CopyofDissertation.doc

UNDERSTANDING MATHEMATICAL CONCEPTS THE CASE OF THE LOGARITHMIC FUNCTION by SIGNE E KASTBERG B A Keene State college M A The University of Georgia A Dissertation Submitted to the Graduate Faculty of the University of Georgia in Partial Fulfillment of the Requirements of the Degree DOCTOR OF PHILOSOPHY ATHENS GEORGIA 2001 UNDERSTANDING MATHEMATICAL CONCEPTS THE CASE OF THE LOGARITHMIC FUNCTION by SIGNE E KASTBERG Approved Major Professor James W Wilson Committee Edward Azoff Shawn Glynn Jeremy Kilpatrick Roy Smith Electronic Version Approved Gordhan L Patel Dean of the Graduate School The University of Georgia August 2001 2001 Signe E Kastberg All Rights Reserved iv TABLE OF CONTENTS CHAPTER I DEFINING A PROBLEM AND CRAFTING A SOLUTION PATH 1 RATIONALE 1 THEORETICAL FRAMEWORK 3 RESEARCH QUESTIONS 12 CHAPTER II DISCUSSION OF RELEVANT LITERATURE 13 THEORIES OF UNDERSTANDING 14 COMMON ELEMENTS IN THE FOUR THEORIES OF UNDERSTANDING 25 CONSISTENCY OF THE COMMON ELEMENTS WITH MY DEFINITION OF UNDERSTANDING 29 REPRESENTATIONS 31 REMEMBERING AND UNDERSTANDING 32 THE LOGARITHMIC FUNCTION 33 CONCLUSION 38 CHAPTER III METHODOLOGY 39 RESEARCH TECHNIQUES 39 PROCEDURE 51 DATA ANALYSIS 56 CHAPTER IV CASE STUDIES 59 JAMIE 59 RACHEL 96 NORA 127 DEMETRIUS 164 CHAPTER V COMMONALITIES IN PROBLEM SOLVING BEHAVIOR 199 DEFINITIONS OF UNDERSTANDING 199 STUDENTS UNDERSTANDING OF THE LOGARITHMIC FUNCTION 200 UNDERSTANDING AND PROBLEM SOLVING BEHAVIOR 210 CHANGES IN UNDERSTANDING AND THE USE OF STUDENTS WAYS OF KNOWING 213 CHAPTER VI DISCUSSION AND IMPLICATIONS 220 UNDERSTANDING 220 SIGNIFICANCE OF THE STUDY 229 LIMITATIONS OF THE STUDY 229 SUGGESTIONS FOR FUTURE RESEARCH 231 CONCLUSION 233 REFERENCES 234 APPENDIX A INTERVIEW PROTOCOLS 241 PROTOCOL FOR INTERVIEW 1 PREINSTRUCTIONAL PHASE 241 PROTOCOL FOR INTERVIEW 2 PREINSTRUCTIONAL PHASE 244 PROTOCOL FOR INTERVIEW 3 INSTRUCTIONAL PHASE 245 PROTOCOL FOR INTERVIEW 4 POSTINSTRUCTIONAL PHASE 246 PROTOCOL FOR INTERVIEW 5 POSTINSTRUCTIONAL PHASE 247 PROTOCOL FOR INTERVIEW 6 POSTINSTRUCTIONAL PHASE 248 PROTOCOL FOR INTERVIEW 7 POSTINSTRUCTIONAL PHASE 250 PROTOCOL FOR INTERVIEW 8 POSTINSTRUCTIONAL PHASE 251 PROTOCOL INTERVIEW 9 POSTINSTRUCTIONAL PHASE 251 APPENDIX B 253 1 CHAPTER I DEFINING A PROBLEM AND CRAFTING A SOLUTION PATH Rationale According to Hiebert and Carpenter 1992 one of the most widely accepted ideas within the mathematics education community is the idea that students should understand mathematics p 65 This fundamental assumption was the basis for this study I too believe that students should understand mathematics I came to this belief by way of my teaching A teacher often assumes that a student understands the concept presented and then finds in a subsequent class that the student cannot recall the concept One example of this phenomenon is the logarithmic function While teaching college algebra at a community college for eight years the most frustrating concept for me to teach was the logarithmic function Even those students I felt understood the concept could not remember or use the properties of the function in subsequent courses This absence of memory about the objects motivated me to ask why Why didn t my students remember what they had seemed to know so well just a few months earlier Did they really understand the concept in the first place As a teacher I had to assume that something was wrong and I wanted to know what that was and how to attack the problem This report is the result of a question that I posed as a mathematics teacher Why can t my students remember the definition properties of and how to use the logarithmic function As a doctoral student I tried to understand my question Generally I believed that if my students understood the logarithmic function they would be able to remember it Based on this belief I began searching for curriculum that might help students understand the logarithmic function I reasoned if I taught the concept differently then students would understand A search of curriculum used to teach the logarithmic function and problems that could be solved using the logarithmic function uncovered a variety of methods in addition to the traditional logarithmic function as the inverse of the 2 exponential function approach Presentations that relied on the historical development of the logarithmic function Toumasis 1993 Katz 1995 and one that relied on an area under the curve definition of the logarithmic function SMSG 1960 were both novel and appeared promising My focus during this period was on finding the curriculum that would produce understanding in my students Having read and analyzed the historical development of logarithms over the course of several months I suddenly realized that the focus of my research was not the logarithmic function but students understanding of the logarithmic function This shift in focus gave my research the base I had been looking for My question became what does it mean for students to understand the logarithmic function Now I needed to be more precise I turned to the literature for definitions and theories about students understanding of mathematical concepts Several theories of understanding seemed helpful Hiebert Pirie Sierpinska 1994 Skemp 1987 Each was developed out of researchers interpretations of students actions during problem solving The location of understanding is in the mind of the individual In addition two of the theories Sierpinska 1994 Skemp 1987 explicitly state the individual can at times consciously control his or her understanding Despite locating the locus of control for understanding within the individual none of the researchers asked the individuals what they understood I reasoned if understanding was occurring within the individual then the individual could tell me about his or her own understanding This assumption proved to be a fairly large one I realized students might not share my definition of understanding but I initially failed to consider they might