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

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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

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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.

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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 APOS Theory 21 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

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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 vi

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

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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

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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

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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

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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:

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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 , b is the logarithm of the power . (Euler, 1810, p.

91)

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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.

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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?

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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)

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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.

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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

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(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).

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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.

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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.

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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).

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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.

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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 researcher- developed 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.

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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.

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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.

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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” (Mamona-

Downs & 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).

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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,

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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 sense- making 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).

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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

1 gets infinitely large, what happens to the value of ? 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

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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

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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 second- order 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-as- given 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.

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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).

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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)

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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: pre- function, 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.

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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.

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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, if and only 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, .

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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:

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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

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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.

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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 and only if . 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 if and only if

? Since no procedure exits for solving 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

63 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

64 envision the result of processes as permanent entities” (p. 194) as reification.

Specifically, reification implies that the student “understands as the number that is the product of x factors of b and 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

67 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 and uses this information to find the value of 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

68 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,

70 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

71 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

73 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

74 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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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

77 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

78 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

80 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

81 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

83 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

84 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

86 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

87 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

88 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

89 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

90 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

91 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

92 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

93 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

OBJECTS PROCESSES 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

MathematicalOBJECTS 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

97 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

98 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 whole- number 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

99 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

100 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 if and only 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

Instructions/Activities/Tasks Observable Skills/ Students’ Understandings as Evidence of Outlined in APOS Theory Understanding Basic operations with whole number Explain why their Action level of understanding of exponents solutions make sense exponentiation Pretest, Part A, B

Extend this to include all real numbers as Explain why the rules Still at an action level, but students exponents work are beginning to generalize to a Pretest, Part C, D larger set of inputs for the exponent

Using the graph of we will fill in Correctly complete Process level understanding begins to the “holes” by including integer values, worksheet and make develop when the student can rational and irrational numbers in the the table, draw the generalize about overall domain, move to generalizations about graph for each characteristics of the function the function ( ) function on the same without actual values for the base and Task 1, Part A set of axes the exponent

Understand the limitations of the set of Response to Still at a process level of inputs/outputs for the function, questions posed by understanding; however, when asked differentiate between parameters and researcher to explore further, they begin to variables move to next level of understanding Task 1, Part B

Explain what it means to reverse They can use a graph Object level understanding of exponentiation, solicit conversations to find the value of exponents and exponential functions about how to accomplish this task the exponent Task 1, Part B

Ask students to describe the role of the Looking for Trying to move to a process level parameter and the variables, do they something like: understanding of the notation even recognize the difference? means x is the associated with logs Task 2, Part B product of how many factors of a

Subjects will be asked to complete Complete a table of Action level; however, subjects several tables using exponents and logs values, verbalize any should be attempting to make Task 2, Part B connections between connections between exponents and exponents and logs current work with logs

Introduce the formal definition of the Correctly complete Action level, subjects have a rule to logarithmic notation, ask subjects to the exercises follow to complete the activity perform several transformations from logs to exponents and the reverse Task 3, Part A

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Table 1 Continued

Subjects should evaluate the Correctly complete Process level understanding: the expression without the use of a the exercises format has changed and technology is calculator. Students are being prohibited; subjects are no longer asked to coordinate what they looking at an equation and just figuring know about logarithmic out how to rearrange the “pieces.” notation and exponents They need to understand what each Task 3, Part B,C “symbol” in the expression represents

Students are asked to solve Correctly complete By guessing answers, students exponential equations the exercises demonstrate they have not interiorized Task 4, Part A logarithmic rules, operating at action level. If they verify responses using logarithmic notation, classify as process level understanding

Students are being asked to use I would consider this to be process the two ideas in combination level understanding if explanations with each other, flexibly contain ideas that demonstrate ability moving from one form to the to move freely between representations other to make sense of notation Task 4, Part B

Development of logarithmic Complete the tables Action level if only able to complete rules and offer some type table; process level if a proof is Task 5, Part A of generalizations formulated

Have students complete an Can answer a series Process level if justification includes activity demonstrating of true-or-false comments like “You add exponents understanding of how to apply questions and justify when you multiply with the same base logarithmic rules responses value” Task 5, Part B

Complete the table without a Can complete Process level understanding if they can calculator, answer questions without the apply previously learned material to using their table assistance of complete exercises Task 6, Part A technology or a rule

