class of functions referred to as exponentials because the variable quantity was the
exponent itself. More specifically, the function has the form ( )
of b can be any positive real value except 1.
where the value
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Participants were asked to develop a table of discrete values for an exponential
function and use the ordered pairs to graph the exponential function ( )
. This
was done to extend the definition of exponents to include all real numbers. After the
participants had successfully graphed
( )
, the researcher asked them to make
observations about the domain of this function. Tom responded, “There it is, [points to
his graph], all real numbers.” To encourage further discussion about Tom’s comment the
researcher posed the following question: “If this is true, could you use the graph to
approximate the value for (√ )?” Doug responded, “So (√ ) would be
√
, the
square of 3 is what? It would be - , no . . .” He then said, “You can’t do it because it’s
not a real number.” When asked to clarify what he meant he said, “Well it won’t be a
nice number we can locate.” Earl stated, “We are trying to say what y equals when x is
the square root of 3.” Probing further, Jim responded, “Why would you use a graph? I
would just do this [ (√ )] with a calculator.” Reluctant to use the graph, Earl
commented that √ is bigger than 1 but less than 2, but still did not approximate the
value of the expressions. When prompted by the researcher, “Could you approximate the
value?” Tom asked, “You mean give a possible range?”
It was apparent that while subjects were competent in the mechanics of graphing,
they were unable to appreciate the graph’s significance. They quickly concluded that the
domain for this exponential function was all real numbers, but it was evident they did not
know how to make sense of this. They encountered an irrational number as an input but
seemed to miss a fundamental connection: “A point is on the graph of the line L if and
only if its coordinates satisfy the equation of L” (Moschkovich, Schoenfeld, & Arcavi,
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1993 as cited in Knuth, 2000, p. 501). Were the students unable to see that the point
(√
(√ )), was on the graph of ( )
? The nature of the algebra curriculum is
such that the problems we offer students are for the most part limited to those that can be
readily solved within the framework of symbolic representations alone. “As a result,
visual representation is not perceived as necessary by most students when engaged in
mathematics problem solving” (Yerushalmy & Schwartz, 1993 as cited in Knuth, 2000,
p. 505). In the next segment of the data, Tom’s remarks are enlightening:
I: From the graph, can you approximate the value?
T: [Looking at his graph] How do you tell where the line is? [Meaning: for the
input √ ]
J: You need a value for the √ .
E: We are trying to say what y equals when x is the square of three [He is
pointing at the graph he has drawn.]
I: What was your approximation for √ ?
E: 1.4
However, the input may have suggested to the students that they needed a level of
precision that they could not attain by reading the graph and as a result did not think that
the graphical solution method was valid. The fact that the input itself was an irrational
number may have been in conflict with their generalized idea of an exponent. Tom was
still concerned that he needed a range of values to describe the output for an input of √
and seemed unwilling to make a more precise estimate. Jim finally asked, “You mean
you want us to eyeball it?” Doug responded, “It will be halfway between 3 and 9.” He
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then said, “No, maybe somewhere between 5 and 9.” Tom then narrowed his result down
to “somewhere between 6 and 7.” When asked if he could be more precise, Tom
responded, “Well that was the problem. This [√
is not exact because it goes on
infinitely.” Doug responded, "No, it is exact. We just can’t get to the end of it.” When
asked what type of number this was, Tom responded, “One we are not equipped to deal
with.” When asked then what possible values for x could be used as inputs, Tom
responded, “They could be infinite.” Doug echoed this when he said, “So that means x
could be any real number.”
When asked to describe the range for this function, Jim responded, “All real
numbers,” only to be corrected by Doug, who replied, “Look at your graph. The range is
zero to infinity.”
I: Does the range represent a set of values?
T: Yes, the y’s, the outputs.
D: So the range is zero to infinity.
I: Does that mean we include zero in this set?
D: No, it looks like this (
).
T: I don’t see my range being equal to zero.
D: No, it won’t touch zero.
I: Why not?
J: It would be 1 over 3 raised to infinity, which is not zero.
This conversation implies that the subjects were attempting to generalize the domain and
range of exponential functions based on the visual representation of the graph.
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Learning logarithmic concepts depends on the ability to reverse the
exponentiation process; therefore, students must possess a process or emerging-object
conception of exponentiation and exponential functions. To encourage students to
explore the nature of exponential functions, they were asked to complete a table for
several exponential functions. Tom asked, “Are we supposed to be drawing conclusions
as we complete the tables?” Others seemed to ignore his questions and continued to
evaluate the functions for the given inputs. As they all worked to complete the table of
values (See Appendix A for complete description of all Tasks), Tom conjectured that
once he evaluated half the functions the other half would “come for free.”
Doug often seemed confused. When trying to evaluate ( )
he said, “So y equals one half raised to the x is the same as
( ) for
to the negative three
power is 125.” His response implies several things. Doug missed details frequently and
only saw 5 raised to the third power, which does equal 125. For example, in spite of
earlier interventions, he still asked about ( )
and immediately replaced the fraction
with its decimal equivalent, resulting in an incorrect solution. Earl told him the answer
was 8, not 125. Turning to Doug, he asked him, “( ) is , right? So this means ( )
1 over
is
which is 8.” Doug responded that he needed to see this worked out
mathematically with what he knew, indicating he still needed a concrete image, whereas
Earl was able to see this result in his mind. Earl could manipulate the symbolism without
having to put pencil to paper, indicating he was moving toward a process understanding
of exponentiation.
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As the group continued to complete the table, Doug stated that the results of
( )
( ) appear to be similar to
( )
but in reverse order. Admitting he was
not absolutely certain of this fact, he stated, “I need to work these out to see if they do go
backwards.” He had saved ( )
for last because in his eyes, he explained, this
was “different” from the others. When asked how he thought about this function, Doug
replied, “I didn’t do it yet.” Jim said, “I saw ( )
( ) and used this format.”
Jim saw a pattern while completing the tables, but for him it was just numbers.
He said, “I just started copying the answers because I knew they were the same thing.”
Jim completed ( )
( )
and he explained to Tom, “When you see a negative
exponent you flip them and make the exponent positive. Then it is just the same answer
as the first 3, just flipped.” At this point, Jim’s actions are still guided by an external cue:
the exponent itself. If an individual has a process conception of exponentiation, they can
reflect on, describe, or even reverse the steps without actually performing the
manipulation. According to Asiala et al. (1996), “In contrast to an action, a process is
perceived by the individual as being external, and under one’s control, rather than as
something one does in response to external cues” (p.11).
The goal of the graphing exercise was to broaden the participants’ understanding
of exponential functions. On the same set of axes, the subjects graphed the points they
had just obtained from their tables and then described any patterns they noticed. As they
graphed each equation, Jim asked if he was supposed to see a pattern. Tom added, “As I
move down from where I started [on the graph of ( )
], it looks like the graphs
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are shifting downwards and they are all passing through the point (0, 1).” When asked to
explain further, he motioned to the graph of
( )
and said, “
and
will be
.” These statements indicated he could think about what was going to happen
below
without actually completing the graphs for each function. Jim echoed this belief when he
added, “
and ( ) are exact opposites,” motioning with his hands to show how they
would look. Tom agreed and added, “They kind of mirror image each other, I mean
across the y-axis they kind of would look the same,” making a flip-flop motion with his
hand. Earl quietly completed the graph of each function while the others described what
the curve should look like and where it should fall relative to ( )
before actually
completing the graphs. Doug, still trying to make sense of the functional properties of
exponentials, asked, “Can we say the lower limit of the function is either positive or
negative infinity?”
I: What to do you mean by the lower limit?
D: There is no lower limit, is there? It approaches zero.
I: Yes, it approaches zero.
D: I want to describe that approaching zero. I would like to use the term infinity.
J: Well I guess you could use 1 over infinity because that is a small number.
D: Is that what it means?
I: Well, infinity itself is a construct, but 1 over a very large number is very close
to but not equal to zero.
Upon completion of the activity, the participants summarized characteristics of
exponential functions in writing. Their responses suggested an ability to perceive
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exponential functions as entities possessing various global properties. Specifically, they
all pointed to the generalized case of what would happen to the function if the value of b
changed. Tom stated, “If b is a positive whole integer, the graph increases from left to
right; if b is a positive fraction the graph decreases from left to right.” Earl wrote, “When
b > 1, the graph is increasing, as b increases the curve steepens; when b <1 and > 0 the
graph is decreasing; and when b = 1 the graph is a straight line at y =1.” Furthermore, all
participants agreed the function possessed a horizontal asymptote; however, at times their
attempt to use precise language to describe functional properties was weak, indicating
instability in their conceptions. For example, Doug wrote, “If the base < 0 the line curves
down, the greater the denominator the sharper the curve but it doesn’t cross zero. For
example, if you have , interpret it as
and my explanation holds.” The participants
having completed this activity, the groundwork was set for them to reflect on a set of
possible inputs in relation to a set of outputs, a necessary condition for understanding
logarithmic functions.
Developing a Relationship between Exponential and
Logarithmic Functions: Task 2
Thompson (1994) states that when students build a process conception of
function, “They do not feel compelled to imagine actually evaluating an expression in
order to think of the result of its evaluation” (p. 7). In other words, students can describe
and predict the behavior of the function without actually completing the calculation.
“Breidenbach et al. (1992) claimed that a process conception of function provides an
entryway into an object-oriented understanding of function” (as cited in Slavit, 1997, p.
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262). Slavit echoed this belief (1997), stating, “Students are more able to comprehend
properties such as 1-1, onto, and invertibility once a process conception is achieved” (p.
262). At this level, a process conception of functions strengthens notions of reversibility
as students continually develop increasingly sophisticated understandings for domain and
range of exponential functions.
Task 2 began with an exercise in which students first had to find inverse functions
for several elementary functions such as ( )
( )
( )
in
order to strengthen their overall concept image of functions. Initially students verbalized
the role of an inverse function before they attempted to find an equation to represent the
inverse function. Prior to this activity, students ended Task 1 by estimating the value for
x in several exponential equations. They were also asked to go home and think about
what it means to reverse the process of exponentiation. Doug reported that because the
exponential involved multiplying, its inverse must be some type of division. Jim and Earl
reported that it meant to take the xth root of the answer. Tom, on the other hand, actually
looked up what it meant to reverse the process of exponentiation and reported to the
group that you use the notation
, and successfully rewrote several problems in
this format—but admitted he was not sure what the notation meant other than that this is
what you use when you need to solve for an exponent. When challenged to explain what
he meant by this he replied, “I’m not sure, other than this is the rule I found in the
textbook to use and then there was some other rule to get it into your calculator.” Doug,
while he indicated some type of division was involved, said he used Excel and successive
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approximations to solve the problem
, stating that because he knew the exponent
was between 1 and 2, he started in the middle, finding that 1.5 was too large.
D: I just kept doing it. When 1.5 was too large then I knew it was between 1 and
1.5 so I tested 1.25 and if it was still too large I went down; if it was too small,
I went up. It is just the principle I thought of; I don’t know how to do it
manually.
T: Is that the technique, guess and check, guess and check until you get the
answer?
I: That is a good starting point.
T: That could take forever!
D: It seems we are approaching a limit of what it is.
Earl had been sitting quietly, thinking about how he could undo the process of
exponentiation. He stated that you need to take the xth root but struggled with how to
complete the mathematics implied in this statement. He suggested that if he could
understand what 3 raised to non-integer values represents, then he could devise a plan to
reverse the process of exponentiation. He then asked:
E: I understand that 3 raised to the third power is 3 times 3 times 3 and 32.41 is 3
times 3 times something, but how do you show .41 factors of 3? I can’t see
this. What does it mean?
D: What did you do?
E: I just wrote this out because the exponent acts as a counter telling how many
times to take the base as a factor.
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D: Yes, I guess that was what I was doing with Excel.
E: I mean I can plug it into my calculator and get an answer, but it doesn’t make
sense. I can draw the curve of ( )
with the points 33 and 32 [he
motions with his hands to show what the curve would look like] and guess
where it is on the curve, but I don’t know what the value actually is. How do
you know what it is?
I: So how—or can—you represent this piece, 41 hundredths of a factor of 3?
√ , so could we do something similar?
E: We know this:
I: Could we? What is the alternate notation for square root?
T:
power. So there is something here. Are we saying there is a way to do this?
I: Could you write this as
?
D: But we just can’t find it if we get a number that is nonterminating.
I: What do you mean by this? 0.41 is 41 hundredths, a terminating decimal.
D: I mean if the exponent is not a whole number the output will be
nonterminating.
J: We need some sort of function that could keep going and going and you get
longer and longer numbers that are close to what you need. I remember doing
something similar in a programming class for cube roots.
I: OK, but how do you evaluate an
rewritten as √
root? This one,
comes out “even,” so if the
when
root is 2 then
D: How do you know when something is not 5? Is that the question?
I: What do you mean, is not 5?
.
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D: I mean it doesn’t come out to a whole number.
I: OK.
T: OK, so let’s think backwards.
D: Well you got to take the two given numbers and somehow extrapolate what
the exponent is from the two given numbers. So let’s go with 2 and 32. How
do we get 5?
T: Let me think about that.
D: Take the 32, divide it by 2, and keep doing it until you end up with . . .
J: The root? It’s kind of like a square root.
D: It counts how many times 2 is a factor of 32.
I: What if it doesn’t divide evenly?
D: Its still theoretically is what is happening.
T: OK, so divide 5 by 3.
I: What are you going to do with the remainder? You can’t get another whole
factor 3 again.
D: Well, wait a minute. Maybe that is it. It’s one and something, and then if you
divide that one and something by 3 and then where do you stop? You don’t
stop, because 3 is a nonterminating decimal; I mean 3 to that x power.
E: Except that some value will give you five.
T: I got a formula.
D: Well, there has to be one.
T: The xth root of y equals the base.
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I: OK, but we still need to solve for x.
As the group began to complete the preliminary activity, it was obvious that Jim
and Earl remembered the procedure for finding an inverse function. They worked
quietly. Observations revealed they both began by first replacing
the equation for x. Next they replaced x with
( ) with y and solved
( ) and replaced y with x. When asked
to explain why this works, Earl responded with an example.
for
E: So if
( )
the beginning of
and
, we want to find out how to get back to
.
D: I don’t see what you are getting back to. I can see what you are doing, but it
seems a bit arbitrary.
E: Well it’s the rule she [Professor R] taught us.
I: Without following this procedure, could you tell me what the inverse function
is for ( )
D:
?
, but what is the definition of inverse that fits both of these
operations?
I: How about simply undo what you did?
D: That’s not very mathematical, is it?
T: So the undoing, if you are raising the power of the base to the exponent to
undo that you have to take the exponent and divide it by the base . . . , it’s not
really divide is it? What is the opposite of exponenting? Is it logging?
Roots? Square roots?
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I: You extract roots if you know the exponent. [For example,] if you know the
exponent is 2, to undo the processing of squaring you [take the] square root.
T: How do we undo a
root? That [referring to the
root] doesn’t do
anything.
Moving on, when asked if they remembered any other properties of inverse
functions, Jim responded, “Vaguely.” When asked to elaborate, Jim indicated they had
learned the procedure for finding the equation and she (the instructor) had mentioned
some other properties. Doug added, “If we did learn other things, it didn’t stick.” It was
unclear whether they had spent any time in class exploring other relationships between
inverse functions; such as graphical representations or properties of inverse functions.
While this material was covered in Chapter 1 of the course, two of the four participants
could not even remember working with inverse functions. The other two participants
seemed to be operating at an action level of understanding: they could follow a procedure
they had apparently committed to memory.
Part A, Task 2, required the participants to graph ( )
and ( )
log
on
the same set of axes. Since participants were using TI-84s equipped with the latest
operating system, they were able to complete this activity without relying on the changeof-base rule to enter the logarithmic expression into the equation editor. However, prior
to this activity they were asked to make observations about the graphs of ( )
its inverse function ( )
and
. Doug reported that the x and y axes were juxtaposed.
When asked to explain what he meant by this, Earl said, “When looking at the graphs
[using his hands] if you flip this one like this, it flips onto the other curve.” The graph of
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the line y = x was added to the two graphs previously displayed to clarify the “flipping”
motion Earl had described and the idea of the x and y axes interchanged that Doug had
referenced. Sensing that all participants could identify pairs of inverse functions when
they were graphed on the same set of axes, the researcher had the group graph ( )
and ( )
log
on the same set of axes. Doug commented that the two functions
were mirror images of each other when reflected across the line
. Furthermore, he
said you could see that x and y are switched.
D: So the inverse of the exponent function is the log function. Did I say that
right?
I: Yes.
J: What does the log do? I mean, what is in the word log that allows it to do
what it does without the word log? That is what I want to see.
E: That would make it clear.
D: So log is not a mathematical operation?
I: Log is a mathematical operation.
J: I want to see that.
D: So what kind of method, what kind of arithmetic is being done when you do
logs?
I: Well, it reverses the process of exponentiation.
The instructor explained that both of these functions belong to a class of functions
referred to as transcendental; therefore, “normal” algebraic rules may not necessarily
apply. In other words, these functions transcend algebra, in the sense they cannot be
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expressed in terms of a finite sequence of the algebraic operations of addition,
multiplication, and root extraction (Barnett, Ziegler, Byleen, & Sobecki, 2009). With the
functions graphed, the students calculated a table of values for each, using instructordesigned inputs. Rather than using decimal approximations, the instructor told the
students to use fractional notation whenever possible. They quickly recognized a pattern:
inputs for the exponential function were outputs for the logarithmic function. When
asked to generalize their observations in regard to the domain and range for both sets of
functions, both Earl and Doug responded that the domain of f1 becomes the range of f2,
while Jim wrote “the base of the
clarify, he wrote, “
and the exponent a are the same.” When asked to
the log of x is the answer to the exponent c.”
Tom wrote, “(x, y) becomes (y, x).” However, it was unclear what Tom intended when
he wrote this. When asked to clarify what he meant by this notation, he responded that
he knew this was not the order he would use when plotting points; rather, he was just
trying to describe what happens succinctly.
Directing the students to work now with just tables and the log button, the
instructor asked participants to summarize their thoughts on what the log button was
doing. This activity was an attempt to strengthen the construction of processes and
objects in the minds of the participants. According to Dubinsky (1991), “The
construction of a process begins with actions on objects which are organized, possibly as
a procedure, and interiorized with the awareness of a coherent totality” (p. 167). Tom
summarized the group’s thoughts about this topic when he reported, “I see what you are
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trying to get at. Somewhere in that log there is some kind of operation going on that is
undoing exponents, but we really don’t know what that is.”
T: It is undoing it [exponentiation] but the process . . .
J: How, that is what I want to see.
D: Yeah, then maybe it would make more sense, to me anyways.
T: The process of how it is doing it is the question because log must mean
something! I mean, I guess when we are undoing multiplication with division
we are not really freaking out about division because it’s acceptable.
I: What do you mean by acceptable?
T: It is just something that we can visualize; I mean we know the process
involved, but with logs, I don’t know how it is finding the exponent.
What exactly does it mean to evaluate a logarithmic function? Even if students
can rearrange a single logarithm term into its equivalent exponential format, this ability
does not imply understanding. The group admitted that this material seemed challenging
but at the same time seemed similar to finding square roots, yet different. It was clear
that all participants had constructed an internal process in response to the question, what
exactly is a logarithm? Earl communicated that square roots undo powers of 2, which is
not the same as
Jim explained that a root is not really the opposite of exponents,
necessarily, because in order to find a root you need to know what exponent you are
“undoing.” Tom claimed the root is in there somewhere, but he just was not sure what or
how to apply this to logarithms. At this stage, subjects were attempting to reason about
this operation called log. This reasoning was evident in the final activity of Task 2, when
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the instructor asked participants to conjecture on the value of
without using a
calculator. Doug had indicated that the log button reverses exponentiation, adding, “It
allows you to see what power the base was raised to, to give you the answer,” but he
failed to respond to the final question other than by adding a check mark. If this check
mark indicated he could conjecture on a possible value, he did not attempt one. Earl and
Jim both started by writing out powers of 4 and realized the value they were looking for
had to be between 3 and 4. Earl added that he was looking for 4 to some power that gives
you 80 and the result would be 4 raised to the third plus something with the value closer
to 3 than to 4, while Jim said it would be somewhere between 3.1 and 3.2. Tom, on the
other hand, said definitively he knew exactly what the value would be and wrote the
ordered pair (80,
). Properties of logarithms had not yet been
discussed, but Tom thought that if he broke 80 into two values whose logarithms where
known he could come up with an exact value for
. Dubinsky (1991) might
describe this response as Tom’s attempt to construct a process understanding for
logarithms; however, he was not yet able to generalize this construction process. Tom’s
current cognitive structures that permitted this construction were weak.
The constructive process of learning requires the learner to reflect on their
actions. “The modification of existing mental structures so that a novel input fits them is
called an accommodation,” and, according to von Glaserfeld (1995), “all
accommodations are triggered by perturbations” (as cited in Battista, 1999c, p. 13).
However, Dubinsky (1991) reported, “It can be difficult to decide whether a particular
activity of a subject is a procedure or an internal process” (p.167). The next task
149
attempted to guide students’ cognitive activity to a more sophisticated understanding by
providing the necessary perturbation to strengthen the participants’ process understanding
of logarithmic concepts.
Strengthening the Understanding of Logarithmic Concepts: Task 3
The instructor asked participants to think over the weekend about what it means
to reverse the process of exponentiation and how one might do this mathematically. Tom
began the session by recapping what the activities were trying to promote.
T: I did some thinking about this over the weekend and I kind of get what we
have been talking about here, at least I think I do. Let me talk this out. Logs
are the inverse representation of exponent problems. Not just an exponent but
also the whole problem, the base and the exponent itself.
I: Yes, we completed the table [see Task 2 Part B] to reflect on the idea of
reversing exponentiation.
T: There it is. That is what the log is.
I: I am not sure why the word log was used but it does indicate reversing
exponentiation.
T: Got ya!
I: This is asking [referring to the problem
]: you are looking for an output
given an input of 81, the implied question is 3 raised to some power, in this
case the output that will produce the input 81.
D: The question is how we do this manually.
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I: We do not do these types of calculations “manually” anymore. You rely on
tabulated values or the calculator.
D: Before computers, someone had to tabulate the values.
T: Someone did make the tables.
J: Giant books of logs.
I: John Napier began this process.
T: How did he do this?
I: He used ratios, first with powers of 10.
D: Like successive approximations.
I: Something like that.