not be using the definition they gave me to identify what they did and did not understand Indeed how someone defines understanding might not involve personal action But as Bruner 1990 explained meaning lies somewhere between a person s actions during an experience and their explanations of their actions Hence if I gathered students definitions of understanding their reflections on understanding and not understanding mathematical concepts and saw them in action I Comment A1 Page 3 3 might be able to discover what they meant when they used the term understanding In turn the students meanings and reflections could help me interpret their actions and the understanding that resulted from them From my observations and interpretations I could then build descriptions of students understanding of the logarithmic function After discovering I wanted to ask the students about their own understanding I still did not know what I meant by the term All of the theories in the literature seem plausible but no single theory seemed to explain what it meant for a student to understand a mathematical concept When I began thinking about students understanding of the logarithmic function my goal was to find a way to teach the logarithmic function so students would remember it As I pursued the goal I took various paths and arrived here I am no longer studying curriculum looking for what might work Instead I am studying students their ways of knowing and their explanations of their ways of knowing in a mathematical context The purpose of the study was three fold to develop descriptions of students understanding of the logarithmic function of changes in their understanding of the function and of ways of knowing students use to investigate problems involving the logarithmic function Theoretical Framework Before I present my definition of understanding I would like to clarify the assumptions on which the study was based First I assumed that the goal of mathematics teaching is student understanding Second I assumed that a student s understanding of a mathematical concept exists in his or her mind Third I was aware that I could not know precisely what was in a student s mind but assumed that I could infer the workings of the mind from external evidence Goldin 1998a Skemp 1987 Fourth I assumed that when students tried to solve mathematics problems they were self referencing I believed that they would try to make sense to themselves Finally I assumed that a student s understanding is qualitatively richer and quantitatively larger than external evidence and ultimately my descriptions could indicate Hence although my descriptions may not 4 match students understanding they provide useful information for those who teach the logarithmic function design curriculum to be used in the teaching of the logarithmic function and those who research students understanding of mathematical concepts Understanding Understanding can change This is certainly a statement on which all mathematical educators would agree A student s understanding of a mathematical concept may become either more or less consistent with standard mathematical views of the concept but the most probable mediator of understanding is a student s prior knowledge One observation that assumes near axiomatic status in cognitive science is that student s prior knowledge influences what they learn and how they perform Hiebert Glaser 1984 we are able to see differences in what the two groups view as important whether the study focuses on teaching or learning If theories are different understanding will be different Schoenfeld 1988 found students theories about geometry were largely the result of their experiences with the subject For example they believed most geometry proofs could be done in a very few minutes The students experience became the basis for the theory they acted on If they could not do a proof in a few minutes they gave up A student s understanding of a mathematical concept is much the same his or her collection of theories about a concept are what he or she uses to decided when if and how a concept is used Thus it is our general theories about mathematics and our specific theories about concepts that govern our learning process and form our understanding of concepts 6 Categories of Evidence I will base my inferences about students privately held theories on four categories of evidence conception representation connection and application A conception is a student s conscious beliefs about the concept A representation is a symbol the student uses to communicate the concept A connection is a relationship between representations An application is a use of the concept to solve a problem After defining and giving an example of each of these categories of evidence I will explain why they are indications of students theories about a mathematical concept Conception A student s conception of a mathematical concept is limited to his or her expressly communicated feelings and ideas about the concept For example a student may describe the logarithmic function as a collection of letters This description is a conception If the student describes the logarithmic function as frustrating he or she is also expressing a conception of the function This assessment may be the result of various factors including but not limited to a student s goals for his or her mathematical activity A student s conception of a mathematical concept can certainly impact his or her future attempts to learn more about or apply the concept Sierpinska 1992 Skemp 1987 It is certainly human nature to attempt to categorize objects that we perceive Mathematical objects are no exception to this rule When a student sees a mathematical object such as a function he or she will try to make sense of it based on his or her past experiences with mathematical objects Research on students classifications of function illustrates this point If a student believes that all functions are when faced with a coordinate axes on which