Complete a similar table, less Can answer the Process level understanding moving information given, looking for questions posed toward object level—students see that more conceptual understanding about the given table when asked to evaluate a log of a of when these laws of logs will prime number they can only give an not apply approximate answer without Task 6, Part B 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

107 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

108 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 ± Agreement ± 1

Tom 6 of 9 items 1 of 9 items 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 1 of 11 items Jim 6 of 11 items 3 of 11 items 2 of 11 items Earl 7 of 11 items 3 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

111 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

112 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 pre- object 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

113 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

114 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

115 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

116 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

117 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

118 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

120 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)

123 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 real- number 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

125 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 5 8 7 10 2 (Questions 1, 2, 5, 6, 10, 11, 14, 18)

Mathematics promotes deeper understandings 0 0 4 4 8 (Questions 3, 8, 9, 13)

Mathematics is about right answers 7 15 2 7 1 (Questions 4, 7, 12, 15, 16, 17, 19, 20)

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 but added, “This is just a guess.” Tom explained for this

problem, , “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,

128 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 as its result.

Earl responded by writing . Tom and Doug did something similar, while Jim

responded with . 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

130 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 ( ) where the value of b can be any positive real value except 1.

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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

134 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 ( ) ( ) for

he said, “So y equals one half raised to the x is the same as 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 ( ) is

1 over 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

137 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 below .” These statements indicated he could think about what was going to happen 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

138 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

140 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?

E: We know this: √ , so could we do something similar?

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 root? This one, when

rewritten as √ comes out “even,” so if the 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 ( ) with y and solved the equation for x. Next they replaced x with ( ) and replaced y with x. When asked to explain why this works, Earl responded with an example.

E: So if for ( ) and , we want to find out how to get back to

the beginning of .

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 change- of-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 ( ) and

its inverse function ( ) . 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

145 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

146 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 instructor- designed 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 and the exponent a are the same.” When asked to

clarify, he wrote, “ 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

147 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

148

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-to-

one 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 . As he started Activity 3A-

1b he wrote, “ ” and initially wrote “ ” 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 ( ). Three of the four subjects correctly wrote 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

156 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.

157

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 ] Does this have an inverse?

E: 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?

158

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 1 2 3

S2 1 t 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.

3 I: So let’s call the value on the S2 line v, so v 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,

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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!

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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.

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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

3 4 explain why is between 3 and 4, Doug volunteered that since 2 is 8 and 2 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: , Doug

immediately commented that logs could not be negatives.

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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.]

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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 ?

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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

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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 log As they worked, it was noticed that Earl had written “No because the log 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 crossed out log and wrote above it log ( ). When asked to respond to a similar task, “Does log log log ” Jim wrote, “No it should be equal to this; log ( ) log ” 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 ( )) is 36, but I thought we said that log ( )

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 log log log log log log log representation S1 0 1 2 3 4 5 6 S2 1 2 4 8 16 32 64 Alternative 22 23 24 25 26 representation

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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.

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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

itself.” Next, they were asked if log 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 Is that equal to 4?

E: Times log .

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.

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I: OK, well, from this expression, 12 log , this exponent of 12 is multiplying

this exponent, log .

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.

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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 APOS level of understanding insightful

Pretest Simplify exponential “Other than the rule I really Tom was able to use the CDE; expressions don’t see it. I just know how however, he determined that the involving integer to use the rule.” Tom was laws of exponents simplified the exponents. familiar with the laws of amount of “work” he needed to do. exponents. He generally did He could not offer plausible not use the common definition reasons for their validity. Process of exponents (CDE) to level understanding: he interiorized simplify the given expression; the action but was not able to rather he was relying on encapsulate parts of this concept. memorized rules.

Task 1 Graph the When asked to comment on Tom reflected on the physical exponential function the domain of this function graph itself and was able to ( ) to extend Tom indicated [pointing to his determine that the smooth graph of the meaning of graph] “There it is, all real this function indicated that any real exponents to include numbers.” number could be used as an input, all rational values. but was reluctant to use the graph Using the graph, When asked to use the graph to estimate the output, perhaps due participants were to find (√ ) after some to a lack of deep understanding of asked to approximate prompting, Tom suggested the decimal exponents. Still at a the value of (√ ). value of √ is something less process level, he has yet to form a than 2, then inquired, “You generalized idea of what an mean you want us to find a exponent represents outside the possible range for that value?” context of counting the number of He added the problem was the factors. input was not an exact value.