J: In order to make sense of this, there should be some mathematical way on how
to do this instead of this.
I: There really isn’t much printed material on this. Approximations are involved,
and there is a method for how to accomplish this. What he [Napier] was
trying to do was create some type of relationship between an arithmetic and a
geometric sequence. For example, consider the two sequences: [I write this
on the board]. Can you rewrite S2 as powers of a single number?
S1 0 1 2 3
4
5
6
S2 1 3 9 27 81 243 729
T: Sure, powers of 3: 30 31 32 33 34 35 36.
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D: OK. So locate 1.5 on the arithmetic scale. Would that result in a distance
midway between 3 and 9?
I: No.
D: Of course not!
I: Because one scale is arithmetic and the other is geometric, there is a
relationship; it is a type of proportionate relationship. It is just not that one-toone correspondence that you want it to be.
D: So for the same reason that’s why it is very difficult to raise 3 to the 1.5
power.
I: [I write “(
)” and ask if they are OK with this. No response. I then write
this in an equivalent format, “ √
”, and ask if this makes sense].
T: Well, it is even easier to write 3 raised to the power.
D: Oh yeah, that will work too, but it doesn’t get us any closer to a number.
I: No, but there is more of a procedure or process that is familiar: you know
what a cube is and you know what square root means.
T: I know what probably I am going to say, well I’m going to say it anyways and
see what it is. I was working on the problem
and I did some things
that came out to give me 5.01. I was thinking inversely, so I raised 5 to the
power of one-third and that gave me about 1.67. Then I took my original
base, 3, and then I multiplied. I went 3 times 1.67 and came up with 5.01.
D: Well, if your calculator was perfect . . .
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T: Well, what I was doing there, because what freaked me was that it looked
good to me because everything was logically making sense to me. I am not
doing logs—what am I doing?
E: So basically, you found the cube root of 5 and multiplied by 3 and you got
close to 5. So if you did the cube root of 27 you would get 3, then times 3 you
get 9, not 27.
T:
, there is no point in doing that. We know it is 3.
E: I just picked another number besides 5 to see if your method worked.
I: OK, so how about if we change the answer to 25 instead of 5?
T: OK, now that makes it hard. So going back to what I was saying, 25 raised to
the one-third power is 8.3 something and times 3 is 24.9 something, it is
working, it’s working!
I: OK, so what is the exponent?
Tom was so intent on getting back to the answer of 25 that he had lost sight of
what he was trying to find: an exponent such that when 3 was raised to that power it
would produce a result of 25. He reiterated again that his process seemed to be working.
I then asked the group to think about the following:
. Tom reasoned that 10
raised to power is 5 and 5 times 2 is 10. I asked again what the implied question is
when we are using a fractional power such as , and the group responded that it was
equivalent to a square root.
I: What is 10 to the power?
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T: Let me do it again. I got 5.
I: Are you sure?
T: Oh! I got 3.16 something times 2 and we find the error in what I was doing.
When I don’t use parentheses around [the exponent], I get my answers, so I
wasn’t working magic here.
I: How would this procedure figure out the value of the exponents?
D: To finish what I was thinking, I thought we were trying to say something like
(
)
That’s what I thought it sounded like we were saying a few
minutes ago.
T: OK, let’s go back to the original world [meaning day 1] and discover what x
is. Are we getting close [to finding the value of x]?
Once the group was introduced to the formal definition of the inverse function of
an exponential,
if and only if
, they were asked to use this rule to
transform various logarithmic and exponential expressions. Each participant successfully
completed Part A in Task 3 without any difficulty; however, Tom did something a bit
different. Under each exponential equation he wrote
1b he wrote, “
” and initially wrote “
. As he started Activity 3A” but added, “It did not look
right.” When asked to explain what he meant, he looked to the printed rule on the
worksheet and at what he had written, and then realized he had incorrectly followed the
rule. Jim added that the easiest way to look at this is: if b raised to the e equals v, then
log b of v equals e.
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To further expand their knowledge of what it means to evaluate a logarithmic
expression when the rule or corresponding exponential graph is no longer available, the
subjects were asked to evaluate 10 logarithmic expressions without using a calculator.
Tom and Jim both correctly evaluated 6 of the 10 expressions, but did not justify their
work. Doug and Earl, on the other hand, rewrote each expression in its equivalent
exponential form. Only Earl stated explicitly what “x” would equal in each expression.
He correctly answered 9 out of 10, while Doug correctly evaluated 5 out of 10. No
correct responses were initially recorded for log (
correctly wrote
). Three of the four subjects
and then indicated they were not sure what to do next. When
asked to explain what the above equation represents, Doug responded, “You can’t have
imaginary powers, right?” When asked to clarify this statement he said, “I can’t use ‘i’ as
an exponent.”
Two days earlier, the subjects had correctly identified the domain and range for an
exponential function, but they were not connecting this concept with this last
expression,
(
). Earl indicated he was looking for 4 to some power that gives
negative 16. To clarify the issue, the group referred to a pair of graphs created earlier,
and
. Tom indicated that negative 16 has to be happening somewhere
on the graphs but was unclear where. The researcher reminded participants of an earlier
discussion the previous week, in which she had explained that the domain for one
function is the range for the other. Looking at the last expression, the instructor asked
participants to determine where negative 16 would appear.
T: That is my x, right?
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I: Yes, so negative 16 would have to be over here [indicating the left-hand side
of the graph].
T: That’s impossible.
I: What do you mean by impossible?
T: Because the graph indicates that it can’t cross the y-axis [referring to the graph
for the logarithmic function] and if you can’t cross the y-axis then we can’t
explore negative 16, so no solution!
D: Are we looking for negative16 on the x-axis?
I: If you are looking at the logarithmic function—but most of you rewrote this in
an equivalent exponential format—so what would you be looking for in this
format?
T and D: Negative 16 on the y-axis?
I: So does it matter which function (logarithmic or exponential) you use to
develop understanding?
T: So there is a solution for that one? I mean I don’t see one.
J: There is never not a solution. You can always make something up. They
made up imaginary numbers.
T: I guess the reason why we don’t have a solution to this one is because if we
look at the graph there are parts of this coordinate system that have no graph
components on them.
Overall, participant understanding still appeared to be limited. Although
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participants demonstrated they could follow rules, when asked something slightly
different, they struggled. Negative values used as inputs for logarithmic expressions
proved problematic. The idea of an expression having no solution seemed impossible.
Jim reiterated, “There still has to be a solution somewhere.” Doug seemed to be fixated
on the fact that you cannot have “i” in the exponent but was unable to clarify what he
meant by this.
When asked to consider the following two properties of logarithms,
and
, the group seemed engaged with the topic. When asked if “b” could
ever be equal to 1, the group considered several possibilities for both properties.
E: So is log11 everything? Doesn’t that violate this
rule?
I: What do you mean?
E: Just write
That could be zero as well. So wouldn’t it be all real
numbers too?
I: You mean if I did
? This is impossible.
E: Right, but . . .
T: It can only be 1, the input, and that is the reason why you can’t have it as 1.
E: Right.
I: What do you mean?
T: As soon as you say
, no solution.
I: So you are saying we can let the base be 1 when the input is 1?
D: There are infinite solutions to that, but no solutions when the input is
something else.
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I: Why? [The group is quiet.] Well, let’s go back to the idea of a vertical line.
What is the same and what changes?
J: y changes and x stays the same.
I: [I sketch the graph of
E:
] Does this have an inverse?
log 1?
I: Why?
T: Freaky! The answer to that one is always 1.
I: OK, so you are saying this is an exception, when the base can equal 1? But is
this really a logarithmic function?
J: Well, it has a graph.
I: What kind of graph?
T: And this is why we follow this rule [referring to the original two properties] so
we don’t get off on things like this.
D: If you remember the rule, then you don’t have to do the mental calculations
every time you get in an unfamiliar situation.
T: I’m going to tell you this right now: I can’t do anything without the rules. If I
am sitting here looking at something for the first time and I haven’t been
introduced to the rule at all . . .
D: But you solved this [pointing to the properties written on the board] without
rules.
T: The reason why I am saying that is because, throw some notation in that you
have never seen before and you are done! You follow me?
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I: Yes, but at some point, you need to make sense of the meaning behind the rule
or definition; you don’t need to see the rule to use it.
T: Things have to be defined to you first and then you internalize it.
J: You have proved the concept to yourself, so why don’t we just teach proofs to
kids?
D: Proof is an act of communication, so it is not proving to you but proving to the
world.
E: I’ve proved very little of this to myself. I get the rules written on the board or
in the book, and I’ll do a couple of problems and then I am able to recognize
similar problems, then move on. I very rarely will go back and say OK, I
want to prove this to myself. Not many college students will.
T: We only have so much time in the day to be thinking about anything, so we
are not going to spend time thinking deeply. You take what you need and be
done with it!
As they completed the next activity, Task 3 Part C, which asked them to compare
two logarithmic terms without the use of a calculator, it was unclear whether deeper
understanding had developed. Doug and Earl started by writing the equivalent value of
each logarithmic expression, presumably so they could place the correct symbol between
the terms. In a similar manner, Tom and Jim simply placed a symbol between the two
terms but did not indicate or justify why they responded as they did. However, as a
group, participants successfully evaluated all the problems with the exception of
and
. Tom asked, “Were we supposed to know this? I know they are equal, but . . .”
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To finish the activity, they were asked to write a short paragraph describing their
thinking about the symbol “
” Before they engaged in this writing exercise, class
discussions revisited the idea of what exactly the log button was doing. Although
intuitively the participants could all state “find the exponent,” it was doubtful they had
created any meaning for this symbol. For example, Tom stated, “I think what we are
looking at mathematically is [that] all of our world has been pretty much simple. You
can always find another number to finish an equation, but this is not like anything we
have seen.” Jim added, “We can add, subtract, multiply, and divide like terms,” but as a
group they struggled with the idea that there was not an exact procedure to undo an
exponent. They wanted to know what the “log” was doing mathematically in terms of a
procedure.
Several times over the course of this research, students indicated that if they could
practice this exact method or at least see the development of this procedure, the idea of
logarithms would or could take on a deeper meaning. However, with the wide-scale
availability of scientific calculators, the progression of ideas associated with logarithms
has all but disappeared from the curriculum (Umbarger, 2006).
How to Construct a Table of Logarithms
“Logarithms were, by the 1670s, conceptually well on their way to their crucial
role within modern mathematics, although it took a further seventy years or so for their
presentation to reach the clarity and simplicity imposed by Euler” (Fauvel, 1995, p. 46).
Yet most modern textbooks begin with Euler’s formalized definition of a logarithm.
Authors of modern textbooks rarely talk about the evolution of ideas and concepts that
160
led to the creation of logarithms, and generally they encourage the wide-scale use of
modern calculators to assist in calculations. Because the purpose of this research was to
understand how students acquire an understanding of logarithmic concepts, it seemed
reasonable to pursue the historical development of logarithms as a supplement to the
original planned activities.
Logarithmic tables revolutionized the calculation process. First published by
Napier and then modernized by Briggs, the tables represented exponents for a given base
value. The word logarithm is derived from two Greek words: logos (ratio) and arithmos
(number); thus, a logarithm counts the number of ratios for any given base value
(Umbarger, 2006). In the following two sequences, S1 counts the number of ratios of 2 in
sequence S2.
S1 0 1 2 3 4 5 6
S2 1 2 4 8 16 32 64
The participants quickly caught on to the idea that if S2 could be rewritten as an
exponential expression, S1 could be modified as well using logarithmic notation.
However, this still did not answer the question, “How do you find the exponent that
produces a value of 3 when working in base 2?” In order for them to answer this
question, several concepts required further development. Both the researcher and the
participants noted that when the original tables were first crafted, the only mathematical
tools available were addition, subtraction, multiplication, division, square roots, and later
cube roots. With this in mind we continued this session, first discussing how to make the
sequences denser in the intervals 0 to 1 in S1 and 1 to 2 in S2. We noted that
multiplication in S2 correlates to addition in S1.
161
I: So if I want to do, say 4 times 8, which is 32 . . .
D: Oh, I see, 2 plus 3 is 5 and 2 raised to the fifth power is 32.
I: This is your exponential growth value or output [referring to S2] and S1 is your
exponent or input; however, S1 increases arithmetically while S2 does not.
What I want to do next is make these sequences more dense, because right
now basically we are saying the
is 1,
is 2,
is 3,
is
4, and so on, so this is the logarithm [pointing to S1] of this number [pointing
to S2] with respect to base 2. That is what the exponent is, the logarithm. So I
want to make S1 dense between 0 and 1. [So I write 0 and 1 and locate in
the middle, and underneath 0 I write 20, and under the 1 I write 21.] I’ll just
put in the middle and this is still 20 or 1 and this is still 21 which is 2. I want
to know what this is right in the middle. [I am pointing to a space between 1
and 2 on S2 and I label it “t”.] I just demonstrated that if we add exponents in
S1, this corresponds to multiplying in S2. [Figure 4 clarifies this process.]
T: Can I ask a question? If there is a relationship between the 0 here and the 20
and a relationship over here 1 and 21, why doesn't the same logic follow here
and
? Do you see what I am getting at?
I: OK. You are good. That is exactly what it is.
T: I see this pattern. It might be wrong, but there is a pattern.
I: You are correct. There is a pattern. What we don’t know is this piece right
here [pointing to t].
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T: But we could find it, the square root of 2.
I: We are following that logic there, so t squared is 2, so it is equal to the square
root of 2, or about 1.414.
T: Yeah, because we are asking the square root of two.
I: So this is saying log21.414… is .
T: So the logic is, move in the right pattern.
I: OK, and again this number (1.4141…) is between 1 and 2. I will do
something similar, say, for .
T: Now you are freeing us because now we are going, say, 2 to the , then we are
going to, say, the cube root of 2.
I: Yes [I show them why mathematically], because
and if I let u
equal this then u times u times u is u3, which is equal to 2, so u is the cube root
of 2. [I place this in the correct position and label u on the S2 sequence.]
What about ?
T: So now we are going, the cube root of 2 to the second power?
I: So [I demonstrate why this is correct],
T: So are we saying the cube root of 2 to the second power?
I: Why do you say that?
T: Because I’m still following the same logic. Wait a minute—maybe I’m not. 2
plus 2 to the 1 is 2 to the 1, 2?
163
√
S1
0
S2
1
t
1
2
3
2
4
8
It follows that if addition of integers in S1 corresponds to multiplying in S2, this means
√
Figure 4. Using sequences to develop logarithmic concepts.
E: Yes.
T: So I am following the right logic. I’m basically saying that the bottom
number is what the root value is in that fraction, and then the top number is
whatever is under that radical raised to the top number power.
I: So let’s call the value on the S2 line v, so v3 would equal 4.
T: I’m breaking it down before I get 4. I’m using the pattern I see. You did the
same.
I: Yes, we are working from the top down. But what if you want to know what
is, because right now we are saying
(√ )
? We picked the
exponent and then generated the output or value we needed for that exponent,
164
but what about the reverse process? The inputs we are generating on the
bottom scale (S2) are irrational numbers. How do you get rational numbers to
appear? We are making both sequences “denser,” but it is easier to work from
the arithmetic scale to the geometric scale. But we want to know how to work
in the opposite direction. [Appendix C contains class notes used to develop
this idea with the participants.]
J: You have to reverse the process; you can’t really cheat it like you are doing
with the roots.
I: The handout I just gave you [See Appendix D for this document] talks about
how you can do this.
J: And this will answer all our questions?
I: I think it will. Let’s just briefly look at what is here, specifically non-integer
factoring. So the question is, how would you factor 3 in such a way that you
could use the information developed in the sequence above between 0 and 1?
Then you would use non-integer factoring to complete this process. For
example, you divide the number 3 by the largest value less than 3 (from the
numbers developed from the rooting method just described). So I would
divide 3 by the square root of 2, then I divide that result by the next smallest
number in my chart less than this result. You continue with this method until
the last factor is “sufficiently” close to 1. When done, you would then add all
the exponents used to achieve this factoring. [Appendix E contains the
complete class notes used for the non-integer factoring of 3.]
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E: This is a unique way of division.
I: If we look at page 89 on the handout, there is another example completely
worked out.
J: It’s like a process that keeps reducing itself until you finally get either it,
[meaning the value of the logarithm] or it just keeps going forever [meaning
the division never ends], or you get to the end [meaning a terminating decimal
value].
I: You only get to an “end” for rational numbers.
Sensing some satisfaction now that the group had seen how to reverse the process
of exponentiation, Jim questioned whether exclusion by most textbooks of this historical
account of the development of logarithms was pedagogically sound. He said, “You need
to know where the work came from.” Earl referred to this as “logarithmic magic,” and
collectively the group inquired about the importance of the historical development of
mathematical ideas, wondering whether the knowledge will be lost if we allow
calculators to do all the work without some knowledge of the operations that are being
performed.
As the discussion moved toward the use of logarithmic tables, I explained that the
original tables were first developed using base ten, and subsequent rules rooted in
properties of exponents were then used to evaluate logarithms to bases other than ten.
Continuing along a historical path, participants received a log table to find
.
The value obtained from the table was 4487 but the table did not indicate how to express
the result. When asked how to interpret this number obtained from the table, Tom
166
replied, “It’s some type of decimal, but do you move [the decimal point] one in or two
in?
E: It’s asking what power of 10 gives you 2.81, so 10 to the power 4487—
decimal point somewhere—gives you 2.81.
I: [I write what he has said on the board:
]
J: Well if we go to the beginning, it is under one it . . .
E: . . . is less than 10.
I: So would 10.4487 make sense?
D: Oh, of course, this makes perfect sense.
I: What would you do for
?
T: Now the answer has to be between 1 and 2.
E: 4.487?
T: No, that’s way too large. OK, do that again. We know that the decimal point
was right at the end.
E: So 1.4487?
T: Yes, that makes it sound right.
I: Why?
J: It just goes up to what power it is like, if it is above the power of 1 but below
2 it’s going to be like 1 point something and above 2, then 2 something.
D: So the digits on the left side of the decimal point indicate the whole number
part of the exponent and the right-hand side is the fractional part. However,
you have to infer the whole number part; the table only gives you the
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Figure 5. Table of common logarithms. Adapted from “Explaining Logarithms: A
Progression of Ideas,” by D. Umbarger, 2006, p. 84.
D: (continued) fractional part.
I: So, in other words, we can think of it like this:
. With this in mind, what is the
T: 3.4487
I: So is there a pattern?
T: Yes, once they figured out for so many numbers I guess this pattern just
started to pop up.
J: They still can’t put it in a formula. You know a nice neat little package?
I: Not really.
?
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D: So by requiring that you calculate the whole number piece on your own, the
table saves a whole lot of space. At this point, I would appreciate some
practice. I think I understand the concept [of using the table], but unless I
work with it, it won’t stick. If we are trying to understand, we need to
practice to ground ourselves.
J: If you know the “why” it is easier to remember the “hows,” and if you
remember the “why” you can figure out the “how” every time if you have
forgotten it. The calculator is great, but until you are at the point where you
need to jump to that next step, you need to be able to see what is going on.
Doug and Tom both agreed that what we had been trying to do is to understand
the thinking that led to the creation of the logarithmic tables, reasoning that unless we can
see the mathematical procedure for “undoing” exponents, it is nothing more than button
pushing. The group as a whole had rationalized that division of some sort had to be
involved with reversing the operation of exponentiation, but they were unclear how this
was possible if the exponent was not an integer value.
The participants had been asked to describe their thinking about the symbol
as they completed Task 3. Tom responded, “It implies multiplying b times itself x
amount of times to get N.” When asked to explain further, he said, “Because I see
as the inverse function of
” then followed up with a concrete example. Both Jim
and Earl indicated that they thought b to what power gives x. Jim further added that he
needed to do this in order “to undo the exponential process and better understand
something in a linear fashion.” Doug saw the symbols as a way to estimate logs;
169
however, when asked to clarify his thinking, he added, “If the bases are the same, the
larger number yields the larger exponent. If not the same, find the exponent that yields
the number immediately below the target and then find the exponent that yields the
number immediately above the target. This technique works with the target given, but if
the target involved is a fraction, then the algorithm wouldn’t work.” When asked what he
meant by the target value, he indicated the input for the logarithmic expression.
Deepening the Exponential-Logarithmic Connection: Task 4
Task 4 was used to promote the development of an exponential-logarithmic
connection by asking students to use the basic definition of a logarithmic function to
solve exponential equations and, more importantly, to explore when it would be
appropriate to use logarithmic notation to solve algebraic equations. The participants
were now considering how a series of actions could be organized to construct a coherent
understanding of the exponential and logarithmic connection. The ideas of reversal,
generalization, and coordination are pivotal to this understanding. As the participants
developed higher levels of reflective abstraction, their thinking became increasingly
flexible. For example, in Task 4 Part A question 1, participants solved four exponential
equations, some of which had been solved earlier using the graph only. All participants
completed the task without error; however, it is suspected that they were just following a
memorized procedure since they had been given the rule detailing how to change an
exponential equation into a logarithmic equation. When asked how he had figured this
out, Tom replied, “Well, I just laid it out as log
, entered it in the calculator,
and bam! There it is, the answer.” Shortly after his breakthrough, Tom announced “Oh!
170
I see what you are getting at now, because before, without really understanding the logs, I
couldn’t get that [referring to the decimal number]. I couldn’t even get that concept in
my head [meaning from the graph] but now I can.” Jim added, “Before, it was more of a
guess, but now it is more accurate.”
The participants then attempted to solve a nonroutine application problem for
(
which the equation was given:
) . Earl reverted to increasing $1000
by 7% yearly, similar to his response on the pretest, adding that he “was kind of at a loss,
except to keep trying values for x because it goes up by 7% each year,” instead of
attempting to solve the equation given for the exponent. Tom, on the other hand, quickly
reported the answer to be 10.24 years; however, when asked to explain what he had done,
he responded, “I went log
and it gave me a number.”
I: Why did you use 2 in your expression?
T: I was thinking double and double is 2.
I: OK, but, reading through the problem and looking at the given exponential
equation, how did you arrive at this logarithmic expression?
J: I was thinking we are looking at compound interest. What is the log here for?
D: Because there is a power of x involved.
I: What does the power represent?
T: The total number of years.
D: So the base is 1070?
I: Can you do 1000 times 1.07?
T: Not before the exponent.
171
D: Oh, it’s the log of the base 1.07.
I: But the question is how we got from here:
(
) , to here:
log
T: Can I divide? [He does.] I’m back to 2 again, right where I was!
