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STUDENTS‟ CONCEPTUALIZATIONS OF MULTIVARIABLE LIMITS By BRIAN CLIFFORD FISHER Bachelor of Science in Mathematics Oklahoma State University Stillwater, Oklahoma 2000 Master of Science in Mathematics Oklahoma State University Stillwater, Oklahoma 2002 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2008 ii STUDENTS‟ CONCEPTUALIZATIONS OF MULTIVARIABLE LIMITS Dissertation Approved: Dr. Douglas B. Aichele Dissertation Adviser Dr. John Wolfe Dr. James R. Choike Dr. Alan Noell Dr. William Warde Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENTS It is not often that I get to publicly thank all those who have made such a large impression on my life. I am grateful for this opportunity, and I hope my words and my life do justice to the investment that so many people have made in me. I know I will miss some people who have worked very hard for me, so I apologize in advance for those omissions. First, to my wife, Kimberly, I will not attempt to recount all the ways you have been an inspiration to me. Let me only say „thank you‟ for always believing in me, even when I did not. Evelyn, you are too young to remember this process, but know that whenever I am tired and frustrated your smile always lights up my day and helps me put life into perspective. Thank you to my committee for all the effort you put into me. Dr. Aichele, thank you for pushing me and having confidence in me. Thank you for investing so much time and effort in me, and thank you for always finding the right question to straighten me out when I needed direction. Dr. Choike, thank you for all the guidance you have given me over the years. Much of who I am as a teacher is because of the time we spent talking in your office. Dr. Wolfe, it was in your class almost four years ago that I found my interest in mathematics education research, and it was during my time spent in Geometric Structures that you finally sold me on inquirybased teaching methods. Dr. Noell, thank you for joining my committee even though I asked you late – I still believe I learned iv more in your complex analysis course than any other in grad school. Dr. Warde, thank you for the time to read and respond to every document I sent your way, your comments and suggestions were always helpful and appreciated. I would like to thank the entire math department at Oklahoma State University. To the faculty, thank you for giving me so much individual attention over the years. Thank you for stopping me in the hall and checking on my progress. Thank you for always having high expectations of me. To the staff, thank you for putting up with all my requests and never complaining. To the graduate students, thank you for always being encouraging and making the department a fun place to work and learn. Let me not forget to say “thank you” to all the students who took the time to take part in my study. I am especially grateful towards the seven students who sacrificed their time to set through three separate interviews with me. You all sacrificed your time for me and answered every question I asked with sincerity. Without you all this study could never have taken place – thank you! Let me thank my friends and family for all the good memories over the years. Your faith and encouragement were ever present and always needed. Thank you, Mom and Dad, for raising me to cherish thinking and learning. Thank you, Becky, for being much more than a sister. Thank you to the entire Burns family for celebrating and struggling with me and making me one of your own. Thank you, Dustin, for opening up your house for me when I needed a place to rest. Thank you, Matt, for always keeping me focused on what is important. Thank you to the entire Edmond Church of Christ and especially our young married class for teaching me what it means to have a spiritual family. Finally, a tremendous “thank you” goes to everyone who sincerely asked me v what I was studying and listened while I prattled on about mathematics, regardless of whether or not you found it interesting. Lastly, I want to say publicly what I say regularly in private. Thank you, God, for putting all these people in my life to encourage me, uplift me, and keep me focused on the things that really matter. Thank you for taking me great places despite my weaknesses, and thank you for letting me work hard enough to be proud of what You accomplish.vi TABLE OF CONTENTS Chapter Page I. INTRODUCTION ......................................................................................................1 Statement of the Problem .........................................................................................1 Problem Context ......................................................................................................3 History of the Limit Concept .............................................................................3 Curriculum Change and the Limit Concept .......................................................7 II. RELEVANT LITERATURE ..................................................................................15 Dynamic Imagery and Covariational Reasoning ...................................................15 Students‟ Conceptions of the Limit Concept .........................................................18 Theoretical Perspectives ........................................................................................26 Cognitive Obstacles and Cognitive Change ....................................................27 Mental Representation and Conceptual Metaphors .........................................29 Abstraction and Generalization........................................................................31 Infinite Processes .............................................................................................32 III. METHODS ............................................................................................................35 Why a Qualitative Study? ......................................................................................35 Participants and Setting..........................................................................................37 Study Questionnaire ...............................................................................................39 Data Collection ......................................................................................................42 Researcher‟s Perspective .......................................................................................45 IV. DATA ANALYSIS ...............................................................................................49 Questionnaire Results ............................................................................................49 Qualitative Data Analysis ......................................................................................53 Definitions of Limit in Formal Mathematics ....................................................54 Neighborhood Model of Limit ..........................................................................57 Dynamic Model of Limit ..................................................................................59 Topographical Model of Limit..........................................................................61 Textbook Treatment of the Limit Concept .......................................................64 Coding Scheme .................................................................................................65 V. STUDY RESULTS ................................................................................................70 vii Qualitative Study Results .......................................................................................70 The Concept of Limit in Different Settings ...........................................................71 Introductory Calculus.......................................................................................71 Symbolic Manipulation of Multivariable Functions ........................................82 Three Dimensional Graphing of Multivariable Functions ...............................89 Multivariable Functions using Polar Coordinates............................................98 Contour Graphs of Multivariable Functions ..................................................103 Misconceptions about Multivariable Limits ........................................................114 Uncategorized Misconceptions ......................................................................116 Misconceptions Involving the Dynamic Model .............................................122 Misconceptions Involving the Topographical Model ....................................132 VI. DISCUSSION AND CONCLUSION ................................................................158 Visualizing Multivariable Limits .........................................................................161 Amanda‟s Experience with Multivariable Limits ...........................................162 Infinite Processes in Multivariable Limits ...........................................................181 Josh‟s Experience with Multivariable Limits ................................................185 Answering the Study Questions ...........................................................................208 Reflections ...........................................................................................................212 REFERENCES ..........................................................................................................216 APPENDICES ...........................................................................................................225 Appendix A – Study Questionnaire .....................................................................225 Appendix B – Interview 1 Materials ....................................................................227 Appendix C – Interview 2 Materials ....................................................................230 Appendix D – Interview 3 Materials ...................................................................234 Appendix E – Informed Consent Forms ..............................................................239 Appendix F – IRB Approval Form ......................................................................242 viii LIST OF TABLES Table Page Table 1: Questionnaire Results (Cumulative) ..........................................................49 Table 2: Questionnaire Results (Percentage) ...........................................................50 Table 3: Questionnaire Results (Binomial) ..............................................................50 Table 4: Questionnaire Results (Binomial Percentage) ...........................................50 Table 5: Question #6 Results (Cumulative) .............................................................51 Table 6: Question #6 Results (Percentage) ..............................................................51 Table 7: Questionnaire Results, Interview Volunteers ............................................51 Table 8: Question #6 Results, Interview Volunteers ...............................................51 Table 9: Questionnaire Results, Volunteer Comparison ..........................................51 Table 10: Question #6 Results, Volunteer Comparison...........................................51 Table 11: Questionnaire Results, Interview Participants .........................................52 Table 12: Question #6 Results, Interview Participants ............................................52 Table 13: Three Models of Limit .............................................................................65 Table 14: Instances for Each Coded Category .........................................................69 Table 15: Recorded Instances by Setting .................................................................71 Table 16: Calculator Usage Relative to the Three Models of Limit ........................77 Table 17: Student Misconceptions of Multivariable Limits ..................................115 Table 18: Misconceptions Categorized According to Limit Models .....................116 ix LIST OF FIGURES Figure Page Figure 1: Questionnaire Used by Williams ............................................................23 Figure 2: Three Dimensional Surface Examined by Amanda ................................88 Figure 3: Figure 2 as Viewed from the Positive z  axis ........................................88 Figure 4: Two Views of the Surface Discussed in Excerpt 21 ...............................90 Figure 5: Paths Amanda Creates with the Cursor ..................................................91 Figure 6: Comparison of the Function Amanda and Jennifer Experienced to a Parabloid ......................................................................94 Figure 7: Mike‟s Use of Different “Directions” ...................................................104 Figure 8: Three Dimensional Graph Discussed in Excerpt 57 .............................126 Figure 9: Contour Graph Discussed in Excerpt 59 ...............................................128 Figure 10: Contour Graph Discussed in Excerpt 60 .............................................129 Figure 11: Graph Used for Single Variable Limit Problems ................................135 Figure 12: Two Graphs Shown to Ashley ............................................................136 Figure 13: The Two Graphs Discussed in Excerpt 73 ..........................................141 Figure 14: Figure Drawn by Amanda ...................................................................164 Figure 15: Josh‟s Two “Perspectives” ..................................................................193 Figure 16: Josh‟s Use of Paths on a Contour Graph ............................................199 Figure 17: Josh‟s Use of „Closeness‟ on a Contour Graph ..................................201 Figure 18: Josh Uses „Closeness‟ to Explain a Limit that Does Not Exist ..........2021 CHAPTER I INTRODUCTION Statement of the Problem It is well accepted that the limit concept plays a foundational role in presentday calculus education. At the same time, there is widespread agreement among both educators and researchers that most students struggle to develop a solid understanding of the limit concept (for example: Vinner, 1991). This may be due to the actual depth of concept. Tall (1992) refers to Cornu (1983) and states that "this is the first mathematical concept that students meet where one does not find the result by a straightforward mathematical computation. Instead it is 'surrounded with mystery,‟ in which 'one must arrive at one's destination by a circuitous route'" (Tall, 1992, p.501). The importance of limits in undergraduate calculus, combined with the difficulty students experience in grasping the concept has resulted in much attention from mathematics education researchers. Several researchers have worked to understand popular misconceptions about the limit concept (Davis and Vinner, 1986; Williams, 1991). It has been suggested by several researchers that a well developed notion of limit could be constructed using the metaphor of motion (Carlsen et al., 2002; Kaput, 1979; Monk, 1992; Tall, 1992; Thompson, 1994b). Furthermore, Williams (1991) found that a 2 significant number (30%) of second semester calculus students contained a dynamic view of limit, and that this dynamic viewpoint was extremely resistant to change. For these reasons, I have decided to examine how students who hold a dynamic view of limit conceptualize the multivariable limit concept. For functions of two variables, motion must take place on a surface instead of along a curve. It is not automatically clear how students will interpret motion in this new setting. Furthermore, the application of motion in multivariable calculus has the potential to create an epistemological obstacle in the sense of Cornu (1991) and require students to restructure their understanding of limits. I expect this restructuring to take place in a form of generalization similar to that described by Harel and Tall (1989). With this in mind I have created the following problem statement for this study: Describe how students with a dynamic view of limit generalize their understanding of the limit concept in a multivariable environment. As the problem statement suggests, this is a qualitative research study which results in a description of student behavior. I will focus the description on the following goals: 1. Describe what type of generalization students in this setting tend to experience with respect to the schema outlined by Harel and Tall (1989) which emphasizes three modes of generalization: expansive generalization, reconstructive generalization, and disjunctive generalization. 2. Describe the role of motion in students‟ understanding of multivariable limits. Does it create a cognitive obstacle, or are students able to apply this imagery to the new multivariable situation? 3 3. Describe how students respond to studying multivariable limits in four different contexts: traditional symbolic manipulation, symbolic manipulation involving polar coordinates, threedimensional graphing, and contour graphing. Of particular interest is whether some of these contexts tend to allow students to reconstruct their understanding of the limit concept to more closely resemble the formal definition. Problem Context History of the Limit Concept The concept of limit can trace its history back to ancient Greece. The Greek mathematicians spent most of their energy solving geometry problems. The solutions to many of these problems involved limiting concepts. One of the earliest such solutions was provided by Hippocrates of Chios (not to be confused with the famous doctor, Hippocrates of Cos). He proved that “the ratio of two circles is equal to the ratio of the squares of their diameters” (Edwards, 1979, p. 7). He accomplished this by inscribing polygons inside the circle and showing that the relationship is true for all such polygons. He then concluded that since this is true for all such polygons, it must also be true for a circle. However, Hippocrates had no limit concept capable of finishing his argument. In general, the Greek mathematicians were bothered by the infinite ideas inherent in the limit concept, and soon they began developing methods that could be used to avoid the “horror of the infinite.” Mathematicians such as Eudoxus, Archimedes, and Euclid began using the method of exhaustion to perform calculations such as that of Hippocrates. This method used contradiction to rigorously prove a statement. It depended on the principle that any magnitude can be made smaller than a second magnitude by repeatedly 4 dividing the first magnitude in half. Using these methods, the Greeks (especially Archimedes) were able to solve many modern day calculus problems. Ultimately, these notions gave rise the formation of calculus as we know it today. However, it is important to note that the ancient Greeks contained no explicit concept of limit. In addition they were unable to generalize their methods, and instead chose to start from scratch to solve each problem they faced. Additionally, they were unable to make the connections between problems of areas and tangents which gave rise to modern day calculus (Baron, 1969; Edwards, 1979). For hundreds of years after the era of Greek mathematics, mathematicians were unable to approach the ideas of calculus as understood by the Greeks. Prior to the sixteenth century, the works of the Greek mathematicians were “not always generally accessible and never fully mastered” (Edwards, 1979, p. 98). However, there were many important developments prior to the sixteenth century that made it possible for later mathematicians to approach a new way of understanding limit. Among those developments were various graphical representations of what we would call functions. These ideas were introduced in the fourteenth century by Nicole Oresme (Edwards, 1979; Babb, 2005). However, these graphical representations were not intended to be thought of as a set of corresponding values, like a modern day function graph. Instead, Oresme intended for each vertical height of his graph to represent the „intensity‟ of a quantity (Edwards, 1979; Thompson, 1994a; Babb, 2005). As he wrote in his Treatise on the Configuration of Qualities and Motions, “every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some 5 point of the space or subject of the intensible thing,” (Grant, 1974, as quoted in Edwards, 1979, p.88). By the middle of the seventeenth century these graphical representations had developed quite a bit. Many noted mathematicians of the time used an idea of motion to understand these representations. In fact, Newton “regarded the curve, f(x,y) = 0 as the locus of intersection of two moving lines, one vertical and the other horizontal,” (Edwards, 1979, p. 191). Newton‟s use of motion was no doubt influenced by his mentor, Isaac Barrow. While Newton was a student under Barrow‟s guidance, Barrow gave an important series of lectures on time and motion. Barrow perceived a line as a “trace of a point moving forward… the trace of a moment continuously flowing” (Baron, 1969, p. 240). This use of the metaphor of motion to understand graphical representations characterized much of Newton‟s work. In fact, Bardi (2006) states that “Newton‟s big breakthrough was to view geometry in motion…” (p. 30), and in one of Newton‟s first attempts to compile his early works, To Resolve Problems By Motion in 1666, he “deliberately elects to make the concept of motion the fundamental basis” of his work (Baron, 1969, p. 263). Using these ideas of motion allowed Newton to solve many problems in the development of the calculus; however, a precise definition of limit was still several hundred years away. It was not until the nineteenth century‟s increased focus on mathematical rigor that the formal limit definition as we know it today came into being. One issue that had to be confronted was the notion of infinitesimals. The idea was not a new one. In fact, Fermat came very close to modern limit calculations when he substituted x + e for the 6 variable x and after simplification removed e from the expression (Baron, 1969; Edwards, 1979). It is important to note that Fermat did not consider the value e to approach zero or even become zero (he did not even imply that e should be small), instead he simply removed expressions containing e. It should be noted that even at the time this was questioned by such mathematicians as Rene Descartes (Baron, 1969). However, these ideas and the use of infinitesimal values continued to be popular for hundreds more years. Finally, it was Cauchy who developed “the first comprehensive treatment of mathematical analysis to be based from the outset on a reasonably clear definition of the limit concept” (Edwards, 1979, p. 310). Cauchy‟s notion of limit was based on an infinitely small variable, which he also called an infinitesimal. This is different from the view that an infinitesimal is an infinitely small quantity; instead, according to Cauchy it is a variable whose value decreases indefinitely. Even at this time, the limit concept was “tinged with connotations of continuous motion” (Edwards, 1979, p. 333). The close of the nineteenth century saw the precise construction of the real number system, and with it Weierstrass was able to develop the definition of limit that is commonly used today. His disapproval of the dynamic view of limits led him to create a static formulation in terms of ε and δ which became popular throughout the twentieth century. In summary, I would like to observe several themes which ran through the historical development of the limit concept: The Greeks had problems “passing to the limit.” They much preferred static arguments that did not contain notions of the infinite. 