Task 1 Develop the need for Tom had taken the time over Process level understanding of the using logarithms or the weekend to research operational characteristics of some other alternate logarithms. When asked to exponents; however; he has yet to mathematical estimate the value for x, he encapsulate the entirety of the procedure by asking correctly wrote but concept. For example, it did not students to solve admitted that is just what the occur to him to use the graph to book said. He said he would approximate an answer. He did need to do something else use the definition of logarithms because the base was not 10. correctly; however, for him it was Unable to figure out how to without meaning; it was just use his calculator, he remarked another rule to follow. “Is that the technique to use, guess and check, guess and check? That could take forever!"

188

Table 5 Continued

Task 2 Graph both “I get what you are getting at. Object conception for exponential and Somewhere in that log there is some exponential functions; logarithmic functions kind of operation going on that is however, according to Asiala on the same set of undoing exponents, but we really et al. (1996), “In the course axes and describe don’t know what that is.” He then of performing an action or any relationships. goes on to say, “The process of how process on an object, it is it is doing it is the question because often necessary to de- the log must mean something, I encapsulate the object back mean I guess when we are undoing to the process from which it multiplication with division we are came in order to use its not really freaking out about properties in manipulating it” division because it is acceptable.” (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 “I did think about this over the Action moving toward a for logarithms, weekend and I kind of get what we process conception of students first rewrite are talking about here, or at least I logarithms. When he exponential and think I do. Let me talk this out, logs incorrectly rewrote one of the logarithmic are the inverse representation of logarithmic equations as an expressions, then exponent problems, not just the exponential equation, he evaluate logarithmic exponent but the whole problem, the quickly realized he had a expressions without a base, answer and the exponent problem. He knew there was calculator, using only itself.” an inverse relationship the alternate between exponents and representation for the [Speaking of rules] “Over and over logarithms, but he had not yet logarithmic or again you made sense of it, now it is interiorized the symbolic exponential form. part of you.” notation needed to represent this relationship.

Task 3 Write a short This implies multiplying b times Process conception, but the paragraph that itself N number of times assuming participant seems to need to describes your that because I see this as write out the rule each time thinking about the the inverse function of he needs to simplify symbols as indicating he may not have you completed each interiorized the action. portion of this activity.

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Table 5 Continued

Task 4 Develop the need and “Well I just laid it out as Process understanding of a recognize when an , Oh I see what you single logarithmic term. alternate are getting at now because before Once the rule had been representation is without really understanding the committed to memory, Tom needed to solve logs I couldn’t get that (referring to successfully solved several either an exponential the decimal approximation for the numeric examples and an or a logarithmic log), I couldn’t even get that application problem. equation. concept in my head (meaning estimate from the graph) but now we can…because now you can set it up then use the calculator.”

On the application problem Tom Without looking at the instinctively wrote , which written formula, he was able was correct, but he was unable to to solve this problem; provide solid reasoning why he however, he had difficulty solved this way other than he knew explaining his actions and it was doubling and the rate was connecting his ideas to the 1.07. After an all-group discussion, printed material. he responded, “But I got there before I even did all this. I don’t know how I did it, but I did it.”

Task 4 As a group, they When asked to explain if ( ) Process conception moving were asked to toward an object conception ( ) is decreasing or increasing, explain/evaluate of exponential functions. He various logarithmic Tom incorrectly guessed increasing, further asked, “What if x is a and /or exponential but when asked to explain why he fraction?” indicating he was expressions or changed his mind he said, “When I attempting to expand his equations and state picture the graph it is going to understanding. Jim added, any relevant decrease,” adding that on Day 1 the “It only changes direction if characteristics graph “began high in quadrant II the exponent is negative.” noticed. and falls steeply to the right because Apparently, this clarified his the base is fractional.” As we confusion. To draw closure moved to a logarithmic function, Tom stated, “I got to say this ( ) he was not surprised to myself: the exponential that the curve, according to him, functions reflect across the y- “was on the other side of the graph” axis and the logarithmic and noted that ( ) was functions reflect across the x- its mirror image. axis.”

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Table 5 Continued

Task 4 Given 6 different types No one had difficulty with While no application problems of equations, this task. Participants knew were given, all were successful. It participants were asked when to use a logarithm in is doubtful that this would have to explain in detail the their solution process. They occurred if an application problem solution method used to all recognized that first each had been presented. This indicates solve each. side of the equation a process understanding. needed to be divided by 3 before they could solve for x.