Tom instinctively knew how to complete this problem but complained that he had
to do this long process to get to the answer. In his mind, he was able to visualize the
structure of the doubling formula and was able to coordinate this existing schema to
obtain the exponential to logarithmic structure. When asked what he would do if given a
problem without context, such as
( ) , he indicated, “OK, if I see that I might kind
of spaz out.”
D: But if you recognize this [pointing to the variable in the exponent], you know
to use a log.
T: So right there you are saying I have to divide by 5 on both sides before I log?
I: Yes. What if the exponent was a variable expression like
D: Well, solve for the
?
, then subtract 2 from that answer.
So log ( )
I: So you get that number. [I circle the logarithmic expression.]
T: Yeah, oh! Then subtract 2, so cool!
To determine the strength of the participants' process conception of logarithmic
concepts, they were asked to consider a variety of different situations involving both
logarithmic and exponential functions and/or equations. The tasks completed thus far
were orchestrated in an attempt to foster the specific mental constructions proposed by
172
the genetic decomposition. Specifically, the questions were designed to provoke an
image in the minds of the subjects without reliance on visual cues. When asked to
explain why
is between 3 and 4, Doug volunteered that since 23 is 8 and 24 is 16
and 14 is between these, the result then has to be some number between 3 and 4,—an
indication that he was capable of reversing the process of finding a logarithm by using its
equivalent exponential form. It was unclear if the group collectively had a strong process
conception of exponential functions, but they all seemed to be moving in that direction.
Further evidence of a strengthening process conception was evident when participants
were asked to explain how they could find
and how to find
given
log
J: I’d make my own table of powers of 5, find where it is in between, and then
make a guess.
E: It might be an exact value because there is that nice-looking 25 on the end.
J: Even if it wasn’t, I could find out what it is in between and narrow it down.
T: It's 7.
I: What about this: find the log3729 given
T: Well, it is saying 9 to the third power is 729.
D: And 9 is 3 squared.
J: So the answer would be 6.
I: Why?
E: Because 3 squared cubed is 3 to the sixth power.
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When asked if the function ( )
( ) was increasing or decreasing Tom
incorrectly guessed increasing, but corrected himself, stating, “Oh, but when I picture the
graph, it is going to decrease.” Jim agreed, using his hands to indicate how the graph
would look if graphed on paper or with technology and added that it would change
direction only if the exponent were negative. Although Doug indicated he was still not
clear on the differences between increasing and decreasing functions, when he was
shown several graphs of increasing and decreasing exponential functions he said, “Oh!
That way decreasing and increasing.” It is not clear what he meant by this comment. He
may have just been thinking about increasing inputs without regard to the corresponding
output. Earl assigned concrete values to the variable to verify that as x increased y
decreased;—an indication of his preference for working with concrete images.
Clearly, all participants seemed comfortable working with logarithms and
exponents when rational numbers were used. Furthermore, they were able to discuss
similarities and differences between the graphs of exponential and logarithmic functions.
Participants were curious about the transformations of these functions and seemed eager
to discuss the properties. At the completion of this task, all participants received six
different types of equations to solve and were asked to explain their solution methods.
All four participants, although not offering a written explanation, executed flawless
mathematical computations for each equation.
When asked how one would solve the following problem:
immediately commented that logs could not be negatives.
, Doug
174
D: Wait, that has a negative in it. Didn’t we say something about logs couldn’t
be negatives?
J: You can’t take the log of a negative.
D: Oh, but you can have negative as a result.
J: Yes.
D: Oh. OK.
I: Think of the graph of a logarithmic function; is this a decreasing or increasing
function?
D: Since the base is one-half, I guess decreasing.
T: [He has been working on solving this problem.] Can I tell you how I set it up?
I: Sure.
T: I went raised to the power of negative 6 equals my x. [I wrote this on the
board to verify this is what he wrote.] Then I put 1 over 3 over 2 but I think I
might be wrong. [He now looks at the equation written on the board.] Well, I
looked at that [he points to the equation
] and I saw that that was
negative 6. Oh, now I see where I went wrong. I should have just reduced
that.
E: To 64.
T: [Not hearing this comment from Earl.] It should have just been negative 3.
[He is not viewing the negative 6 as an exponent.]
175
I: [I added parentheses around the entire fraction ( )
hoping to clarify that
negative 6 was the exponent and not a factor but when I looked at what he had
written, , I am not sure what he was thinking. To Earl:] You said 64. Why
did you say that?
E: Well, 1 over to the sixth power . . .
D: So 2 raised to the sixth power.
E: 64.
Tom was still struggling to see the error in his solution. He had been listening but
at the same time concentrating on his written work. He finally agreed that he had had the
procedure right initially but lost sight of the meaning of the negative exponent. He
added, “Where I went wrong, instead of seeing negative six as my exponent, even though
I said it was an exponent, I didn’t hold on to that in my mind. I changed it to something
else.” Doug attempted to further clarify the meaning of negative exponents when he
asked, “So a fraction raised to a negative power becomes a whole number. Is that what it
is? ”
I: Well, based on that statement, what is the value of this: ( ) ?
D: [Without hesitation] ( )
I: Is this a whole number?
D: OK, they exchange places.
I: How do you see this problem:
log
?
176
D: [Without hesitation] Well, there is a 3 in there somewhere. x is 3, no, I mean
negative 3. Just throwing out words!
I: Tell what you were thinking.
D: I see 3 cubed is 27.
I: Where did the cube come from?
J: I saw 27 to the 1 over 3 then 3 is the . . . , so it would be 27 to the 1 power,
cube root; the denominator is the root value, so 3.
D: Twenty-seven base to the one-third power equals….
T: Three.
Are the participants beginning to see a logarithm as an alternate representation of
a number? Has the problem structure itself become the object of their reflection? In
APOS Theory, Dubinsky (1991) postulates that “When the subject has a high degree of
awareness of a process in its totality, this process can be encapsulated to obtain an object”
(p.181). While the subjects seem comfortable with a single logarithmic expression and/or
an equation involving a single logarithmic term, what happens when they encounter more
than one logarithmic term in a single expression?
Exploring Properties of Logarithms: Task 5
The four components of APOS Theory detail a hierarchically ordered list;
however, learning does not necessarily proceed in a linear fashion. According to
McDonald and Dubinsky (2001):
What actually happens; however, is that an individual will begin by being
restricted to certain specific formulas, reflect on calculations and start thinking
177
about a process, go back to an action interpretation, perhaps with more
sophisticated formulas, further develop a process conception, and so on. In
other words, the construction of these various conceptions of a particular
mathematical idea is more of dialectic than a linear sequence. (p. 277)
Task 5 explored properties of logarithms and their connection to exponential
operations to determine whether subjects were able to access prior knowledge structures
and use them purposefully to make the desired mental constructions envisioned by the
task. Detailed analysis of these tasks will be given in this section.
In an attempt to move student thinking forward, participants were asked to
respond to the following question: Does log
log
it was noticed that Earl had written “No because the log
log
As they worked,
and log
”
However, he did not offer any other type of reasoning, presumably because it was
obvious that 12 does not equal 7. Jim wrote, “No, adding logs is the same as multiply
what you are taking the log of.” Not sure what he meant by this comment. However, he
and wrote above it log (
crossed out log
similar task, “Does log
this; log (
)
log
log
log
). When asked to respond to a
” Jim wrote, “No it should be equal to
” This indicates he knew that you do not add
logarithmic terms, but it was unclear whether he knew how to decompose
into
two separate terms. Doug wrote “no” for both statements, first indicating the value for
each logarithmic term, then explaining that when dealing with logarithms,
Tom’s comments were different; he wrote “no” and then gave a numeric account. He
.
178
saw each logarithmic term as an exponential, writing
and
underneath the original statement; therefore,
D: Well, log
is 6, and 6 plus 6 is 12, and 12 is the log
, 2 to the twelfth
power. [He begins to rattle off powers of 2.]
I: OK, so we all agree that log
is 6, and 6 plus 6 is 12?
A: Yes.
I: Is the log
J: No, it’s not that at all, it should be 7.
D: Yeah, but why?
T: Well, exponentially it doesn’t work out.
I: Well, each term separately represents an exponent.
D: So the law of addition doesn’t apply?
T: Wait a minute, say that again.
I: OK, if the result of the right-hand side is 12, what should the result on the left
represent?
T: There would have to be a number over there, 212.
D: Or don’t add them.
J: It needs to be 4096.
T: 212, whatever that is, wouldn’t have to be right here. [Meaning the left-hand
side].
D: So 12 to the power . . . wait a minute.
I: Why?
179
T: Because it has to be some number that is 212.
E: Which is 64 squared.
I: So are we saying that this log
should be replaced with log (
[The original statement was rewritten as log (
)
log
)
log
]
Tom replied, “OK,” but he seemed rather unsure of himself. He turned to the rest
of the group as if he were looking for verification. Doug was confused as well,
asking, “So it’s 2128?” Jim answered, “No, we need the 128 to be replaced with 4096.” It
appeared that both Jim and Earl were able to justify the results using multiple
representations. Neither Tom nor Doug had picked up his calculator to verify that 212
was equal to 4096 and that 642 was equivalent to this value as well.
T: So are we saying 64 times 64 should be 4096? So instead of this being
addition [pointing to the right-hand side of the equation] . . .
I: So if we are saying that log (
)
log
log
then how should
the log of a product break down?
D: It didn’t break out over here. That is a different result.
E: Well, it is just showing that log base 2 of something plus itself is log base 2 of
the square of that.
T: Is it log 2 of 6 plus log 2 of 6? [I write this as he stated it,
No, no give me log
D: And 26 is 64.
T: OK, give me log
again.
I: OK. What do you want to do with these two terms, add them?
.]
180
T: No, multiply it.
I: Why?
T: Well, isn’t that [pointing to log (
)] what it says?
I: Well, what is the value of log
T: 6.
I: So 6 times 6 [which I write under the expression Tom has told me to write]
under log (
log
)) is 36, but I thought we said that log (
)
, which is equivalent to 12.
T: Erase that log
log
.
D: What is 64 squared?
E: 4096.
D: OK, so the answer is log base 2 of 212 is the same as log base 2 of 64 times log
base 2 of 64, so change that plus sign to a multiplication. [I have now erased
everything except the following: log
log
log
, and Doug wants
the addition sign to be replaced with multiplication.] And it doesn’t work.
J: So isn’t adding of two logs the same as multiplying whatever their value is?
I: OK, let’s go back to the sequences we were working with yesterday. We said
we could use logs to represent the exponents.
Alternative
representation
S1
S2
Alternative
representation
log
log
log
log
log
log
log
0
1
1
2
2
4
3
8
4
16
5
32
6
64
22
23
24
25
26
181
I: What if I write log (
)?
D: Which is equal to log
, and that equals 5.
I: OK, so how does the right-hand side have to break down?
D: Oh, log base 2 of 4 and log base 2 of 8.
I: OK, but . . .
J: You need a plus sign between those two pieces. [I had written down what
Doug said underneath.]
I: Why?
D: Well, let’s see.
J: You are going to add to logs, multiplying breaks out in the definition.
D: 2 and 6, multiply!
J: No! You are going to add those.
T: [To Doug] No, you are going to get a 2 over here and a 3 over here, and 2 plus
3 is 5. You got to add, so you add!
D: You add? Yeah I see it. You are right.
I: But why would you add?
D: When you multiply with exponents, you add with logs?
I: Why?
T: Because you are adding the exponents. So basically you take the two results,
the two exponent results, then add them, and raise the base to that power and
that is really what that number should be.
182
In Task 5 Item 3b, the subjects wrote down their thoughts on any patterns they
might have noticed. Earl and Doug wrote, “log
log (
log
).” Jim added,
“When you add two logs with the same base you multiply what you are taking the log
of.” Tom, on the other hand, reverted to exponential form, writing only
It was not clear at this point how Tom was making sense of this information. As the
discussion progressed, we came back to expression log
. I asked how you would find
the value of this expression. Tom said he would first have to find the value for 212. Doug
said, “No, the answer is 12 because the log of the exponent’s exponent is the exponent
log
itself.” Next, they were asked if
is equivalent to log
Doug immediately
said, “Well 2 to the log 2 is itself or 1.” Continuing with this same reasoning, I asked
does log
log
log
?
J: They are all equivalent.
D: So your last statement, is it predicated on both of those guys always being the
same, 3 and 3? It wouldn’t work with, say, log
?
I: I am not sure what you are asking.
D: What I’m asking is, suppose I have log
E: Times log
Is that equal to 4?
.
I: Yes.
D: Oh, so that is a rule.
I: What does this imply?
D: Multiply the exponents. Is that what you are getting at? That’s the part I am
having trouble wrapping my head around.
183
I: OK, well, from this expression, 12 log
this exponent, log
, this exponent of 12 is multiplying
.
D: I see 12 as a coefficient. I don’t see it as an exponent.
I: It is a coefficient, but . . .
D: It could mean something a little different as well.
E: Could we write log
log
?
I: What do you think?
E: I think yes.
Task 5 Item 4 asked participants to label a series of statements either true or false,
based on our previous discussions. Earl avoided this task and moved on to Part B. Was
this because the statement in Task 5 Item 4 used bases other than 2? For example,
subjects had to respond to the following question: “True or false? Does log
log
log
” Did the fact that the right-hand side of this statement could not be
simplified into integer values cause Earl to avoid such problems? Tom clearly indicated
that this was false since the left-hand side was equivalent to 2 and the right-hand side
could be rewritten as
plus
which he then simplified to
.
He did not see how the right-hand side could produce an exponent of 2. Apparently,
when he tried to make sense of this new information, it evoked conflicting images. Jim
and Doug, however, immediately recognized this as a true statement, indicating that
adding on one side is the same as multiplying on the other; they each wrote “True” since
4 times 9 is 36.
184
As they completed the day’s work, it appeared that Jim was capable of applying
the properties associated with logarithmic expressions without much difficulty. Doug, on
the other hand, needed to spend a considerable amount of time thinking about the actions
he needed to perform in order to achieve the desired result. In other words, he had not
formed a process understanding, but was making strides in that direction. Earl operated
efficiently when the exponents or logarithms in an expression represented integer or
fractional values; however, it was unclear how he thought about logarithmic terms when
each individual term did not produce a rational value. Tom, on the other hand, still
seemed fixated on a single expression and had yet to realize how he could link individual
pieces using basic operations.
The final activity developed for Task 5 was to have the participants write a proof
for the addition, subtraction, and multiplication properties for logarithms. However, time
constraints did not allow subjects to complete this activity. They did, however,
demonstrate informally that in the process of adding logarithmic terms multiplication is
needed. Sensing a connection to the properties of exponents, they further explained that
subtraction of multiple logarithmic terms should involve division. The participants also
seemed to understand that if a single logarithmic term was written in the form of an
exponent, the result could be simplified. For example, Tom wrote:
All participants did exhibit an action understanding of logarithmic concepts associated
with basic properties of logarithms, but not all were able to internalize this information to
form new cognitive structures.
185
The last task developed for this teaching experiment, Task 6, was not completed.
Because participants’ responses had dictated, to some degree, the direction and pace of
the daily activities, time constraints did not allow participants to explore the use of
properties of logarithms and their connections to laws for exponents.
Summary of Chapter 4
Impact of the Teaching Experiment on Students’ Knowledge of Logarithmic
Concepts
To describe the development of student thinking, APOS Theory provided the
theoretical analysis based on a set of mental constructions a student might make.
Dubinsky and others refer to this as genetic decomposition. The genetic decomposition
postulates certain mental constructions, which the instruction should foster, providing the
foundational knowledge for this study. In the following section of this chapter, the
researcher has summarized the overall impact of the teaching experiment on the
knowledge of each student individually.
These summaries may show what mental images or constructs the students may
have constructed during the course of this project. Initially based on the researcher’s
understanding of the concept, a set of mental constructions that a student might make as
they attempt to understand logarithmic concepts was proposed. This genetic
decomposition formed the basis for analysis of student thinking.
Summary of Tom’s Performance. Table 5 summarizes Tom’s responses to these tasks
designed to evoke the development of new cognitive structures. Initially, Tom admitted
that he knew how to use the laws of exponents but did not really understand their
186
meaning. It appeared that once he memorized a rule, he could use it efficiently, but not
without error. This was evident when he was asked to evaluate a logarithmic expression.
He first wrote out the definition:
if and only if
. He then wrote:
then commented that it “didn’t look right.” When reminded of the rule he
had written down at the top of his paper he quickly recognized his error.
Tom indicated that this was how he memorized rules, writing them repeatedly
until they were committed to memory; however, it is unclear what type of understanding
this developed. For example, he stated, “I can’t do anything without the rules. The
beginning is definitions, things have to be defined to you first, then you internalize it.
Now, as you practice over and over again, you made sense of it; now it is part of you.”
He then added that today’s typical college student will not spend much time trying to
form any deep understandings, adding, “You take what you need and move on.”
Tom recognized there was a connection between the graphs of logarithmic and
exponential functions but was not sure how to represent this relationship algebraically.
When told that the symbol log was the inverse operation, he responded with “I kind of
get what we are talking about here, at least I think I do. Let me talk this out. Logs are
the inverse representation of exponent problems, not just an exponent but the whole
problem, the base and the exponent itself.” Tom attempted to accommodate this new
information into his existing schema for exponential functions; however, for him this is
difficult since there is not a physical representation of the mathematical procedure.
Dubinsky (1991) speculated that interiorization might be difficult for students when this
representation is not readily available; therefore, it is essential to provide concrete
187
Table 5
Summary of Tom’s Performance
Tasks
Activities
Issues encountered that were
insightful
APOS level of understanding
Pretest
Simplify exponential
expressions
involving integer
exponents.
“Other than the rule I really
don’t see it. I just know how
to use the rule.” Tom was
familiar with the laws of
exponents. He generally did
not use the common definition
of exponents (CDE) to
simplify the given expression;
rather he was relying on
memorized rules.
Tom was able to use the CDE;
however, he determined that the
laws of exponents simplified the
amount of “work” he needed to do.
He could not offer plausible
reasons for their validity. Process
level understanding: he interiorized
the action but was not able to
encapsulate parts of this concept.
Task 1
Graph the
exponential function
( )
to extend
the meaning of
exponents to include
all rational values.
Using the graph,
participants were
asked to approximate
the value of (√ ).
When asked to comment on
the domain of this function
Tom indicated [pointing to his
graph] “There it is, all real
numbers.”
Tom reflected on the physical
graph itself and was able to
determine that the smooth graph of
this function indicated that any real
number could be used as an input,
but was reluctant to use the graph
to estimate the output, perhaps due
to a lack of deep understanding of
decimal exponents. Still at a
process level, he has yet to form a
generalized idea of what an
exponent represents outside the
context of counting the number of
factors.
Develop the need for
using logarithms or
some other alternate
mathematical
procedure by asking
students to solve
Tom had taken the time over
the weekend to research
logarithms. When asked to
estimate the value for x, he
correctly wrote
but
admitted that is just what the
book said. He said he would
need to do something else
because the base was not 10.
Unable to figure out how to
use his calculator, he remarked
“Is that the technique to use,
guess and check, guess and
check? That could take
forever!"
Task 1
When asked to use the graph
to find (√ ) after some
prompting, Tom suggested the
value of √ is something less
than 2, then inquired, “You
mean you want us to find a
possible range for that value?”
He added the problem was the
input was not an exact value.
Process level understanding of the
operational characteristics of
exponents; however; he has yet to
encapsulate the entirety of the
concept. For example, it did not
occur to him to use the graph to
approximate an answer. He did
use the definition of logarithms
correctly; however, for him it was
without meaning; it was just
another rule to follow.
188
Table 5 Continued
Task 2
Graph both
exponential and
logarithmic functions
on the same set of
axes and describe
any relationships.
“I get what you are getting at.
Somewhere in that log there is some
kind of operation going on that is
undoing exponents, but we really
don’t know what that is.” He then
goes on to say, “The process of how
it is doing it is the question because
the log must mean something, I
mean I guess when we are undoing
multiplication with division we are
not really freaking out about
division because it is acceptable.”
Object conception for
exponential functions;
however, according to Asiala
et al. (1996), “In the course
of performing an action or
process on an object, it is
often necessary to deencapsulate the object back
to the process from which it
came in order to use its
properties in manipulating it”
(p. 11). Tom sees that the log
is undoing exponentiation but
does not understand how to
de-encapsulate this concept.
Task 3
Using the definition
for logarithms,
students first rewrite
exponential and
logarithmic
expressions, then
evaluate logarithmic
expressions without a
calculator, using only
the alternate
representation for the
logarithmic or
exponential form.
“I did think about this over the
weekend and I kind of get what we
are talking about here, or at least I
think I do. Let me talk this out, logs
are the inverse representation of
exponent problems, not just the
exponent but the whole problem, the
base, answer and the exponent
itself.”
Action moving toward a
process conception of
logarithms. When he
incorrectly rewrote one of the
logarithmic equations as an
exponential equation, he
quickly realized he had a
problem. He knew there was
an inverse relationship
between exponents and
logarithms, but he had not yet
interiorized the symbolic
notation needed to represent
this relationship.
Write a short
paragraph that
describes your
thinking about the
symbols
as
you completed each
portion of this
activity.
This implies multiplying b times
itself N number of times assuming
that
because I see this as
the inverse function of
Task 3
[Speaking of rules] “Over and over
again you made sense of it, now it is
part of you.”
Process conception, but the
participant seems to need to
write out the rule each time
he needs to simplify
indicating he may not have
interiorized the action.
189
Table 5 Continued
Task 4
Task 4
Develop the need and
recognize when an
alternate
representation is
needed to solve
either an exponential
or a logarithmic
equation.
As a group, they
were asked to
explain/evaluate
various logarithmic
and /or exponential
expressions or
equations and state
any relevant
characteristics
noticed.
“Well I just laid it out as
, Oh I see what you
are getting at now because before
without really understanding the
logs I couldn’t get that (referring to
the decimal approximation for the
log), I couldn’t even get that
concept in my head (meaning
estimate from the graph) but now
we can…because now you can set it
up then use the calculator.”
Process understanding of a
single logarithmic term.
Once the rule had been
committed to memory, Tom
successfully solved several
numeric examples and an
application problem.
On the application problem Tom
instinctively wrote
, which
was correct, but he was unable to
provide solid reasoning why he
solved this way other than he knew
it was doubling and the rate was
1.07. After an all-group discussion,
he responded, “But I got there
before I even did all this. I don’t
know how I did it, but I did it.”