several points are plotted and asked to draw as many function as possible through the points what will the student draw Lines A student s conception of a concept impacts how it is applied Hence his or her conception is evidence of his or her understanding of the concept 7 Representation A student s representation of a mathematical concept consists of the symbols the student uses to think about and or communicate the concept to others In the study I focused on four modes of representation written pictorial tabular and oral Briefly a written representation is a collection of letters and numerals a pictorial representation consists of an image a tabular representation is a compilation of numerical data in a table and an oral representations is spoken A student is likely to use a combination of these four modes when thinking or communicating about a concept Written representations are notations that students use to think about and communicate a mathematical concept in writing The written representations discussed in this report are names notations maxims and descriptions Names are terms that refer to mathematical objects procedures and collections of objects or procedures One example of a name is the term base Notations are definitions properties and examples of mathematical concepts written using mathematical symbols log2 1 0 and logaa 1 are examples of notation Maxims are short statements that are meant to serve as mathematical rules or guides One example is the common logarithms are exponents Descriptions are accounts of procedures outcomes of procedures mathematical objects and relationships that are intended to explain how they work A function is a collection of letters and numbers is one example of a description Pictorial representations are images that students use to think about and communicate a mathematical concept visually An example of a pictorial representation that is often used in the exploration of the logarithmic function is the graph of y log2 x as shown in Figure 1 Figure 1 Graph of y log2 x 8 Tabular representations are tables of numerical data that students use to think about and communicate a mathematical concept An example of a tabular representation of y log2 x is shown below x12 log2 x 2 101 Oral representations are spoken words and expressions that students use to talk about a mathematical concept As with written representations the oral representations discussed in this report are names notations maxims and descriptions The definitions for these terms remain the same with the exception that they are spoken not written An example of an oral representation is log of one is zero Representations play a role in all mathematical communication We use representations to convey to others an approximation of what we see in our mind s eye All of the theories of understanding cited in this report Hiebert Pirie Sierpinska 1994 Skemp 1987 incorporate representation Although the understanding of a mathematical concept exists in the mind of the individual external symbols used to represent the concept are evidence of a student s private theories about the concept For example if a student in an attempt to approximate log3 2 graphs the function then this is evidence of the student s theory that a logarithm is associated to the graph of the logarithmic function On the other hand if the student uses his or her calculator and the change of base formulas then we might conjecture that the student sees a logarithm as an algebraic computation Students uses of representations are indications of their understanding of a mathematical concept Connection If a student translates a representation from one mode to another or a transforms a representation to another in the same mode Lesh Post Poincar 1946 The most that I can claim about a student s understanding of a mathematical concept is that it is likely to change Changes in understanding about a concept will be reflected in the changes that occur in a student s theories about the concept The second purpose of the study is to identify changes in understanding that occur during the course of the study Ways of Knowing When a student does a problem he or she does not always approach the problem the way I would In fact in many cases I have been surprised at the approaches a student takes For example given the sequence 1 2 4 8 some students explain the action in the sequence as multiplying each term by two to get the next term These students do not see this sequence as powers of two Identifying this sequence in this way made it impossible for the students to see the standard map between this geometric sequence and the arithmetic one 0 1 2 3 Instead the map that they described was coordinated action multiply by two on the geometric sequence and add one on the arithmetic one Describing the relationship between the two sequences as a map of one action to another allowed the students to predict the term in the arithmetic sequence that corresponded to the geometric one but was not flexible enough to allow the student to make predictions about terms that I inserted into the geometric sequence such as I will call these nonstandard approaches ways of knowing Hence students ways of knowing are defined as operations and strategies they use to investigate problems they are asked to solve The students ways of knowing can be used as the basis for future understanding but they can also be constraining If a student sees the relationship between the two sequences as a map where multiplication by two in one sequence corresponds to addition 12 by one in the other he or she may have an extremely difficult time reversing the relationship to find the term that maps to Despite the constraints a students way of knowing may create these ways of knowing are sources of meaning for them As sources of meaning the constraints have the potential to be what Sierpinska 1994 calls the basis See Chapter 2 for the student s understanding For me identifying the student s ways of knowing is i