Pre-task 5 Given logarithmic “I went raised to the power Process understanding moving

equations, towards an object understanding of of negative 6 equals my x a single logarithmic term. At times and could and then I put 1 over 3 over Tom seems to forget how to

you solve for x? 2 ( ) but I think I might evaluate certain exponential ⁄ be wrong. OK, I had all the expressions. In his words, “I procedures right but where I didn’t hold on to it in my mind,” went wrong instead of suggesting an incomplete seeing 6 as my exponent I construction of an object saw it as something else.” conception.

Task 5 Develop an “I went here and said [he Initially Tom did not have any idea understanding for the flipped back to the previous how the ideas were connected, relationship between page that had values for log perhaps he had never entertained the properties of base 4], well, I can’t decide the notion that someone the exponents and how what is in the middle here properties for exponents should be they apply to [indicating that did connected in some way to logarithmic appear on the previous logarithms. expressions. 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 Out of the group Tom was the only ( )).” participant not able to connect this knowledge with logarithms of Tom still thinking of each multiple terms. For example, he term individually and has responded that not interiorized this action. is true because When asked to discuss the , but could not make validity of the following sense of the following: statement, since the two terms he writes 2, 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 . It’s

just memorization, but let me try a different approach. I’ll divide and use my rule

but doesn’t fit with the CDE, but 1 over 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 APOS level of understanding insightful

Pretest Simplify When asked: “If you have this Moving toward a process level exponential expression by itself, 20, how understanding, he was attempting expressions would you evaluate this?” Doug to interiorize his actions to create a involving integer responded, “That doesn’t fit into structural image of the concept but exponents. my mind. It is 1, I know that, but admitted he was unable to explain I don’t know why.” why any of the rules made sense. When reminded of the laws of He indicated it was just exponents, he used these in memorization on his part and that combination with CDE to deepen sometimes he was not sure if he his understandings. was remembering correctly.

Task 1 Graph the After completing the graph of a Still at a process level, but Doug exponential function smooth curve, Doug was able to was attempting to coordinate the ( ) to describe both the domain and operational characteristics of extend the meaning range for this function; however, exponential expressions with the of exponents to he was quick to answer that you structural characteristics of include all rational could not use the graph to exponential functions. values. Using the approximate the result of (√ ) graph, participants since √ is irrational. were asked to approximate the value of (√ ).

Task 1 Develop the need Doug would use the graph if it Process level conception moving for using logarithms were large enough to show more toward an object level conception or some other details; however, he stated that of exponential functions. He alternate he knew it would be between 1 seems to be thinking about how to mathematical and 2, “So you cut this in half reverse this process using some procedure by asking and raise it to the 1.5 power. type of mathematical procedure. students to solve When 1.5 was too large then I He is not sure what it is exactly, knew it would be between 1 and but seems to think there has to be a 1.5 and I tested it that way.” He limiting process as you narrow later added, “It seems we are down your result. Doug further approaching some type of limit.” 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 Doug writes, “They are inverse Process conception. He could exponential and functions. The x and y axes are recognize the relationship between logarithmic switched, I mean the domain of exponential and logarithmic functions on the f1 becomes the range of f2. The functions, yet when it came to same set of axes log button reverses the process completing a table of values for and describe any of exponentiation; it allows each pair of functions, he was relationships. you to see what power the base unable to trust his intuitions; he was raised to, to give you the needed to calculate the result each answer.” time.

Task 3 Using the Doug asks, “So what kind of Process conception. He had definition for method, what kind of interiorized the definition but still logarithms, arithmetic is being done when does not seem to be able to make students first you do logs?” sense of the operational rewrite exponential characteristics of the logarithmic and logarithmic function that appears to be critical expressions, then for all participants. evaluate logarithmic expressions without a calculator, using only the alternate representation for the logarithmic or exponential form.

Task 3 Write a short “You can estimate the logs. If Did not really address the issue of paragraph that the bases are the same, the the symbols but did indicate that describes your larger number yields the larger you can find the integer exponent thinking about the exponent.” above and below the target value, symbol as then estimate. This does indicate you complete each process understanding; he has portion of this interiorized the action of changing activity. from one format to the other.

Task 4 Develop the need After the question was asked, Action level, possible process level and recognize “Why is a log involved?” conception. When asked to solve when an alternate Doug responded, “Because several exponential equations using representation is there is a power of x involved.” what he now knew about logs, needed to solve However, Doug did not know participant was successful; either an how to begin this exercise however, when asked to solve an exponential or a prior to this discussion. application problem which did logarithmic contain an exponential equation, he equation. was not sure how to begin until prompted.