7 Newton (and many others) found the metaphor of motion to be a powerful tool in understanding the concepts of calculus. However, these ideas of motion were unable to provide a rigorous definition of limit, and were eventually replaced with Weierstrass‟s static definition. For many centuries different mathematicians struggled with the meaning of infinitesimals. These ideas usually contained some sense of infinitely small quantities until Cauchy used a dynamic view of infinitesimal to create a more coherent meaning of the concept of limit. In conclusion, it should be pointed out that the limit concept was understood in many ways throughout history. Each of these ways of thinking made it possible for mathematicians to understand limits in a useful manner, but each way of thinking also created a barrier towards understanding the limit concept in the way we know it today. In that respect, these ways of thinking created epistemological obstacles in the sense of Cornu (1991). Curriculum Change and the Limit Concept The first half of the twentieth century saw calculus generally reserved for undergraduate education and rarely discussed in high school settings. This time period was marked by an emphasis on twotrack high school mathematics programs (Jones, 1970). With the Great Depression came decreased college enrollment, and educators responded by focusing on functional competence as the key objective of high school mathematics. Often times mathematics classes became electives, and as a result there was a trend for colleges to lessen their mathematics requirements for admission (Jones, 1970). The affect of this atmosphere on teaching the limit concept is not entirely clear; 8 however, it might be reasonable to conclude that approaches to limits, as well as other calculus concepts, mirrored that of other subjects in their focus on functional competence. In that sense, we would assume that the limit concept was taught primarily as a procedure by which a certain result could be obtained. After World War II the educational climate in the United States began to change dramatically. Technological advances made during and immediately following World War II revitalized the status of mathematics and science in the country. Colleges saw an increase in enrollment partially due to returning soldiers attending college on the “GI Bill” (Jones, 1970). At the same time, the country began recognizing that its population was not prepared to meet the demands of a new technological society. Accompanying all this with the feeling that America was beginning to fall behind the rest of the world scientifically, emphasized by the Soviet launch of Sputnik in 1957. The United States began a period of reexamining the way mathematics was taught at all levels throughout the country. The resulting period from the early 1960s to the mid 1970s became known as the “New Math Era” and was marked by an increased focus on abstraction and mathematical rigor (Bosse, 1995). An early introduction to key mathematical ideas also marked this period resulting in a push for calculus to be introduced to students while in high school. The effect this had on teaching calculus and the limit concept was significant. Calculus was approached in a more rigorous manner than before and became a more common element in a student‟s education. One of the most controversial reports coming from this time was the Cambridge Conference on School Mathematics (1970). The Cambridge Conference was considered an ambitious goal set out to challenge the mathematics education community on what can 9 be accomplished (Adler, 1970). This report called for calculus to be taught using “precise formulations” rather than what it refers to as “loose calculus,” which “deals with „variables‟ (in a Leibnizian sense) rather than functions,” (Cambridge Conference on School Mathematics, 1970, p. 40). The Cambridge Conference set forth two proposed curricular programs, both of which featured a rigorous treatment of calculus in the final two years of high school. However, there was some disagreement whether calculus should first be introduced at an earlier time on a more intuitive basis. The argument against an introduction was that The student who has already developed some taste for mathematical rigor will be dissatisfied with only half the story in calculus when the fundamental concepts are not carefully defined and precisely used. Because he cannot carry his arguments back to welldefined concepts, he will not fully understand what calculus is about. Finally, one often forms wrong impressions in an intuitive approach which are hard to “unlearn” later, and the luster is worn off the subject when one has to return to it later to tie together loose ends (Cambridge Conference on School Mathematics, 1970, p. 47). On the other hand, the Cambridge Conference recognized the historical significance of the calculus and wanted all students to be able to appreciate it whether or not they completed the final years of the mathematics program. In the late 1960s and early 1970s a backlash against the “New Math Era” began. This resulted in several different movements, including the “Back to the Basics” movement. Importantly, most reform movements after the mid1970s called for a decreased emphasis on mathematical rigor. In their publication, Agenda for Action, The 10 National Council of Teachers of Mathematics (NCTM) laid out its recommendations for school mathematics in the 1980s. Among other things, this publication called for “the use of imagery, visualization, and spatial concepts” (1980, p. 3) to understand mathematical ideas. This is clearly a different emphasis than the precise definitions used by the Cambridge Conference. In addition, there became a question as to the need and relevance of calculus. In the same publication, the NTCM challenges mathematics educators and college mathematicians to “reevaluate the role of calculus” (ibid, p. 21) in school curriculum. Importantly, it was suggested that perhaps calculus should not be the focal point of college preparatory mathematics and that other branches of the mathematical sciences should be encouraged in its place. A few years later, Shirley Hill made the case for a new curriculum that suggested an alternative path for capable students which “would stress statistics and computer science rather than calculus” (Hill, 1982, p. 116). By the late 1980s and early 1990s educators began to focus again on calculus as a foundation of mathematical learning. In A Call for Change, the Mathematical Association of American (MAA) set out recommendations for teacher preparation. In this they called for teachers to model real world problems using calculus and to explore the concepts of calculus both on an intuitive basis and in depth (Leitzel, ed., 1991). From the perspective of this publication, the emphasis is clearly on gaining an intuitive understanding of calculus. It writes, Historically, while investigating continuous processes, many of the ideas and techniques of calculus were developed and used on an intuitive basis before the theory was made rigorous… By building an intuitive base for analyzing 11 continuous processes, these teachers might be more willing to take intellectual risks in their own classrooms. The actual material covered is less important than developing conceptual understanding of the ideas (ibid, p. 35). In 1989, the NCTM released their Curriculum and Evaluation Standards for School Mathematics. In this publication, the NCTM “does not advocate the formal study of calculus in high school for all students or even for collegeintending students. Rather, if calls for opportunities for students to systematically, but informally, investigate the central ideas of calculus” (NCTM, 1989, p. 180). Therefore, by the early 1990s the trend in mathematics education was to teach calculus on an informal, intuitive basis rather than using the precise formulations and rigor of the “New Math Era.” The calculus reform movement is generally considered to have begun in 1986 with the Tulane calculus conference (Schoenfeld, 1995). This conference resulted in The MAA‟s publication of Toward a Lean and Lively Calculus (Douglas, 1986) which aimed to slim down the calculus curriculum by teaching fewer topics but covering them in greater depth. This began a period marked by numerous projects all aimed at reforming the calculus curriculum. There was significant variation between these different projects, but most incorporated an increased use of technology, an emphasis on applications, and the use of multiple representations (Ganter, 1999). The increased emphasis on technology in mathematical teaching and learning was nearly inevitable with the increased availability of technology in society. In many ways, this emphasis of technology spurred on the other major changes during the calculus reform movement. The use of technology in mathematical learning allowed students to encounter problems in real world settings that would have been impossible before. This 12 allowed the reform movement to place an emphasis on application problems, often projects spanning over multiple class periods. In addition, the use of technology allowed for easy transitions between symbolic, numerical, and graphical representations of functions. This ease of transition allowed the reformers to place a greater emphasis on multiple representations in the classroom. This emphasis on multiple representations is of a particular interest to this study. This is one of the foundations of the Harvard Consortium‟s hallmark textbook (HughesHallett, et al. 1994). This text emphasized the “Rule of Three,” which pushed for all concepts, in particular function concepts, to be studied in graphical, numerical, and analytical settings. The group later reformed this concept to the “Rule of Four” which added verbal representations to the list (Schoenfeld, 1995). This notion of multiple representations found its way into publications beyond just those of the calculus reform movement. For example, A Call for Change, published by The Mathematical Association of America, was written as a recommendation for the curriculum of teachers of mathematics (Leitzel (Ed.), 1991). This publication called for teachers to be able to “represent functions as symbolic expressions, verbal descriptions, tables, and graphs and move from one representation to another” (p. 31). This push for representing functions in multiple ways brought with it a notion of function that was broader than before. Instead of restricting the notion of function to its definition, there is now an emphasis on thinking of functions in a wide variety of manners. The purpose of the present study is to understand how students connect and generalize these different representations of the limit concept. In particular, how do they generalize a graphical notion of dynamic 13 motion in multivariable settings and how does the representation of the multivariable function impact this generalization? Importantly, we see that calculus education has undergone several major changes during the past century. The major characteristics of each period were: PreWorld War II was marked by an emphasis on “functional mathematics” The “New Math Era” came after World War II and encouraged an increased focus on mathematical rigor and precision and an earlier introduction to mathematical topics. The backlash to the “New Math Era” resulted in calls to return to the basics in teaching math. This resulted in a decreased emphasis in mathematical rigor. During this time there also came a reevaluation of the role of calculus in education with some experts calling for programs which emphasize statistics and computer science over calculus. During the 1990s, the central ideas of calculus again became a center piece of mathematics education. Experts called for these central ideas to be approached, at least throughout high school, though informal intuition rather than with a formal calculus course. The Calculus Reform Movement began in 1986 and was marked by an increased emphasis on technology, applications, and multiple representations. In particular, the emphasis on multiple representations called for students to be able to connect symbolic, numeric, and graphical representations of a function. In conclusion, it should be noted that the precise, formal definition was once a foundation of calculus education during the “New Math Era.” However, today, formal definitions 14 have been replaced by intuition and informal understanding using multiple representations. It is in this spirit that I will explore these informal notions of limit that are developed by students and how these notions manifest themselves in a multivariable environment. 15 CHAPTER II RELEVANT LITERATURE Dynamic Imagery and Covariational Reasoning One common issue in research on students‟ understanding of calculus is the use of dynamic imagery to represent functions. It was observed that students struggled to understand the function concept in the traditional correspondence manner. As a result, researchers began studying the use of a dynamic understanding of function. Monk (1992) labeled these modes of thinking as “pointwise” and “acrosstime.” He observed that for some problems it was advantageous for students to use “acrosstime” thinking to make sense of the situation. As a result of more study, a number of scholars including Kaput (1994) began pushing for a more dynamic view of function in school curriculum. As summarized by Thompson (1994a), “today‟s static picture of function hides many of the intellectual achievements that gave rise to our current conceptions.” (p. 29) Over the next few years more studies in the vein of Monk‟s 1992 study were conducted and several authors began referring to this type of “acrosstime” thinking about functions as covariational reasoning. Confrey and Smith (1994, 1995) wrote some of the first publications referring to covariational reasoning. They mixed their notion of covariation, which was for students to “coordinate values in two different columns” 16 (1995, p.78), with an emphasis on multiplication as “splitting” instead of repeated addition. They noted that using covariation to understand functions “makes the rate of change concept more visible and at the same time, more critical” (1994, p.138). This connection to rate of change is central throughout the research on functions as covariation. Thompson (1994b) studied the relationship between students‟ understanding of the fundamental theorem of calculus and their concepts of rate of change. He suggested that student‟s difficulties with the fundamental theorem of calculus are rooted in their poor understanding of rate of change and their inability to develop an image of function as covariation. During this time the notion of covariation evolved from the idea of coordinating the values in two columns of data to one of holding two values of a function in mind simultaneously. In 1998, Saldanha and Thompson further explained their view of covariation by noting that “In early development one coordinates two quantities‟ values – think of one, then the other, then the first, then the second, and so on. Later images of covariation entail understanding time as a continuous quantity, so that, in one‟s image, the two quantities‟ values persist” (p. 298). In this sense, understanding function as covariation is more than a special way to look at a table or graph. It is a way of thinking that includes two different changing values which simultaneously depend on each other. Cottrill et al. (1995) also noticed these simultaneous changing values while exploring how students come to understand the limit concept. They used the theoretical perspective of APOS theory (to be explained in the “theoretical perspectives” section) to create a description of how people come to learn about the concept of limits. From their perspective, one of the key difficulties in coming to understand the limit concept lies in 17 the complexity of the concept itself. They argue that successfully constructing a limit schema involves coordinating two different processes together through complicated existential and universal quantifiers, which is a task that remains inaccessible to most students. So, similar to Saldanha and Thompson‟s (1998) view that students‟ understanding of covariation must simultaneously coordinate two changing values in their minds, Cottrill et al. (1995) found that understanding the limit concept requires coordinating two simultaneous processes. In the study done by Saldanha and Thompson (1998) the researchers observed an eighth grade student as he dealt with covarying quantities. Two important elements came from this study. The first element was that coming to understand functions as covariational quantities is a nontrivial task. The second was that the notion of covariation is developmental. It is in that vein that Carlson et al. (2002) developed a framework for studying functions as covariation. In this study they developed five mental actions and five corresponding levels of reasoning which could be associated with varying degrees of understanding functions as covarying quantities. In the lowest levels of understanding, students are able to coordinate the change in one variable with change in another, but with little understanding of the degree to which changing one quantity will affect another. Meanwhile the highest levels of understanding require that a student can hold in his/her mind the instantaneous rate of change of one quantity with respect to another and realize this as a continuously changing rate as the value of the independent variable changes. It was found in this study that many students understood functions as covarying quantities on a lower level, but few achieved a high level of covariational 18 understanding. It was also suggested that this type of dynamic reasoning might be an example of transformational reasoning as described by Simon (1996). Several studies have looked at the existence of such dynamic function concepts with respect to the concept of limit. Williams (1991) found that a dynamic view of limit was common among students and that it was extremely resistant to change. Many other authors tend to agree “that cognitively the strongest images are the dynamic ones,” (MamonaDowns, 2001, p. 264). On the other hand Oehrtman (2002, 2003) classified motion as a “weak metaphor.” He found that while students frequently refer to situations using the language of motion, they are not describing something which is actually moving. These two studies will be discussed in greater detail in the next section, and it is one of the subfocuses of this study to examine the nature and strength of such dynamic images in students while they encounter multivariable limits. Students’ Conceptions of the Limit Concept Over the past twentyfive years, there have been many studies on how students understand the limit concept. They vary in many ways. Some focus on limits as understood in an introductory calculus class (for example, Williams, 1991) while others focus on limits of sequences and series (for example, Alcock and Simpson, 2004). Some attempt to characterize common misconceptions (for example, Davis and Vinner, 1986) while others attempt to describe how a students comes to understand the concept (for example, Cottrill, et al., 1995). However all studies share the findings that the limit concept is difficult for students to grasp and a complete understanding of the limit concept is rare. 19 Many authors have emphasized the language used when referring to limits. It is common in beginning calculus courses to use the phrase approaching when referring to a limit; however, it has been pointed out that in everyday language phrases such as “approaches,” “tend towards,” or “gets close to,” carry a connotation that the point is never actually reached (Schwarzenberger and Tall, 1978). In this sense, the words used to refer to limit concepts often carry everyday meanings that are in conflict with their mathematical meanings (Monaghan, 1991). Davis and Vinner (1986) suggest that this influence of language is an unavoidable obstacle towards understanding the limit concept. Beyond the obstacle of language, Davis and Vinner (1986) attempted to characterize several predominant misconceptions found among students studying limits of sequences and series. Several of these misconceptions are clearly related to understanding limits in beginning calculus. Among the related misconceptions are: A sequence can never reach its limit; a limit is a bound on the sequence; and a sequence must have a final term. The first two of these misconceptions were studied by Williams (1991). He found that 70% of the students in his study agreed that “a limit is a number or point the function gets close to but never reaches,” and 33% agreed that “a limit is a number or point past which a function cannot go,” (p. 221). From this we see that these misconceptions, and in particular the misconception of limit as being unreachable, are prevalent among students studying limits in an introductory calculus course. It has also been shown that students often hold to naïve beliefs about limits. Tall (1992) describes the “generic limit property” which is a belief that if every term of a sequence contains a common property, it can be assumed that the limit of that sequence will also contain that property. As Tall (1992) notes, this belief has its roots in the 20 history of mathematics, “as in Cauchy‟s belief that the limit of continuous functions must again be continuous” (p. 502). He contends that this naïve belief is a cause of the popular misconception that 0.99999… is strictly less than one. Cornu (1991) looked at limits through the lens of epistemological obstacles. From his perspective “it is useful to study the history of the concept to locate periods of slow development and the difficulties which arose which may indicate the presence of epistemological obstacles” (p. 159). From analyzing the history of the limit concept, Cornu located four such obstacles: 1. “The failure to link geometry with numbers” (p. 159). This was evidenced by the Greeks‟ study of the limit concept. They were able to use very sophisticated geometric limiting arguments to solve a variety of interesting problems. However, in their studies, each problem was approached in its own geometrical context. They were only able to apply these ideas to magnitudes, not numbers, and therefore they were unable to generalize their efforts to a unifying concept of limit. 2. “The notion of the infinitely large and infinitely small” (p. 160). Many great mathematicians, including Isaac Newton and AugistinLouis Cauchy, struggled with the notion of infinitesimal quantities. The idea of an infinitely small quantity was freely used by Leonhard Euler to solve a variety of interesting problems. However, it was not until Cauchy described the infinitesimal as a variable which tends to zero, and Weierstrauss developed a static definition of limit that the current limit concept came to being. 21 3. “The metaphysical aspect of the notion of limit” (p.161). The limit concept has often times more closely resembled a philosophical subject rather than a mathematical one. Many great mathematicians, from the Greek mathematicians to Joseph Louis Lagrange and the European mathematicians of the 18th century, expressed horror at the metaphysical aspects of limit. Cornu states that many students today find themselves in a similar situation when they are able to compute using limits but fail to understand it as “real” mathematics. 4. “Is the limit attained or not?” (p. 161). This question was debated for centuries among top mathematicians. Some believed that a quantity can only be made as close as we like to its limit, while others believed that at some point the infinitely small quantities “vanished” allowing the quantity to actually achieve its limit. As observed by Davis and Vinner (1986) and Williams (1991) this is still a question among students. Although, it should be said that these two studies viewed it from the vantage point that the limit may be attained (such as for the limit of a constant function) while Cornu viewed the phrase “attaining the limit” to mean that the limit must be attained as the limit point is approached. Both Sierpinska (1987) and Williams (1991) also studied epistemological obstacles related to limits. Sierpinska noted four sources of epistemological obstacles: scientific knowledge, infinity, function, and real number. Through her study she attempted to cause cognitive conflict in students who have various cognitive obstacles in an effort to help each student overcome his/her obstacles. She found that none of the obstacles had been completely overcome, however some cognitive conflict did take place. In her 22 opinion, attitudes towards scientific and mathematical knowledge created another obstacle which was difficult for these students to overcome. In another study about students‟ attitudes, Szyldlik (2000) studied the relationship between students' understanding of the limit concept and their sources of conviction. She found that students who hold internal sources of conviction are more likely to see calculus as logical and consistent. These students are better able to use definitions and logic to make sense of mathematics, more likely to have a coherent understanding of the limit concept, and more likely to hold a static conception of limit throughout the interview. On the other hand, she found that students who hold external sources of conviction tend to view mathematics as a collection of procedures and rules to be memorized and applied in the appropriate situations. To these students, mathematical theory, including definitions, proofs, and counterexamples, is unlikely to play an important role in their understanding. These students are more likely to give incomplete or contradictory explanations of the limit concept, more likely to hold common misconceptions about limits, and less likely to have the ability to explain the procedures they are using. In perhaps the most comprehensive study to date regarding the understanding of the limit concept, Williams (1991) compared students‟ limit models to six limit characterizations and explored a variety of materials intended to cause cognitive conflict within the students. He did this by asking students to complete a questionnaire containing several common beliefs as shown in prior research (see Figure 1). 23 Figure 1. Questionnaire used by Williams (1991, p. 221). According to Williams, “statements 1 – 6 can be characterized as describing limit respectively as (a) dynamictheoretical, (b) acting as a boundary, (c) formal, (d) unreachable, (e) acting as an approximation, and (f) dynamicpractical” (p. 221). From the results of this questionnaire, he found “that students often describe their understandings of limit in terms of two or more of these informal ideas” (p. 225). In addition, the most popular characterizations from his questionnaire were: 1. the dynamictheoretical model, selected as “true” by 80% of respondents and selected the best description by 30% of respondents; 2. the unreachable model, selected as “true” by 70% of respondents and selected the best description by 36% of respondents; and 3. the formal model, selected as “true” by 66% of respondents and selected the best description by 19% of respondents. A. Please mark the following six statements about limits as being true or false: 1. T F A limit describes how a function moves as x moves toward a certain point. 2. T F A limit is a number or point past which a function cannot go. 3. T F A limit is a number that the yvalues of a function can be made arbitrarily close to by restricting the xvalues. 4. T F A limit is a number or point the function gets close to but never reaches. 