Without looking at the
written formula, he was able
to solve this problem;
however, he had difficulty
explaining his actions and
connecting his ideas to the
printed material.
When asked to explain if ( )
Process conception moving
toward an object conception
of exponential functions. He
further asked, “What if x is a
fraction?” indicating he was
attempting to expand his
understanding. Jim added,
“It only changes direction if
the exponent is negative.”
Apparently, this clarified his
confusion. To draw closure
Tom stated, “I got to say this
to myself: the exponential
functions reflect across the yaxis and the logarithmic
functions reflect across the xaxis.”
( ) is decreasing or increasing,
Tom incorrectly guessed increasing,
but when asked to explain why he
changed his mind he said, “When I
picture the graph it is going to
decrease,” adding that on Day 1 the
graph “began high in quadrant II
and falls steeply to the right because
the base is fractional.” As we
moved to a logarithmic function,
( )
he was not surprised
that the curve, according to him,
“was on the other side of the graph”
and noted that ( )
was
its mirror image.
190
Table 5 Continued
Task 4
Pre-task 5
Given 6 different types
of equations,
participants were asked
to explain in detail the
solution method used to
solve each.
No one had difficulty with
this task. Participants knew
when to use a logarithm in
their solution process. They
all recognized that first each
side of the equation
needed to be
divided by 3 before they
could solve for x.
While no application problems
were given, all were successful. It
is doubtful that this would have
occurred if an application problem
had been presented. This indicates
a process understanding.
Given logarithmic
equations,
“I went raised to the power
of negative 6 equals my x
and then I put 1 over 3 over
Process understanding moving
towards an object understanding of
a single logarithmic term. At times
Tom seems to forget how to
evaluate certain exponential
expressions. In his words, “I
didn’t hold on to it in my mind,”
suggesting an incomplete
construction of an object
conception.
and
could
you solve for x?
2(
⁄
) but I think I might
be wrong. OK, I had all the
procedures right but where I
went wrong instead of
seeing 6 as my exponent I
saw it as something else.”
Task 5
Develop an
understanding for the
relationship between
the properties of
exponents and how
they apply to
logarithmic
expressions.
“I went here and said [he
flipped back to the previous
page that had values for log
base 4], well, I can’t decide
what is in the middle here
[indicating that
did
appear on the previous
table] but when I looked at
these two they added up to
80, so I said, well, if I can
add these and they equal 80
then I can represent it this
way, and that was my
thinking. My thinking was
somehow these two add to
80 (he had written
(
)).”
Tom still thinking of each
term individually and has
not interiorized this action.
When asked to discuss the
validity of the following
statement,
he writes 2,
Initially Tom did not have any idea
how the ideas were connected,
perhaps he had never entertained
the notion that someone the
properties for exponents should be
connected in some way to
logarithms.
Out of the group Tom was the only
participant not able to connect this
knowledge with logarithms of
multiple terms. For example, he
responded that
is true because
, but could not make
sense of the following:
since the two terms
on the right-hand side of the
equation did not have integer
values.
191
representations.
Tom demonstrated that while he could solve a logarithmic equation for x by
rewriting in its equivalent exponential format, he experienced cognitive difficulties when
asked to evaluate this expression. For example, when asked to solve
for x
he asked, “Can I tell you how I set it up? I went raised to the power of negative 6
equals my x, then I put 1 over 3 over 2, but I think I might be wrong.” He indicated 6
was his exponent and knew something about one over something but did not evaluate it
as an exponential. Is this because he is relying on memorized rules without
understanding? His difficulty in working with exponential expressions became more
apparent when he was asked why he thought
could be rewritten as
. He responded, “I went here [referring to a table he had constructed
for logarithms base 4] and I said, well, I can’t really decide what is in the middle here.
But when I looked at these two they added up to 80, so I said, well, if I can add these and
they equal 80 then I can represent it this way, and that was my thinking. My thinking
was, somehow these two add to 80.”
When working with logarithmic terms students, needed to consider how the laws
of exponents could be applied. Tom was looking for rules to justify his work, yet he
seemed to be missing an essential connection between logarithms and exponents.
Operationally, he could work with a single logarithmic term; however, when asked to
consider how multiple terms could be equivalent to a single logarithmic term, he
continually relied on the one rule he had committed to memory:
if and only if
. His written work suggested he was unable to move beyond an operational
192
understanding and consider the structural nature of exponentials and their connection to
logarithms. In other words, Tom was unable to coordinate his knowledge of exponents
with the symbolic notation associated with logarithms. For example, when asked to
explain how to add two logarithmic terms, Tom stated, “So you take the two results, the
two exponent results, and add them, then raise the base to that power and that is really
what that number should be.” In Tom’s mind, this explanation holds only if the two
logarithmic terms themselves are rational numbers. If the logarithmic terms produced
irrational results, Tom was unable to see how to perform the operation. For example,
asked to validate the truthfulness of the statement
, Tom was
unable to reconcile his concept image with his concept definition. He said that this
statement was false; however, his reasoning was flawed. In previous class discussions,
Tom either neglected to consider the alternate representations presented, was content with
his own explanation, or just moved on.
Summary of Doug’s Performance. Table 6 summarizes significant events that
encouraged Doug to reconsider his existing knowledge in his quest for an understanding
of logarithmic concepts. Doug seemed motivated to understand rote algorithmic
manipulations associated with exponential expressions and built a more general
conception for function or, more specifically, one not tied to a narrow definition as a
relation between two sets. He readily admitted that he followed rote procedures and
hoped that he remembered them correctly. He stated that he did not understand the
mathematical underpinnings associated with the laws of exponents. Working with the
common definition of exponents (CDE), Doug began to develop an understanding for
193
integer exponents. For example, he said, “There again I don’t know why
just memorization, but let me try a different approach. I’ll divide
but
doesn’t fit with the CDE, but 1 over
. It’s
and use my rule
does.” Doug needed a visual
representation to validate a rule he had remembered. He later questioned whether all real
numbers could be used as exponents—wondering if a number like .2395 could be an
exponent. Tom responded, “Yes, it’s similar to the first power of 2, only smaller.”
In Task 1, the participants were asked to graph an exponential function by
creating a table of values for discrete values of x. All participants constructed a smooth
continuous graph for the given function. When asked to identify the domain and range
for this function, Doug was quick to correct Jim when he incorrectly reported the range as
the set of all real numbers. Doug said, “Based on the graph, it would be zero to infinity.”
(He correctly wrote this in interval notation as (
) ) When asked if the graph could be
used to approximate the value of (√ ), he said “No because it is not a real number”;
however, when asked what he meant by this statement he responded, “Well, it’s not a
nice number” and then went on to ask the group, “The square root of 3 is what?” Not
getting an immediate response, he answered that it would be
. Doug frequently gave
responses that were initially off base. This may indicate that his mathematical knowledge
up to this point consisted of rote procedures. As the group completed Task1 Part A
number 3, for several different exponential functions, Doug continued to struggle with
negative exponents. When working with the function ( )
( ) Doug said, “So
194
Table 6
Summary of Doug’s performance
Tasks
Activity
Issues encountered that were
insightful
APOS level of understanding
Pretest
Simplify
exponential
expressions
involving integer
exponents.
When asked: “If you have this
expression by itself, 20, how
would you evaluate this?” Doug
responded, “That doesn’t fit into
my mind. It is 1, I know that, but
I don’t know why.”
When reminded of the laws of
exponents, he used these in
combination with CDE to deepen
his understandings.
Moving toward a process level
understanding, he was attempting
to interiorize his actions to create a
structural image of the concept but
admitted he was unable to explain
why any of the rules made sense.
He indicated it was just
memorization on his part and that
sometimes he was not sure if he
was remembering correctly.
Task 1
Graph the
exponential function
( )
to
extend the meaning
of exponents to
include all rational
values. Using the
graph, participants
were asked to
approximate the
value of (√ ).
After completing the graph of a
smooth curve, Doug was able to
describe both the domain and
range for this function; however,
he was quick to answer that you
could not use the graph to
approximate the result of (√ )
since √ is irrational.
Still at a process level, but Doug
was attempting to coordinate the
operational characteristics of
exponential expressions with the
structural characteristics of
exponential functions.
Task 1
Develop the need
for using logarithms
or some other
alternate
mathematical
procedure by asking
students to solve
Doug would use the graph if it
were large enough to show more
details; however, he stated that
he knew it would be between 1
and 2, “So you cut this in half
and raise it to the 1.5 power.
When 1.5 was too large then I
knew it would be between 1 and
1.5 and I tested it that way.” He
later added, “It seems we are
approaching some type of limit.”
Process level conception moving
toward an object level conception
of exponential functions. He
seems to be thinking about how to
reverse this process using some
type of mathematical procedure.
He is not sure what it is exactly,
but seems to think there has to be a
limiting process as you narrow
down your result. Doug further
added that the procedure is similar
to roots but some type of repeated
division is needed. He was unsure
how to proceed since it only
divided evenly 1 time and he did
not know what to do with the
remainder, but added, “It’s still
theoretically what is happening.”
195
Table 6 Continued
Task 2
Graph both
exponential and
logarithmic
functions on the
same set of axes
and describe any
relationships.
Doug writes, “They are inverse
functions. The x and y axes are
switched, I mean the domain of
f1 becomes the range of f2. The
log button reverses the process
of exponentiation; it allows
you to see what power the base
was raised to, to give you the
answer.”
Process conception. He could
recognize the relationship between
exponential and logarithmic
functions, yet when it came to
completing a table of values for
each pair of functions, he was
unable to trust his intuitions; he
needed to calculate the result each
time.
Task 3
Using the
definition for
logarithms,
students first
rewrite exponential
and logarithmic
expressions, then
evaluate
logarithmic
expressions
without a
calculator, using
only the alternate
representation for
the logarithmic or
exponential form.
Doug asks, “So what kind of
method, what kind of
arithmetic is being done when
you do logs?”
Process conception. He had
interiorized the definition but still
does not seem to be able to make
sense of the operational
characteristics of the logarithmic
function that appears to be critical
for all participants.
Task 3
Write a short
paragraph that
describes your
thinking about the
symbol
as
you complete each
portion of this
activity.
“You can estimate the logs. If
the bases are the same, the
larger number yields the larger
exponent.”
Did not really address the issue of
the symbols but did indicate that
you can find the integer exponent
above and below the target value,
then estimate. This does indicate
process understanding; he has
interiorized the action of changing
from one format to the other.
Task 4
Develop the need
and recognize
when an alternate
representation is
needed to solve
either an
exponential or a
logarithmic
equation.
After the question was asked,
“Why is a log involved?”
Doug responded, “Because
there is a power of x involved.”
However, Doug did not know
how to begin this exercise
prior to this discussion.
Action level, possible process level
conception. When asked to solve
several exponential equations using
what he now knew about logs,
participant was successful;
however, when asked to solve an
application problem which did
contain an exponential equation, he
was not sure how to begin until
prompted.
196
Table 6 Continued
Task 4
As a group, they
were asked to
explain/evaluate
various logarithmic
expressions or
equations.
Asked to explain why
is
between 3 and 4, Doug quickly
responded, “Because 3 to the
second—no, no, I mean 2 to the
third power is 8, and 2 to the
fourth power is 16.”
Process conception moving toward
an object conception of a single
logarithmic term, once he heard what
he had said, he immediately
recognized this as incorrect.
Task 4
Given 6 different
types of equations,
participants were
asked to explain in
detail the solution
method used to
solve each.
No one had difficulty with this
task. Participants knew when to
use a logarithm in their solution
process. They all recognized that
first each side of the equation
needed to be divided
by 3 before they could solve for x.
While no application problems were
given, all were successful. This
indicates a process understanding.
Pre-task 5
Given logarithmic
equations,
and
Referring to the second equation:
“Well, there is a 3 in there
somewhere, x is 3, no negative 3,
just throwing out words.” When
asked to explain his thinking he
responded, “I see 3 cubed is 27”
but then admitted that the problem
itself suggested it. He said, “If
you gave us something bizarre I
wouldn’t be able to solve it.”
Tom then told him, “If you follow
through all the steps, you should
be able to solve it, but not all of
them are going to jump out like
that.”
According to Asiala et al. (1996), “In
the course of performing an action or
a process on an object, it is often
necessary to de-encapsulate the
object back to the process from
which it came in order to use its
properties in manipulating it” (p. 11).
Based on his admission that he would
not be able to solve it unless the
result was readily apparent, Doug
was not working with a fully
interiorized concept; he initially
reacted in response to external cues
as opposed to reorganizing the
material into a more structured
format.
participants were
asked to solve each
for x.
Task 5
Develop an
understanding for
the relationship
between the
properties of
exponents and how
they apply to
logarithmic
expressions.
As Doug worked to justify why
he
referred to two sequences
involving powers of 4. First, he
multiplied 16 and 64 and came up
with 1024, then he added 2 and 3
and came up with 5. That led him
to generalize
. He later added
that if the bases were not the same
you would have to find a
relationship between the first and
the second base before you could
do anything. He also was able to
recognize
Moving toward a process level
conception. It appears that the class
discussions had made sense and
Doug was able to determine the
validity of 4 different statements, but
no mention was made of how to
connect this to the laws of exponents.
He seemed to be able to link several
different processes together, but
could not encapsulate this process
into an object. The explanations
given at this point were procedurally
oriented.
197
( ) is 1 over .5 to the 3 power which is 125, yeah!” Earl answered, “No, ( )
is 8.”
Doug asked for clarification and Earl responded with “Well, ( ) is , so ( )
is 1 over
which is 8.” Doug then added, “I need to see that worked out mathematically with what
I know.” When Jim told him to look for the patterns, Doug replied, “I don’t trust myself.
I need to work these out to see if they do go backwards from ( )
.” As the
participants attempted to generalize characteristics of exponential functions, Doug was
clear on the domain and range for exponential functions. He added, “If
curves up and if
the graph
the graph curves down,” then added that the greater the
denominator, the sharper the decline would be. Doug seemed to have a process
understanding of exponential functions, but continued to struggle operationally with
exponential expressions. The idea of operational and structural duality of exponential
expressions and functions seemed to limit Doug’s ability to move to more advanced
mathematical thinking.
As the participants moved on to the next task, they were asked to find the inverse
function for several different linear functions. As they began to work, Doug asked,
“Well, that is my question. What does ‘Find the inverse function’ mean?” Both Jim and
Earl recited the procedure used to find an inverse, but were unable to describe the role of
the inverse function. After a discussion about properties of inverse functions, Doug
offered, “They undo each other” as the role of the inverse. When asked to find the
inverse function for ( )
, he answered, “ ( )
√ ,” giving an indication that he
was attempting to symbolize the inverse operation of raising a given base to any real
198
number. As participants completed several graphs for pairs of exponential and
logarithmic functions, Doug reflected on the visual representation. He stated, “So the
inverse of an exponent function is a log function.” He then asked, “If log is a
mathematical operation, what kind of method is being done when you do logs?” He
appeared to be making an effort to accommodate the new information; however, he
needed to see how the log finds this exponent. He conceded that some type of repeated
division must be involved, but the symbol log did not offer any insight for him into the
mathematical process.
The historical development of logarithms provided Doug with a glimpse of the
mathematical procedure used to develop the modern table of logarithms. He recognized
the need to first estimate the number of “whole” ratios. He stated, “So you have to infer
the whole number part; the log table only gives the fractional part.” As the group worked
through the designated activities, it appeared that Doug was beginning to internalize what
it means to find the logarithm of a given value. According to Sfard and Linchevski
(1994), “The operational way of thinking dictates the actions to be taken to solve the
problem at hand, while the structural approach condenses the information and broadens
the view” (p. 203). Doug’s written work indicated that he knew when to use logarithms
to solve a given equation; however, if the format of the equation did not match the given
formal definition, some type of adaptation was needed. Once he was able to navigate this
obstacle, it appeared that he was moving toward an object conception of logarithms—
understanding when to use appropriate logarithmic notation to solve an application
problem involving exponentials. As the group continued to work on Task 5, Doug
199
seemed at this point to be reacting to visual cues, indicating a need to encapsulate the
processes.
When first exposed to expressions that contained multiple logarithmic terms,
Doug said that addition was not allowed with logarithms. When asked to explain what he
meant by this statement, he said, “The usual law of addition of like terms does not
apply.” He added, “It didn’t break out over here. That is a different result” [referring to
the question: Does
new information he asked, “So the answer
As Doug struggled to assimilate the
is the same as
?”
After he had written this out for himself he added, “It doesn’t work.” Jim explained,
“The adding of two logs is the same as multiplying whatever their result is.” On the last
assessment; however, Doug failed to apply this newly acquired knowledge, but admitted
he was just reacting to the statements and not really taking the time to think things out—
suggesting that he was moving toward a process understanding but had not yet
interiorized the meaning of this newly acquired information.
Summary of Jim’s Performance. Table 7 summarizes Jim’s responses to the
tasks. While not as vocal as Tom or Doug, Jim seemed to have a more sophisticated
understanding of both the operational and structural characteristics of exponential
expressions and functions. For Jim, it appeared as if the problems presented were merely
strings of symbols to which well-defined procedures were applied. Although he was able
to use the CDE, he instead relied on laws of exponents (LOE) to simplify exponential
expressions. His concept image of an exponent did not seem to require more than the
symbolic form in order for it to make sense. Rational exponents in particular were not
200
problematic; he saw the denominator as the root and the numerator as the power, so, for
him,
meant: find the square root of nine to the first power. Jim was mentally able to
transform the laws of exponents into generalized properties of exponents without
experiencing any cognitive difficulties.
When asked to use the graph to estimate the value of ( )
√ , Jim
for
did not see how the graph would be useful. He commented, “You need a value for √
before you can begin,” so for him it seemed natural to just use the calculator to
approximate the value. Jim noted that the overall steepness of the curve changed relative
to the size of the base. He stated, “When
it is an increasing function. When
it is a decreasing function, where the value of the exponent can take on any value.”
When asked to explore the connection between pairs of functions, Jim was able to
recognize the similarities between the structural notations of each. He readily committed
the definition of a logarithm to memory. However, when asked to evaluate
he hesitated, not sure how to respond. He stated, “This is equal to
(
)
, but I don’t
have an answer.” As the group discussed the meaning of this equation, they finally
agreed, when they looked at the graph of ( )
, that
was not in the range of
the function. Still reluctant to state “no solution,” Jim added, “There is never not a
solution. You can always make something up. They made up imaginary numbers,”
indicating his propensity for seeing mathematics as a rigid system of concepts where
definitions and rules play an integral part in the presentation of new material.
Although Jim was able to use both exponential and logarithmic notation in what
appeared to be a flexible manner, it was interesting to note his comment on using
201
Table 7
Summary of Jim’s Performance
Tasks
Activity
Issues encountered that
were insightful
APOS level of understanding
Pretest
Simplify
exponential
expressions
involving integer
exponents.
Active use of the common
definition of exponents to
justify his responses, but
then stating it was much
easier to just apply the laws
of exponents.
Jim had a clear notion of the common
definition of exponents and was able to
extend this definition to include both
zero and negative exponents. He
clearly understood the operational
nature of fractional exponents. Process
level understanding, but was unable to
voice a generalized notion of
exponents. For him it appeared as if
the symbols were the concept.
Task 1
Graph the l function
( )
to
extend the meaning
of exponents to
include all real
values. Using the
graph participants
were asked to
approximate the
value of (√ ).
Could articulate
generalizations about
exponential functions, but
could not explain why the
graph would be useful for
determining the value of
(√ ) given the graph of
( )
. In his own
words he asked, “You
mean find the value by just
eyeballing it?”
Could articulate generalized properties
of exponential functions, but it seemed
as if he did not need to consider deeper
meanings of rational number exponents
when computationally they were not
difficult to perform with a calculator.
Process level understanding moving
toward an object conception, since in
his mind there was no need to explain
the wider notion of exponents in a
meaningful way.
Task 1
Develop the need
for using logarithms
or some other
alternate
mathematical
procedure by asking
students to solve
“We need [to find] some
sort of function that could
keep just going and going
and you get longer and
longer numbers that are
closer to what you need.”
Object level conception: he knows that
the process of exponentiation can be
reversed and seems to remember this is
where logarithms are used, but does not
remember how the rules for logarithms
work. Jim seemed to be aware of this
process as a totality and realized that
actions and/or processes can act on the
entire expression itself.
Task 2
Graph both
exponential and
logarithmic
functions on the
same set of axes
and describe any
relationships.
While the formal definition
for logarithms had not been
given, when asked to
summarize similarities
between the functions, Jim
wrote, “If
then
and tried to
clarify this by adding, “the
log of x is the answer to the
exponent c.”
Process conception. Jim apparently
remembered something from high
school mathematics, but did not really
attempt to form any deep connections.
He worked well following rules and
recognizing patterns, knew the two
functions are inverses of each other, but
did not put much thought into the
relevant properties shared by both
functions.
202
Table 7 Continued
Task 3
Using the definition
for logarithms,
students first
rewrite exponential
and logarithmic
expressions,
evaluate
logarithmic
expressions without
a calculator using
only the alternate
representation for
the logarithmic or
exponential form.
“What does the log do?
What is in log that allows it
to do what it does, without
the word log?” That is
what I want to see.
Right now the way I see it,
the easiest way to look at it
is
”
Process conception. He has
interiorized the action, transforming
from one format to the other
effortlessly. He also realized that
before hand-held calculators the only
way to evaluate logarithms, if it was
not integer value, was by using the
table of logarithms.
Task 3
Write a short
paragraph that
describes your
thinking about the
symbols
as
you completed each
portion of this
activity.
“It means what b raised to
some power is equal to,
what you are taking the log
of. In order to undo the
exponential process and
better understand
something in a linear
fashion.”
Process conception, moving toward an
object conception; however, there
appear to be some gaps in knowledge.
While there are no mathematical
inconsistencies, the need to explain the
concepts from an intuitive point of
view is secondary.
Task 4
Develop the need
and recognize when
an alternate
representation is
needed to solve
either an
exponential or a
logarithmic
equation.
When asked to use logs to
solve several numeric
examples, Jim was
successful, although he
asked how you would
convert to other bases (this
rule was not needed: the
group was using the latest
TI-OS system that allows
evaluating logs of any
base). When asked to solve
an application problem,
however, he stated, “I was
thinking we are looking at
compound interest. Why is
log in here?”
Action conception, moving toward a
process conception. Unable to use
existing knowledge to solve application
problem even when the variable
appeared in the exponent.
203
Table 7 Continued
Task 4
As a group,
participants were
asked to
explain/evaluate
various
logarithmic
expressions or
equations.