196

Table 6 Continued

Task 4 As a group, they Asked to explain why is Process conception moving toward were asked to between 3 and 4, Doug quickly an object conception of a single explain/evaluate responded, “Because 3 to the logarithmic term, once he heard what various logarithmic second—no, no, I mean 2 to the he had said, he immediately expressions or third power is 8, and 2 to the recognized this as incorrect. equations. fourth power is 16.”

Task 4 Given 6 different No one had difficulty with this While no application problems were types of equations, task. Participants knew when to given, all were successful. This participants were use a logarithm in their solution indicates a process understanding. asked to explain in process. They all recognized that detail the solution first each side of the equation method used to needed to be divided solve each. by 3 before they could solve for x.

Pre-task 5 Given logarithmic Referring to the second equation: According to Asiala et al. (1996), “In equations, “Well, there is a 3 in there the course of performing an action or and somewhere, x is 3, no negative 3, a process on an object, it is often

just throwing out words.” When necessary to de-encapsulate the

asked to explain his thinking he object back to the process from participants were responded, “I see 3 cubed is 27” which it came in order to use its asked to solve each but then admitted that the problem properties in manipulating it” (p. 11). for x. itself suggested it. He said, “If Based on his admission that he would you gave us something bizarre I not be able to solve it unless the wouldn’t be able to solve it.” result was readily apparent, Doug Tom then told him, “If you follow was not working with a fully through all the steps, you should interiorized concept; he initially be able to solve it, but not all of reacted in response to external cues them are going to jump out like as opposed to reorganizing the that.” material into a more structured format.

Task 5 Develop an As Doug worked to justify why Moving toward a process level understanding for he conception. It appears that the class the relationship referred to two sequences discussions had made sense and between the involving powers of 4. First, he Doug was able to determine the properties of multiplied 16 and 64 and came up validity of 4 different statements, but exponents and how with 1024, then he added 2 and 3 no mention was made of how to they apply to and came up with 5. That led him connect this to the laws of exponents. logarithmic to generalize He seemed to be able to link several expressions. . He later added different processes together, but that if the bases were not the same could not encapsulate this process you would have to find a into an object. The explanations relationship between the first and given at this point were procedurally the second base before you could oriented. do anything. He also was able to recognize

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 the graph curves up and if 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 As Doug struggled to assimilate the

new information he asked, “So the answer 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 ( ) for √ , Jim 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

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Table 7

Summary of Jim’s Performance

Tasks Activity Issues encountered that APOS level of understanding were insightful

Pretest Simplify Active use of the common Jim had a clear notion of the common exponential definition of exponents to definition of exponents and was able to expressions justify his responses, but extend this definition to include both involving integer then stating it was much zero and negative exponents. He exponents. easier to just apply the laws clearly understood the operational of exponents. 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 Could articulate Could articulate generalized properties ( ) to generalizations about of exponential functions, but it seemed extend the meaning exponential functions, but as if he did not need to consider deeper of exponents to could not explain why the meanings of rational number exponents include all real graph would be useful for when computationally they were not values. Using the determining the value of difficult to perform with a calculator. graph participants (√ ) given the graph of Process level understanding moving were asked to ( ) . In his own toward an object conception, since in approximate the words he asked, “You his mind there was no need to explain value of (√ ). mean find the value by just the wider notion of exponents in a eyeballing it?” meaningful way.

Task 1 Develop the need “We need [to find] some Object level conception: he knows that for using logarithms sort of function that could the process of exponentiation can be or some other keep just going and going reversed and seems to remember this is alternate and you get longer and where logarithms are used, but does not mathematical longer numbers that are remember how the rules for logarithms procedure by asking closer to what you need.” work. Jim seemed to be aware of this students to solve process as a totality and realized that actions and/or processes can act on the entire expression itself.

Task 2 Graph both While the formal definition Process conception. Jim apparently exponential and for logarithms had not been remembered something from high logarithmic given, when asked to school mathematics, but did not really functions on the summarize similarities attempt to form any deep connections. same set of axes between the functions, Jim He worked well following rules and and describe any wrote, “If then recognizing patterns, knew the two relationships. functions are inverses of each other, but and tried to clarify this by adding, “the did not put much thought into the log of x is the answer to the relevant properties shared by both exponent c.” functions.