5. T F A limit is an approximation that can be made as accurate as you wish. 6. T F A limit is determined by plugging in numbers closer and closer to a given number until the limit is reached. B. Which of the above statements best describes a limit as you understand it? (Circle one) 1 2 3 4 5 6 None C. Please describe in a few sentences what you understand a limit to be. That is, describe what it means to say that the limit of a function f as xs is some number L. 24 After administering the questionnaire, Williams selected a small group of students to take part in the second phase of the study which consisted of tasks designed to create cognitive conflict within students possessing these informal models of limit. Of most interest to my study are the results of the students identified as possessing the dynamictheoretical model of limit. None of the students containing dynamictheoretical models changed their view of limit during the course of the study. When these students encountered functions that contradicted their current model of limit, often they would dismiss the contradiction as irrelevant – an anomaly that does not pertain to most situations. In this way, Williams noticed that students‟ attitudes towards mathematical truth played a key role in determining their reactions towards the study‟s tasks. Several students made statements that they do not believe a general description of limit exists. As stated by one of the study participants, “I don‟t think there is a definition that is going to fulfill every function there is” (p. 232). Williams found several aspects of models that students valued that might have contributed to their resistance to change their viewpoints when faced with cognitive conflicts. Two of these aspects discussed are expediency and simplicity. Where Williams found that the dynamictheoretical model of limit was common among calculus students and relatively resistant to change, Oehrtman (2002, 2003) classified motion as a “weak” metaphor, stating that language referring to motion was frequently not intended to be a description of something actually moving. In one example which is particularly relevant to this study, students were asked to “Explain what it means for a function of two variables to be continuous” (Oehrtman, 2003, p. 399). In response to this prompt, six of the twentyfive participants actually discussed an object in 25 motion, while another eleven students used motion language without applying that language to an actual object. For the six students who described something as moving, Oehrtman argues “that motion tended to be simply superimposed on another conceptual image that actually carried the structure and logic of their thinking” (p. 402). For example, one student describes a continuous two variable function as a board in which a mouse can run around on without falling through. In this case, the motion of the mouse was not the primary imagery; rather, the primary image was that of a board without any holes in it. Much of Oehtman‟s 2003 work was based on the use of metaphors studied by Lakoff and Nunez (2000). The relevance of this theory for the present study will be discussed in the section “mental representations and conceptual metaphors.” From this perspective, Nunez (1999) points out that there are inherent differences between two conceptualizations of continuity, which he refers to as natural continuity and CauchyWeierstrauss continuity. From Nunez‟s perspective, natural continuity arises from a natural metaphor “a line IS the motion of a traveler tracing that line” (p. 56). From this perspective, the motion creates the line, and continuity is the result of fluid motion. Contrasting this view of continuity is the CauchyWeierstrauss view of continuity, which is the result of the 19th century formal mathematics. From Nunez‟s perspective, CauchyWeierstrauss continuity is built upon three conceptual metaphors: “A line IS a set of points; Natural continuity IS gaplessness; Approaching a limit IS preservation of closeness near a point” (p. 57). To Nunez, these three metaphors create a separate conceptualization of continuity that contrasts significantly with the conceptualization of natural continuity. 26 Just as Nunez observed a cognitive difference between natural continuity and the formal definition of continuity, other authors have noticed significant difference between students‟ understanding of limits and the formal definition of limit. One significant difference is described by Kyeong Roh Hah (2005) as reversibility or reverse thinking. These words describe one inherent difference in the formal limit approach to the intuitive approach often used by students. Introductory calculus often teaches the limit concept as the result of the function as the independent variable gets closer and closer to the limit point. Notice, in this case it is the independent variable which is made close to the limit point and the value of the dependent variable is observed. This contrasts the formal definition of limit which requires that the dependent variable can be made arbitrarily close to the limit value for all values of the independent variable within some neighborhood of the limit point. In this case it is the dependent variable that is being made close to the limit value instead of the independent variable being made close to the limit point. Roh Hah studied this type of reverse thinking in the context of infinite sequences and described it as “the ability to think of the infinite process in defining the limit in terms of the index and simultaneously to reverse the process by finding an appropriate index in terms of an arbitrarily chosen error bound” (pp. 2021). Theoretical Perspectives In the 1950s and 1960s, the "new math" movement brought with it an increased emphasis on clear definitions and mathematical rigor. However, by the late 1970s and early 1980s, mathematics education researchers began observing sharp differences between mathematical concepts as they were taught in class and the concepts as they were understood by the students. In particular, several authors noted key difficulties 27 understanding such concepts as limits of functions (Ervynck, 1981; Sierpinska, 1987) and limits of sequences and series (Davis and Vinner, 1986). At this time researchers began distinguishing between mathematical ideas as they are presented in formal mathematics and those same ideas as they are understood by students. The terms concept image and concept definition were created to describe this difference (Tall and Vinner, 1981; Vinner and Hershkowitz, 1980). The term concept image was introduced to describe the "total cognitive structure that is associated with the concept, which includes mental pictures and associated properties and processes" (Tall and Vinner, 1981, p.152). On the other hand, the concept definition was created to refer to a formal definition of a concept, such as a definition found in a textbook. In line with constructivist learning theories, this idea focuses learning and understanding on the individual and his or her conceptions rather than on the formal words used to describe a concept. Cognitive Obstacles Other researchers showed that large parts of students‟ concept images are built on intuition and experiences gained outside the formal teaching of a subject. Cornu (1991) refers to these conceptions of an idea obtained from daily experience prior to formal instruction as spontaneous conceptions. These conceptions can be quite powerful and do not disappear when formal ideas are presented. Instead, these spontaneous conceptions and any new knowledge obtained from instruction may coexist independently or they may intermingle to form new conceptions in the student. This occurs even if the different ideas conflict with each other. According to Papert (1980), "Sometimes the conflicting pieces of knowledge can be reconciled, sometimes one or the other must be abandoned, 28 and sometimes the two can both be 'kept around' if safely maintained in separate mental compartments" (Papert, 1980, p.121). Importantly, it was observed that learning a new idea, in itself, was not enough to change a students' prior conceptions. Instead, the student might simultaneously hold on to both ideas and then select which one to use in any given situation. Students may even retrieve combinations of the two ideas, with detrimental results (Davis and Vinner, 1986). In response to these observations, researchers began looking for models of cognitive change which might describe how a student may conceptually reorganize a concept. One such model requires that three criteria must be met before a student will be willing to undergo conceptual reorganization. First, the student must be dissatisfied with the current organization of a concept. Second, an alternative conception must be available which the student finds both reasonable and understandable. Third, the student must come to view this alternative conception as useful or valuable (Posner, Strike, Hewson and Gertzog, 1982). In a similar vein, Nussbaum and Novick (1982) propose an instructional method which will allow students to create conceptual change. They propose that, first, the student take part in an exposing event created to help students become acquainted with their own current conceptions. The student is then exposed to a discrepant event created to cause dissatisfaction towards the student's current conceptions. Finally a resolution is provided which gives the student an opportunity to interact with new, alternative conceptions. From these ideas of cognitive change, several researchers in mathematics education began studying cognitive obstacles. A cognitive obstacle can be described as a conception that creates a barrier to further student understanding. Cornu (1991) describes 29 several different types of obstacles: "genetic and psychological obstacles which occur as a result of the personal development of the student, didactical obstacles which occur because of the nature of the teaching and the teacher, and epistemological obstacles which occur because of the nature of the mathematical concepts themselves" (Cornu, 1991, p.158). Epistemological obstacles are of particular interest to this study since they tend to be conceptions that prove to be quite useful in one domain but create an obstacle when translated into another, similar domain. Furthermore, epistemological obstacles are often unavoidable and essential to learning, and they are frequently found in the historical development of the concept (Cornu, 1991). Mental Representations and Conceptual Metaphors It is important to consider how students come to understand mathematics. One common theoretical perspective is that of mental representations (Williams, 2001). This viewpoint holds that mathematical learning places ideas, facts, and procedures as part of an internal network of mental representations, and the depth of understanding is determined by connections with other representations within this network (Hiebert and Carpenter, 1992). From this perspective, a mathematical idea is understood when its mental representation has a large number of strong, robust connections with other representations. In contrast to this theory is the idea of conceptual metaphors developed by Lakoff and Nunez (2000). From the perspective of these authors, there is an “intimate relation between cognition, mind, and living body experience in the world” (Nunez, 1999, p. 49). In this way, all mathematics is considered to be a result of our embodied experiences, and 30 the meaning of mathematics is built upon conceptual metaphors which project meaning onto new, abstract domains from previously understood, more concrete concepts. It is from this second perspective that this study will be conducted. This would emphasize the belief that knowledge is not only based on connections between related concepts, but that students will actually create meaning for new, abstract concepts based on their understanding of other wellunderstood ideas. In relation to this study on students' understanding of limits in threedimensional calculus, we would expect the students to have previously developed a strong understanding of the limit concept in single variable calculus and they would attempt to project this understanding onto the new, multivariable limit problem. One powerful metaphor used to understand limits is the fictive motion metaphor (Talmy, 1988). This provides a metaphorical means of conceptualizing a static curve as the result of dynamic motion. This metaphor is common throughout the English language. A statement such as “the trail goes to the peak of the mountain” uses dynamic language to capture essence of a static object, a trail. In this way, the fictive motion metaphor is used in mathematics to perceive a graph not as a set of points but a path created by dynamic motion. Since this study focuses on students‟ use of dynamic imagery in multivariable calculus, the fictive motion metaphor has the potential to play an important role in the description of students‟ conceptualization of the multivariable limit concept. Metaphorical thinking has been studied in regard to limits in particular and within mathematics education as a whole. Of particular interest are the studies by Oehrtman (2002, 2003) in which he analyzed the written and verbal language of firstyear calculus 31 students' reasoning about limits. In his research he divides metaphors into weak and strong metaphors. Strong metaphors are ones that "force the relevant concepts involved to change in response to one another" (Oehrtman, 2003, p.398). These metaphors are active and support creative thinking in new domains. Abstraction and Generalization The concept of reflective abstraction was introduced and discussed by Piaget (see, for example Piaget, 1985) to describe the development of logicomathematical structures in a child during cognitive development. Reflective abstraction is considered to be entirely internal, as opposed to empirical abstraction and pseudoempirical abstraction which derive from properties of objects and actions of those objects, respectively. Piaget considered four different kinds of mental constructions that could take place during reflective abstraction: interiorization, or the construction of an internal process in order to represent a perceived phenomena; coordination of two or more processes into a single new one; encapsulation of a dynamic process into a static object; and generalization, or the application of existing knowledge to new phenomena. Generalization is discussed in Harel and Tall (1989) and he distinguishes three different types of generalization that may occur. Expansive generalization extends an individual's thinking from one domain to another without changing the original ideas. Reconstructive generalization extends an individual‟s thinking while at the same time reconstructing the existing concepts in order to make the generalization reasonable. Harel and Tall also describe disjunctive generalization in which new ideas are created without an attempt to connect them with prior understanding. This is generalization in the sense that the student is familiar with a larger range of concepts, but this could not be 32 considered a mental reconstruction of the student‟s knowledge in the way that Piaget discusses it. Infinite Processes Many difficulties in understanding the limit concept are found in the infinite nature of limits. The limits concept is often conceptualized as an infinite process which can never be completed in its entirety. However, even though this process cannot be completed, mathematicians are capable of speaking about the limit process as a coherent whole, and they are capable of using the result of a limit as an object to create more sophisticated processes. This means of understanding a process as an object is not unique to the limit concept, but plays a vital role in all of mathematics. Tall et al. (2000) give a thorough description of several authors‟ descriptions of the cognitive processes involved in converting a mathematical process to an encapsulated object. I will briefly describe several of these viewpoints below. Dubinsky (1991) and his colleagues developed a theory of conceptual development based on the creation of actions, processes, objects, and schemas. This theory has become known as APOS theory. In this model of student learning, the student first understands a mathematical concept as an action to be performed. After some experience with the action, the student is able to perceive the action as a process and speak of its result without being required to perform the action. Eventually this process will be encapsulated into an object which can in turn be used to create more sophisticated mathematical actions. The student then gathers these related actions, processes and objects into a coherent collection called a schema. The concept of schema is similar to 33 that of a concept image, with the exception that a schema is required to be coherent while a concept image is not. From the perspective of APOS theory, an action is a stepbystep mathematical procedure. In order for a concept to progress from an action to process, the individual must become aware of the various steps involved in the action and have the ability to reflect on them. It is this ability to think about the action without actually performing it that distinguishes a process from an action. The primary difference between a process and an object is the ability to conceptualize the process as a whole and perform actions with it. Sfard (1991) prefers to the use the word reification over encapsulation to describe the process of understanding a mathematical process as an object. For Sfard the act of reification is a movement from an operational understanding to a structural understanding. Sfard‟s description of the transformation of a process to an object takes place in three steps. She describes the adoption of a familiar process as the interiorization of that process. Once interiorized, the process can be compacted and understood as a whole; which she refers to as condensation. To Sfard a condensed process is still operational and the individual will interact with the process in an operational manner. It is the process of reification that transitions the individual from dealing with an operational process to a structural object. To Sfard it is precisely this transition from operational to structural understanding that signifies the transition from a condensed process to a reified object. Eddie Gray and David Tall (1994) used the word procept to describe their understanding of how a mathematical concept can take the form of both a process and an 34 object simultaneously. From their perspective the important development is the creation of a symbol to represent both the process and the object is an essential part of the development of a procept. Many of these authors developed their theories using finite procedures, such as counting or addition. The infinite nature of the limit concept makes it particularly challenging for an individual to progress from viewing the concept as a process to viewing it as an object. Tall et al. (2000) discussed this difficulty and pointed out that in “the peculiar case of the limit concept where the (potentially infinite) process of computing a limit may not have a finite algorithm at all […] a procept may exist, which has both a process (tending to a limit) and a concept (of limit), yet there is no procedure to compute the desired result” (p. 226). 35 CHAPTER III METHODS The methods of this study were developed with the problem statement in mind: “Describe how students with a dynamic view of limit generalize their understanding of the limit concept in a multivariable environment.” In particular, the study was designed to accomplish the following objectives; the study should: Identify students with a dynamic view of limit. Provide an opportunity to analyze participants‟ prior understanding of the limit concept. Allow participants to encounter the limit concept in multivariable environments. Provide an opportunity to observe participants‟ generalization of the limit concept in these multivariable environments. In this chapter I will begin by explaining why qualitative research methods were used to design this study. I will then describe the methods of the study, including the participants, settings, interviewee selection method, and data collection methods. Finally, I will conclude this chapter with a discussion about the researcher‟s role and perspective. Why a Qualitative Study? Data collection for the study took place during the fall 2007 semester at a large state university. It was determined that a series of indepth interviews using qualitative analysis would be required to adequately respond to the study‟s problem statement. 36 Maxwell (1996) describes five research purposes which are specifically suited to qualitative analysis. “Understanding the meaning, for participants in the study, of the events, situations, and actions that are involved with and of the accounts that they give of their lives and experiences” (p. 17). “Understanding the particular context within which the participants act, and the influence that this context has on their actions” (p. 17). “Identifying unanticipated phenomena and influences, and generating new grounded theories about the latter” (p. 19). “Understanding the process by which events and actions take place” (p. 19). Developing causal explanations” (p. 20). An examination of each of these five research purposes creates a strong argument for this study to be qualitative in nature. The purpose of this study is to create a description of students‟ conceptualizations of the limit concept. Internal conceptualizations are, by their nature, not observable. Therefore, observable data must take the form of written responses, mathematical calculations, and verbal descriptions of mathematical concepts. The primary interest of the study is not in reporting the observable data, but rather in understanding the meaning of the observable data in terms of the students‟ internal conceptualizations of the limit concept. Because of this, the context of the observable data and the process which students use to create mathematical conclusions play an essential role in understanding the meaning of the observable data. 37 The fact that this is a first study on students‟ conceptualizations of multivariable limits is an important factor in choosing a qualitative study. The lack of previous research reports on the topic requires an ability to respond to unanticipated events. A quantitative study cannot be prepared to deal with unanticipated behavior, but a qualitative study using grounded theory techniques can describe unanticipated behavior in the context in which it happens. Finally, the purpose of this study asks a causal question that requires qualitative methods to answer. The causal question in this study is “what role does prior understanding of the limit concept have in students‟ conceptualizations of multivariable limits?” This question is qualitative in the sense that it seeks to describe the influence of certain cognitive events on other events. This contrasts a quantitative question which seeks to explain current events in terms of the variance of a previous set of events. For the above reasons a qualitative study was developed that allowed the researcher to observe and interview students interacting with multivariable limits. The study was developed to contain three key components, a) an initial questionnaire which provided a means of selecting interview participants and comparing those participants to the student population as a whole, b) an interview probing students‟ understanding of single variable calculus, and c) a series of two interviews involving multivariable limits in four different settings. Participants and Setting The goal of this study was to analyze the changes in student thinking about the limit concept as they encounter multivariable limits. For this reason, it was important to observe students who are familiar with both multivariable functions and the limit concept 38 but have yet to study multivariable limits. The university where this study took place teaches calculus as a threepart sequence with multivariable calculus being a main focus of the third semester in this sequence. Therefore, participants for the study were chosen from this third semester calculus course. Participants for the study were chosen using a process of purposeful selection. The selection of a purposeful sampling seeks to choose uniquely qualified individuals capable of providing information necessary to answer the study‟s research questions as described by Maxwell (1996). For a smallscale study, purposeful selection is often preferred to a random sample. Random samples are necessary to externally generalize the findings of the study; however, since external generalization requires a sufficiently large sample size, in the case of a small sample size it is preferred to purposefully select participants likely to provide useful information towards answering the study‟s research questions. It is important to note that the sample chosen for this study was not what is often called a „convenience sample.‟ Rather than choosing participants based on convenience, participants were chosen using the predetermined set of guidelines outlined in the next section. The purposeful selection process of this study sought to find students who tend to conceptualize the limit concept in a dynamic manner. Since the goal of this project is to describe student‟s cognitive behavior, preference was given to students who demonstrated a strong ability to express themselves in a clear manner. For these reasons, a questionnaire was given to all willing students who participated in third semester calculus in the fall of 2007. This questionnaire had two useful purposes. Most importantly, it provided an opportunity to analyze a large number of students‟ 39 understanding of the limit concept, allowing those students with the preferences described above to be selected for the interviews. It also created a description of the entire population of students enrolled in this course, indicating how well our selected students represented the course‟s population as a whole. Study Questionnaire The development of the questionnaire (see Appendix A) was inspired by Williams (1991). Part A consisted of six definitions each representing a different theoretical model of limit commonly held by students studying calculus. It is important to note that actual models of limit held by students are extremely complex cognitive structures, and they are not expected to precisely line up with any theoretical model given here. For this reason, I will use the phrase theoretical model to refer to those theoretical models of limit that I believe a student may possess, and I will use the phrase personal model to refer the actual model of limit held by an individual student. Similar to Williams, the beginning questionnaire attempted to gauge to what degree students agree with various theoretical models. Each of these models was inspired by either research on students‟ understandings of the limit concept or an historical development of the limit concept. The model represented by each question is given below: Question 1. “A limit describes how a function moves as you approach a given point.” Dynamic Model. This model is based on the dynamic imagery that the graph of a function is the path of a point swept out over time. This imagery was used by Newton when he developed the calculus (Edwards, 1979) and has been shown to be common among calculus students studying the limit concept (Williams, 1991). 