When asked how to
evaluate
without a graph or a
calculator, Jim stated he
would make his own table
of powers of 5 and hope
78125 showed up on the
table. Earl added he
thought it would since
there was 25 on the end.
Jim countered with “Even
if it wasn’t I could narrow
it down.”
Process conception, moving toward an
object conception of a single logarithmic
expression and/or function.
Task 4
Given 6 different
types of
equations,
participants were
asked to explain
in detail the
solution method
used to solve
each.
No one had difficulty with
this task. Participants
knew when to use a
logarithm. They all
recognized that first, each
side of the equation
needed to be
divided by 3 before they
could solve for x.
All were successful. It is doubtful that
this would have occurred if an application
problem had been presented; however,
since they had seen one completed earlier,
presumably they all could complete one—
since Jim and Earl both indicated that
once they have been shown a solution
method they remember the solution
method. This indicates a process
understanding.
Pre-task 5
Given logarithmic
equations,
and
Jim explained, “I saw 27
to the 1 over 3 and then
raised so it would be 27 to
the 1 power cube root.
The denominator is the
root value (for rational
exponents).”
Procedures, committing them to memory,
and using them effectively, seemed to
pose no difficulties for Jim. He was able
to reflect on operations applied to a
particular process, a characteristic of an
emerging object conception of a single
logarithmic equation.
When asked if
Process level, moving toward object
conception. Since properties of
logarithms were not formally presented,
Jim was able to extract a procedure from
the group discussion and apply it
consistently; however, it is unclear if he
was able to connect this to laws of
exponents other than his comment that
“Adding with logs is the same as like
multiplying whatever their value is.” He
moved freely between both forms and
demonstrated a solid understanding of the
laws of exponents, but would need more
time to see how properties of logarithms
are derived from the laws of exponents.
could
you solve for x?
Task 5
Develop an
understanding for
the relationship
between the
properties of
exponents and
how they apply to
logarithmic
expressions.
is a true
statement, Jim replied,
“No, when you add two
logs with the same base
you multiply what you are
taking the log of.”
204
logarithmic notation to solve an application problem, because apparently, he was not
accessing prior knowledge to solve a problem presented in slightly different format. As
the group worked, Tom immediately recognized what to do to solve this problem. Earl
said, “You asked this on the pretest. I was kind of at a loss except to keep trying values
because it goes up by 7% every year,” while Jim stated, “I was thinking we are looking at
compound interest. What is log even here for?” When asked, “Why is a log involved in
this problem?” Doug answered, “Because there is power of x involved.” This response
seemed to satisfy all.
Harel and Kaput (1991) posit, “Some mathematical symbols cannot be understood
via the symbol” (p. 92). This statement implies that the symbol
names a
mathematical concept but “without denoting specific aspects of the structure of what is
named” (p. 92). However, when its referent exponential format has been encapsulated
into a conceptual entity,
thinking about the symbol
can be more meaningful. When asked to describe his
Jim wrote, “It means what ( ) raised to some power is
equal to what you are taking the log of, in order to undo the exponential.” This indicates
that his concept image for exponential functions now included some level of
understanding for how logarithmic notation interacts with exponential expressions.
A brief account of the historical development of logarithms provided Jim with
enough “proof” to verify the mathematical existence of logarithms. As we attempted to
calculate
without the assistance of technology or logarithm tables, the laws of
exponents and a little algebra were used to “fill-in the gaps” between 0 to 1 in an
arithmetic sequence which represents the exponent in a similar geometric sequence. Jim
205
rationalized, “This works to reverse the process, but to find our value you can’t really
cheat it like you are doing with the roots.” All participants agreed that the result of
is 1 and a fractional piece. Using a process called non-integer factoring, we found a
result. (See Appendix E for a detailed account of this process.) Jim commented, “It’s a
process that keeps reducing itself until you finally get either it [meaning the division
process comes out even]or it [meaning the division process] just keeps going forever until
you get close to one.” While not elegant, the method referred to as non-integer factoring
served to document for the group the notion that some type of repeated division was
involved in undoing the process of exponentiation.
Summary of Earl’s Performance. Earl, while admitting that he really had not
put much, if any, thought into understanding the origins of mathematical concepts, made
an effort to explore his thinking as he completed the activities. Often quiet, when he did
interact with the group he questioned the legitimacy of others’ reasoning. Table 8
summarizes his performance as he attempted to enhance his understanding.
Earl demonstrated successful procedural knowledge in working with exponential
expressions, but did not offer any generalized observations. It was apparent that Earl was
trying to extend the common definition of exponents into a broader notion. When asked
to use the graph of
( )
to evaluate (√ ), he instinctively knew what to do,
indicating that memorizing procedures was not problematic. He was able approximate an
answer, but could not reconcile this image. He stated, “I understand that the expression
looks like this:
and
but how do you show 0
? I can’t
see this, what does it mean?” He continued, “I mean, I can plug it into my calculator and
206
get an answer and guess where it is on the curve, but I don’t know what it actually is.
How do you know what this is?” As the discussion continued about alternate
representations for exponential expressions, he stated, “We know this:
√ , so could we do something similar?” His thinking indicated he was attempting to
align his concept image with the concept definition for exponential functions—
suggesting he was moving toward an object understanding of exponential expressions
and functions.
Asked to estimate the value of
Earl was able to use his knowledge about
exponential expressions to suggest an interval that would contain the variable. He stated,
“So I know 7 to something is 54. I know it is between 2 and 3, but I have no idea
where.” As the discussion moved on, it was suggested that it is necessary to find the
root of the answer, but all participants agreed they did not know how to do this. Doug
suggested that some type of repeated division by the base value might work, but wavered
when the division produced remainders. Although in agreement that this method might
work, Earl was unsure how to proceed, since this would not follow the “usual” rules for
division. Working with existing schemas, Earl was attempting to understand how to
undo the process on exponentiation, indicating a move toward an object level conception.
Earl successfully demonstrated his understanding of general characteristics of
exponential functions. His written description of the effect of the base on the overall
shape of the function, along with a description of both the domain and range for this
function, was accurate. When asked to graph both the exponential and logarithmic
functions on the same set of axes and complete a table of values for both, Earl wrote the
207
Table 8
Summary of Earl’s performance
Tasks
Activity
Issues encountered that
were insightful
Pretest
Simplify exponential
expressions involving
integer exponents.
Earl simplified the
expression first by using
the common definition of
exponents given, but then
reverted to using the laws
of exponents. He did not
provide any written
justification for his work.
Process level understanding; however,
it is unclear how he has made sense of
these rules. He did indicate that with
more practice he could become more
fluent with the laws of exponents.
Graph the exponential
function ( )
to
extend the meaning of
exponents to include
all rational values.
Using the graph
participants were
asked to approximate
the value of ( )
when
√ .
Earl told the group, “We
are trying to find what y
equals when x is the
square root of 3.” He
stated he knew what 32
looked like and even
Process understanding: Earl was
attempting to form a generalized
understanding of exponents, but part of
the encapsulation was left unfinished.
He could visually represent integer
exponents and was comfortable with
the idea of expressing terminating
decimals as fractional powers and then
roots, since he was familiar with the
process of finding square or cube roots.
He was able to work with irrational
exponents numerically, but the idea of
how to physically represent irrational
numbers was problematic, indicating he
had not completely sacrificed the
notion of exponent as a counter.
Develop the need for
using logarithms or
some other alternate
mathematical
procedure by asking
students to solve
“I can draw the curve of
3x with 32 and 33 and 1
through zero and guess
where it’s at on the
curve, but I don’t know
what it actually is. How
do you know what this
is?”
.
Task 1
Task 1
what
represented, but
could not visually
represent exponential
expressions with decimal
or irrational exponents.
APOS level of understanding
According to Dubinsky, coordination
and reversal are two important aspects
in the construction of processes and
objects. While Earl was operating at a
process level conception, he was
having difficulty trying to
accommodate the structural
characteristics of exponential functions.
While he might even possess an object
level conception, it is uncertain whether
he could de-encapsulate the object back
to the process.
208
Table 8 Continued
Task 2
Graph both
exponential and
logarithmic functions
on the same set of
axes and describe any
relationships.
When asked to
generalize, Earl wrote,
“the domain of ( )
equals the range of
( )
the range
of ( )
equals the
domain of ( )
” He further added
that he believed the log
button on the calculator
takes the base and checks
powers until it gets the x
value.
Process conception, moving toward an
object conception. Difficult to say with
certainty since participant was not very
vocal yet could work through the
handouts with relative ease.
Task 3
Using the definition
for logarithms,
students first rewrite
exponential and
logarithmic
expressions, evaluate
logarithmic
expressions without a
calculator using only
the alternate
representation for the
logarithmic or
exponential form.
Successfully completed
the activity without much
interaction. Encountered
difficulty evaluating
(
) He wrote, “4
to the power of what
gives you negative16?”
Process conception, moving toward
object conception. While there were no
mathematical inconsistencies in his
work, he was unable to successfully
deal with the expression noted since it
did not have a real-number solution.
He knew the result was impossible, but
was at a loss for how to evaluate the
given expression since in his mind you
could not. Earl was unable to see a
more global perspective regarding
logarithmic functions.
Task 3
Write a short
paragraph that
describes your
thinking about the
symbols
as you
completed each
portion of this
activity.
“When I see this, I know
to ask myself, “b to what
power gives x.”
Process understanding of a single
logarithmic term. Earl would narrow
the result between two integer values,
but was reluctant to approximate.
Descriptions were generally
procedurally oriented.
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Table 8 Continued
Task 4
Develop the need and
recognize when an
alternate
representation is
needed to solve either
an exponential or a
logarithmic equation.
.
When asked to use what
he now knew about logs
to solve numeric
examples, Earl
successfully completed
each exercise. However,
when an application
problem was presented in
which the group was and
the equation used to
model the situation was
given, Earl did not
consider using logs to
solve; rather he solved
using a trial-and-error
method.
Process understanding. Earl admitted
earlier to not attempting to justify
mathematical procedures. He stated, “I
get the rules written on the board or in
the book. I’ll do a couple of problems
and then I am able to recognize similar
problems, then move on. I very rarely
go back and say, OK, I want to try to
justify. Not many college students
will.” Earl did not see how to solve
this problem mathematically; therefore,
he resorted to trial-and-error instead of
recognizing that the variable in the
exponent signals the need to use
logarithms to find the solution.
Task 4
As a group,
participants were
asked to
explain/evaluate
various logarithmic
expressions or
equations.
When asked if ( )
is increasing or
decreasing, Earl
explained, “increasing,
because if x were equal
to 1, then y is zero, and if
x is 2, y is 1, so
increasing.”
Process conception. Earl consistently
verified/explained concepts with
numeric results rather than offering
generalizations.
Task 4
Given 6 different
types of equations,
participants were
asked to explain in
detail the solution
method used to solve
each.
No one had difficulty
with this task.
Participants knew when
to use a logarithm in their
solution process. They
all recognized that first
each side of the equation
needed to be
divided by 3 before they
could solve for x.
All were successful. It is doubtful that
this would have occurred if an
application problem had been
presented; however, since they had
seen one completed earlier, presumably
they all could complete one—since Jim
and Earl both indicated that once they
have been shown a solution method
they remember the solution method.
This indicates a process understanding.
Pre-task 5
Given logarithmic
equations,
As Tom was attempting
to reorganize the first
problem, Earl responded
with “64” and explained
that he saw 1 over to
the sixth power.
Procedures, committing them to
memory, and using them effectively
seem to pose no difficulties. While not
overly explicit about his thinking, Earl
is able to reflect on operations applied
to a particular process, a characteristic
of an emerging object conception of a
single logarithmic equation.
and
solve for x.
210
Table 8 Continued
Task 5
Develop an
understanding for the
relationship between
the properties of
exponents and how
they apply to
logarithmic
expressions.
When asked if
was a
true statement, Earl
wrote it is false. He
added “It should read
(
)
because
and 2+3 is 5.
following: “Domain of ( )
range
is 5
( ), range of
followed with a concrete example, using ( )
asked to evaluate
(
Earl was able to generalize a pattern for
the addition of logs. He wrote,
( )
; however,
it is unclear if he knows why this is true
other than he is following a pattern. He
also sees that
but
does not offer an explanation. Not sure
if he is aware of the connection
between the laws of exponents and
properties of logs. Moving toward an
object conception, he can move flexibly
between the different formats, sees the
relevant patterns, and can use them but
without a solid understanding of why
these properties for logarithms are true.
Earl’s understanding of the process as a
totality falls short.
( )
( ),” then
domain
and ( )
However, when
), he was not sure how to respond.
Earl’s written responses to questions regarding the symbol
and the
corresponding log button on the calculator suggested that Earl was moving toward an
object conception of logarithms. He no longer required the formal definition in order to
transform either an exponential or a logarithmic expression into an equivalent format.
Furthermore, Earl was aware that logarithm is the inverse operation for exponents. He
stated, “When I see the symbol
I ask myself,
to what power gives
added, “The log button takes the base and checks powers until it gets the
” He then
value. If it
passes [the value], then it returns to the previous whole exponent and does some type of
sorting until it gets an approximate value.” This thinking indicates he has successfully
internalized the process. He was aware of how to decide what to do next in terms of a
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specific step to take, he can move back and forth between both formats, easily reversing
the mental activity, but his lack of generalized knowledge about specific characteristics
of both functions indicated he had not yet achieved an object conception as defined by
APOS Theory.
Given a series of different types of equations, Earl successfully demonstrated the
correct procedure needed to solve each. However, when he encountered an application
problem, he did not see how logarithms could be used to solve this problem. Even
though the equation was given, his solution method consisted of trial and error. He
stated, “I was kind of at a loss except to keep trying values because it goes up every year
by 7%,” indicating a lack of cognitive awareness. He later admitted, “The only reason
why I’m taking these math classes is so I can take the next math class. I never intend to
apply any of this to anything in real life; it is just so I can look at a math text or problem
and understand it. I don’t need any real-world applications.” He stated that in the past,
for him to master a topic all he needed to see was a few examples. He added, “I get the
rules written on the board or in the book, I’ll do a couple of problems, and then I’m able
to recognize similar problems, then move on.” Presumably, he had not seen how to solve
an application involving compound interest and was unable to reconcile his concept
image to accommodate this situation.
When presented with the information on how to calculate the logarithm of any
number without the use of tables or technology, Earl seemed to be able to follow the
procedure outlined by the researcher and indicated that if he read the handout presented
in class by the researcher, it would probably make sense to him. His thinking throughout
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this teaching experiment seemed rooted in his ability to take an action and internalize it;
however, at times his understanding was limited to the ability to perform calculations.
He quickly recognized the numerical values of negative exponential expressions and
could explain his thought process clearly. However, if the process required more than
manipulating a string of symbols, Earl failed to demonstrate versatility in his thinking.
When working with more complex logarithmic expressions, Earl was able to
recognize that addition of two logarithmic terms corresponds to multiplication of those
same values. For example, when asked to simplify
that
he first indicated
(
, and therefore he wrote:
)
When asked to further explain, he did not verify this statement using the
laws of exponents, but he was able to write the following generalizations:
It was evident that Earl could use these
properties flexibly. When asked to validate the truthfulness of several logarithmic
equations, he was able to do so without difficulty—suggesting development of
mathematical knowledge. He clearly exhibited a process understanding, but it is unclear
whether he had encapsulated the process, since no reference was made to laws of
exponents and their influence on properties of logarithms.
Growth in Knowledge of Logarithmic Concepts
At the beginning of this study, all participants had an action understanding of
whole number exponents and seemed to rely on memorized rules when dealing with
integer and fractional exponents, at times answering correctly and at other times not. No
one participant was initially able to justify why the laws of exponents are valid, except
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for reciting the rules. Other researchers have noted that a procedural understanding of
exponents is too limited to provide a framework on which to build the knowledge of
logarithms (Berezovski, 2004; Kenney, 2005; Weber, 2002a, 2002b). To build on what
previous research suggested, instructional tasks were designed to promote conceptual
growth in the meaning of the exponent and completed before the introduction of
logarithmic concepts.
Upon completion of the first two tasks, all participants had significantly
strengthened their understanding of exponents and exponential functions. They were able
to verbalize why the laws of exponents could be extended to include all real numbers.
However, they were reluctant to use a graph to estimate the value for an exponential
function when given an irrational exponent, preferring instead to rely on calculator
approximations for irrational inputs. In short, there appeared to be an overarching
connecting structure that participants used to explain how all the diverse definitions and
features of exponentials could be brought together to integrate exponential concepts.
According to Dubinsky (1991), the construction of processes and objects is a
spiral process, with students developing object conceptions of particular concepts, only to
use these “new” objects to build new processes. Further, he claims that five key
components are essential to the construction of processes and objects outlined in APOS
Theory: interiorization, encapsulation, coordination, reversal, and generalization. Task 3
was structured to help participants coordinate their existing knowledge structures and
develop a procedure for the reversal of the exponentiation process.
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All participants agreed that some type of repeated division was involved, but had
difficulty accepting that the log button on their calculator performed this operation for
them. They were beginning to reflect on the mechanisms needed to reverse the process
of exponentiation; however, they needed concrete evidence of a mathematical procedure
to validate exactly what the log button on the calculator does.
After a brief look at the historical development of logarithmic tables, the
participants seemed satisfied that a numerical procedure does exist for finding logarithms.
All participants agreed the process was not something they would consider doing on a
regular basis; however, they would consider using logarithmic tables because this forced
them to estimate the characteristic portion of the logarithm, since every number N can be
written in the form
where k represents the characteristic part of the logarithm N.
The tabulated values for a given base, referred to as the mantissa, represents the fractional
portion of a logarithm. In other words, to use logarithmic tables one first needs to
estimate the integer value and then use the table in combination with the estimate to
determine the proper placement of the decimal point in the result.
Task 4 further reinforced the idea that logarithms represent exponents. To
complete the development of the process of finding the logarithm for a base other than
10, a process called non-integer factoring was introduced. However, it was unclear how
the participants made sense of this material since this process relied on the properties of
logarithms.
Finally, subjects considered how the laws of exponents could be applied to
logarithms. They made significant gains in understanding a single logarithm as an
215
exponent; however, not all participants were able to move beyond an action
understanding of how to apply logarithmic properties. Table 9 provides a brief summary
of the cognitive growth of all four participants. Further detail and analysis of these
findings are presented in Chapter 5.
Conclusion
At the beginning of this study, all participants admitted that they needed to review
the “rules” associated with exponential expressions. It was also apparent that they had
spent little, if any, time exploring the issues concerning the isomorphism between the
structures involving exponents and exponential expression, as participants were hesitant
to explain their understandings beyond the rules. Using APOS analysis as they
completed each task, coupled with her understanding and observation of the students, the
researcher arrived at a better understanding of the cognitive conflicts students can
encounter.
According to Dubinsky and McDonald (2001), “The theoretical analysis points to
questions researchers may ask in the process of data analysis and the results of this data
this data analysis indicate both the extent to which instruction has been effective and
possible revisions to the genetic decomposition” (p. 279). Refinement of the genetic
decomposition as originally proposed by the researcher reflects the methods that this
particular group of students needed to use in order to make sense of the new material.
The results of this study should lead curriculum specialists to revise or rethink the current
epistemology of the concept of logarithmic functions presented in most mathematical
textbooks.
216
Table 9
Summary of Growth in Understanding
Activity
Growth Noted
Pretest
Broadened view of
exponents to include zero
and negative values
Students were asked to explain
why the laws of exponents are
valid for zero and negative
exponents
Day 1 transcriptions:
Participants expressed their
understanding of exponents
as rule based, with one stating
“other than the rule I don’t
see it.[the meaning]”
Task 1
Subjects were able to
extend the meaning of
function to include a new
class of functions called
exponentials
Students were asked to create
graphs for several exponential
functions
Participants were able to
graph exponential functions
and identify key features
Task 2
Ability to consider how
to reverse the process of
exponentiation and the
apparent connection to
logarithmic functions
Students were asked to graph both
a logarithmic function and an
exponential function on the same
set of axes
Participants were able to state
similarities and differences
between the two pairs of
functions
Task 3
Were able to consider
alternate representations
of the exponential
expressions using a new
symbol log
Students were asked were asked to
evaluate several logarithmic
expressions without the assistance
of a calculator
Participants recognized
different representations that
allowed successful
completion of comparison
exercises
Group not satisfied with
just accepting that the log
button would find the
exponent
Apparent Impetus
Students were asked to create
logarithmic tables
Evidence
Participants were able to
develop logs for fractional
inputs using understanding
of fractional exponents
Task 4
Were able to recognize
when appropriate to use
logarithms to solve
Unknown
Participants completed a
problem set that involved
multiple types of problems
Task 5
Were able to consider
why the sum of two
logarithms is equivalent
to the product of the
argument of the two
logarithmic terms
Students were asked to verify the
following statement as either true
or false:
log
log
log
All participants agreed it
could not be true
217
On completion of this unit, all participants had significantly strengthened their
understanding of exponential concepts. Additionally, they realized that if a variable
quantity appears in the exponent, logarithmic concepts provide a means to operationalize
the solution process. Growth was most profound in the area of logarithmic concepts for
three of the four participants. It appeared that, for three participants, their understanding
extended beyond a procedural process. Since the usual properties of logarithms had not
been presented prior to instruction, participants had to construct a concept image based
on their past experience.
In summary, it appears that the notation
was not problematic for
participants. Rather, it was the lack of understanding of the underlying algorithmic
properties of the notation itself that posed problems in understanding. Without a clear
understanding of the steps to take to evaluate a logarithm, other than using a calculator or
recalling a set of memorized properties, participants felt that significant gaps in
understanding would remain. In other words, they said they would remember things for
an exam but not be able to recall these facts without a more detailed concept image.
Further detail and analysis of these findings follows in Chapter 5.
CHAPTER V
DISCUSSION AND IMPLICATIONS
Introduction
Because the original motivation for the teaching and learning of logarithms has all
but disappeared from today’s mathematics curriculum, students—and teachers—are left
wondering what logarithms are used for, and why they are still on the syllabus. But,
exponential and logarithmic functions are pivotal in the development of advanced
mathematical concepts. Because of their esoteric nature; however, this is a difficult topic
for students to understand. The development of tools to make computation easier, more
accurate, and faster has created the need for a change in the approach to teaching this
topic. For most students, however, logs remain a mysterious button on their calculator.