202

Table 7 Continued

Task 3 Using the definition “What does the log do? Process conception. He has for logarithms, What is in log that allows it interiorized the action, transforming students first to do what it does, without from one format to the other rewrite exponential the word log?” That is effortlessly. He also realized that and logarithmic what I want to see. before hand-held calculators the only expressions, Right now the way I see it, way to evaluate logarithms, if it was evaluate the easiest way to look at it not integer value, was by using the

logarithmic is table of logarithms. expressions without ” a calculator using only the alternate representation for the logarithmic or exponential form.

Task 3 Write a short “It means what b raised to Process conception, moving toward an paragraph that some power is equal to, object conception; however, there describes your what you are taking the log appear to be some gaps in knowledge. thinking about the of. In order to undo the While there are no mathematical symbols as exponential process and inconsistencies, the need to explain the you completed each better understand concepts from an intuitive point of portion of this something in a linear view is secondary. activity. fashion.”

Task 4 Develop the need When asked to use logs to Action conception, moving toward a and recognize when solve several numeric process conception. Unable to use an alternate examples, Jim was existing knowledge to solve application representation is successful, although he problem even when the variable needed to solve asked how you would appeared in the exponent. either an convert to other bases (this exponential or a rule was not needed: the logarithmic group was using the latest equation. 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?”

203

Table 7 Continued

Task 4 As a group, When asked how to Process conception, moving toward an participants were evaluate object conception of a single logarithmic asked to without a graph or a expression and/or function. explain/evaluate calculator, Jim stated he various would make his own table logarithmic of powers of 5 and hope expressions or 78125 showed up on the equations. 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.”

Task 4 Given 6 different No one had difficulty with All were successful. It is doubtful that types of this task. Participants this would have occurred if an application equations, knew when to use a problem had been presented; however, participants were logarithm. They all since they had seen one completed earlier, asked to explain recognized that first, each presumably they all could complete one— in detail the side of the equation since Jim and Earl both indicated that solution method needed to be once they have been shown a solution used to solve divided by 3 before they method they remember the solution each. could solve for x. method. This indicates a process understanding.

Pre-task 5 Given logarithmic Jim explained, “I saw 27 Procedures, committing them to memory, equations, to the 1 over 3 and then and using them effectively, seemed to and raised so it would be 27 to pose no difficulties for Jim. He was able

the 1 power cube root. to reflect on operations applied to a could The denominator is the particular process, a characteristic of an you solve for x? root value (for rational emerging object conception of a single exponents).” logarithmic equation.

Task 5 Develop an When asked if Process level, moving toward object understanding for is a true conception. Since properties of the relationship statement, Jim replied, logarithms were not formally presented, between the “No, when you add two Jim was able to extract a procedure from properties of logs with the same base the group discussion and apply it exponents and you multiply what you are consistently; however, it is unclear if he how they apply to taking the log of.” was able to connect this to laws of logarithmic exponents other than his comment that expressions. “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.

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, can be more meaningful. When asked to describe his thinking about the symbol 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

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Table 8

Summary of Earl’s performance

Tasks Activity Issues encountered that APOS level of understanding were insightful

Pretest Simplify exponential Earl simplified the Process level understanding; however, expressions involving expression first by using it is unclear how he has made sense of integer exponents. the common definition of these rules. He did indicate that with exponents given, but then more practice he could become more . reverted to using the laws fluent with the laws of exponents. of exponents. He did not provide any written justification for his work.

Task 1 Graph the exponential Earl told the group, “We Process understanding: Earl was function ( ) to are trying to find what y attempting to form a generalized extend the meaning of equals when x is the understanding of exponents, but part of exponents to include square root of 3.” He the encapsulation was left unfinished. all rational values. stated he knew what 32 He could visually represent integer Using the graph looked like and even exponents and was comfortable with

participants were what represented, but the idea of expressing terminating asked to approximate could not visually decimals as fractional powers and then the value of ( ) represent exponential roots, since he was familiar with the when √ . expressions with decimal process of finding square or cube roots. or irrational exponents. 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.

Task 1 Develop the need for “I can draw the curve of According to Dubinsky, coordination using logarithms or 3x with 32 and 33 and 1 and reversal are two important aspects some other alternate through zero and guess in the construction of processes and mathematical where it’s at on the objects. While Earl was operating at a procedure by asking curve, but I don’t know process level conception, he was students to solve what it actually is. How having difficulty trying to do you know what this accommodate the structural is?” 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 When asked to Process conception, moving toward an exponential and generalize, Earl wrote, object conception. Difficult to say with logarithmic functions “the domain of ( ) certainty since participant was not very on the same set of equals the range of vocal yet could work through the axes and describe any ( ) the range handouts with relative ease. relationships. 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.