40 This is the theoretical model we are most interested in and represents what Williams refers to as a dynamictheoretical model. Question 2. “A limit can be found by plugging in a number infinitely close to a point.” Infinitesimal Model. This model is based on the existence of infinitely small quantities. Cornu (1991) discussed this as an epistemological obstacle towards the development of a formal limit concept. Question 3. “A limit is a number that a function can be made arbitrarily close to by taking values sufficiently close to a certain point.” Formal Model. This model is based on the modern, CauchyWeierstrass definition of limit, and closely mimics the definition of limit given in many introductory calculus textbooks. Question 4. “A limit is a number or point the function gets close to but never reaches.” Unreachable Model. This model is based on the popular misconception that a limit can never be attained (Davis and Vinner, 1986). Question 5. “A limit is an approximation that can be made as accurate as you wish.” Approximation Model. Based on the notion of limit as an approximation. This was used in the study by Williams (1991). As observed by Williams, students rarely possess a personal model of limit closely aligned to one of these theoretical models. Instead, students‟ personal models of limit tended to be complex combinations of these theoretical models. For this reason, classification of students into distinct categories is extremely difficult. Questions 6  8 were included to better distinguish which theoretical model (or models) best represented each student‟s personal model of limit (see Appendix A). 41 In our process of purposeful selection, the following criteria were used to select which students would take part in the remainder of the study. 1. Students should select “somewhat agree” or “strongly agree” to question 1 of the questionnaire. 2. Students should circle the number 1 on question 6 of the questionnaire. 3. Students should use dynamic language in their responses to questions 1, 7, and 8. Dynamic language is considered to be language that emphasizes the use of motion in understanding the limit concept. Such language might include key phrases such as “moves towards,” “approaches,” or “gets closer to.” 4. Students should provide written descriptions that demonstrate an ability to express themselves in a clear manner. From these four criteria, students were invited to participate in the interview portion of the study in the following manner: Questionnaires were collected from all students indicating interest in participating in the interviews. Questionnaires that failed to meet #1 above were removed from consideration. The remaining questionnaires were analyzed and those that did not meet #2 or #3 above were removed from consideration. The remaining questionnaires were analyzed along both #3 and #4 above, and students were judged as to how strongly they met each of the criteria. Students who were judged to have used strong dynamic language in their responses as described in #3 above and who were judged to demonstrate a strong ability to express their thinking as describes in #4 above were invited to participate in the interviews. In total nine students were invited to participate in the interviews. Seven of the nine students agreed to participate in the interviews and all seven completed interview process. 42 Data Collection Recalling that the goal of this study is to create a description of student behavior, it was decided that a qualitative study using taskbased interviews would be the best method for data collection. Thomas (1998) described various interview strategies used in qualitative studies. The interviews in this study contained questions of two types: general questions discussing the meaning of the limit concept and taskbased questions centered on specific limit problems. The general questions can be described as loose questions in the sense that their goal is to “reveal the variable ways respondents interpret a general question” (Thomas, 1998, p. 129). Taskbased questions can be described as responseguided questions in the sense that they “consist of the interviewer beginning with a prepared question, then spontaneously creating followup queries relating to the interviewee‟s answer to the opening question” (p. 132). In the case of this study, the initial questions take the form of a mathematics problem and followup questions are asked to clarify meaning about students‟ responses while solving the mathematical problem. Data from the interviews took three forms: written work in response to mathematical limit problems, verbal responses to questions throughout the interview, and observation of student behavior throughout the interview (important behaviors might include pointing at a graph or the use of hand gestures). In order to capture both verbal and observational data throughout the course of the interviews, it was decided that videotaping would be a primary source of data collection. Each video segment was stored on DVD disks and viewed only by those involved in overseeing the study. 43 In order to analyze students‟ prior understanding of the limit concept, the first interview session (see Appendix B) focused solely on the students‟ understanding of limits in a traditional introductory calculus course. Participants for this study were carefully chosen so that they possessed a personal model of limit that involves some element of dynamic imagery. Question 1 and question 2 of the first interview session were designed to evaluate the strength and nature of this dynamic imagery in each student‟s personal model of limit. Question 1: Review your answers to the questionnaire given earlier. Would you like to change any of your answers? Are there any answers that you would like to clarify? Question 2: When describing a function as “approaching” or “getting close to” a point, this idea would best be explained as: a) Evaluating a function at different numbers over time with those numbers successively getting closer to the point in question. b) Mentally envisioning a point on a graph moving closer and closer to the limit point. Question 1 reviewed with each student his/her responses to the questionnaire, providing an opportunity for students to explain their responses in detail. Question 2 was designed to evaluate the dynamic nature of each student‟s conception of limit by providing them with two options, one which involves examining a function at various points over time and a second which involves mentally envisioning a point in motion along a graph. Williams (1991) would refer to those students who select (a) as having a dynamicpractical model of limit and those selecting (b) as having a dynamictheoretical model of limit. The remainder of the first interview session was designed to observe students‟ behavior on traditional introductory calculus limit problems in order to 44 determine how the students‟ personal model of limit is put into practice on actual problems (see Appendix B). This portion of the interview involved both symbolic and graphical limit problems from introductory calculus, and a calculator was made available to students for use with these problems. The remaining interview sessions were designed to provide students with the opportunity to encounter multivariable limits in four different settings: traditional symbolic manipulation, symbolic manipulation involving polar coordinates, threedimensional graphing, and contour graphing. These were designed as four separate treatments (see Appendices C and D). Each treatment introduces students to the multivariable setting in question, asks students to describe how to determine whether a multivariable limit exists or not in that setting, asks students to explain why they believe their method should work, and observes students using this method on problems from this setting. It was decided that for purposes of time, these four treatments would take place over the course of two interview sessions, with each session containing a symbolic and graphing portion. The first session contained traditional symbolic manipulation and threedimensional graphing while the second session contained symbolic manipulation involving polar coordinates and contour graphing. In order to allow the students to interact with a larger number of multivariable functions, a computer was used to experience graphs during the three dimensional graphing portion of the interviews and computer generated graphs were printed out and provided during the contour graphing portion of the interview. The mathematical software package Maple 11© was used to create all the graphs used in this study. 45 In addition to these four treatments, students were asked several interview questions following each multivariable limit experience designed to help them reflect on their experiences with multivariable limits. This portion of the interviews was created to provide students with an opportunity to discuss their overall understanding of the limit concept and the connection between single variable and multivariable limits. Researcher’s Perspective In qualitative research it is necessary that the researcher be actively involved in the setting of the research study, and as a consequence the collection and interpretation of the data will be effected by the role the researcher plays. Holliday (2002) explains, “The presence of the researcher in the research setting is unavoidable and must be treated as a resource” (p. 173). Because of this fact, the perspective that I bring with me into the study should be assessed in order to put the data from the interviews in context. For this reason, I will spend the remainder of this section discussing my beliefs and expectations prior to the collection of data for the study. In this study I took the position that students would enter the study with an initial personal model of the limit concept. I use the word model much in the same sense as Williams (1991) to be a collection of cognitive structures which has an internal meaning to the student and carries with it some predictive qualities. I use the phrase personal model in contrast to the phrase theoretical model which captures a hypothetical conceptualization of the limit concept. Throughout the study I expected students to be able to coherently express their beliefs about their personal model of limit and use the model to determine the truthfulness of related mathematical statements. I also expected the students to be able to use this model to make sense of related mathematical ideas; in 46 particular, I expect the students to try to use their personal model of limit to make sense of limit problems in a multivariable setting. I anticipated that the students‟ models of limit would be relatively static, until confronted with a discrepant event, in the sense of Nussbaum and Novick (1982). At the same time, I expected students involved in the interviews to be in the early developmental stages in terms of their personal model of multivariable limits. I expected this notion of multivariable limits to be less coherent and less consistent than their personal model of single variable limits. However, I anticipated that students would use their single variable limit model to interpret multivariable limits, providing a basis for their actions and statements in the new context. It is worthwhile to note that I use the term model to mean something very similar to Dubinsky‟s (1991) use of the word schema. The primary difference is my emphasis on the predictive qualities of a students‟ model of limit, while Dubinsky emphasizes the collection of actions, processes and objects which are contained in a limit schema. However, both these notions are more specific than Tall and Vinner‟s (1981) notion of a student‟s concept image. Both a model and a schema are intended to be coherent in the sense that students are, to some degree, aware of these structures and able to use them in productive ways. Meanwhile, the term concept image is the collection of all cognitive structures connected with the concept. This concept image may not be coherent and a student may have little awareness of it or ability to use it productively. Many past studies have looked at students‟ abilities to use the imagery of motion to understand the limit concept (Monk, 1992; Thompson, 1994b; Carlson et al, 2002). Williams (1991) found that 30 percent of the students in his study contained what he 47 called a dynamictheoretical view of limit, while 80% of the students believed that a dynamictheoretical definition of limit was true. This dynamictheoretical view of limit is marked by a student‟s use of motion to understand the limit concept. Students involved in this study were carefully chosen to have a personal model of limit similar to the dynamictheoretical model of limit and I expected this aspect of their thinking to influence the way they conceptualize multivariable limits. In a multivariable setting, the idea of motion easily assists showing that a limit does not exist, since a limit such as 2 2 2 2 ( , ) (0,0) lim x y x y x y can be shown to not exist by observing that when moving along the yaxis the limit tends to 1, while when moving along the xaxis, the limit tends to 1. Since these two directional limits are unequal, the limit does not exist. This should be somewhat familiar to students; since, in the twodimensional case you can show a limit does not exist by showing that lim f (x) lim f (x) x a x a . However, unlike in the twodimensional case, the multivariable limit is not solved by simply showing that the two directional limits are equal. In fact, in the threedimensional case there are uncountably many directional limits, following any possible path to the point (a,b), that must all be equal for lim ( , ) ( , ) ( , ) f x y x y a b to exist. Therefore, the imagery of motion towards a point is insufficient to show that a multivariable limit exists. Because this use of motion is insufficient to completely understand the multivariable limit problem, I expected students to encounter a cognitive obstacle from their application of motion into multivariable limits. From there, I anticipated an effort 48 on the part of the students to reconstruct their understanding of the limit concept to allow for a complete understanding of multivariable limits. The nature of this anticipated reconstruction process is one of the primary focuses in this study. These expectations color the way I interacted with students during the interviews. As I engaged in responseguided questioning, they affected the types of questions I asked and the manner in which these questioned were presented. I do not believe my role in the interviews should be perceived as a negative aspect of the study design; rather, I believe that my expectations enabled me to guide the interviews towards a line of discourse that would be profitable for answering the study‟s research questions. 49 CHAPTER IV DATA ANALYSIS In this chapter I will describe how the collected data was analyzed. I will begin with a description of the questionnaire results which show how the seven selected interview participants compare to the entire calculus III student population. Then I will describe how the interview data was analyzed qualitatively. Finally, I will describe how this analysis of the transcripts along with an analysis of formal mathematics led to the development of three models of limit: neighborhood, dynamic, and topological. These three models will be used to code the interview data and shape the results provided in chapter V. Questionnaire Results During the first week of the fall semester, the study questionnaire was distributed in all five sections of calculus III offered by the university. All willing students completed the questionnaire at this time and a total of 208 students returned their responses. The results on questions one through five are given in Tables 1 and 2 below. Strongly Somewhat Somewhat Strongly Likert Average Agree Agree Neither Disagree Disagree (4 = strongly agree) Statement 1 105 68 3 22 10 3.13 Statement 2 46 97 19 33 13 2.62 Statement 3 63 93 28 14 5 2.96 Statement 4 84 51 20 30 23 2.69 Statement 5 32 66 35 37 37 2.09 Table 1: Questionnaire Results (Cumulative) 50 Strongly Somewhat Somewhat Strongly Agree Agree Neither Disagree Disagree Statement 1 50% 33% 1% 11% 5% Statement 2 22% 47% 9% 16% 6% Statement 3 31% 46% 14% 7% 2% Statement 4 40% 25% 10% 14% 11% Statement 5 15% 32% 17% 18% 18% Table 2: Questionnaire Results (Percentage) As opposed to a Likert Scale analysis, the result can also be viewed as binomial data, as in tables 3 and 4 below. Agree Disagree Statement 1 173 32 Statement 2 143 46 Statement 3 156 19 Statement 4 135 53 Statement 5 98 74 Table 3: Questionnaire Results (Binomial) Agree Disagree Statement 1 83% 15% Statement 2 69% 22% Statement 3 77% 9% Statement 4 65% 25% Statement 5 47% 36% Table 4: Questionnaire Results (Binomial Percentage) It can be observed from the tables above that the respondents have a strong tendency towards agreeing with the statements as presented. This corresponds with the finding from Williams (1991) that students are often capable of believing several models of limit simultaneously. This result, however, should be treated carefully due to the known phenomenon of acquiescence bias that students tend to agree with statements as presented. Due to this known fact, students were asked which model best described the way they understood the limit concept and the results are presented in tables 5 and 6 below. 51 Statement: 1 2 3 4 5 None 78 15 37 58 5 8 Table 5: Question #6 Results (Cumulative) Statement: 1 2 3 4 5 None 39% 7% 18% 29% 2% 4% Table 6: Question #6 Results (Percentage) From the 208 students who completed the study questionnaire, 36 agreed to take part in the interview portion of the study. Their responses follow in tables 7 and 8. Strongly Somewhat Somewhat Strongly Likert Average Agree Agree Neither Disagree Disagree (4 = strongly agree) Statement 1 20 12 1 3 0 3.36 Statement 2 7 15 6 7 1 2.56 Statement 3 10 17 3 2 3 2.83 Statement 4 13 7 5 6 5 2.47 Statement 5 7 11 3 6 9 2.03 Table 7: Questionnaire Results, Interview Volunteers Statement: 1 2 3 4 5 None 14 4 6 9 0 1 Table 8: Question #6 Results, Interview Volunteers These responses are closely aligned with the responses of the student population as a whole, as tables 9 and 10 show. Agree Disagree 36 Volunteers 208 Students 36 Volunteers 208 Students Statement 1 89% 83% 8% 15% Statement 2 63% 69% 23% 22% Statement 3 84% 77% 16% 9% Statement 4 65% 65% 35% 25% Statement 5 67% 47% 56% 36% Table 9: Questionnaire Results, Volunteer Comparison Statement 1 2 3 4 5 None 36 Volunteers 42% 12% 18% 27% 0% 3% 208 Students 39% 7% 18% 29% 2% 4% Table 10: Question #6 Results, Volunteer Comparison 52 As the above tables show, with a few slight variations the two groups responded to the questionnaire in a similar fashion. The only exception was found in the response to question 5 which was found to be significantly different (p < .05). Since the emphasis of this study is on students with a dynamic understanding of limit, this difference in the students‟ opinions of the approximations model of limit was deemed to be insignificant in light of the study‟s goals. Therefore, this difference was not further explored in this study. From these 36 volunteers 9 were contacted to take part in the individual interview sessions, and of those 9, 7 participated in the interviews. All participants who began the interview process completed all 3 interviews. A total of three males and four females took part in the interview portion of the study. These seven interview participants will be given the pseudonyms Mike, Jessica, Jennifer, Amanda, Josh, Ashley, and Chris for the remainder of this study. These seven students were chosen using a method of purposeful selection so that their questionnaires showed a tendency towards understanding the limit using dynamic imagery. These seven students‟ responses to the questionnaire are shown in tables 11 and 12. Strongly Somewhat Somewhat Strongly Agree Agree Neither Disagree Disagree Statement 1 5 2 0 0 0 Statement 2 2 3 0 1 1 Statement 3 1 3 1 0 2 Statement 4 2 2 1 1 1 Statement 5 1 1 0 2 3 Table 11: Questionnaire Results, Interview Participants Model: 1 2 3 4 5 None 6 0 1 0 0 0 Table 12: Question #6 Results, Interview Participants 53 The above tables demonstrate the clear preference for participants who agree with statement 1 on the questionnaire. As shown by the entire class results, these seven students represent a sizeable portion of the class. From table 4, 83% of the students agree with statement 1 and from table 6, 38% of the students believe statement one best describes how they understand the concept of limit. Both of these numbers were the highest recorded for any of the five statements. Qualitative Data Analysis Qualitative data was analyzed using a grounded theory approach to data analysis. Maxwell (1996) describes grounded theory when he says, “The theory is grounded in the actual data collected, in contrast to a theory that is developed conceptually and then simply tested against empirical data” (p. 33). The fact that little previous research had been recorded on multivariable limits requires that a grounded theory approach be employed. This process of theory development led to the realization that the initial theoretical models of limit created for the study questionnaire were inadequate to describe students‟ interactions with the multivariable limit concept. This led to the creation of three new theoretical models of limit which were observed as part of students‟ descriptions of multivariable limits. These models grew out of the observation that students tend to conceptualize limits using either a) a sense of „closeness,‟ b) a dynamic process, or c) an examination of external features. In addition these models of limit were closely tied to the formal mathematics of the limit concept. 54 Investigating these three models led to a coding scheme for examining how students understand multivariable limits. This coding scheme was used to analyze the data and provide a description of student behavior throughout the interviews. In the remainder of this chapter, the development of the three models of limit will be described, resulting in a coding scheme which shall be used to bring clarity to the interview data. Definitions of Limit in Formal Mathematics In the effort to describe how students understand the multivariable limit concept, it is worthwhile to consider the formal mathematics behind the concept of limit. In this section, we will discuss how the concept of limit is developed formally and what cognitive structures might be necessary to understand this formal development. In formal mathematics, there are two ways to define the concept of limit in ℝn. The traditional definition requires the use of universal and existential quantifiers. Formal Limit Definition: If there exists a value L such that for every positive number, ε > 0, there exists a value, δ > 0 such that f (x) L whenever 0 x a , then we say that the limit of the function, f, as x approaches a is L, and we write f x L x a lim ( ) . In contrast to this formal definition there is a definition based on sequences. Sequential Limit Definition: If for every sequence (x : x a) n i with x a n n lim( ) we have that f x L n n lim ( ) , then we say that the limit of the function, f, as x approaches a is L, and we write f x L x a lim ( ) . 55 For a function, f: ℝn →ℝm, these two definitions can be proven to be equivalent. However, in practice, using each definition involves inherently different cognitive processes. To compare the two definitions, I will describe the possible cognitive processes necessary to develop an understanding of each definition. A possible description of the cognitive structures required for understanding the formal limit definition is given below: 1. Mentally construct a neighborhood of values around the point L. 2. Mentally construct a corresponding neighborhood of values around the point a. a. Coordinate these two constructions such that the neighborhood around a is mapped into the neighborhood around L. 3. Construct a process of reducing the size of the neighborhoods around L while maintaining corresponding, coordinated neighborhoods around a. Note that coordinating these two neighborhoods requires reverse thinking as described by Roh Hah (2005). In contrast to the formal definition, a possible description of the cognitive structures required for understanding limits using the sequential definition of limit is given below: 1. Construct a schema for evaluating the limit using a single sequence. a. This involves mentally constructing a sequence, (xn), that approaches the point a. b. Then constructing the resulting sequence, ( f (xn) ). c. Finally, evaluating the result of this infinite sequence, the point L. 56 2. Mentally construct an infinite process of evaluating the limit of these sequences successively. 