To address this issue, this study has asked the following questions:

How do students acquire an understanding of logarithmic concepts?
o How do students assign meaning to the symbolic notation associated with
logarithms?
o What are the critical events that contribute to a total cognitive understanding of
logarithmic concepts?
This chapter presents a discussion of the results and the significance of the study in terms
of the relevant literature reviewed in Chapter 2. Detailed findings for each research
question, implications for instruction, recommendations for future research, and
comments on the limitations of this study are included.
218
219
How Do Students Acquire an Understanding of Logarithmic Concepts?
To answer this question, the researcher identified two major themes and
developed sub questions in an attempt to understand how students’ construct knowledge
about logarithms. The first question characterizes the study participants’ growth in the
understanding of exponential expressions and functions, and how that growth was
connected to logarithmic notation. Specifically, the analysis explains how students
assigned meaning to the symbol
focusing on their progress. Additionally,
summative analysis of the cognitive obstacles each participant experienced was used to
interpret participants understanding of logarithmic notation, first of a single logarithm,
then of multiple logarithmic terms.
The second section details critical events that contributed to the participants’
progress in their effort to understand logarithmic concepts. The theoretical analysis of
these events indicated where revisions to the genetic decomposition might have to be
made to meet the cognitive demands of the participants.
Question 1a: How Do Students Assign Meaning to the Symbolic Notation
Associated with Logarithms?
To answer this question, it is necessary to synthesize several related ideas and
constructs. Directed initially by the proposed genetic decomposition, the researcher
began the analysis of the data by “asking the question: did the proposed mental
constructions appear to be made by the students as they completed each task?” (Asiala,
Cotrill, Dubinsky & Schwingendorf, 1997, p. 2). Four different themes or stages were
220
identified as critical in the development of logarithmic concepts. To track students’
movement through the various layers of understanding, inferences were made about
specific mental constructions they might have articulated during each phase.
Understanding of exponents and exponential expressions. The fact that the
participants saw no apparent conceptual connection between the common definition of
exponents and the notion of exponents as real numbers did not seem to concern them. “I
just know how to use the rules,” was said several times, indicating that procedural
fluency equaled understanding in the minds of the participants. Earl stated that the
pretest showed him he needed to review the rules for exponents, suggesting he
recognized that symbol manipulation alone is not sufficient to promote deep
understanding. For Earl and the others, remembering the rules equated with
understanding. At this point in the teaching experiment, all students were clearly
operating at a process level conception; that is, the depth of understanding appeared to be
limited to thinking about a concept exclusively within a procedural context.
The researcher’s initial conjecture asserted that participants should possess at
least a process level understanding of exponents in order to build a more substantive
understanding of logarithmic concepts. Weber (2002b) supports this conjecture, finding
that most students who participated in his study were unable to view exponentiation as a
process, thereby hindering any progress they might have made in understanding advanced
mathematical concepts such as logarithms.
221
In order to extend the meaning of exponent, the researcher had the students graph
several different exponential functions. Using discrete points, each participant
successfully created a continuous graph of an exponential function; however, when asked
to use the graph to explain the existence of (√ ) all participants were reluctant to do
so. This suggests that a fundamental shift in thinking is needed in order for students to
understand the isomorphic relationship between exponents and exponential expressions,
and that this shift may not occur without significant cognitive conflict. Confrey (1991)
echoes similar findings in her work, suggesting that “instruction dominated by a formal
series of definitions and rules followed by extensive practice in symbol manipulation
masks the broader systematic qualities of the relationship” (p. 127). She stated “one must
study the genesis and evolution of a mathematical idea . . . and document the pathway
students traverse in gaining insight into the idea” (p. 127) in order to develop
instructional strategies to promote relational understandings.
The unwillingness to use a graph to estimate the value of ( ) when x is irrational
indicates a possible disconnect between the equation-to-graph and graph-to-equation
connection. While the participants agreed that for each value of x the function has a
corresponding y, the physical representation of a smooth continuous graph whose domain
is continuous for both rational and irrational numbers did not seem to be aware of the fact
that the graphical representation itself offers a means for determining a solution. In a
study conducted by Knuth (2000), he found the following:
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Students do not develop the ability to flexibly employ, select, and move between
algebraic and graphical representations. In fact, many students either perceived
the graphical representation as unnecessary or used it as a means to support their
algebraic-solution methods rather than a means to a solution in and of itself. (p.
506)
Furthermore, as students attempted to make this shift in understanding using the graph,
they had difficulties with the absence of a parallel representation in terms of repeated
multiplication of the base using the exponent as counter for the number of factors. This
observation by the researcher suggests that the isomorphism needed to make this
fundamental shift in meaning does not occur by chance. It has to be developed; if not,
students tend to rely on memorized rules.
Sensing a genuine desire on the part of the participants to extend the meaning of
exponents beyond the notion of a counter, the researcher introduced a procedure for
constructing decimal exponents with powers and roots. Incorporating roots and powers
into the structure and context of exponential expressions is the first step toward
developing a deeper understanding of an exponent as a real number. The use of rational
numbers to represent the process of finding roots and powers of roots did not appear to be
problematic for participants; however, it was not clear exactly what sort of understanding
had occurred. All agreed that
an expression such as
√
; the participants also established the idea that
could be rewritten as
√
. This
heightened awareness of the role of the exponent offered participants some insight into
223
the fact that exponents does not necessarily imply counter. However, time restrictions
prohibited further exploration of the meaning of
when x is an irrational number.
Pedagogical strategies that focus on developing student thinking are critical if the
goal of instruction is for students to know both what to do, and why. With little more
than instrumental understanding, most students will tend to forget the “tricks” they used
that got them past the unit test; in other words, they may fail to encapsulate the process of
exponentiation. “The problem of curriculum development is therefore to present the
student with contexts in which cognitive growth is possible, leading ultimately to
meaningful mathematical thinking in which formalism plays an appropriate part” (Tall,
1991, p. 18).
Table 10 lists each participant’s APOS conception at the completion of Task 1,
where the focus was on extending the understanding of exponential functions and
creating a need for the development of some type of symbolic notation to reverse the
process of exponentiation.
Each participant strengthened his conception of exponents after Task 1; however,
the evidence from this study is insufficient to claim that after completion of a single unit
on graphing exponential functions, all students were able to make a fundamental shift in
their understanding of the meaning of exponent. Opportunities for the participants to
experience the exponent as an unknown, as a number, and in functional relationships
emphasized the different uses of the term, which did encourage all participants to
entertain the notion of an exponent as a multifaceted concept. This experience
224
Table 10
APOS Conception of Exponential Expressions and Functions after Task 1
________________________________________________________________________
Action
Process
Emerging Object
Object
________________________________________________________________________
Tom
*
Doug
*
Jim
*
Earl
*
Note. Emerging object conception implies that the individual is in transition from process to
object level understandings.
prompted them to expand their conception of exponents beyond procedural fluency.
While the notation for the inverse function may be problematic initially, if students are
able to interiorize the process of exponentiation, they are more likely to recognize the
need to reverse this operation and coordinate this knowledge with the formal definition of
the inverse function for exponentials.
Development of the inverse exponential function. The second theme identified
as critical in determining how students assign meaning to the symbolic notation
associated with logarithmic concepts was the importance of inverse function and its
development. Because exponential functions are increasing or decreasing functions, they
will always have different y values for unequal x values. Therefore, an exponential
function is a one-to-one function and thus possesses an inverse. Researchers recently
reported that “the traditional way of teaching inverse functions puts obstacles in the way
225
of the learner,” with the root of the problem embedded in the procedure of switching
variables (Wilson, Adamson, Cox, & O’Bryan, 2011). Results from this study support
this premise. When initially asked to describe the role of an inverse function, participants
in this study could explain the procedure for computing an inverse function, but they did
not understand why it worked or how it could be useful.
After a brief classroom discussion, the participants decided that an inverse
function “undoes what you just did.” Still not convinced they understood that the formal
role of the inverse function,
, is to map y back into x, the researcher asked them to
complete a table of values for instructor-defined inputs for pairs of functions, for
example, ( )
and ( )
. The participants quickly recognized a pattern:
the inputs for the exponential function produced the outputs for the corresponding
logarithmic function, and the outputs for the exponential functions were the inputs for the
corresponding logarithmic function. The subjects quickly ascertained that the principle
for interchanging
graph of
and
to find the inverse function gives a method for obtaining the
from the graph of
Using graphing technology, the subjects created
several pairs of inverse functions. Doug stated, “So they look like they are mirror images
of each other across that line [indicating ( )
] and you can see by looking at the
ordered pairs that the x and y’s are switched.” He then added, “The inverse of the
exponent function is the log function.”
NCTM (2000) has long advocated for the use of technology to promote deeper
understanding. In its 2008 position statement, NCTM claims that technology has the
226
potential to develop students’ understanding, stimulate interest, and increase proficiency.
Typically, exponential functions are introduced using base two. Students are then asked
to generalize characteristics of this function. However, when asked to consider its
inverse, they are unable to develop a cohesive concept image, due in part to the restrictive
nature of technology typically in use in today’s classrooms. By using the latest hand-held
technology, which does allow the user to evaluate logarithms to bases other than 10 or e,
students can work with graphs, tables, and equations before the formal introduction of
definitions and properties, using the different representations provided by graphing
technology to support their thinking. Used as an instructional tool rather than a
computational device, the technology enabled the participants in this study to analyze and
discuss the similarities and differences between exponential and logarithmic functions
before the introduction of formalized definitions. They were able to articulate that
switching the order of the coordinates moves a point to the opposite side of the line
( )
In other words, an exponential and logarithmic equation would be
superimposed, highlighting both graphical and tabular characteristics of these two
functions. This in turn provided a framework that had potential for minimizing
dependence on memorization and helped students make sense of logarithmic concepts in
a meaningful way. It allowed them to build a connection between the symbols and the
underlying mathematical structure of the symbolic notation. Without opportunities to
build meaning for these ideas, aside from memorization of rules and definitions, students
are left with an impoverished concept image, as described by Tall and Vinner (1981). In
227
other words, student understanding is limited to an action conception devoid of sound
mathematical reasoning.
The total cognitive structure needed to work effectively with logarithmic concepts
is far greater than the evocation of the symbol
. Deep understanding is not found
in the ability to manipulate symbols, but in the ability to recognize the symbols as a
complete cognitive structure. Sfard and Linchevski (1994) conjecture that an essential
feature of algebraic representations is that meaningfulness comes from the ability to see
the abstract ideas beneath the symbols. In other words, symbolic representations give the
learner the power to perform important operations without thinking about each
transformation, an essential characteristic of advanced mathematical thinking.
Furthermore, research suggests that no symbol should be introduced until the learner is
ready to appreciate its usefulness (Tall, 1992; Van Oers, 1996). Development of the need
for an alternate symbolic representation to undo the process of exponentiation appears to
be critical in the first step toward building a cohesive image for logarithmic concepts.
How do students assign meaning to
? The third theme identified as
critical in answering how students assign meaning to the symbolic notation associated
with logarithms was the symbol
itself and the image it invoked in the minds of the
participants. Given the rule
if and only if
all students were successful
in completing a series of exercises in which they had to switch from one notation to the
other. Tom still needed to write the formal definition each time before he successfully
completed this activity, indicating he had not yet internalized this process. He was able
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to recognize when something did not “look right” and corrected his error without
assistance from the instructor. He indicated that if he kept writing down the rule each
time, he would be able to remember it. All other participants appeared to have
committed the formal definition to memory and could recall it at will. The fact that all
participants had memorized this definition is an indication that all were operating at least
at an action conception as defined by APOS Theory.
As instruction progressed, the group evaluated a series of logarithmic terms. For
Earl offered the group his interpretation of the notation. He tersely offered that
he needed to find how many powers of 3 are in 81. Doug and Jim both stated that they
would create a table of powers of three. Doug added, “It will be either exact or it won’t,
and if not, then you got to go fractionally.” Tom simply stated, “What’s on the left has to
equal the right,” then added, “so you can use the log tables but only after a point.” This
statement seemed to confuse him because the instructions asked the group to evaluate a
series of expressions without tables or technology. It appeared that he still was relying on
a mechanical procedure to evaluate logarithmic expressions. He had difficulty expressing
the implied question embedded in the notation itself. For example, when asked to
evaluate
, he first said “5,” then changed his response to . When pressed to
clarify, he explained, “It’s saying what is [sic], what number do I have to [sic], log it
[sic], no, how many times do I need this ‘a’ to get
?”
Doug attempted to make sense of these expressions, sometimes successfully and
at other times not. For example, when asked to evaluate
, he wrote
,
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indicating he had memorized the procedure, but when pressed to solve for x, he first
wrote
, and did not complete the statement. Later, he correctly identified
as being equivalent to
; however, he first rewrote the expression as
(
)
. The
inconsistencies seen in Doug’s responses may indicate that while he was able to switch
from logarithmic form to exponential form, his understanding of integer exponents was
weak. For example, Doug correctly transformed
but then said, “ to the power of
negative 8.” After hearing the class discussion, Doug admitted he first saw and 4, and
said, “I knew there was a negative exponent involved but I mistakenly multiplied 2 and 4
when I should have been thinking differently.” Pressed to clarify what he meant, he said
he knew there was a reciprocal involved due to the negative exponent. Then, to find it,
he simply multiplied, only to realize that he should not have done this. This was typical
behavior for Doug. He would complete a calculation, then, when he reflected on his
work or listened to responses from the group, he recanted his original answer. In
engaging in student-to-student and teacher-student interactions, Doug was able to
reorganize his understanding and develop an increasingly sophisticated conceptual
understanding; however, his overall ability to identify the correct value of a negative
exponent remained weak.
Evaluating the relative size of two different logarithmic expressions was
somewhat successful; all participants answered at least six out of nine questions correctly
on Task 3 Part C. However, it was still unclear what type of understanding had
230
developed. The written work submitted suggested that all participants were operating at a
process level conception. This inference is suspect for two reasons. First, students were
no longer using technology to evaluate the logarithmic expressions. Secondly,
participants did not write down equivalent exponential notation for each exercise,
indicating that they had to rearrange the expression mentally to generate an approximate
numeric value. For example, Task 3 asked participants to compare the relative size of
several different logarithmic expressions. Earl completed the exercises by writing the
numerical equivalent for each term, then compared the relative sizes. For example, when
he encountered
and
, Earl simply wrote this underneath the expression:
Jim, on the other hand, did not even write the equivalent numeric value for each
expression as he completed this task, opting instead to place the correct symbol between
the terms without any hesitation. When Jim was asked to elaborate on his single
incorrect response to Task 3, he added, “I missed the 4 in front, so it would be
, which is 4 not 1; so it should be
.” One might argue
that the subject was simply randomly selecting a symbol as he completed exercises;
however, Jim’s explanation for the item above indicated that he had not done so.
It seems reasonable to infer that both students were at a process level conception
for comparing the relative size of logarithmic terms. They were both able to think of the
action needed to transform the given expressions without actually doing it and to
combine this with their previous knowledge about exponential expressions.
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On this same task, Tom and Doug were less successful, but still were operating
above an action level conception. When asked to review and justify their written work,
both were able to work through previous difficulties. On the pretest, both had
experienced difficulties working with negative exponents because of their lack of
understanding of exponents as real numbers; this difficulty seems to be a contributing
factor. If given an expression such as
, both were able to correctly evaluate it;
however, Doug needed to rewrite this in its equivalent exponential format, whereas Tom
just wrote
. On the other hand, when faced with a fractional value as either a
parameter or an input, Tom skipped these items. His avoidance of unfamiliar problems
suggests that Tom was unwilling to make sense of material that was not readily part of
his cognitive framework. If he did not “recognize” the exponent, he simply did not
complete the problem. For example, when confronted with the expression
(
)
Tom could not conceptualize how many factors of 4 he needed to produce (.25). He did
not see this action as a familiar procedure. Unable to reconstruct this expression
mentally, Tom simply did not solve the problem, providing evidence that he had partial
or incomplete cognitive structures related to exponential expression. As he started this
task, he first asked if he could use a calculator, an indication that he might not yet
abstracted meaning from the symbols. In other words, he was not able to consider the
symbol
as a mathematical object; for him it signified a procedure.
When they were asked to write a short paragraph describing their thinking about
the symbol
all participants indicated that the symbolic notation implied that you
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needed to find some numeric value such that when b is raised to that value, x is the result.
Jim wrote, “It means what ( ) raised to some power is equal to what you are taking the
log of.” He continued, . . . “in order to undo the exponential process and better
understand something in a more linear fashion.”
Additionally, Doug said that one could estimate the relative size of a logarithm by
narrowing down the result between two distinct integer values. This same student also
noted that if the base values were the same, the larger input yielded the larger exponent.
He wrote, “To estimate [the value of a logarithm], find the exponent that yields the
number immediately below the target then find the exponent immediately above the
target. This technique works well with the targets given in the previous exercise, but if
the target involved fractions the algorithm wouldn’t work.” This thinking indicates that
Doug lacked an object conception of exponential expressions. He was capable of
working with whole number exponents, but, when asked to extend his thinking to include
all real numbers, he lacked a cohesive concept image for rational exponents.
At the completion of Task 3, it was clear that students were thinking of the
symbol
more as a numeric value than as a functional relationship. Their responses
indicated that if given the value of the variable, they should be able to find the equivalent
numeric output of this expression. However, when the output was not an integer value,
students were required to access their previous understandings about exponentiation and
de-encapsulate their limited object conception of exponentiation back to its underlying
processes and construct transformations that could be applied to this concept (Weller et
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al., 2000). This ability to de-encapsulate their understanding of exponents appeared to be
problematic for participants when the “answer” was not readily available. Participants
appeared to be searching for an algorithmic-driven procedure that could reverse the
process of exponentiation. The participants were trying to connect their concept image
for exponentiation, which for the most part manifested itself as repeated multiplication of
a given factor, with some type of repeated division process.
This finding extends Kenney’s (2005) work, in which she noted students’ inability
to identify the equivalence or nonequivalence of five pairs of logarithmic expressions
after they had completed a unit on exponential and logarithmic functions. She suggested
that students saw no meaningful relationship in the log symbols themselves. The current
study also builds on the work of Chesler (2006), who explored student understanding of
logarithmic and exponential functions by administering a ten-question questionnaire as an
announced quiz at the completion of instruction on exponential and logarithmic concepts.
The questionnaire was intended to provide insight within an APOS framework on
students’ level of knowledge. Chesler found little or no evidence that the students
possessed more than an action understanding of exponents. Furthermore, he found that
students had a sense that a relationship existed between exponents and logarithms—but
could not communicate the relationship precisely.
Research has found that symbols typically indicate to the student a mathematical
process or operation to be carried out. Being able to think about the symbolism as an
entity in itself allows the learner greater flexibility when he or she is asked to “do”
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mathematics. The learner is no longer constrained by a single procedure, but is able to
select from a broad spectrum of definitions, imagery, and actions which essentially all
have the same effect. According to Tall (1998), symbols evoke a very special meaning.
Specifically, he writes:
Many of them evoke both a process to be carried out and a concept which is the
output by that process. In many (but not all) instances, the dual use of symbol as
a process and concept usually begins by becoming familiar with the process and
routinizing it, to carry it out with less attention to specific details. (p. 4)
Without a clear understanding of the mathematical procedure to be carried out,
other than “to find the exponent” when the symbol log is encountered, the students are
less likely to develop advanced understandings of logarithmic concepts.
Table 11 indicates the participants’ understanding of the notation
at the
completion of Task 3. Based on the written and verbal responses’ the data suggests that
all students were capable of remembering and correctly using the formal definition of
logarithms; however, it is uncertain whether this memorized procedure was sufficient to
promote movement to the next level of understanding.
It is difficult to conjecture on the meaning each participant assigned to the symbol
. Participants were aware that log represents an “operation,” but the operation
lacked a concise paper-and-pencil algorithm for simplifying an expression. Exponential
and logarithmic functions are not algebraic; they belong to a class of transcendental
functions, since they cannot be expressed in terms of a finite sequence of the algebraic
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Table 11
Understanding of the Notation
________________________________________________________________________
Emergent process
Tom
Process
Emergent object
*
Doug
*
Jim
*
Earl
*
Note. Emergent indicates that responses by the participants cannot be classified as possessing a
clear process (or object) conception at this point.
operations of addition, subtraction, multiplication, and division. Without a computational
process to give meaning to the symbol,
must be viewed as an abstract object.
Many theoretical statements suggest that operational conceptions precede the structural
(Sfard & Linchevski, 1994). Traditional curricular models assume that students can
develop an object conception in the absence of operational characteristics. In teaching
about logarithms, the instructor presents a definition, and students are assumed to develop
a rich concept image as they practice countless exercises, moving back and forth between
the logarithmic and exponential formats. Tall and Vinner (1981) hypothesize that in
order to develop appropriate cognitive structures for new concepts, one needs a welldeveloped concept image, and not a concept definition. Tall (1988) writes, “When
students meet an old concept in a new context, it is the concept image, with all the
implicit assumptions abstracted from earlier contexts, that responds to the task” (p. 3).
Logarithms are just that: a new context derived from its inverse function, the exponential.
236
Results indicated that not all participants were able to treat the symbol
as an
abstract object without having some insight into the operational characteristic of the
operator log. They knew that
produced a numeric value that was equivalent to an
exponent, but each participant expressed an unwillingness to accept this notation without
first understanding the numerical procedure embedded in the notation. In other words,
participants were attempting to mentally interiorize the processes involved in evaluating
logarithmic expressions.
Task 4 was used to strengthen each participant’s understanding of logarithmic
concepts. Working with exponential equations, students were now asked to solve the
given equations using their knowledge of logarithms. However, when they encountered
(
an application with a slightly different format,
participant to recognize the need to first simplify, (
format,
)
) , Tom was the only
then use the alternate
, and a calculator to answer the question, “How long will it take
$1000 to double?” However, when asked to explain his thinking, he hesitated, stating,
“Hold it, let me back up and see. I think I just threw something in. I didn’t read the
whole problem, but I think I am on the right path. Oh, it has to double, that is what I was
thinking, and double is two.” Tom may not have used the equation given; rather, he read,
“Find a doubling time for an interest rate of 7%.” When asked to consider the equation,
he paused and listened to the ensuing discussions. He then stated, “But I got there before
I even did all of this; I don’t know how I did it, but I did it.” From this statement, one
could infer that Tom had not yet reflected on the mathematics itself, but rather was
237
relying on prior knowledge about doubling time. However, to solve the problem, he
coordinated this knowledge with his understanding of logarithms.