Task 3 Using the definition Successfully completed Process conception, moving toward for logarithms, the activity without much object conception. While there were no students first rewrite interaction. Encountered mathematical inconsistencies in his exponential and difficulty evaluating work, he was unable to successfully logarithmic ( ) He wrote, “4 deal with the expression noted since it expressions, evaluate to the power of what did not have a real-number solution. logarithmic gives you negative16?” He knew the result was impossible, but expressions without a was at a loss for how to evaluate the calculator using only given expression since in his mind you the alternate could not. Earl was unable to see a representation for the more global perspective regarding logarithmic or logarithmic functions. exponential form.

Task 3 Write a short “When I see this, I know Process understanding of a single paragraph that to ask myself, “b to what logarithmic term. Earl would narrow describes your power gives x.” the result between two integer values, thinking about the but was reluctant to approximate. symbols as you Descriptions were generally completed each procedurally oriented. portion of this activity.

209

Table 8 Continued

Task 4 Develop the need and When asked to use what Process understanding. Earl admitted recognize when an he now knew about logs earlier to not attempting to justify alternate to solve numeric mathematical procedures. He stated, “I representation is examples, Earl get the rules written on the board or in needed to solve either successfully completed the book. I’ll do a couple of problems an exponential or a each exercise. However, and then I am able to recognize similar logarithmic equation. when an application problems, then move on. I very rarely problem was presented in go back and say, OK, I want to try to . which the group was and justify. Not many college students the equation used to will.” Earl did not see how to solve model the situation was this problem mathematically; therefore, given, Earl did not he resorted to trial-and-error instead of consider using logs to recognizing that the variable in the solve; rather he solved exponent signals the need to use using a trial-and-error logarithms to find the solution. method.

Task 4 As a group, When asked if ( ) Process conception. Earl consistently participants were is increasing or verified/explained concepts with asked to decreasing, Earl numeric results rather than offering explain/evaluate explained, “increasing, generalizations. various logarithmic because if x were equal expressions or to 1, then y is zero, and if equations. x is 2, y is 1, so increasing.”

Task 4 Given 6 different No one had difficulty All were successful. It is doubtful that types of equations, with this task. this would have occurred if an participants were Participants knew when application problem had been asked to explain in to use a logarithm in their presented; however, since they had detail the solution solution process. They seen one completed earlier, presumably method used to solve all recognized that first they all could complete one—since Jim each. each side of the equation and Earl both indicated that once they needed to be have been shown a solution method divided by 3 before they they remember the solution method. could solve for x. This indicates a process understanding.

Pre-task 5 Given logarithmic As Tom was attempting Procedures, committing them to equations, to reorganize the first memory, and using them effectively

problem, Earl responded seem to pose no difficulties. While not and with “64” and explained overly explicit about his thinking, Earl

solve for x. that he saw 1 over to is able to reflect on operations applied to a particular process, a characteristic the sixth power. of an emerging object conception of a single logarithmic equation.

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Table 8 Continued

Task 5 Develop an When asked if Earl was able to generalize a pattern for understanding for the was a the addition of logs. He wrote, relationship between true statement, Earl ( ) ; however, the properties of wrote it is false. He it is unclear if he knows why this is true exponents and how added “It should read other than he is following a pattern. He

they apply to ( ) also sees that but logarithmic does not offer an explanation. Not sure expressions. because is 5 if he is aware of the connection and 2+3 is 5. 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.

following: “Domain of ( ) range ( ), range of ( ) domain ( ),” then

followed with a concrete example, using ( ) and ( ) However, when asked to evaluate ( ), 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 ” He then added, “The log button takes the base and checks powers until it gets the 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

211 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

212 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 he first indicated that , 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

213 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.