3. Capture this infinite process into a coherent understanding of the limit of all possible sequences simultaneously. Capturing infinite processes is an inherently difficult task as described in chapter IV, “Infinite Processes.” Just as Nunez and his colleagues (1999) found that the concept of continuity has two cognitively different conceptualizations, the two notions of limit based on the two formal definitions above appear to be built on inherently different conceptualizations. The formal definition must first be grounded on a notion of „closeness‟ that can be used to create mathematical neighborhoods. This conceptualization is static and relies heavily on the conceptualization of the real number system. On the other hand, the sequential definition is built upon a process of examining points along a sequence. The conceptualization of this process is significantly different from that of mathematical neighborhoods in that it is both dynamic in nature and infinite. Although students rarely understand the limit concept using one of these formal definitions, the two definitions do provide a blueprint for different cognitive structures that can be used to develop an understanding of the limit concept. The idea of „closeness‟ used in the formal definition will form the basis for the neighborhood model of limit, and the dynamic process used in the sequential definition will form a basis for the dynamic model of limit. In the following sections, I will describe these models along with a third model, the topographical model, which is not based in formal mathematics. 57 The Neighborhood Model of Limit It is well documented that students struggle to understand the formal definition of limit. This is, in part, due to the need for “reverse thinking” (Roh Hah, 2005). A description of limit using reverse thinking was observed in only one student throughout the course of the study. This discussion occurred in response to question 5 on the questionnaire which posed an approximation view of the limit concept. Excerpt 1 JESSICA: Yes, because if you get closer to it, like, if you like, x, well you can make it as close to 2 as you want. I mean, yeah, it could take you like years to figure it out but you could get as close as you wanted to depending on what the number that you put in. Jessica‟s use of the phrases “you can make it as close to 2 as you want” and “you could get as close as you wanted to depending on what the number that you put in” both show some evidence of reverse thinking. They show that Jessica has an awareness that goes beyond simple closeness to an understanding that this closeness can be controlled. Oehrtman (2003) has argued that a metaphor of approximation can be used to help students better understand the definition of limit, and Jessica‟s response seems to support this notion. However, this was the only incident of her using language that demonstrated the use of reverse thinking, and it seems unlikely that this notion plays a vital role in her understanding of limits. Statement 3 on the questionnaire was designed to illustrate the formal definition of limit; however, no students used reverse thinking when responding to this statement. Amanda shows her confusion towards the problem in the excerpt below 58 Excerpt 2 AMANDA: Alright, number three, „A limit is a number that a function can be made arbitrarily close to by taking values sufficiently close to a certain point.‟ This one I read over and over several times and I was like, „Huh?‟ I know it‟s given other people problems. INTERVIEWER: Yeah. AMANDA: See, the arbitrarily close to, isn‟t that like, you can make a function close to certain point? Like, to me it‟s, it looks like you‟re choosing where the function is going. But, I don‟t really know. See, I don‟t think you can really choose where you‟re… where you‟re limit is. But, maybe I interpreted that wrong? Amanda takes the notion of reverse thinking and interprets it as “choosing where the function is going.” To her, the dynamic nature of the limit concept requires that the independent variable be considered first, and the dependent variable is analyzed as a result. This perspective shuts out the ability to apply reverse thinking to the limit concept. I argue that “reverse thinking” is only part of the cognitive structure needed to develop an understanding of the formal definition of limit. It is also important for students to develop a sense of „closeness‟ that can lead to the construction of mathematical neighborhoods. In Nunez‟s (1999) description of continuity, he points out that a key metaphor for CauchyWeierstrauss continuity is “limit IS preservation of closeness near a point” (p. 58). Using this metaphor, dynamic language such as the words “approaching” or “tending to” lose their meaning and are replaced by the static elements of a neighborhood consisting of values near the limit point. This idea of “closeness” will be the basis for the neighborhood model of limit discussed throughout this study. The phrase “neighborhood thinking” will refer to the 59 use of „closeness‟ by a student to describe his/her conceptualizations of the limit concept. In practical terms, neighborhood thinking can manifest itself in both graphical and symbolic settings. In a graphical sense, students “look around” the function to analyze the behavior of its graph near the limit point to create arguments about the value of the limit. In a symbolic way, students may calculate values very near the limit point to draw conclusions about the limit itself. Mathematicians use inequalities to create symbolic arguments involving „closeness‟; however, there was no evidence that students associated the use of inequalities with neighborhood thinking during this study. The Dynamic Model of Limit There is very little chance that students involved in this study had previously encountered the sequential definition of limit. Even though the study of infinite sequences was developed the prior semester, there is little in the curriculum to connect the ideas of sequences with those of finding traditional limits in calculus one. However, the related infinite process is a concept that is encountered by almost every student in introductory calculus. This process tends to manifest itself in two ways, symbolically using a process of evaluating the function at successive points and visually using a metaphor of motion. The symbolic process of analyzing limits is generally introduced early in an introductory calculus course. This process is often conceptualized in the form of a table of values in which the independent variable gets successively closer to the limit point and the corresponding dependent variable is analyzed. This process is intended to be both dynamic and infinite in nature, with the independent variable getting closer and closer to the limit point with each iteration of the process. In action, however, this is never treated 60 as an infinite process, but rather the process is ended after some finite number of steps and a conclusion is drawn about the limit of the function. In this way, as noted by Tall and his colleagues (2000) the concept of limit comes to be understood as both a process of approaching a point and the conclusion of that infinite process, while the process itself can never be carried out to its conclusion. Visual representations of this dynamic motion have been referred to by others as the „fictive motion metaphor‟ (Talmy, 1988; Lakoff and Nunez, 2000). Using this metaphor, the static curve of a graph actually represents the results of motion. Colloquial language supports the use of static objects embodying motion. For example, the phrase “this road goes to the lake” describes a static object, a road, as conveying a sense of motion. The road itself is not in motion, but motion is visualized on top of this structure. In a similar way, students may visualize motion in the static graph of a function. In the case of a single variable limit, this motion takes place from either the right or the left, creating the right and left hand limits. This visualization is different from the process of analyzing a function using a sequence of points in the fact that the visualization does not carry the same infinite nature as the process. Instead, the fluid motion from either side can be conceptualized as a single action and the conclusion can be visualized without using an infinite process. Similar to the symbolic process, the term „limit‟ can refer to both the visualized motion towards the limit point and the conclusion of that motion. Although the two manifestations above are conceptualized in significantly different ways, they both describe a dynamic process, either symbolic or visual, which results in the value of the limit. This process forms the basis of the dynamic model of limit discussed throughout this study. The phrase “dynamic thinking” will refer to 61 students‟ use of a dynamic process, either symbolically or visually, which is intended to result in the limit value. The Topographical Model of Limit A student in calculus I can be quite successful solving most limit problems encountered by only looking at the surface features of a function and never developing a sense of limit related to that of either the formal definition or the sequential definition of limit. This way of thinking will be referred to as the topographical model of limit. It begins with an ability to recognize and classify discontinuities in a function, and through this classification procedure, it is possible to know the resulting limit value. A possible description of the cognitive structures required for developing topographical thinking in calculus I is given below: 1. The function must be viewed as an object with inherent characteristics and properties. 2. Points must be classified as either continuous or discontinuous. If it is discontinuous, then the type of discontinuity (removeable, jump, infinite) must be classified. 3. The result of the limit must be deduced from the classification process. a. A continuous function‟s limit is given by evaluating the function at the point. b. A removable discontinuity must be „removed‟ before the limit can be evaluated. That is, using a Gestalttype viewpoint, a point must be identified which can fill in the hole on the function. c. A jump discontinuity has no limit. 62 d. An infinite discontinuity must be examined to determine if a sign change takes place. The term “function” in stage 1 is used in a loose sense. It is quite possible for students to treat external elements of the function, such as an equation or a graph, as the object being encountered. Even though for some students these external elements are not necessarily connected to the concept of function, they may still be used in a productive manner for solving most calculus I limit problems. However, it should be noted that this method does not cover every possible function that a student could encounter. For example, the function x f (x) sin 1 cannot be classified under the above system at x = 0. This means of conceptualizing the limit concept, by itself, potentially weakens the students‟ ability to understand important limiting situations in calculus. For example, Carlsen, Oehrtman, and Thompson (2007) argue that a topographical understanding of limits does not provide an understanding of the limiting processes necessary to understand differentiation and integration. It can also be observed that this conceptualization may lead to difficulties in understanding limits at infinity, which require an inherently different classification system in order to be understood topographically. It is also important to observe that the topographical understanding of limit directly contradicts with the limit concept as introduced in formal mathematics. In formal mathematics, the notion of continuity is defined using the definition of limit as its basis. The topographical understanding of limit described here requires an understanding 63 of continuity prior to the concept of limit, which is opposite that of the formal mathematics. Throughout this study, the term “topographical thinking” will refer to the use of external characteristics of the function to make decisions about the value of the limit. These external characteristics tend to be visual in the form of a graph; however, it is also possible for students to use the external characteristics of a symbolic expression, for example the value of a function at a single point, to draw conclusions about the limit. This type of topographical thinking has several characteristics that distinguish it from both the neighborhood and dynamic model. Topographical thinking is static in the sense that it refers to the external features of a function as opposed to dynamic thinking which envisions motion involved in a function, and topographical thinking also places an emphasis on classifying functions based on external characteristics. This classification process need not be well defined; rather it can be based on loose feelings about the external characteristics. However, it is different from dynamic and neighborhood thinking in the fact that it aims to classify and not analyze the features of the function. Textbook Treatment of the Limit Concept Students involved in this study used the textbook Calculus by James Stewart (2003). This textbook introduces the concept of limit with the following definition. 64 Definition 1: We write f x L x a lim ( ) and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. (p. 71) Immediately following this definition, the text states that “roughly speaking, this says that the values of f (x) get closer and closer to the number L as x gets closer to the number a (from either side of a) but x a ” (p. 71). These two statements seem to be an attempt to connect students with the two formal definitions of limit discussed previously. The words “arbitrarily” and “sufficiently” appear to be an attempt to capture the notion of „closeness‟ used in the formal definition of limit. The words “closer and closer” appear to be an attempt to capture the dynamic nature of the sequential definition. Students using this textbook are introduced to both the neighborhood and the dynamic model of limit. The implication from the written textual material is that the neighborhood model of limit should be considered the „official‟ understanding of limit, while the dynamic model is described as a “rough” description of the neighborhood model. There is no mention of understanding the limit using conceptualizations connected to the topographical model, and there is no implication that the two models of limit described are, in any way, cognitively different. 65 Coding Scheme The three models of limit described were used to create a coding scheme for analyzing the qualitative interview data. Interview statements were analyzed, and when applicable, categorized into one of the three categories in Table 13. Table 13: Three Models of Limit The analysis of the transcripts frequently revealed language relfelcting cognitive structures inherent in the three models of limit. Applicable text was sorted into „instances‟ of language supporting one of the three categories. An „instance‟ is understood to be a statement describing one complete thought. Statements were divided into two or more instances when the interviewee appeared to switch the focus of his/her description from one thought to another. For example in Ashley‟s statement below while referring to a multivariable contour graph. Her statement was considered to have two separate instances. Excerpt 3 ASHLEY: I‟m trying to think back to the pattern idea I had with those, so there‟s a very distinct movements and this one seems to have a more distinct movement. Let‟s see… This one probably exists, just because again it has that upward sloping of values and similar line patterns. I would say this one doesn‟t exist. Category Description Dynamic Uses motion to develop meaning in a function. Neighborhood Uses a sense of „closeness‟ to develop meaning in a function. Topographical Uses the shape of a graph to develop meaning in a function. 66 The first instance is her description of how motion plays a role in her understanding of multivariable limits. The phrase “there‟s a very distinct movements and this one seems to have a more distinct movement” was judged to be an instance of dynamic thinking since the statement emphasized the “movement” of the graph. However, it was judged that in the second part of her statement she switched her focus from describing her use of motion in general to describing her thought process on one particular problem. So it was judged that the statement, “because again it has that upward sloping of values and similar line patterns” constituted a second instance which was classified as topographical thinking because its central focus is that of exterior features of the contour graph. In some cases, however students used a very long statement to describe a single thought. For example, Chris made the following statement when reflecting on multivariable limits. Excerpt 4 CHRIS: It depends which direction you‟re looking at as to where the limit is coming from, where it‟s going, what kind of thing, how is moving or approaching, using those two words that are confusing. But, yeah, in the first two, from a different direction, from a different x or a different y direction it changes like where it is or where it‟s moving to. But on this one, on the third one, from every direction it‟s the same. This statement was judged to be a single instance since Chris maintained focus on describing his thought about which direction the limit was coming from throughout his statement. Even though he mentioned three separate problems, it was judged that he was not switching his focus from one problem to the next, but rather he was using the three problems as examples to illustrate his thinking. For these reasons, the above text was 67 considered one instance of dynamic thinking since the focus of the entire passage was on the use of motion from different directions on the graph. Statements which were not easily associated with one of the three primary categories were not recorded as an instance. Josh‟s statement below is an example of a noninstance. Excerpt 5 JOSH: Ok, First one (limit describes how a function moves as you approach a given point) the way my calc I teacher […] described it was just like, it was the behavior of the function. That you may have a discontinuous function that has a point above but if you look at the limit your analyzing the behavior on each side, you‟re not just evaluating a value at that given point. So that‟s why I thought this one was the most accurate description of it because it kind of described the behavior. Josh‟s statement above had elements of all three coding categories in it. He used motion, closeness, and the shape of the graph to describe his thinking; however, the main emphasis of the description surrounded the word “behavior” which was used ambiguously in this statement. Since the statement did not contain a strong description of any of the three categories, it was considered a noninstance and was not recorded as part of the coding scheme. An important issue when coding the interview results is that of conventional mathematics language usage. In calculus, common language about limits involves words such as “approaches” or “goes towards.” Although these words carry with them a connotation of dynamic motion, the meaning a student may give to them can vary significantly. For this reason, the use of common mathematical language alone is not considered sufficient evidence that a student is using dynamic reasoning. Below is an example of Amanda‟s description of a multivariable limit 68 Excerpt 6 AMANDA: Ok, so a limit from a line is kind of like how the line approaches the point. So, say zero, if the line 2x is how it approaches zero that just, you just have to worry about the x and y variables, same with in a curve, like a parabola or something, same thing, you just have to worry about the x and y coordinates and you can easily see just the x and y coordinates on a graph. But in multivariable, you have several different variables that are approaching the same point, and so you can‟t exactly see how that happens on a graph, easily, so you have to take several slices and look at those curves at the slices and put it all together and analyze it that way. In her description, Amanda frequently used the word “approaches,” but it is not clear that Amanda was actually evoking a sense of motion to describe the multivariable limit. Instead, it is quite plausible that Amanda was using the word “approaches” as similar to the word “behavior” which does not necessarily connote dynamic motion. In general, a conscious effort was made during the coding process to err on the side of exclusion rather than inclusion. In that sense, a statement such as Excerpt 4 above might actually refer to dynamic motion, but was excluded because reasonable doubt of its appropriateness remained. Upon completion of the initial coding process, all coded text was reexamined to determine accuracy in coding. Instances from different interviews and involving different students were compared to determine that the coding scheme was implemented in a consistent basis. Any discrepancies in the coded instances were addressed and changed as necessary. In all, 283 total instances were recorded with the majority describing dynamic thinking. Table 14 below shows the number of recorded instances for each coded category. 69 Coded Category Instances (Percentage) Dynamic Thinking 160 (56.5%) Neighborhood Evaluation 41 (14.5%) Topographical Examination 82 (29.0%) Total 283 Instances Table 14: Instances for Each Cod
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Title  Students' Conceptualizations of Multivariable Limits 
Date  20080701 
Author  Fisher, Brian Clifford 
Keywords  Mathematics; Education; Calculus; Multivariable Calculus; Limits; Multivariable Limits 
Department  Mathematics 
Document Type  
Full Text Type  Open Access 
Abstract  The objective of this study was to describe how students with a dynamic view of limit generalize their understanding of the limit concept in a multivariable setting. This description emphasizes the type of generalization that takes place among the students (Harel and Tall, 1989) and the role of motion among students' conceptualizations. To achieve these goals, a series of taskbased interviews were conducted with seven students enrolled in multivariable calculus. These interviews were analyzed and a coding scheme was developed to describe the data. This coding scheme arose from analysis of the data combined with the role of limit in formal mathematics. It emphasizes three models for understanding the limit concept, the dynamic model, the neighborhood model, and the topographical model. After analyzing the coded data, two important interactions between the three models of limit were described. First, it was found that students superimposed dynamic imagery on top of existing topographical structures in order to understand multivariable limits, and a weak topographical understanding of multivariable limits contributed to students struggling to understand the multivariable limit concept. Second, it was found that students implementing dynamic imagery in the context of multivariable limits confronted an infinite process of analyzing motion along an infinite number of paths. It was found that students' struggles to understand the multivariable limit were connected to their struggles to understand this infinite process. Additionally, it was found that the condensation of this infinite process led several students towards the neighborhood model of limit. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  STUDENTS‟ CONCEPTUALIZATIONS OF MULTIVARIABLE LIMITS By BRIAN CLIFFORD FISHER Bachelor of Science in Mathematics Oklahoma State University Stillwater, Oklahoma 2000 Master of Science in Mathematics Oklahoma State University Stillwater, Oklahoma 2002 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2008 ii STUDENTS‟ CONCEPTUALIZATIONS OF MULTIVARIABLE LIMITS Dissertation Approved: Dr. Douglas B. Aichele Dissertation Adviser Dr. John Wolfe Dr. James R. Choike Dr. Alan Noell Dr. William Warde Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENTS It is not often that I get to publicly thank all those who have made such a large impression on my life. I am grateful for this opportunity, and I hope my words and my life do justice to the investment that so many people have made in me. I know I will miss some people who have worked very hard for me, so I apologize in advance for those omissions. First, to my wife, Kimberly, I will not attempt to recount all the ways you have been an inspiration to me. Let me only say „thank you‟ for always believing in me, even when I did not. Evelyn, you are too young to remember this process, but know that whenever I am tired and frustrated your smile always lights up my day and helps me put life into perspective. Thank you to my committee for all the effort you put into me. Dr. Aichele, thank you for pushing me and having confidence in me. Thank you for investing so much time and effort in me, and thank you for always finding the right question to straighten me out when I needed direction. Dr. Choike, thank you for all the guidance you have given me over the years. Much of who I am as a teacher is because of the time we spent talking in your office. Dr. Wolfe, it was in your class almost four years ago that I found my interest in mathematics education research, and it was during my time spent in Geometric Structures that you finally sold me on inquirybased teaching methods. Dr. Noell, thank you for joining my committee even though I asked you late – I still believe I learned iv more in your complex analysis course than any other in grad school. Dr. Warde, thank you for the time to read and respond to every document I sent your way, your comments and suggestions were always helpful and appreciated. I would like to thank the entire math department at Oklahoma State University. To the faculty, thank you for giving me so much individual attention over the years. Thank you for stopping me in the hall and checking on my progress. Thank you for always having high expectations of me. To the staff, thank you for putting up with all my requests and never complaining. To the graduate students, thank you for always being encouraging and making the department a fun place to work and learn. Let me not forget to say “thank you” to all the students who took the time to take part in my study. I am especially grateful towards the seven students who sacrificed their time to set through three separate interviews with me. You all sacrificed your time for me and answered every question I asked with sincerity. Without you all this study could never have taken place – thank you! Let me thank my friends and family for all the good memories over the years. Your faith and encouragement were ever present and always needed. Thank you, Mom and Dad, for raising me to cherish thinking and learning. Thank you, Becky, for being much more than a sister. Thank you to the entire Burns family for celebrating and struggling with me and making me one of your own. Thank you, Dustin, for opening up your house for me when I needed a place to rest. Thank you, Matt, for always keeping me focused on what is important. Thank you to the entire Edmond Church of Christ and especially our young married class for teaching me what it means to have a spiritual family. Finally, a tremendous “thank you” goes to everyone who sincerely asked me v what I was studying and listened while I prattled on about mathematics, regardless of whether or not you found it interesting. Lastly, I want to say publicly what I say regularly in private. Thank you, God, for putting all these people in my life to encourage me, uplift me, and keep me focused on the things that really matter. Thank you for taking me great places despite my weaknesses, and thank you for letting me work hard enough to be proud of what You accomplish.vi TABLE OF CONTENTS Chapter Page I. INTRODUCTION ......................................................................................................1 Statement of the Problem .........................................................................................1 Problem Context ......................................................................................................3 History of the Limit Concept .............................................................................3 Curriculum Change and the Limit Concept .......................................................7 II. RELEVANT LITERATURE ..................................................................................15 Dynamic Imagery and Covariational Reasoning ...................................................15 Students‟ Conceptions of the Limit Concept .........................................................18 Theoretical Perspectives ........................................................................................26 Cognitive Obstacles and Cognitive Change ....................................................27 Mental Representation and Conceptual Metaphors .........................................29 Abstraction and Generalization........................................................................31 Infinite Processes .............................................................................................32 III. METHODS ............................................................................................................35 Why a Qualitative Study? ......................................................................................35 Participants and Setting..........................................................................................37 Study Questionnaire ...............................................................................................39 Data Collection ......................................................................................................42 Researcher‟s Perspective .......................................................................................45 IV. DATA ANALYSIS ...............................................................................................49 Questionnaire Results ............................................................................................49 Qualitative Data Analysis ......................................................................................53 Definitions of Limit in Formal Mathematics ....................................................54 Neighborhood Model of Limit ..........................................................................57 Dynamic Model of Limit ..................................................................................59 Topographical Model of Limit..........................................................................61 Textbook Treatment of the Limit Concept .......................................................64 Coding Scheme .................................................................................................65 V. STUDY RESULTS ................................................................................................70 vii Qualitative Study Results .......................................................................................70 The Concept of Limit in Different Settings ...........................................................71 Introductory Calculus.......................................................................................71 Symbolic Manipulation of Multivariable Functions ........................................82 Three Dimensional Graphing of Multivariable Functions ...............................89 Multivariable Functions using Polar Coordinates............................................98 Contour Graphs of Multivariable Functions ..................................................103 Misconceptions about Multivariable Limits ........................................................114 Uncategorized Misconceptions ......................................................................116 Misconceptions Involving the Dynamic Model .............................................122 Misconceptions Involving the Topographical Model ....................................132 VI. DISCUSSION AND CONCLUSION ................................................................158 Visualizing Multivariable Limits .........................................................................161 Amanda‟s Experience with Multivariable Limits ...........................................162 Infinite Processes in Multivariable Limits ...........................................................181 Josh‟s Experience with Multivariable Limits ................................................185 Answering the Study Questions ...........................................................................208 Reflections ...........................................................................................................212 REFERENCES ..........................................................................................................216 APPENDICES ...........................................................................................................225 Appendix A – Study Questionnaire .....................................................................225 Appendix B – Interview 1 Materials ....................................................................227 Appendix C – Interview 2 Materials ....................................................................230 Appendix D – Interview 3 Materials ...................................................................234 Appendix E – Informed Consent Forms ..............................................................239 Appendix F – IRB Approval Form ......................................................................242 viii LIST OF TABLES Table Page Table 1: Questionnaire Results (Cumulative) ..........................................................49 Table 2: Questionnaire Results (Percentage) ...........................................................50 Table 3: Questionnaire Results (Binomial) ..............................................................50 Table 4: Questionnaire Results (Binomial Percentage) ...........................................50 Table 5: Question #6 Results (Cumulative) .............................................................51 Table 6: Question #6 Results (Percentage) ..............................................................51 Table 7: Questionnaire Results, Interview Volunteers ............................................51 Table 8: Question #6 Results, Interview Volunteers ...............................................51 Table 9: Questionnaire Results, Volunteer Comparison ..........................................51 Table 10: Question #6 Results, Volunteer Comparison...........................................51 Table 11: Questionnaire Results, Interview Participants .........................................52 Table 12: Question #6 Results, Interview Participants ............................................52 Table 13: Three Models of Limit .............................................................................65 Table 14: Instances for Each Coded Category .........................................................69 Table 15: Recorded Instances by Setting .................................................................71 Table 16: Calculator Usage Relative to the Three Models of Limit ........................77 Table 17: Student Misconceptions of Multivariable Limits ..................................115 Table 18: Misconceptions Categorized According to Limit Models .....................116 ix LIST OF FIGURES Figure Page Figure 1: Questionnaire Used by Williams ............................................................23 Figure 2: Three Dimensional Surface Examined by Amanda ................................88 Figure 3: Figure 2 as Viewed from the Positive z  axis ........................................88 Figure 4: Two Views of the Surface Discussed in Excerpt 21 ...............................90 Figure 5: Paths Amanda Creates with the Cursor ..................................................91 Figure 6: Comparison of the Function Amanda and Jennifer Experienced to a Parabloid ......................................................................94 Figure 7: Mike‟s Use of Different “Directions” ...................................................104 Figure 8: Three Dimensional Graph Discussed in Excerpt 57 .............................126 Figure 9: Contour Graph Discussed in Excerpt 59 ...............................................128 Figure 10: Contour Graph Discussed in Excerpt 60 .............................................129 Figure 11: Graph Used for Single Variable Limit Problems ................................135 Figure 12: Two Graphs Shown to Ashley ............................................................136 Figure 13: The Two Graphs Discussed in Excerpt 73 ..........................................141 Figure 14: Figure Drawn by Amanda ...................................................................164 Figure 15: Josh‟s Two “Perspectives” ..................................................................193 Figure 16: Josh‟s Use of Paths on a Contour Graph ............................................199 Figure 17: Josh‟s Use of „Closeness‟ on a Contour Graph ..................................201 Figure 18: Josh Uses „Closeness‟ to Explain a Limit that Does Not Exist ..........2021 CHAPTER I INTRODUCTION Statement of the Problem It is well accepted that the limit concept plays a foundational role in presentday calculus education. At the same time, there is widespread agreement among both educators and researchers that most students struggle to develop a solid understanding of the limit concept (for example: Vinner, 1991). This may be due to the actual depth of concept. Tall (1992) refers to Cornu (1983) and states that "this is the first mathematical concept that students meet where one does not find the result by a straightforward mathematical computation. Instead it is 'surrounded with mystery,‟ in which 'one must arrive at one's destination by a circuitous route'" (Tall, 1992, p.501). The importance of limits in undergraduate calculus, combined with the difficulty students experience in grasping the concept has resulted in much attention from mathematics education researchers. Several researchers have worked to understand popular misconceptions about the limit concept (Davis and Vinner, 1986; Williams, 1991). It has been suggested by several researchers that a well developed notion of limit could be constructed using the metaphor of motion (Carlsen et al., 2002; Kaput, 1979; Monk, 1992; Tall, 1992; Thompson, 1994b). Furthermore, Williams (1991) found that a 2 significant number (30%) of second semester calculus students contained a dynamic view of limit, and that this dynamic viewpoint was extremely resistant to change. For these reasons, I have decided to examine how students who hold a dynamic view of limit conceptualize the multivariable limit concept. For functions of two variables, motion must take place on a surface instead of along a curve. It is not automatically clear how students will interpret motion in this new setting. Furthermore, the application of motion in multivariable calculus has the potential to create an epistemological obstacle in the sense of Cornu (1991) and require students to restructure their understanding of limits. I expect this restructuring to take place in a form of generalization similar to that described by Harel and Tall (1989). With this in mind I have created the following problem statement for this study: Describe how students with a dynamic view of limit generalize their understanding of the limit concept in a multivariable environment. As the problem statement suggests, this is a qualitative research study which results in a description of student behavior. I will focus the description on the following goals: 1. Describe what type of generalization students in this setting tend to experience with respect to the schema outlined by Harel and Tall (1989) which emphasizes three modes of generalization: expansive generalization, reconstructive generalization, and disjunctive generalization. 2. Describe the role of motion in students‟ understanding of multivariable limits. Does it create a cognitive obstacle, or are students able to apply this imagery to the new multivariable situation? 3 3. Describe how students respond to studying multivariable limits in four different contexts: traditional symbolic manipulation, symbolic manipulation involving polar coordinates, threedimensional graphing, and contour graphing. Of particular interest is whether some of these contexts tend to allow students to reconstruct their understanding of the limit concept to more closely resemble the formal definition. Problem Context History of the Limit Concept The concept of limit can trace its history back to ancient Greece. The Greek mathematicians spent most of their energy solving geometry problems. The solutions to many of these problems involved limiting concepts. One of the earliest such solutions was provided by Hippocrates of Chios (not to be confused with the famous doctor, Hippocrates of Cos). He proved that “the ratio of two circles is equal to the ratio of the squares of their diameters” (Edwards, 1979, p. 7). He accomplished this by inscribing polygons inside the circle and showing that the relationship is true for all such polygons. He then concluded that since this is true for all such polygons, it must also be true for a circle. However, Hippocrates had no limit concept capable of finishing his argument. In general, the Greek mathematicians were bothered by the infinite ideas inherent in the limit concept, and soon they began developing methods that could be used to avoid the “horror of the infinite.” Mathematicians such as Eudoxus, Archimedes, and Euclid began using the method of exhaustion to perform calculations such as that of Hippocrates. This method used contradiction to rigorously prove a statement. It depended on the principle that any magnitude can be made smaller than a second magnitude by repeatedly 4 dividing the first magnitude in half. Using these methods, the Greeks (especially Archimedes) were able to solve many modern day calculus problems. Ultimately, these notions gave rise the formation of calculus as we know it today. However, it is important to note that the ancient Greeks contained no explicit concept of limit. In addition they were unable to generalize their methods, and instead chose to start from scratch to solve each problem they faced. Additionally, they were unable to make the connections between problems of areas and tangents which gave rise to modern day calculus (Baron, 1969; Edwards, 1979). For hundreds of years after the era of Greek mathematics, mathematicians were unable to approach the ideas of calculus as understood by the Greeks. Prior to the sixteenth century, the works of the Greek mathematicians were “not always generally accessible and never fully mastered” (Edwards, 1979, p. 98). However, there were many important developments prior to the sixteenth century that made it possible for later mathematicians to approach a new way of understanding limit. Among those developments were various graphical representations of what we would call functions. These ideas were introduced in the fourteenth century by Nicole Oresme (Edwards, 1979; Babb, 2005). However, these graphical representations were not intended to be thought of as a set of corresponding values, like a modern day function graph. Instead, Oresme intended for each vertical height of his graph to represent the „intensity‟ of a quantity (Edwards, 1979; Thompson, 1994a; Babb, 2005). As he wrote in his Treatise on the Configuration of Qualities and Motions, “every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some 5 point of the space or subject of the intensible thing,” (Grant, 1974, as quoted in Edwards, 1979, p.88). By the middle of the seventeenth century these graphical representations had developed quite a bit. Many noted mathematicians of the time used an idea of motion to understand these representations. In fact, Newton “regarded the curve, f(x,y) = 0 as the locus of intersection of two moving lines, one vertical and the other horizontal,” (Edwards, 1979, p. 191). Newton‟s use of motion was no doubt influenced by his mentor, Isaac Barrow. While Newton was a student under Barrow‟s guidance, Barrow gave an important series of lectures on time and motion. Barrow perceived a line as a “trace of a point moving forward… the trace of a moment continuously flowing” (Baron, 1969, p. 240). This use of the metaphor of motion to understand graphical representations characterized much of Newton‟s work. In fact, Bardi (2006) states that “Newton‟s big breakthrough was to view geometry in motion…” (p. 30), and in one of Newton‟s first attempts to compile his early works, To Resolve Problems By Motion in 1666, he “deliberately elects to make the concept of motion the fundamental basis” of his work (Baron, 1969, p. 263). Using these ideas of motion allowed Newton to solve many problems in the development of the calculus; however, a precise definition of limit was still several hundred years away. It was not until the nineteenth century‟s increased focus on mathematical rigor that the formal limit definition as we know it today came into being. One issue that had to be confronted was the notion of infinitesimals. The idea was not a new one. In fact, Fermat came very close to modern limit calculations when he substituted x + e for the 6 variable x and after simplification removed e from the expression (Baron, 1969; Edwards, 1979). It is important to note that Fermat did not consider the value e to approach zero or even become zero (he did not even imply that e should be small), instead he simply removed expressions containing e. It should be noted that even at the time this was questioned by such mathematicians as Rene Descartes (Baron, 1969). However, these ideas and the use of infinitesimal values continued to be popular for hundreds more years. Finally, it was Cauchy who developed “the first comprehensive treatment of mathematical analysis to be based from the outset on a reasonably clear definition of the limit concept” (Edwards, 1979, p. 310). Cauchy‟s notion of limit was based on an infinitely small variable, which he also called an infinitesimal. This is different from the view that an infinitesimal is an infinitely small quantity; instead, according to Cauchy it is a variable whose value decreases indefinitely. Even at this time, the limit concept was “tinged with connotations of continuous motion” (Edwards, 1979, p. 333). The close of the nineteenth century saw the precise construction of the real number system, and with it Weierstrass was able to develop the definition of limit that is commonly used today. His disapproval of the dynamic view of limits led him to create a static formulation in terms of ε and δ which became popular throughout the twentieth century. In summary, I would like to observe several themes which ran through the historical development of the limit concept: The Greeks had problems “passing to the limit.” They much preferred static arguments that did not contain notions of the infinite. 7 Newton (and many others) found the metaphor of motion to be a powerful tool in understanding the concepts of calculus. However, these ideas of motion were unable to provide a rigorous definition of limit, and were eventually replaced with Weierstrass‟s static definition. For many centuries different mathematicians struggled with the meaning of infinitesimals. These ideas usually contained some sense of infinitely small quantities until Cauchy used a dynamic view of infinitesimal to create a more coherent meaning of the concept of limit. In conclusion, it should be pointed out that the limit concept was understood in many ways throughout history. Each of these ways of thinking made it possible for mathematicians to understand limits in a useful manner, but each way of thinking also created a barrier towards understanding the limit concept in the way we know it today. In that respect, these ways of thinking created epistemological obstacles in the sense of Cornu (1991). Curriculum Change and the Limit Concept The first half of the twentieth century saw calculus generally reserved for undergraduate education and rarely discussed in high school settings. This time period was marked by an emphasis on twotrack high school mathematics programs (Jones, 1970). With the Great Depression came decreased college enrollment, and educators responded by focusing on functional competence as the key objective of high school mathematics. Often times mathematics classes became electives, and as a result there was a trend for colleges to lessen their mathematics requirements for admission (Jones, 1970). The affect of this atmosphere on teaching the limit concept is not entirely clear; 8 however, it might be reasonable to conclude that approaches to limits, as well as other calculus concepts, mirrored that of other subjects in their focus on functional competence. In that sense, we would assume that the limit concept was taught primarily as a procedure by which a certain result could be obtained. After World War II the educational climate in the United States began to change dramatically. Technological advances made during and immediately following World War II revitalized the status of mathematics and science in the country. Colleges saw an increase in enrollment partially due to returning soldiers attending college on the “GI Bill” (Jones, 1970). At the same time, the country began recognizing that its population was not prepared to meet the demands of a new technological society. Accompanying all this with the feeling that America was beginning to fall behind the rest of the world scientifically, emphasized by the Soviet launch of Sputnik in 1957. The United States began a period of reexamining the way mathematics was taught at all levels throughout the country. The resulting period from the early 1960s to the mid 1970s became known as the “New Math Era” and was marked by an increased focus on abstraction and mathematical rigor (Bosse, 1995). An early introduction to key mathematical ideas also marked this period resulting in a push for calculus to be introduced to students while in high school. The effect this had on teaching calculus and the limit concept was significant. Calculus was approached in a more rigorous manner than before and became a more common element in a student‟s education. One of the most controversial reports coming from this time was the Cambridge Conference on School Mathematics (1970). The Cambridge Conference was considered an ambitious goal set out to challenge the mathematics education community on what can 9 be accomplished (Adler, 1970). This report called for calculus to be taught using “precise formulations” rather than what it refers to as “loose calculus,” which “deals with „variables‟ (in a Leibnizian sense) rather than functions,” (Cambridge Conference on School Mathematics, 1970, p. 40). The Cambridge Conference set forth two proposed curricular programs, both of which featured a rigorous treatment of calculus in the final two years of high school. However, there was some disagreement whether calculus should first be introduced at an earlier time on a more intuitive basis. The argument against an introduction was that The student who has already developed some taste for mathematical rigor will be dissatisfied with only half the story in calculus when the fundamental concepts are not carefully defined and precisely used. Because he cannot carry his arguments back to welldefined concepts, he will not fully understand what calculus is about. Finally, one often forms wrong impressions in an intuitive approach which are hard to “unlearn” later, and the luster is worn off the subject when one has to return to it later to tie together loose ends (Cambridge Conference on School Mathematics, 1970, p. 47). On the other hand, the Cambridge Conference recognized the historical significance of the calculus and wanted all students to be able to appreciate it whether or not they completed the final years of the mathematics program. In the late 1960s and early 1970s a backlash against the “New Math Era” began. This resulted in several