Earl and Jim questioned the need to use logarithms since compound interest was
involved. Earl indicated that he remembered something similar to this problem on the
pretest and admitted he did not know what to do then, so he resorted to a trial-and-error
solution method. Jim and Doug did not attempt to solve this problem; however, Doug
did recognize the need to use a logarithm, but was unsure how to deal with the coefficient
of 1000.
Earl and Jim both agreed that since they had not “seen” how to solve this type of
problem, they hesitated to start—suggesting that both of these students rely heavily on a
traditional approach to instruction in which the focus is on instrumental learning rather
than relational understandings (Skemp, 1977). Tom and Doug both saw the need to use
logarithms since the variable appeared in the exponent; however, Doug’s algebraic skills
were weak, perhaps due to the lag time between this course and his last mathematics
course. While Earl and Jim produced more correct responses to the numerical problems
posed than the others and were able to recognize patterns easily, they were reluctant to
think about their own thinking. This reluctance seems to indicate more about their belief
systems than about their mathematical ability. They both seemed to find security in the
“familiar operational notion of a built-in finite computation to give an answer” (Tall et
al., 2001, p. 95).
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Table 12 summarizes each participant’s conception of logarithmic and
exponential concepts at the completion of Task 4. For a more detailed look at each
participant’s understanding, the reader is referred to the tables in Chapter 4, which detail
each participant’s individual performance.
Table 12
Understanding of a Single Logarithmic Term and Exponential Concepts at the
Completion of Task 4
________________________________________________________________________
Emergent process
Process
Emergent object
Tom
*
Doug
*
Jim
*
Earl
*
The influence of the laws of exponents on multiple logarithmic terms. The
last theme to emerge as a result of planned activities conducted during the teaching
experiment was the effect of the laws of exponents on students’ understanding of
multiple logarithmic terms. Task 5 asked students to consider the truthfulness of the
statement
Considerable time was spent in developing the
relationship between an arithmetic and a geometric sequence. Students easily recognized
that multiplication/division in a geometric sequence corresponds to addition/subtraction
in the arithmetic sequence. Furthermore, students understood that if a logarithm
represented an exponent, then
could be represented as 7. In addition, this
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implied that
7
could be rewritten as
, which then could be simplified as
. Tom seemed fixated on this fact and failed to develop a more generalized
understanding of the implications this held for the question, “Does
?” which, all agreed, could not be true. He decided that
the right-hand side of the equation should reflect the addition of two numbers whose sum
is 7. For example, he wanted to replace
with
He
failed to recognize that the product of 2 and 64 is 128; he was looking for two different
expressions that when simplified to a numeric value could be added to produce the
desired result. His understanding seemed limited to discrete integer values for
logarithms.
On Doug’s first attempt to make sense of
suggested that the law of addition does not apply since
, he
. When pressed to
explain what he meant, he indicated that the terms could not be treated as like terms in a
polynomial expression. In other words, he realized that while
in the
world of logarithms something was different.
At first, the group felt the need to find numeric results for each term before they
could perform the operation of addition or evaluate the validity of a given statement.
This need to find a discrete value is an indication that they were not coordinating their
previous knowledge regarding the laws of exponents with logarithmic terms. Initially, no
one was able to make the connection that multiplication is called for when we add
exponents. Earl interjected that the left-hand side should be
, while Jim saw the
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left-hand side as
. Both then agreed that “log base 2 of something plus itself is
the log base 2 of the square of that.” These two participants had developed a procedure
that would work every time, given that the base values where identical. Realizing that
is equal to 4096, Jim then asked, “So isn’t the adding of two logs the same as like
multiplying whatever their value is?” This understanding was an indication that both Jim
and Earl were reorganizing their knowledge structures to accommodate “new”
information. Doug also appeared to be able to make this connection at the completion of
Task 5 Part A.
While Jim, Earl, and Doug appeared to be comfortable working with the
logarithmic format and seemed to have made the connection between laws of exponents
and logarithms, Tom seemed to be fixated on the exponential form of the logarithm.
When asked to generalize or describe any patterns they noticed, all participants with the
exception of Tom, wrote that
(
) or something similar.
Although it cannot be stated with certainty that, with the exception of Tom, all other
participants recognized the connection between logarithmic properties and laws of
exponents, they did recognize that addition with logarithms is related to multiplication of
the argument of the logarithmic terms. Thus, these student participants appeared to have
made some movement in their growth of understanding of logarithmic concepts.
It appears that Tom’s previous experience in manipulating single logarithmic
terms was in conflict when he was asked to consider multiple logarithmic terms
simultaneously. He could comfortably use both the exponential form and its equivalent
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logarithmic form when a single term was involved, but when more than one term was
used, he struggled. As the activity continued, this problem became more apparent. For
example, when asked to evaluate the truthfulness of the statement
Tom failed to recognize that 4 times 9 produces a product of
36, thus rendering
a truthful expression. He was able to
recognize the right-hand side of the statement as being numerically equivalent to 2 but
could not decompose the 36 into an appropriate format. He wrote that
equivalent to
and
was equivalent to
was
but was unable to reconstruct
this knowledge in a meaningful way. His understanding was restricted to a fixed
procedure that worked only if the logarithmic term was decomposed in a manner that
produced the desired numeric value—an indication he was relying on rote-learned facts.
His understanding limited his ability to see the expression as a numerical object that was
available for further manipulation.
Harel and Kaput (1991) state, “The notation’s perceived connection with prior
knowledge takes the form of perceived features that reflect features of the prior
knowledge” (p. 92). Apparently, Tom had failed to make this connection, while the other
participants were able to build a new cognitive structure to accommodate this
information. Harel and Kaput (1991) also claim, “Students should be given opportunities
to build their own notational expressions of their ideas, which then can be guided in the
direction of the standard ones. In this way, one builds both notations and conceptions
simultaneously, rather than building one or the other first then attempting to connect the
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two” (p. 94). Participants were purposely not given any properties of logarithms. Rather,
they were given opportunities to construct a cohesive concept image of logarithmic
concepts prior to the formal presentation of the properties of logarithms. Table 13
summarizes the participants’ understandings at the completion of Task 5. While six tasks
were initially proposed, significant events that occurred during the teaching experiment
reduced the available time for participants to complete Task 6.
Table 13
Understanding of Logarithmic Concepts at the Completion of Task 5
________________________________________________________________________
Emergent process
Tom
Process
Emergent object
*
Doug
*
Jim
*
Earl
*
At the completion of this teaching experiment, the researcher noted that the
understanding of exponents and exponential and inverse functions is the critical first step
in developing a mathematically sound understanding of logarithms. Bayazit and Gray
(2004) showed that learners with a procedural understanding of inverse functions are
less likely to be successful in contexts where procedural rules are absent, as is the case
with exponential and logarithmic functions. Participants in the current study
demonstrated this: for them to understand the mathematics embedded in the symbol
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, they needed more than the loose definition “It counts the number of ratios of a
given base.”
Question 1b: What Are the Critical Events That Contribute to the Total Cognitive
Understanding of Logarithmic Concepts?
The instructional approach used in this study fostered the development of certain
mental constructions that were originally proposed by the genetic decomposition by deemphasizing the traditional lecture format in favor of a more exploratory, cooperative
learning environment. Using a sequence of instructional tasks designed intentionally to
evoke disequilibrium, the researcher used qualitative methods to identify the
constructions that participants appeared to have made, were making, or failed to make
during the teaching experiment. In addition to an object or emergent object conception of
exponential expressions, participants in this study were convinced that if they understood
the development of logarithms from a historical viewpoint, they would be more inclined
to “remember” the meaning of the expression
. The participants mentioned on
several occasions that if they knew the mathematics embedded in the notation, they
would be more likely to build relational understandings.
Historical development of logarithms. Students often admit that they have seen
logarithmic expressions and/or functions but they are not sure just what they are. Is this
because the traditional study of logarithms has been eliminated from the curriculum,
hence depriving our students of knowledge of the evolution of such ideas and concepts,
which would tend to lead to deeper understandings? Each participant in this study
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questioned the numeric procedure that was being used to find the logarithm when the
result did not produce an integer value. The consensus by participants was that if they
knew how to find a logarithm “by hand,” they would be more likely to develop a deeper
understanding for logarithms; otherwise, they felt the symbol
and its subsequent
value would lose its significance. Jim stated, “ If you know the ‘why,’ it’s easier to
remember the ‘hows’ . . . and when you get to that point where that extra step is
hindering you moving on, then, yeah, the calculator is great. You have to understand it
before you can use the calculator.”
Without the rich connections provided by a historical account, students may be
unable to grasp the utility of logarithms. Furthermore, when students do not fully
understand their own actions, it is unlikely that they will progress to an object conception.
This statement suggests the primacy of actions over objects. Repeatedly, the participants
asked about the table of logarithms. They expressed an interest in learning how the table
of logarithms was created. They were not necessarily interested in the why, but in the
how. They also were interested in the algorithm used to compile the table of logarithms.
Participants believed that if they were familiar with the procedure for finding a logarithm,
they would be more likely to remember its meaning.
APOS Theory is predicated on the idea that repeatable actions become
internalized to form processes; however, an action typically requires a definite recipe,
such as the steps in an algorithm, which students then follow by rote. This statement
implies that students “can carry out a transformation by reacting to external cues which
245
give precise details on what steps to take” (Asiala et al., 1996, p. 10). Is the common
practice of having students use the formal definition of a logarithm to complete exercises
that stress how to switch from logarithmic notation to exponential notation, or vice versa,
sufficient to develop an action understanding of logarithms? In essence, traditional
instruction at the college level is predicated on the notion that students are able to abstract
meaning from formal definitions. According to Dreyfus (1991), the ability to abstract
meaning from formal definitions does not work for the vast majority of students. This
approach may leave the most gifted with a compilation of disconnected mathematical
information; for the less gifted, the results may be disastrous. What they do learn is “to
carry out a large number of standardized procedures cast precisely in defined formalisms”
(Dreyfus, 1991, p.25); however, they have not gained any insight into the processes that
led mathematicians to create these standardized procedures. As participants in this study
completed tasks aimed at raising questions in their minds about how to reverse the
process of exponentiation using some type of algorithmic-driven procedure, they were
introduced to the formal definition of a logarithm. This definition provided the
participants with the perturbation required for them to consider that the construction of a
new mathematical concept was needed to make sense of this particular problem situation.
While all participants could accept the fact that the value of a logarithmic
expression is equivalent to the exponent in an exponential expression with the same base,
all agreed it was hard to accept this explanation without understanding the algorithmic
procedure embedded in the word log. This assertion was an indication that participants
246
were attempting to expand their cognitive structure without changing their current view
of the logarithm as an exponent. According to Tall (1991), this expansive generalization
is “necessary to be able to deal with a wider class of applications without having to go
through too much stressful cognitive change” (p. 12). In order to move beyond an action
conception of logarithms, participants needed to develop a stronger cognitive awareness
of the step-by-step numerical process implied in the symbol log. In other words, the
depth of understanding that could be derived from the definition was insufficient for the
individual to move to the next level of understanding.
During the course of this study, participants asked repeatedly to “see” the
algorithmic process that was being performed by the log button on their calculator. They
were not content with the response that the log feature of the calculator was keeping track
of the number of ratios of b in x, and they admitted that if they could see more of the
actual steps that were used to calculate the logarithm, they would be more likely to
remember what the symbol
represents. According to Tall (1991), the goal of
instruction should be to “present the student with contexts in which cognitive growth is
possible, leading ultimately to meaningful mathematical thinking in which formalism
plays an important part” (p. 18).
In response to this request to review the historical development of logarithms, the
researcher used instructional activities designed to strengthen the connection between
exponentials and logarithms using numeric sequences. The problem inherent in finding
the logarithm occurred when the desired output was not an integer value. Therefore, a
247
method for making the arithmetic and geometric sequences that represent the inputs and
outputs for an exponential function sufficiently “dense” between 0 and 1 on the
arithmetic scale, and 1 and b on the geometric scale, was developed. Using the laws of
exponents and basic algebra, participants were able to identify a pattern that quickly
emerged.
As we began to numerically develop quantities between 0 and 1 and 1and b for a
designated sequence, Tom was the first to recognize the pattern, stating, “If there is a
pattern between the 0 here and the
and a relationship over here between the 1 and
why doesn’t the same logic follow here, or does it, between the and
? Do you see
what I am getting at?” The instructor confirmed that this pattern does exist and can be
verified using algebraic procedures; the participants seemed satisfied with this response.
Based on the ensuing group discussion, it appeared as if students were able to look at
every object in the arithmetic sequence and apply a mental process to obtain
corresponding elements in the geometric sequence. This ability allowed the students to
reflect on the reversal process as they explained that the irrational numbers obtained in
the geometric sequence were simply the result of finding roots for the rational exponents
in the arithmetic sequences. (See Appendix C for class notes with complete details.)
This explanation suggests that students were attempting to coordinate this knowledge
with existing cognitive structures to extend their understanding of exponentials.
Next, a procedure known as non-integer factoring was introduced. Non-integer
factoring allows the student to produce a nonunique set of factors for any real number.
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For example, it was agreed that the number “3” could be expressed in the following
manner:
However, before
students could appreciate the next part of this discussion, the relationship
had to be developed.
Working with integer values and the numeric sequences developed earlier,
students were asked to verify whether
and, if this proved
to be an unsubstantiated equation, to suggest what could be done to correct the statement.
Activities were then conducted to encourage the participants to make the desired mental
constructions in order to successfully accommodate this information into their existing
knowledge structures. Students needed opportunities to reflect on the idea that the sum
of two logarithms is equivalent to the logarithm of the product of the arguments. If
students developed a generalized understanding of this relationship, they could appreciate
the development of a logarithm for numbers that did not produce rational solutions.
The premise that any number can be expressed as a product of non-integer factors
until the last factor is close enough to one is critical when one compares precalculator
answers with those obtained using technology. Using division involving irrational
numbers, 3 was rewritten as
(
dividing 3 by
Students were next presented with the statement
). It was unclear if the participants understood the significance of
, but they agreed that the above product was equivalent to 3, and as such
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could be rewritten as
( )
. They also seemed to be aware that
is equivalent to . This indicates that activities completed at an earlier date
allowed participants to extend the rules of exponents to logarithmic terms, or, at most,
enabled them to recognize that when the base of the logarithm was in agreement with the
argument, the resulting value was equivalent to the exponent—suggesting an action
conception.
When asked to evaluate the numeric value of
to determine if this quotient was
sufficiently close to one, all agreed that another “division” was needed. Using the values
created from our sequences, we divided
the previous expression. Next,
by
, obtaining a value closer to one than
was rewritten as
(
).
This process would continue until the first factor in this expression was “sufficiently”
close to 1. (See Appendix E for class notes on this process.) Noting that the precision of
the logarithmic value would improve if successive divisions were applied, Jim replied,
“It’s like a process that keeps reducing itself until you finally get either it[meaning the
division comes out even], or it just keeps going forever [meaning the division], or you get
1.” Participants agreed that this method was tedious; because it was an iterative process,
they also agreed that a good computer programmer could write code to accomplish the
same thing. Both Earl and Jim shared that they had been exposed to something similar in
250
an earlier computer-programing class. They both recalled first learning how to calculate
a square root without technology, then writing a program to accomplish this same task.
They indicated that completing this activity had given meaning to the radical symbol
itself and enabled them to work confidently with all types of roots.
With the number 3 written as a product of its factors, the logarithmic
property,
was used to simply the expression. Students found
that if they left the irrational numbers written in exponential form, the resulting
logarithmic expression was easier to evaluate. The instructor noted that while it has not
been explicitly documented, various historical accounts indicate John Napier and Henry
Briggs used a similar method to construct their table of logarithms. Mathematical purists
are referred to Appendix A in Dan Umbarger’s book Explaining Logarithms, retrieved
from www.mathmogarithms.com for further information on this topic. (See Appendix D
for a brief account of this procedure.)
In discussing the development of logarithmic tables, the instructor noted that
historically the first logarithmic tables were developed using powers of 10. To reinforce
the idea that logarithmic tables are used to find the “non-integer” portion of the exponent,
special names have been given to each part of a logarithm. Because each part plays a
special role in relation to the number that the logarithm represents, students must be
given ample practice in estimating logarithms. Estimating the value first develops an
understanding of the “characteristic” portion, or the integer value, of the logarithm.
Estimating forces students to reflect on the inverse relationship that exists between
251
exponents and logarithms. Once students are successful in estimating the value for a
given logarithm, the tabulated values are then used to approximate the decimal portion of
the logarithm. With the characteristic portion determined by inspection, the mantissa, or
decimal-approximation portion of the logarithm, is read from the table.
Traditional instruction on logarithms has been condensed into a series of rules and
definitions. Without the rich mental images needed to move to a higher level of
conception, many students will not achieve the desired level of understanding needed to
be successful in higher mathematics. For students to possess a process level conception,
they must be able to interiorize their actions. When instruction is based on formal
definitions and symbol manipulation without true understanding, students may
experience temporary success, but are unlikely to have interiorized their actions. David
Tall (1992) expresses this in this way: “Algebraic symbolism violates many individuals’
innate understanding of mathematical symbolism which in arithmetic tells them what to
do and signals how to do it” (p. 6). This situation of impoverished understanding of what
symbols signify in algebra is further exacerbated as the student begins to study
transcendental quantities. For the less able (mathematically), this lack of understanding
means that student success is measured by the ability to produce the correct answer. In
other words, the consequence of traditional instruction that permeates mathematics
instruction at the college level is reliance on instrumental procedures at the expense of
sense-making.
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Revised genetic decomposition. Initially, the researcher proposed a model of
cognition of logarithms that seemed to her to be the most accurate as well as most helpful
to the students. This is referred to as the genetic decomposition for the concept of
logarithms. Analysis of the data from the teaching experiment indicated, however, that
the genetic decomposition originally proposed did not give study participants enough
opportunities to construct a well-developed action conception of logarithmic concepts.
Repeatedly, participants suggested that if they could “see” the algorithm that was being
used to find the numeric value of a single logarithmic term, their understanding would
deepen. The researcher originally thought that working with exponential functions alone
would be sufficient to develop an action conception for logarithms. Furthermore, the
researcher felt that if students were asked to consider how to reverse the process of
exponentiation and afterward it was determined that “undoing” of exponentiation was
not exactly the same thing as finding a root, the formal definition would provide students
with enough understanding to construct an internal process. Davis (1984) pointed out
that “When a procedure is first being learned, one experiences it one step at a time, but as
it is practiced the procedure itself becomes an entity and . . . its similarities to some other
procedure can be noted” (pp. 29-30). Having students struggle with the idea of creating
an inverse algorithm for exponentiation, and then telling them the formal definition, did
not provide the step-by-step procedure they needed in order to recognize the symbol log
as a process for undoing exponentiation.
253
All participants understood that division was somehow linked to the operation of
undoing exponents; however, a verbal account of the mathematics used by Napier and
Briggs to create logarithmic tables was insufficient for creating a concept image for a
logarithm. Participants may have been searching for mental structures that could be
compressed into a thinkable concept rather than a memorized definition. According to
Tall, Thomas, Davis, Gray, and Simpson (2000), “An action becomes a process when the
individual can describe or reflect upon all the steps in the transformation without
necessarily performing them” (p. 3). Current instructional practices for logarithmic
concepts assume that students form a rich concept image for the logarithm once they are
given its definition, which then allows them to interiorize their actions. While many
students can successfully complete a unit of logarithmic concepts (Kastberg, 2002), the
depth of understanding is limited to thinking exclusively about logarithms in a procedural
context.
In response to the participants’ concerns that if they could see the procedure that
was being used to find logarithms, they would better understand the concept, the
researcher revised the original genetic decomposition and added a historical unit to the
plan of instruction. Participants said that their concerns in this case were similar to those
they had in learning root extraction. Three of the four participants had had experience
with finding square roots without using a calculator; two of the three had written a
computer program to carry out this calculation. Participants explained that these
254
experiences had promoted a deeper understanding for root extraction and led to an
appreciation of how technology can and should be used to replace tedious calculations.
The data also suggested that understanding the transition from exponent as a
counter to exponent as a real number was problematic. Although participants were able
to create graphs for exponential and logarithmic functions and to describe the appropriate
domain and range for both, they questioned what
agreed on the existence of representations such as
looked like. Participants all
and knew that the
representation meant two full factors of three. However, they were unsure of how to
represent the fractional portion of the exponent, an indication that they were attempting to
form a coherent concept image for exponentiation. Although it was relatively easy to
explain terminating decimals as exponents because they could be represented as roots,
irrational numbers as exponents proved more difficult for the participants to understand.
Reluctant to use the graph of ( )
to evaluate (√ ) participants explained that it
was not necessary to use the graph because a calculator could calculate the result. This
reluctance to use the graph to estimate could be an indication that the fundamental shift in
meaning of the exponent does not occur by chance: it must be cultivated. If these new
ideas are not satisfactorily accommodated in the mind of an individual, cognitive
conflicts arise when he or she attempts to use this knowledge to evaluate logarithmic
expressions and or equations. Because logarithmic concepts are integrally linked to
exponential concepts, instructional programs should include in-depth analysis of
255
individual problems or functions rather than attempt to master a class of exercises that
focuses on symbolic manipulation with only trivial distinctions between the problems.
The implementation of the proposed instructional plan did deviate somewhat from
the original design. As the data suggested, a revision was made to the preliminary
genetic decomposition in an attempt to promote a deeper understanding of logarithmic
concepts. Figure 6 illustrates the revisions made to the proposed genetic decomposition,
which was described in detail in Chapter 3. Based on this revised genetic decomposition
and the researcher’s new understanding of what it means to learn this concept, future
instructional treatment presented in this study should reflect these revisions and the entire
process should be repeated. According to Dubinsky (1994), “The iterations continue as
long as desired to hopefully converge on a better understanding of the student’s
constructions of this particular topic and how instruction can help him or her make that
construction” (p. 234).
Cautions with Interpretations
Research has suggested that many students fail to develop a process conception of
function. For most students, this means that they need a formula that they can use to
calculate an answer (Breidenbach, et al., 1992; Dubinsky, 1994; Dubinsky & Harel,
1992). Dubinsky (1994) argues that a process conception for functions is not achieved
until the learner is able to reflect on his or her actions. He claims, “When the action of a
function can be considered without an explicit algorithm and when the totality of this
action can be thought about, reversed and composed with others, it is considered that it
256
exponent as a
counter
laws of
exponents
exponent as a real
number
exponential
functions and their
properties
The interiorized process described above is then encapsulated to form a single object,
which then becomes the object of further action
evaluating single
logarithmic
expressions
introduction of
to
symbolize
inverse
formal definition
of logarithm as
the inverse
function
Figure 6. Revised genetic decomposition.
historical
development of
logarithmic tables
CONSTRUCTS
CONTRIBUTING
TO THE
ENCAPSULATION
development of
the properties of
logarithms
257
has been interiorized to a process” (p. 238). Although functions are not the focus of this
study, students’ weak understanding of functions may have contributed to difficulties
they encountered as they attempted to make sense of the notation
. As the focus of
this study was more on the operational features of this notation, understanding of the
function concept itself was not investigated.