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Table 9

Summary of Growth in Understanding

Activity Growth Noted Apparent Impetus Evidence

Pretest Broadened view of Students were asked to explain Day 1 transcriptions: exponents to include zero why the laws of exponents are Participants expressed their and negative values valid for zero and negative understanding of exponents exponents as rule based, with one stating “other than the rule I don’t see it.[the meaning]”

Task 1 Subjects were able to Students were asked to create Participants were able to extend the meaning of graphs for several exponential graph exponential functions function to include a new functions and identify key features class of functions called exponentials

Task 2 Ability to consider how Students were asked to graph both Participants were able to state to reverse the process of a logarithmic function and an similarities and differences exponentiation and the exponential function on the same between the two pairs of apparent connection to set of axes functions logarithmic functions

Task 3 Were able to consider Students were asked were asked to Participants recognized alternate representations evaluate several logarithmic different representations that of the exponential expressions without the assistance allowed successful expressions using a new of a calculator completion of comparison symbol log exercises

Group not satisfied with Participants were able to just accepting that the log Students were asked to create develop logs for fractional button would find the logarithmic tables inputs using understanding exponent of fractional exponents

Task 4 Were able to recognize Unknown Participants completed a when appropriate to use problem set that involved logarithms to solve multiple types of problems

Task 5 Were able to consider Students were asked to verify the All participants agreed it why the sum of two following statement as either true could not be true logarithms is equivalent or false:

to the product of the log log log argument of the two logarithmic terms

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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.

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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

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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.

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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 √ ; the participants also established the idea that an expression such as could be rewritten as √ . This heightened awareness of the role of the exponent offered participants some insight into

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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

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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

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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 and to find the inverse function gives a method for obtaining the graph of 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

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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

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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

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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 Process Emergent object Tom * 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 well- developed 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.

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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, ( ) , Tom was the only participant to recognize the need to first simplify, ( ) then use the alternate format, , 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

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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 could be rewritten as , which then could be simplified as

7 . 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 , he suggested that the law of addition does not apply since . 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 was

equivalent to and was equivalent to 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 Process Emergent object Tom * 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 de- emphasizing 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

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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

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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

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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 Students were next presented with the statement

( ). It was unclear if the participants understood the significance of

dividing 3 by , 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 by , obtaining a value closer to one than

the previous expression. Next, 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

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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

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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.

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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

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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 looked like. Participants all agreed on the existence of representations such as 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

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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

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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 historical to symbolize development of inverse logarithmic tables

formal definition of logarithm as CONSTRUCTS development of the inverse CONTRIBUTING function TO THE the properties of ENCAPSULATION logarithms

Figure 6. Revised genetic decomposition.

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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.

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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 or looked like, but could not visualize something like √ . 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.

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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

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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

261

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 Logarithmic function Inputs “x” Outputs “y” Inputs “x” Outputs “y”

-4

-3

-2

-1

0 1 1 3 2 9 3 27 4 81

277

b. and

Exponential function Logarithmic function Inputs “x” Outputs “y” Inputs “x” Outputs “y”

-4

-3

-2

-1

0 1 1 2 2 4 3 8 4 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 Log 101 = 10 =

1 Log10 10 = 10 =

2 Log 10100 = 10 =

3 Log10 1000 = 10 =

4 Log10 10000 = 10 =

0 Log4 1 = 4 =

1 Log4 4= 4 =

2 Log4 16 = 4 =

3 Log4 64 = 4 =

4 Log4 256= 4 =

279

0 log 1 =

=

-1 log =

=

-2 log 4 =

=

-3 log 8 =

=

-4 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 = 2

54 = 625 32 = 9 ec = y

210 = 1024 42 = 16 √

281 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

5 8) loga (a )

9) log4

10) log4(-16)

283

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 l

log 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 and increasing or decreasing function? What about

log Explain your response.

285

c. Can you explain how to find log without using a calculator or graph?

d. Given that log find log

e. Evaluate log

f. Solve for x, explain your method log

g. l 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 ? Why or why not?

2. What about this one log 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 log , still maintaining a “logarithmic” expression in the final result

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 log

b. log log

c. log log log

d. log log

288

The Multiplication Law Investigate the multiplication law by completing the table below. x y log log log log xy log log 10 1000

100 10000

10 100000

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 Log n n Log n

1 0 10 1

2 20 1.301 3 .477 30 4

5 .699 6

7 .845 8

9 .954

a. Using what you know about properties of logs, complete the table without using

a calculator.

b. Find using the table. Explain how you calculated this value

c. Find using the table. Explain how you calculated this value.

d. Find using the table. Explain how you calculated this value.

290

Part B

Consider the following table of values

n Log 3n n Log 3n 1 0 10

2 .631 11 3 12

4 13 5 1.465 14 6 15

7 16 8 17

9 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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