different movements, including the “Back to the Basics” movement. Importantly, most reform movements after the mid1970s called for a decreased emphasis on mathematical rigor. In their publication, Agenda for Action, The 10 National Council of Teachers of Mathematics (NCTM) laid out its recommendations for school mathematics in the 1980s. Among other things, this publication called for “the use of imagery, visualization, and spatial concepts” (1980, p. 3) to understand mathematical ideas. This is clearly a different emphasis than the precise definitions used by the Cambridge Conference. In addition, there became a question as to the need and relevance of calculus. In the same publication, the NTCM challenges mathematics educators and college mathematicians to “reevaluate the role of calculus” (ibid, p. 21) in school curriculum. Importantly, it was suggested that perhaps calculus should not be the focal point of college preparatory mathematics and that other branches of the mathematical sciences should be encouraged in its place. A few years later, Shirley Hill made the case for a new curriculum that suggested an alternative path for capable students which “would stress statistics and computer science rather than calculus” (Hill, 1982, p. 116). By the late 1980s and early 1990s educators began to focus again on calculus as a foundation of mathematical learning. In A Call for Change, the Mathematical Association of American (MAA) set out recommendations for teacher preparation. In this they called for teachers to model real world problems using calculus and to explore the concepts of calculus both on an intuitive basis and in depth (Leitzel, ed., 1991). From the perspective of this publication, the emphasis is clearly on gaining an intuitive understanding of calculus. It writes, Historically, while investigating continuous processes, many of the ideas and techniques of calculus were developed and used on an intuitive basis before the theory was made rigorous… By building an intuitive base for analyzing 11 continuous processes, these teachers might be more willing to take intellectual risks in their own classrooms. The actual material covered is less important than developing conceptual understanding of the ideas (ibid, p. 35). In 1989, the NCTM released their Curriculum and Evaluation Standards for School Mathematics. In this publication, the NCTM “does not advocate the formal study of calculus in high school for all students or even for collegeintending students. Rather, if calls for opportunities for students to systematically, but informally, investigate the central ideas of calculus” (NCTM, 1989, p. 180). Therefore, by the early 1990s the trend in mathematics education was to teach calculus on an informal, intuitive basis rather than using the precise formulations and rigor of the “New Math Era.” The calculus reform movement is generally considered to have begun in 1986 with the Tulane calculus conference (Schoenfeld, 1995). This conference resulted in The MAA‟s publication of Toward a Lean and Lively Calculus (Douglas, 1986) which aimed to slim down the calculus curriculum by teaching fewer topics but covering them in greater depth. This began a period marked by numerous projects all aimed at reforming the calculus curriculum. There was significant variation between these different projects, but most incorporated an increased use of technology, an emphasis on applications, and the use of multiple representations (Ganter, 1999). The increased emphasis on technology in mathematical teaching and learning was nearly inevitable with the increased availability of technology in society. In many ways, this emphasis of technology spurred on the other major changes during the calculus reform movement. The use of technology in mathematical learning allowed students to encounter problems in real world settings that would have been impossible before. This 12 allowed the reform movement to place an emphasis on application problems, often projects spanning over multiple class periods. In addition, the use of technology allowed for easy transitions between symbolic, numerical, and graphical representations of functions. This ease of transition allowed the reformers to place a greater emphasis on multiple representations in the classroom. This emphasis on multiple representations is of a particular interest to this study. This is one of the foundations of the Harvard Consortium‟s hallmark textbook (HughesHallett, et al. 1994). This text emphasized the “Rule of Three,” which pushed for all concepts, in particular function concepts, to be studied in graphical, numerical, and analytical settings. The group later reformed this concept to the “Rule of Four” which added verbal representations to the list (Schoenfeld, 1995). This notion of multiple representations found its way into publications beyond just those of the calculus reform movement. For example, A Call for Change, published by The Mathematical Association of America, was written as a recommendation for the curriculum of teachers of mathematics (Leitzel (Ed.), 1991). This publication called for teachers to be able to “represent functions as symbolic expressions, verbal descriptions, tables, and graphs and move from one representation to another” (p. 31). This push for representing functions in multiple ways brought with it a notion of function that was broader than before. Instead of restricting the notion of function to its definition, there is now an emphasis on thinking of functions in a wide variety of manners. The purpose of the present study is to understand how students connect and generalize these different representations of the limit concept. In particular, how do they generalize a graphical notion of dynamic 13 motion in multivariable settings and how does the representation of the multivariable function impact this generalization? Importantly, we see that calculus education has undergone several major changes during the past century. The major characteristics of each period were: PreWorld War II was marked by an emphasis on “functional mathematics” The “New Math Era” came after World War II and encouraged an increased focus on mathematical rigor and precision and an earlier introduction to mathematical topics. The backlash to the “New Math Era” resulted in calls to return to the basics in teaching math. This resulted in a decreased emphasis in mathematical rigor. During this time there also came a reevaluation of the role of calculus in education with some experts calling for programs which emphasize statistics and computer science over calculus. During the 1990s, the central ideas of calculus again became a center piece of mathematics education. Experts called for these central ideas to be approached, at least throughout high school, though informal intuition rather than with a formal calculus course. The Calculus Reform Movement began in 1986 and was marked by an increased emphasis on technology, applications, and multiple representations. In particular, the emphasis on multiple representations called for students to be able to connect symbolic, numeric, and graphical representations of a function. In conclusion, it should be noted that the precise, formal definition was once a foundation of calculus education during the “New Math Era.” However, today, formal definitions 14 have been replaced by intuition and informal understanding using multiple representations. It is in this spirit that I will explore these informal notions of limit that are developed by students and how these notions manifest themselves in a multivariable environment. 15 CHAPTER II RELEVANT LITERATURE Dynamic Imagery and Covariational Reasoning One common issue in research on students‟ understanding of calculus is the use of dynamic imagery to represent functions. It was observed that students struggled to understand the function concept in the traditional correspondence manner. As a result, researchers began studying the use of a dynamic understanding of function. Monk (1992) labeled these modes of thinking as “pointwise” and “acrosstime.” He observed that for some problems it was advantageous for students to use “acrosstime” thinking to make sense of the situation. As a result of more study, a number of scholars including Kaput (1994) began pushing for a more dynamic view of function in school curriculum. As summarized by Thompson (1994a), “today‟s static picture of function hides many of the intellectual achievements that gave rise to our current conceptions.” (p. 29) Over the next few years more studies in the vein of Monk‟s 1992 study were conducted and several authors began referring to this type of “acrosstime” thinking about functions as covariational reasoning. Confrey and Smith (1994, 1995) wrote some of the first publications referring to covariational reasoning. They mixed their notion of covariation, which was for students to “coordinate values in two different columns” 16 (1995, p.78), with an emphasis on multiplication as “splitting” instead of repeated addition. They noted that using covariation to understand functions “makes the rate of change concept more visible and at the same time, more critical” (1994, p.138). This connection to rate of change is central throughout the research on functions as covariation. Thompson (1994b) studied the relationship between students‟ understanding of the fundamental theorem of calculus and their concepts of rate of change. He suggested that student‟s difficulties with the fundamental theorem of calculus are rooted in their poor understanding of rate of change and their inability to develop an image of function as covariation. During this time the notion of covariation evolved from the idea of coordinating the values in two columns of data to one of holding two values of a function in mind simultaneously. In 1998, Saldanha and Thompson further explained their view of covariation by noting that “In early development one coordinates two quantities‟ values – think of one, then the other, then the first, then the second, and so on. Later images of covariation entail understanding time as a continuous quantity, so that, in one‟s image, the two quantities‟ values persist” (p. 298). In this sense, understanding function as covariation is more than a special way to look at a table or graph. It is a way of thinking that includes two different changing values which simultaneously depend on each other. Cottrill et al. (1995) also noticed these simultaneous changing values while exploring how students come to understand the limit concept. They used the theoretical perspective of APOS theory (to be explained in the “theoretical perspectives” section) to create a description of how people come to learn about the concept of limits. From their perspective, one of the key difficulties in coming to understand the limit concept lies in 17 the complexity of the concept itself. They argue that successfully constructing a limit schema involves coordinating two different processes together through complicated existential and universal quantifiers, which is a task that remains inaccessible to most students. So, similar to Saldanha and Thompson‟s (1998) view that students‟ understanding of covariation must simultaneously coordinate two changing values in their minds, Cottrill et al. (1995) found that understanding the limit concept requires coordinating two simultaneous processes. In the study done by Saldanha and Thompson (1998) the researchers observed an eighth grade student as he dealt with covarying quantities. Two important elements came from this study. The first element was that coming to understand functions as covariational quantities is a nontrivial task. The second was that the notion of covariation is developmental. It is in that vein that Carlson et al. (2002) developed a framework for studying functions as covariation. In this study they developed five mental actions and five corresponding levels of reasoning which could be associated with varying degrees of understanding functions as covarying quantities. In the lowest levels of understanding, students are able to coordinate the change in one variable with change in another, but with little understanding of the degree to which changing one quantity will affect another. Meanwhile the highest levels of understanding require that a student can hold in his/her mind the instantaneous rate of change of one quantity with respect to another and realize this as a continuously changing rate as the value of the independent variable changes. It was found in this study that many students understood functions as covarying quantities on a lower level, but few achieved a high level of covariational 18 understanding. It was also suggested that this type of dynamic reasoning might be an example of transformational reasoning as described by Simon (1996). Several studies have looked at the existence of such dynamic function concepts with respect to the concept of limit. Williams (1991) found that a dynamic view of limit was common among students and that it was extremely resistant to change. Many other authors tend to agree “that cognitively the strongest images are the dynamic ones,” (MamonaDowns, 2001, p. 264). On the other hand Oehrtman (2002, 2003) classified motion as a “weak metaphor.” He found that while students frequently refer to situations using the language of motion, they are not describing something which is actually moving. These two studies will be discussed in greater detail in the next section, and it is one of the subfocuses of this study to examine the nature and strength of such dynamic images in students while they encounter multivariable limits. Students’ Conceptions of the Limit Concept Over the past twentyfive years, there have been many studies on how students understand the limit concept. They vary in many ways. Some focus on limits as understood in an introductory calculus class (for example, Williams, 1991) while others focus on limits of sequences and series (for example, Alcock and Simpson, 2004). Some attempt to characterize common misconceptions (for example, Davis and Vinner, 1986) while others attempt to describe how a students comes to understand the concept (for example, Cottrill, et al., 1995). However all studies share the findings that the limit concept is difficult for students to grasp and a complete understanding of the limit concept is rare. 19 Many authors have emphasized the language used when referring to limits. It is common in beginning calculus courses to use the phrase approaching when referring to a limit; however, it has been pointed out that in everyday language phrases such as “approaches,” “tend towards,” or “gets close to,” carry a connotation that the point is never actually reached (Schwarzenberger and Tall, 1978). In this sense, the words used to refer to limit concepts often carry everyday meanings that are in conflict with their mathematical meanings (Monaghan, 1991). Davis and Vinner (1986) suggest that this influence of language is an unavoidable obstacle towards understanding the limit concept. Beyond the obstacle of language, Davis and Vinner (1986) attempted to characterize several predominant misconceptions found among students studying limits of sequences and series. Several of these misconceptions are clearly related to understanding limits in beginning calculus. Among the related misconceptions are: A sequence can never reach its limit; a limit is a bound on the sequence; and a sequence must have a final term. The first two of these misconceptions were studied by Williams (1991). He found that 70% of the students in his study agreed that “a limit is a number or point the function gets close to but never reaches,” and 33% agreed that “a limit is a number or point past which a function cannot go,” (p. 221). From this we see that these misconceptions, and in particular the misconception of limit as being unreachable, are prevalent among students studying limits in an introductory calculus course. It has also been shown that students often hold to naïve beliefs about limits. Tall (1992) describes the “generic limit property” which is a belief that if every term of a sequence contains a common property, it can be assumed that the limit of that sequence will also contain that property. As Tall (1992) notes, this belief has its roots in the 20 history of mathematics, “as in Cauchy‟s belief that the limit of continuous functions must again be continuous” (p. 502). He contends that this naïve belief is a cause of the popular misconception that 0.99999… is strictly less than one. Cornu (1991) looked at limits through the lens of epistemological obstacles. From his perspective “it is useful to study the history of the concept to locate periods of slow development and the difficulties which arose which may indicate the presence of epistemological obstacles” (p. 159). From analyzing the history of the limit concept, Cornu located four such obstacles: 1. “The failure to link geometry with numbers” (p. 159). This was evidenced by the Greeks‟ study of the limit concept. They were able to use very sophisticated geometric limiting arguments to solve a variety of interesting problems. However, in their studies, each problem was approached in its own geometrical context. They were only able to apply these ideas to magnitudes, not numbers, and therefore they were unable to generalize their efforts to a unifying concept of limit. 2. “The notion of the infinitely large and infinitely small” (p. 160). Many great mathematicians, including Isaac Newton and AugistinLouis Cauchy, struggled with the notion of infinitesimal quantities. The idea of an infinitely small quantity was freely used by Leonhard Euler to solve a variety of interesting problems. However, it was not until Cauchy described the infinitesimal as a variable which tends to zero, and Weierstrauss developed a static definition of limit that the current limit concept came to being. 21 3. “The metaphysical aspect of the notion of limit” (p.161). The limit concept has often times more closely resembled a philosophical subject rather than a mathematical one. Many great mathematicians, from the Greek mathematicians to Joseph Louis Lagrange and the European mathematicians of the 18th century, expressed horror at the metaphysical aspects of limit. Cornu states that many students today find themselves in a similar situation when they are able to compute using limits but fail to understand it as “real” mathematics. 4. “Is the limit attained or not?” (p. 161). This question was debated for centuries among top mathematicians. Some believed that a quantity can only be made as close as we like to its limit, while others believed that at some point the infinitely small quantities “vanished” allowing the quantity to actually achieve its limit. As observed by Davis and Vinner (1986) and Williams (1991) this is still a question among students. Although, it should be said that these two studies viewed it from the vantage point that the limit may be attained (such as for the limit of a constant function) while Cornu viewed the phrase “attaining the limit” to mean that the limit must be attained as the limit point is approached. Both Sierpinska (1987) and Williams (1991) also studied epistemological obstacles related to limits. Sierpinska noted four sources of epistemological obstacles: scientific knowledge, infinity, function, and real number. Through her study she attempted to cause cognitive conflict in students who have various cognitive obstacles in an effort to help each student overcome his/her obstacles. She found that none of the obstacles had been completely overcome, however some cognitive conflict did take place. In her 22 opinion, attitudes towards scientific and mathematical knowledge created another obstacle which was difficult for these students to overcome. In another study about students‟ attitudes, Szyldlik (2000) studied the relationship between students' understanding of the limit concept and their sources of conviction. She found that students who hold internal sources of conviction are more likely to see calculus as logical and consistent. These students are better able to use definitions and logic to make sense of mathematics, more likely to have a coherent understanding of the limit concept, and more likely to hold a static conception of limit throughout the interview. On the other hand, she found that students who hold external sources of conviction tend to view mathematics as a collection of procedures and rules to be memorized and applied in the appropriate situations. To these students, mathematical theory, including definitions, proofs, and counterexamples, is unlikely to play an important role in their understanding. These students are more likely to give incomplete or contradictory explanations of the limit concept, more likely to hold common misconceptions about limits, and less likely to have the ability to explain the procedures they are using. In perhaps the most comprehensive study to date regarding the understanding of the limit concept, Williams (1991) compared students‟ limit models to six limit characterizations and explored a variety of materials intended to cause cognitive conflict within the students. He did this by asking students to complete a questionnaire containing several common beliefs as shown in prior research (see Figure 1). 23 Figure 1. Questionnaire used by Williams (1991, p. 221). According to Williams, “statements 1 – 6 can be characterized as describing limit respectively as (a) dynamictheoretical, (b) acting as a boundary, (c) formal, (d) unreachable, (e) acting as an approximation, and (f) dynamicpractical” (p. 221). From the results of this questionnaire, he found “that students often describe their understandings of limit in terms of two or more of these informal ideas” (p. 225). In addition, the most popular characterizations from his questionnaire were: 1. the dynamictheoretical model, selected as “true” by 80% of respondents and selected the best description by 30% of respondents; 2. the unreachable model, selected as “true” by 70% of respondents and selected the best description by 36% of respondents; and 3. the formal model, selected as “true” by 66% of respondents and selected the best description by 19% of respondents. A. Please mark the following six statements about limits as being true or false: 1. T F A limit describes how a function moves as x moves toward a certain point. 2. T F A limit is a number or point past which a function cannot go. 3. T F A limit is a number that the yvalues of a function can be made arbitrarily close to by restricting the xvalues. 4. T F A limit is a number or point the function gets close to but never reaches. 5. T F A limit is an approximation that can be made as accurate as you wish. 6. T F A limit is determined by plugging in numbers closer and closer to a given number until the limit is reached. B. Which of the above statements best describes a limit as you understand it? (Circle one) 1 2 3 4 5 6 None C. Please describe in a few sentences what you understand a limit to be. That is, describe what it means to say that the limit of a function f as xs is some number L. 24 After administering the questionnaire, Williams selected a small group of students to take part in the second phase of the study which consisted of tasks designed to create cognitive conflict within students possessing these informal models of limit. Of most interest to my study are the results of the students identified as possessing the dynamictheoretical model of limit. None of the students containing dynamictheoretical models changed their view of limit during the course of the study. When these students encountered functions that contradicted their current model of limit, often they would dismiss the contradiction as irrelevant – an anomaly that does not pertain to most situations. In this way, Williams noticed that students‟ attitudes towards mathematical truth played a key role in determining their reactions towards the study‟s tasks. Several students made statements that they do not believe a general description of limit exists. As stated by one of the study participants, “I don‟t think there is a definition that is going to fulfill every function there is” (p. 232). Williams found several aspects of models that students valued that might have contributed to their resistance to change their viewpoints when faced with cognitive conflicts. Two of these aspects discussed are expediency and simplicity. Where Williams found that the dynamictheoretical model of limit was common among calculus students and relatively resistant to change, Oehrtman (2002, 2003) classified motion as a “weak” metaphor, stating that language referring to motion was frequently not intended to be a description of something actually moving. In one example which is particularly relevant to this study, students were asked to “Explain what it means for a function of two variables to be continuous” (Oehrtman, 2003, p. 399). In response to this prompt, six of the twentyfive participants actually discussed an object in 25 motion, while another eleven students used motion language without applying that language to an actual object. For the six students who described something as moving, Oehrtman argues “that motion tended to be simply superimposed on another conceptual image that actually carried the structure and logic of their thinking” (p. 402). For example, one student describes a continuous two variable function as a board in which a mouse can run around on without falling through. In this case, the motion of the mouse was not the primary imagery; rather, the primary image was that of a board without any holes in it. Much of Oehtman‟s 2003 work was based on the use of metaphors studied by Lakoff and Nunez (2000). The relevance of this theory for the present study will be discussed in the section “mental representations and conceptual metaphors.” From this perspective, Nunez (1999) points out that there are inherent differences between two conceptualizations of continuity, which he refers to as natural continuity and CauchyWeierstrauss continuity. From Nunez‟s perspective, natural continuity arises from a natural metaphor “a line IS the motion of a traveler tracing that line” (p. 56). From this perspective, the motion creates the line, and continuity is the result of fluid motion. Contrasting this view of continuity is the CauchyWeierstrauss view of continuity, which is the result of the 19th century formal mathematics. From Nunez‟s perspective, CauchyWeierstrauss continuity is built upon three conceptual metaphors: “A line IS a set of points; Natural continuity IS gaplessness; Approaching a limit IS preservation of closeness near a point” (p. 57). To Nunez, these three metaphors create a separate conceptualization of continuity that contrasts significantly with the conceptualization of natural continuity. 