Participants appeared to struggle with irrational numbers in general, and more
specifically, irrational exponents that are not algebraic. This fact appears to be a critical
link in the de-encapsulation process. The data suggested that Jim, Earl, and Doug were
able to consider alternative strategies to evaluate logarithmic expressions that did not
produce rational results. However, not enough information was available to ascertain
whether their understanding of irrational numbers that are not algebraic was any different
from Tom’s. What sets these three participants apart from the fourth participant is that
they were able to consider alternative representations in order to move forward in APOS
levels. Tom, on the other hand, unable to form any meaningful connections between
laws of exponents and properties of logarithms, had difficulty coordinating his
understanding of exponentials and logarithms. He was not able to consider alternate
solution methods for logarithmic equations that contained more than one logarithmic
term, indicating that he had a limited understanding of logarithmic concepts, or, more
specifically, of a logarithm as a transcendental number.
258
Implications for Instruction
Previous research has suggested that students with limited understanding of
exponential expressions are unable to move beyond an action conception for logarithms
(Berezovski & Zazkis, 2006; Chesler, 2006; Kenney, 2005; Weber, 2000a, 2000b). The
cognitive transformation needed to move beyond the exponent as a counter has to be
developed. Students are expected to make a seamless shift in the fundamental meaning
of the exponent. This “isomorphic relationship between the exponents and the
exponential expression” (Confrey, 1991, p. 125) becomes the basis for meaning;
therefore, it should be cultivated. By examining the domain for various exponential
functions and developing meaning for exponents as rational and irrational values,
students have a greater chance to accommodate these new ideas.
In this study, it became apparent that students had a hard time abandoning the
intuitive appeal of the initial meaning of exponent when all of the participants failed to
recognize that the graph of the exponential function could be used to approximate the
value of (√ )
√
. Participants questioned the need to use a graph to estimate this
value since a calculator would produce the “correct” value. One of the participants took a
deeper look at the meaning embedded in the notation when he asked how to represent
such an expression. He explained he knew what
visualize something like
√
or
looked like, but could not
. Since many logarithmic values are irrational by nature, it
seems that we need to pay attention to this fundamental shift in the meaning of the
exponent in order to help students make sense of the computational power of logarithms.
259
Before the introduction of any formal instruction on logarithms, instruction
should focus on developing the need to reverse the process of exponentiation and
entertain ideas of how to “undo” this process. The formal definition is then not an
isolated concept. Cognitive growth is then possible; the logarithm is conceptualized as a
number, and hence perceived as a mathematical object. Constructivist philosophy
maintains that learning experiences occur as students actively construct or reconstruct
new schemas. According to Dubinsky (1991), learning is a spiral process “in the sense
that objects are used to construct processes which are then used to construct new objects
from which new processes are formed and so on” (p. 167). As students attempt to
operationalize the notation
, they are aware of some type of process that is “going
on behind the scenes,” yet they may be unable to form a coherent mental image of this
process, thus hindering their understanding of the additive and multiplicative
relationships embedded in logarithmic concepts. Tall (1991) posited that in order to
introduce students to a wider vision of the nature of mathematical thinking, we need to
help them experience some of the struggles that countless mathematicians have
experienced. Presenting information in a neat, polished format perpetuates the notion
that mathematics is not a creative activity.
Textbooks reduce the teaching of logarithms to the somewhat tenuous statement
that refers to a logarithm as an exponent. For most students, the definition is useless
because it reveals no underlying operational characteristics; it is simply mathematical
notation. Using the historical development of logarithms as an epistemological tool can
260
reveal the mathematical thought and operational characteristics of logarithms. We make
an improved conceptual representation possible by presenting the information in a
manner similar to Napier’s original definition. According to Tall (1988), “It is eminently
possible for students to be taught to respond correctly to questions involving the formal
definition” (p. 38); however, when their concept image is at variance with the actual
definition, learning is seriously impeded.
Once a student understands the implied question embedded in the notation
in order to make the transition from an action to a process conception he or she must be
able to do the following:
Move mental objects around, call them into awareness, combine them, compare,
and ignore them . . . all in her or his mind. As a result of this awareness of the
total process, the individual can reflect on the process itself, combine it with other
processes, reverse it and reason about it. (Dubinsky, 1991, p.167).
With a heightened understanding of logarithms, combined with an object conception of
exponential expressions, students are more likely to conceptualize the interpretation of
the verbalization “the sum of the exponents is the product of the powers” as it relates to
logarithmic concepts.
Implications and Recommendations for Future Research
One goal of this investigation was to increase understanding of how students
acquire mathematical knowledge about logarithmic concepts, using APOS Theory to
guide pedagogy, which, according to Asiala et al. (1996), “develops a base of information
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and assessment techniques which shed light on the epistemology and pedagogy
associated with particular concepts” (p. 3). The data gathered in this study warrants the
continued revision of the epistemology of the concept of logarithms. Of particular
interest to participants in this study was the historical account of the development of
logarithms and the use of logarithmic tables. After a brief look at the historical
development, Tom responded, “Oh, I see what you are getting at now, because before,
without really understanding the logs, I couldn’t get that [referring to the decimal
approximation]. I couldn’t even get that concept in my head, but now we can.” Jim then
added, “If you know the whys it’s easier to remember the hows; you just need to know
the logic of it; then you don’t need the rules anymore.” Tom further added, “What was
the thinking that got us to this point? I got no clue. It’s just button pushing, but now it
seems we are trying to regain that consciousness between those two [historical account
and current technologies]. Otherwise, where is the knowledge base?”
The above statements suggest that the introduction of the historical developmental
of logarithms may help students develop their understanding of the concept. The
emphasis should focus on the development of the idea, from Napier’s original work to the
use of logarithmic tables developed by Briggs, and move away from rote memorization
of a series of processes and properties involved with logarithmic functions. Kenney
(2005) observes “The presentation of single logarithmic forms evoked the procedural
response of rewriting the problem in exponential form; however, the addition of a second
log form to the equation no longer prompted students to anticipate a change to
262
exponential form” (p. 7). Her findings indicate that the students might not have been able
to develop any understanding for implied meaning imbedded in the symbolic notation
itself.
In documenting how students at the secondary level learn mathematics, APOS
Theory offers promise for continued research efforts and curriculum development. With
an emphasis on cooperative learning, instructors can interact with and observe students’
success in making the mental constructions proposed by the theory. This approach seems
to generate student enthusiasm, as it moves away from traditional instructional models.
By attempting to understand learning as experienced by the learner, classroom teachers
can become better equipped to recognize and address the difficulties their students face.
Although it is impossible for one individual to know exactly what is going on in
the mind of another, as research similar to this study progresses through iterative cycles,
we need to continue to do qualitative analysis of the cognitive structures students appear
to be making in the process of learning until our understanding begins to converge on
central themes. According to Asiala and others (1996), “Revisions including major
changes in, or even rejection of, a particular genetic decomposition can result from the
process of repeating the theoretical analyses based on continually renewed sets of data”
(p. 31). However, this researcher hopes that successful results over a period of time will
lend credibility to the revised theoretical analysis offered above.
“Without the experiences afforded by teaching, there would be no basis for
coming to understand the powerful mathematical concepts and operations students
263
construct or even for suspecting that these concepts and operations may be distinctly
different from those of researchers” (Steffe & Thompson, 2000, p. 267). Furthermore,
we can compare quantitative analysis of achievement outcomes for students using an
instructional plan aligned with APOS Theory with traditional instructional plans in order
to validate the effectiveness of this tool in enhancing student learning of collegiate
mathematics.
Concluding Remarks and Chapter Summary
Given the esoteric nature of logarithms, it seems clear that we need to devise
different instructional programs in an attempt to alleviate students’ misconceptions and
the belief that mathematics is a rigid system of polished formalism. Today’s curriculum
presents logarithms as a simple exponent relationship; however, the topic of logarithms is
more complex than this, and it has a long and rich history of work and improvements. Its
complexity poses a significant cognitive obstacle for the learner and a challenge for the
educator. When learning about logarithms, students are exposed to a new symbol system
unlike any they have previously seen. They struggle to see how this new information can
“fit” into their existing cognition. The operant is the new word log, with a subscripted
notation that is implied or explicitly stated, and somehow this mathematical
representation is supposed to connect to exponential functions. Traditional classroom
instruction about logarithms supplies students with the notation used to represent an
inverse relationship between exponentials and logarithmic functions, but typically does
not attend to developing appropriate mental referents (Kenzel, 1999; Ursisi & Trigueros,
264
1997, 2004). Without knowing what steps to take next when confronted with the
symbolic notation other than what is provided by the definition, students will likely have
difficulty internalizing this structure. In this study, at least one of the four participants
was unable to make sense of logarithms beyond the notation when confronted with
multiple terms. Explicitly attending to this shift of attention has the potential to support
the development of symbol sense (Bills, 2001; Kenzel, 1999).
Results indicate that use of information on the historical development of
logarithmic concepts can support the understanding of abstract mathematical
relationships. The participants in this study voiced their concern over their lack of
understanding of the numerical algorithm imbedded in the symbol
. If exposed to
the historical processes used by Napier and Briggs, they were more likely to connect its
meaning to exponential concepts. Although the historical development of logarithms was
not included as part of the original instructional plan, it was clear that a definition of
logarithms by itself did not provide this group of students sufficient mental structures to
build a sophisticated understanding. Using arithmetic and geometric sequences as a
springboard for the development of logarithms for irrational values, the students work
simultaneously with properties of exponents and logarithms, thus providing them
alternate ways to conceptualize logarithmic concepts before the specific formal
instruction.
“The overwhelming evidence is that the majority of university students have great
difficulty coping even with elementary (but non-standard) tasks in the advanced
265
mathematical environment” (Mamona-Downs & Downs, 2002, p. 171). Unable to supply
any type of sensible justification for the actions they have taken, students may be able to
get the right answers quickly; but they will not have formed any relational understanding.
In instruction at increasingly abstract levels, as with equations involving more than one
logarithmic term, the more robust and challenging the instructional activities the students
are offered, the more likely they will be able to rely on understandings developed through
these challenges rather than to retreat to their own previous pseudostructural knowledge.
Instructors must create activities that will challenge the existing mental
structures of learners and force them to struggle for meaning; otherwise, the students will
continue with their piecemeal representations of logarithms. Students are often capable
of following routine procedures. But if we hope to increase their participation in higher
mathematics, teachers at the tertiary level must challenge them to develop deeper
understanding of the underlying principles that allow those procedures to work.
APPENDICES
APPENDIX A
INSTRUCTIONAL TASKS
Appendix A
Instructional Tasks
Pretest/Initial Assessment
Part A: A whole number exponent is simply shorthand for repeated multiplication of a
number times itself. For example
. This is the only conceptual
knowledge required. So in general then for any counting number “n”
, which means the product of multiplying together “n” factors of “b”. We call
“b” the base and “n” the exponent.
Part A
1.
2.
3.
4.
5.
6.
(
)
Can you make any conjectures based on your work? Elaborate on your ideas.
268
269
Part B: Next consider the following, using only the information provided in part
A. Simplify these expressions. Justify your work!
1.
2.
How did you think about this? How is this different from problem 1?
3.
4.
How is this different from the problem above? How is this the same?
5.
State all conjectures and/or generalizations you can see based on your solutions
generated for this group. Please elaborate on each.
Now use your generalizations to answer the following:
Part C: Using your generalization or conjecture formed in part B, simplify the
following expressions.
1.
2.
3.
4.
5.
State all conjectures and/or generalizations you can see based on your solutions
generated for this group. Please elaborate on each.
270
Did you have any negative exponents in your results? If so, how did you interpret their
meaning? Please elaborate.
Use your generalizations to interpret
Now write a problem that has as its result
Part D: Can you use your generalizations to simplify these expressions? Show
your work!
1.
2.
3.
4.
5.
6.
*
∙
7.
8.
State all conjectures and/or generalizations you can see based on your
solutions generated for this group. Please elaborate on each.
Summarize what you have just learned by completing these exercises.
271
Part E
1. Evaluate both f(x) and h(x) for the following values of x:
a.
b.
( )
2. Suppose you invest $1000 at 7% interest compounded annually. How much will be in
the account at the end of the first year? Second year? Third year? Can you write the
equation using functional notation that represents the amount A at the end of the nth
year?
Please elaborate on the method you used to calculate how much is in the account after
the first, second, and third years, and how you arrived at your representation for the
amount A after n years.
272
TASK 1
Part A: Objective: Students will graph exponential functions to extend the domain of
the input or independent variable, which in this case is the exponent, to include all real
numbers.
1. Graph the following function by developing a table of values for discrete values of x.
2. From the graph approximate F (√ ). Do you believe your value for F (√ ) is exact?
Why or why not? Does this say anything about values x can be?
273
3. Complete the activity, The Function
Assessment 1: Respond to the following question, “How do you identify characteristics
of the graph of an exponential function?”
274
Part B
4. Together as a group use the graph of y = 10x to answer a series of questions:
a. Can either the input or the output be unknown? Why or why not? Based on your
response can you use the graph of
to answer the following questions:
b.
c.
d.
e.
5. Can you estimate the value of x for each of the following problems? Explain your
response.
a. 3x = 5
b. 2x = 32
c. 7 x = 54
d. 10x = .025
6. How would you solve these same equations and what might the symbolic notation
look like?
7. What does it mean to reverse the process of exponentiation? Why would you need
to be able to do this? What might the symbolic notation look?
275
TASK 2
Preliminary activity
1. Given the function
what is its inverse?
2. Given the function
– , what is its inverse?
3. What is the inverse function of
4. Given the function
what is its inverse?
5. What is the inverse function of
6. What is the role of the inverse function?
7. What is the inverse function of
276
Part A
Using the TI-nspire, students will graph the following pairs of functions. Each pair
will be graphed on the same set of axes.
1. A table of values is calculated for each with instructor-defined inputs for both
functions. Similarities between the two tables and their associated graphs will be
discussed. Students need to be able to verbalize the relationship in terms of inputs
and outputs.
a.
and
Exponential function
Inputs “x”
Outputs “y”
Logarithmic function
Inputs “x”
Outputs “y”
-4
-3
-2
-1
0
1
2
3
4
1
3
9
27
81
277
b.
and
Exponential function
Inputs “x”
Outputs “y”
Logarithmic function
Inputs “x”
Outputs “y”
-4
-3
-2
-1
0
1
2
3
4
1
2
4
8
16
2. Generalize your findings with regard to the domain and range for both sets of
functions and state any similarities between the two pairs of functions that you
noticed.
278
Part B
3. Exploring the Log button on your calculator. Using the TI-nspire, students will
complete the tables illustrated below.
0
10 =
1
10 =
2
10 =
3
10 =
4
10 =
0
4 =
1
4 =
2
4 =
3
4 =
4
4 =
Log 101 =
Log10 10 =
Log 10100 =
Log10 1000 =
Log10 10000 =
Log4 1 =
Log4 4=
Log4 16 =
Log4 64 =
Log4 256=
279
0
-1
-2
-3
-4
=
=
=
=
=
log 1 =
log
=
log 4 =
log 8 =
log 16 =
Summarize your thoughts on what you think the “log” button on the calculator does.
4. Have students summarize their thoughts on “What are logarithms?”
Assessment 2: Conjecture the value of log480 without using the calculator. What is
the implied question in this expression?
280
TASK 3
Part A
1. Practice rewriting exponential equations as log equations and log equations as
exponential equations using the definition
if and only if
a. Rewrite these equations as logarithmic equations using the given definition.
103 = 1000
24 = 16
54 = 625
32 = 9
210 = 1024
42 = 16
=2
ec = y
√
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b. Convert each of the following facts to exponential form using the given definition
1) log10100000 = 5
2) log4 64 = 3
3) log2 32 = 5
4) log
5) log
6) logx y = z
7) log √
282
Part B
1. Evaluate the following logarithmic expressions WITHOUT A CALCULATOR,
justify your response:
1) log2 2
2) log5 1
3) log9 3
4) log121 11
5) log3 81
6) log4 .25
7) log.5 4
8) loga (a5)
9) log4
10) log4(-16)
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Part C
3. Place the correct symbol (> < =) between the terms. Do not use a calculator.
Justify your response.
log
log
log
log
log
log
log
log
log
log
log
log
log
log
log
l
log
Assessment 3: (Individual work) Write a short paragraph (5 -10 sentences) that
describes your thinking about the symbol “logb” as you completed each portion task.
Does it have any meaning, if so can you describe it?
284
TASK 4
Part A
1. Knowing what you now know about logs, can you solve the following
exponential equations? As best as you can, explain what you did and why.
a. a. 10x = 600
b. 10x = 400
c. 2x = 150
d. 3x  20
2. Suppose you invested $1000 at 7% interest rate compounded annually. How
many years would it take your investment to double? So in other words when
does
? Can you think of a way to solve this?
Part B
a. Explain why the log2 14 is between 3 and 4.
b. Is the function
log
and increasing or decreasing function? What about
Explain your response.
285
c. Can you explain how to find log
d. Given that log
without using a calculator or graph?
find log
e. Evaluate log
f. Solve for x, explain your method
g. l
log
log
h. log
log
i. Find 3 different ways to express the value of 4 using logarithms for example,
log
286
Assessment 4: Given the following equations, explain your solution method.
a.
b.
c.
d.
e.
f.
287
TASK 5
Laws of Logs
Part A
1. Does the log
log
log
2. What about this one log
? Why or why not?
log
log
, is this a true statement?
3. If you believe the above statements are false, how can you simply the
log
log
, still maintaining a “logarithmic” expression in the final result?
The first 9 powers of 4 (this might help you out)
Sequence 1 0 1
2
3
4
5
6
7
8
9
Sequence 2 1 4 16 64 256 1024 4096 16384 65536 262144
a.
What about this one log
expression in the final result
log
, still maintaining a “logarithmic”
b. Can you see any possible pattern that might help you? Write down
any thoughts on this idea.
4. OK let’s try a few more before we attempt to generalize the results.
True or false, explain why you answered as you did.
a. log
log
b. log
log
log
c. log
log
d. log
log
log
288
The Multiplication Law
Investigate the multiplication law by completing the table below.
x
y
xy log
log
log
log
log
10
1000
100
10000
10
100000
log
1000 1000000
Any relationships? Please describe any and all that you see.
Proving the Multiplication Law (this is questionable whether or not this information
is supplied, proof is still not complete)
Proving the Multiplication Law:
Rewriting the right hand side using powers
Multiply x and y then rewrite as a logarithmic equation;
log
log
log
log
289
TASK 6
Part A
Consider the following table of values
n
1
2
3
4
5
6
7
8
9
Log n n Log n
0
10 1
20 1.301
.477
30
.699
.845
.954
a. Using what you know about properties of logs, complete the table without using
a calculator.
b. Find
c. Find
d. Find
using the table. Explain how you calculated this value
using the table. Explain how you calculated this value.
using the table. Explain how you calculated this value.
290
Part B
Consider the following table of values
n
1
2
3
4
5
6
7
8
9
Log 3n n Log 3n
0
10
.631
11
12
13
1.465 14
15
16
17
18
a. Complete the table. Can we use other representations to help fill the table?
If so, can describe what this would look like?
b. Using the alternate representations described above, complete the table
without using a calculator.
c. What other information is needed to complete this table? Can you elaborate on
this?
APPENDIX B
MATHEMATICAL BELIEFS AND ATTITUDES SURVEY
Appendix B
Mathematical Beliefs and Attitudes Survey (Yackel, 1984)
1.
Doing mathematics consists mainly of using rules.
2.
Learning mathematics mainly involves memorizing procedures and formulas
3.
Mathematics involves relating many different ideas.
4.
Getting the right answer is the most important part of mathematics.
5.
In mathematics it is impossible to do a problem unless you have first been taught
how to do one like it.
6.
One reason learning mathematics is so much work is that you have to
learn a different method for each new class of problems.
7.
Getting good grades in mathematics is more of motivation than is the satisfaction
of learning the mathematics content.
8.
When I learn something new in mathematics I continue exploring and
developing it on my own.
9.
I usually try to understand the reasoning behind the rules I use in mathematics.
10.
Being able to successfully use a rule or formulas in mathematics is more
important to me than understanding why or how it works.
11.
A common difficulty with taking quizzes and exams in mathematics is that if
you forget the relevant formulas and rules you are lost.
12.
It is difficult to talk about mathematical ideas because all you can really do
is explain how to do specific problems.
13.
Solving mathematics problems frequently involves exploration.
14.
Most mathematics problems are best solved by deciding on the type of problem
and then using a previously learned solution for that type of problem.
292
293
15.
I forget most of the mathematics I learn in a course soon after the course is over.
16.
Mathematics consists of unrelated topics.
17.
Mathematics is a rigid, uncreative subject.
18.
In mathematics there is always a rule to follow.
19.
I get frustrated if I don’t understand what I am studying in mathematics.
20.
The most important part of mathematics is computation.
294
295
296
297
298
299
300
301
APPENDIX C
CLASS NOTES:
USING SEQUENCES TO DEVELOP LOGARITHMIC CONCEPTS
Appendix C
Class Notes: Using Sequences to Develop Logarithmic Concepts
(Adapted from Anderson et al., 2004)
303
APPENDIX D
HOW DID BRIGGS CONSTRUCT HIS TABLE OF COMMON LOGS?
Appendix D
How did Briggs Construct his Table of Common Logs?
305
306
****etc. until the last factor is close enough to1 to give the desired accuracy*****
Finally 5 = 3.16227766
X
1.333521432 X 1.154781985 X 1.018151722 X 1.008457304
take the log of both sides, iff Log Rule
Step 6f:
log (5) =log (3.16227766 x 1.333521432 x l.l54781985 x 1.018151722 x 1.008457304)
307
APPENDIX E
CLASS NOTES: NON-INTEGER FACTORING
Appendix E
Class Notes: Non-Integer Factoring
(Adapted from Umbarger, 2006)
309
310
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