26 Just as Nunez observed a cognitive difference between natural continuity and the formal definition of continuity, other authors have noticed significant difference between students‟ understanding of limits and the formal definition of limit. One significant difference is described by Kyeong Roh Hah (2005) as reversibility or reverse thinking. These words describe one inherent difference in the formal limit approach to the intuitive approach often used by students. Introductory calculus often teaches the limit concept as the result of the function as the independent variable gets closer and closer to the limit point. Notice, in this case it is the independent variable which is made close to the limit point and the value of the dependent variable is observed. This contrasts the formal definition of limit which requires that the dependent variable can be made arbitrarily close to the limit value for all values of the independent variable within some neighborhood of the limit point. In this case it is the dependent variable that is being made close to the limit value instead of the independent variable being made close to the limit point. Roh Hah studied this type of reverse thinking in the context of infinite sequences and described it as “the ability to think of the infinite process in defining the limit in terms of the index and simultaneously to reverse the process by finding an appropriate index in terms of an arbitrarily chosen error bound” (pp. 2021). Theoretical Perspectives In the 1950s and 1960s, the "new math" movement brought with it an increased emphasis on clear definitions and mathematical rigor. However, by the late 1970s and early 1980s, mathematics education researchers began observing sharp differences between mathematical concepts as they were taught in class and the concepts as they were understood by the students. In particular, several authors noted key difficulties 27 understanding such concepts as limits of functions (Ervynck, 1981; Sierpinska, 1987) and limits of sequences and series (Davis and Vinner, 1986). At this time researchers began distinguishing between mathematical ideas as they are presented in formal mathematics and those same ideas as they are understood by students. The terms concept image and concept definition were created to describe this difference (Tall and Vinner, 1981; Vinner and Hershkowitz, 1980). The term concept image was introduced to describe the "total cognitive structure that is associated with the concept, which includes mental pictures and associated properties and processes" (Tall and Vinner, 1981, p.152). On the other hand, the concept definition was created to refer to a formal definition of a concept, such as a definition found in a textbook. In line with constructivist learning theories, this idea focuses learning and understanding on the individual and his or her conceptions rather than on the formal words used to describe a concept. Cognitive Obstacles Other researchers showed that large parts of students‟ concept images are built on intuition and experiences gained outside the formal teaching of a subject. Cornu (1991) refers to these conceptions of an idea obtained from daily experience prior to formal instruction as spontaneous conceptions. These conceptions can be quite powerful and do not disappear when formal ideas are presented. Instead, these spontaneous conceptions and any new knowledge obtained from instruction may coexist independently or they may intermingle to form new conceptions in the student. This occurs even if the different ideas conflict with each other. According to Papert (1980), "Sometimes the conflicting pieces of knowledge can be reconciled, sometimes one or the other must be abandoned, 28 and sometimes the two can both be 'kept around' if safely maintained in separate mental compartments" (Papert, 1980, p.121). Importantly, it was observed that learning a new idea, in itself, was not enough to change a students' prior conceptions. Instead, the student might simultaneously hold on to both ideas and then select which one to use in any given situation. Students may even retrieve combinations of the two ideas, with detrimental results (Davis and Vinner, 1986). In response to these observations, researchers began looking for models of cognitive change which might describe how a student may conceptually reorganize a concept. One such model requires that three criteria must be met before a student will be willing to undergo conceptual reorganization. First, the student must be dissatisfied with the current organization of a concept. Second, an alternative conception must be available which the student finds both reasonable and understandable. Third, the student must come to view this alternative conception as useful or valuable (Posner, Strike, Hewson and Gertzog, 1982). In a similar vein, Nussbaum and Novick (1982) propose an instructional method which will allow students to create conceptual change. They propose that, first, the student take part in an exposing event created to help students become acquainted with their own current conceptions. The student is then exposed to a discrepant event created to cause dissatisfaction towards the student's current conceptions. Finally a resolution is provided which gives the student an opportunity to interact with new, alternative conceptions. From these ideas of cognitive change, several researchers in mathematics education began studying cognitive obstacles. A cognitive obstacle can be described as a conception that creates a barrier to further student understanding. Cornu (1991) describes 29 several different types of obstacles: "genetic and psychological obstacles which occur as a result of the personal development of the student, didactical obstacles which occur because of the nature of the teaching and the teacher, and epistemological obstacles which occur because of the nature of the mathematical concepts themselves" (Cornu, 1991, p.158). Epistemological obstacles are of particular interest to this study since they tend to be conceptions that prove to be quite useful in one domain but create an obstacle when translated into another, similar domain. Furthermore, epistemological obstacles are often unavoidable and essential to learning, and they are frequently found in the historical development of the concept (Cornu, 1991). Mental Representations and Conceptual Metaphors It is important to consider how students come to understand mathematics. One common theoretical perspective is that of mental representations (Williams, 2001). This viewpoint holds that mathematical learning places ideas, facts, and procedures as part of an internal network of mental representations, and the depth of understanding is determined by connections with other representations within this network (Hiebert and Carpenter, 1992). From this perspective, a mathematical idea is understood when its mental representation has a large number of strong, robust connections with other representations. In contrast to this theory is the idea of conceptual metaphors developed by Lakoff and Nunez (2000). From the perspective of these authors, there is an “intimate relation between cognition, mind, and living body experience in the world” (Nunez, 1999, p. 49). In this way, all mathematics is considered to be a result of our embodied experiences, and 30 the meaning of mathematics is built upon conceptual metaphors which project meaning onto new, abstract domains from previously understood, more concrete concepts. It is from this second perspective that this study will be conducted. This would emphasize the belief that knowledge is not only based on connections between related concepts, but that students will actually create meaning for new, abstract concepts based on their understanding of other wellunderstood ideas. In relation to this study on students' understanding of limits in threedimensional calculus, we would expect the students to have previously developed a strong understanding of the limit concept in single variable calculus and they would attempt to project this understanding onto the new, multivariable limit problem. One powerful metaphor used to understand limits is the fictive motion metaphor (Talmy, 1988). This provides a metaphorical means of conceptualizing a static curve as the result of dynamic motion. This metaphor is common throughout the English language. A statement such as “the trail goes to the peak of the mountain” uses dynamic language to capture essence of a static object, a trail. In this way, the fictive motion metaphor is used in mathematics to perceive a graph not as a set of points but a path created by dynamic motion. Since this study focuses on students‟ use of dynamic imagery in multivariable calculus, the fictive motion metaphor has the potential to play an important role in the description of students‟ conceptualization of the multivariable limit concept. Metaphorical thinking has been studied in regard to limits in particular and within mathematics education as a whole. Of particular interest are the studies by Oehrtman (2002, 2003) in which he analyzed the written and verbal language of firstyear calculus 31 students' reasoning about limits. In his research he divides metaphors into weak and strong metaphors. Strong metaphors are ones that "force the relevant concepts involved to change in response to one another" (Oehrtman, 2003, p.398). These metaphors are active and support creative thinking in new domains. Abstraction and Generalization The concept of reflective abstraction was introduced and discussed by Piaget (see, for example Piaget, 1985) to describe the development of logicomathematical structures in a child during cognitive development. Reflective abstraction is considered to be entirely internal, as opposed to empirical abstraction and pseudoempirical abstraction which derive from properties of objects and actions of those objects, respectively. Piaget considered four different kinds of mental constructions that could take place during reflective abstraction: interiorization, or the construction of an internal process in order to represent a perceived phenomena; coordination of two or more processes into a single new one; encapsulation of a dynamic process into a static object; and generalization, or the application of existing knowledge to new phenomena. Generalization is discussed in Harel and Tall (1989) and he distinguishes three different types of generalization that may occur. Expansive generalization extends an individual's thinking from one domain to another without changing the original ideas. Reconstructive generalization extends an individual‟s thinking while at the same time reconstructing the existing concepts in order to make the generalization reasonable. Harel and Tall also describe disjunctive generalization in which new ideas are created without an attempt to connect them with prior understanding. This is generalization in the sense that the student is familiar with a larger range of concepts, but this could not be 32 considered a mental reconstruction of the student‟s knowledge in the way that Piaget discusses it. Infinite Processes Many difficulties in understanding the limit concept are found in the infinite nature of limits. The limits concept is often conceptualized as an infinite process which can never be completed in its entirety. However, even though this process cannot be completed, mathematicians are capable of speaking about the limit process as a coherent whole, and they are capable of using the result of a limit as an object to create more sophisticated processes. This means of understanding a process as an object is not unique to the limit concept, but plays a vital role in all of mathematics. Tall et al. (2000) give a thorough description of several authors‟ descriptions of the cognitive processes involved in converting a mathematical process to an encapsulated object. I will briefly describe several of these viewpoints below. Dubinsky (1991) and his colleagues developed a theory of conceptual development based on the creation of actions, processes, objects, and schemas. This theory has become known as APOS theory. In this model of student learning, the student first understands a mathematical concept as an action to be performed. After some experience with the action, the student is able to perceive the action as a process and speak of its result without being required to perform the action. Eventually this process will be encapsulated into an object which can in turn be used to create more sophisticated mathematical actions. The student then gathers these related actions, processes and objects into a coherent collection called a schema. The concept of schema is similar to 33 that of a concept image, with the exception that a schema is required to be coherent while a concept image is not. From the perspective of APOS theory, an action is a stepbystep mathematical procedure. In order for a concept to progress from an action to process, the individual must become aware of the various steps involved in the action and have the ability to reflect on them. It is this ability to think about the action without actually performing it that distinguishes a process from an action. The primary difference between a process and an object is the ability to conceptualize the process as a whole and perform actions with it. Sfard (1991) prefers to the use the word reification over encapsulation to describe the process of understanding a mathematical process as an object. For Sfard the act of reification is a movement from an operational understanding to a structural understanding. Sfard‟s description of the transformation of a process to an object takes place in three steps. She describes the adoption of a familiar process as the interiorization of that process. Once interiorized, the process can be compacted and understood as a whole; which she refers to as condensation. To Sfard a condensed process is still operational and the individual will interact with the process in an operational manner. It is the process of reification that transitions the individual from dealing with an operational process to a structural object. To Sfard it is precisely this transition from operational to structural understanding that signifies the transition from a condensed process to a reified object. Eddie Gray and David Tall (1994) used the word procept to describe their understanding of how a mathematical concept can take the form of both a process and an 34 object simultaneously. From their perspective the important development is the creation of a symbol to represent both the process and the object is an essential part of the development of a procept. Many of these authors developed their theories using finite procedures, such as counting or addition. The infinite nature of the limit concept makes it particularly challenging for an individual to progress from viewing the concept as a process to viewing it as an object. Tall et al. (2000) discussed this difficulty and pointed out that in “the peculiar case of the limit concept where the (potentially infinite) process of computing a limit may not have a finite algorithm at all […] a procept may exist, which has both a process (tending to a limit) and a concept (of limit), yet there is no procedure to compute the desired result” (p. 226). 35 CHAPTER III METHODS The methods of this study were developed with the problem statement in mind: “Describe how students with a dynamic view of limit generalize their understanding of the limit concept in a multivariable environment.” In particular, the study was designed to accomplish the following objectives; the study should: Identify students with a dynamic view of limit. Provide an opportunity to analyze participants‟ prior understanding of the limit concept. Allow participants to encounter the limit concept in multivariable environments. Provide an opportunity to observe participants‟ generalization of the limit concept in these multivariable environments. In this chapter I will begin by explaining why qualitative research methods were used to design this study. I will then describe the methods of the study, including the participants, settings, interviewee selection method, and data collection methods. Finally, I will conclude this chapter with a discussion about the researcher‟s role and perspective. Why a Qualitative Study? Data collection for the study took place during the fall 2007 semester at a large state university. It was determined that a series of indepth interviews using qualitative analysis would be required to adequately respond to the study‟s problem statement. 36 Maxwell (1996) describes five research purposes which are specifically suited to qualitative analysis. “Understanding the meaning, for participants in the study, of the events, situations, and actions that are involved with and of the accounts that they give of their lives and experiences” (p. 17). “Understanding the particular context within which the participants act, and the influence that this context has on their actions” (p. 17). “Identifying unanticipated phenomena and influences, and generating new grounded theories about the latter” (p. 19). “Understanding the process by which events and actions take place” (p. 19). Developing causal explanations” (p. 20). An examination of each of these five research purposes creates a strong argument for this study to be qualitative in nature. The purpose of this study is to create a description of students‟ conceptualizations of the limit concept. Internal conceptualizations are, by their nature, not observable. Therefore, observable data must take the form of written responses, mathematical calculations, and verbal descriptions of mathematical concepts. The primary interest of the study is not in reporting the observable data, but rather in understanding the meaning of the observable data in terms of the students‟ internal conceptualizations of the limit concept. Because of this, the context of the observable data and the process which students use to create mathematical conclusions play an essential role in understanding the meaning of the observable data. 37 The fact that this is a first study on students‟ conceptualizations of multivariable limits is an important factor in choosing a qualitative study. The lack of previous research reports on the topic requires an ability to respond to unanticipated events. A quantitative study cannot be prepared to deal with unanticipated behavior, but a qualitative study using grounded theory techniques can describe unanticipated behavior in the context in which it happens. Finally, the purpose of this study asks a causal question that requires qualitative methods to answer. The causal question in this study is “what role does prior understanding of the limit concept have in students‟ conceptualizations of multivariable limits?” This question is qualitative in the sense that it seeks to describe the influence of certain cognitive events on other events. This contrasts a quantitative question which seeks to explain current events in terms of the variance of a previous set of events. For the above reasons a qualitative study was developed that allowed the researcher to observe and interview students interacting with multivariable limits. The study was developed to contain three key components, a) an initial questionnaire which provided a means of selecting interview participants and comparing those participants to the student population as a whole, b) an interview probing students‟ understanding of single variable calculus, and c) a series of two interviews involving multivariable limits in four different settings. Participants and Setting The goal of this study was to analyze the changes in student thinking about the limit concept as they encounter multivariable limits. For this reason, it was important to observe students who are familiar with both multivariable functions and the limit concept 38 but have yet to study multivariable limits. The university where this study took place teaches calculus as a threepart sequence with multivariable calculus being a main focus of the third semester in this sequence. Therefore, participants for the study were chosen from this third semester calculus course. Participants for the study were chosen using a process of purposeful selection. The selection of a purposeful sampling seeks to choose uniquely qualified individuals capable of providing information necessary to answer the study‟s research questions as described by Maxwell (1996). For a smallscale study, purposeful selection is often preferred to a random sample. Random samples are necessary to externally generalize the findings of the study; however, since external generalization requires a sufficiently large sample size, in the case of a small sample size it is preferred to purposefully select participants likely to provide useful information towards answering the study‟s research questions. It is important to note that the sample chosen for this study was not what is often called a „convenience sample.‟ Rather than choosing participants based on convenience, participants were chosen using the predetermined set of guidelines outlined in the next section. The purposeful selection process of this study sought to find students who tend to conceptualize the limit concept in a dynamic manner. Since the goal of this project is to describe student‟s cognitive behavior, preference was given to students who demonstrated a strong ability to express themselves in a clear manner. For these reasons, a questionnaire was given to all willing students who participated in third semester calculus in the fall of 2007. This questionnaire had two useful purposes. Most importantly, it provided an opportunity to analyze a large number of students‟ 39 understanding of the limit concept, allowing those students with the preferences described above to be selected for the interviews. It also created a description of the entire population of students enrolled in this course, indicating how well our selected students represented the course‟s population as a whole. Study Questionnaire The development of the questionnaire (see Appendix A) was inspired by Williams (1991). Part A consisted of six definitions each representing a different theoretical model of limit commonly held by students studying calculus. It is important to note that actual models of limit held by students are extremely complex cognitive structures, and they are not expected to precisely line up with any theoretical model given here. For this reason, I will use the phrase theoretical model to refer to those theoretical models of limit that I believe a student may possess, and I will use the phrase personal model to refer the actual model of limit held by an individual student. Similar to Williams, the beginning questionnaire attempted to gauge to what degree students agree with various theoretical models. Each of these models was inspired by either research on students‟ understandings of the limit concept or an historical development of the limit concept. The model represented by each question is given below: Question 1. “A limit describes how a function moves as you approach a given point.” Dynamic Model. This model is based on the dynamic imagery that the graph of a function is the path of a point swept out over time. This imagery was used by Newton when he developed the calculus (Edwards, 1979) and has been shown to be common among calculus students studying the limit concept (Williams, 1991). 40 This is the theoretical model we are most interested in and represents what Williams refers to as a dynamictheoretical model. Question 2. “A limit can be found by plugging in a number infinitely close to a point.” Infinitesimal Model. This model is based on the existence of infinitely small quantities. Cornu (1991) discussed this as an epistemological obstacle towards the development of a formal limit concept. Question 3. “A limit is a number that a function can be made arbitrarily close to by taking values sufficiently close to a certain point.” Formal Model. This model is based on the modern, CauchyWeierstrass definition of limit, and closely mimics the definition of limit given in many introductory calculus textbooks. Question 4. “A limit is a number or point the function gets close to but never reaches.” Unreachable Model. This model is based on the popular misconception that a limit can never be attained (Davis and Vinner, 1986). Question 5. “A limit is an approximation that can be made as accurate as you wish.” Approximation Model. Based on the notion of limit as an approximation. This was used in the study by Williams (1991). As observed by Williams, students rarely possess a personal model of limit closely aligned to one of these theoretical models. Instead, students‟ personal models of limit tended to be complex combinations of these theoretical models. For this reason, classification of students into distinct categories is extremely difficult. Questions 6  8 were included to better distinguish which theoretical model (or models) best represented each student‟s personal model of limit (see Appendix A). 41 In our process of purposeful selection, the following criteria were used to select which students would take part in the remainder of the study. 1. Students should select “somewhat agree” or “strongly agree” to question 1 of the questionnaire. 2. Students should circle the number 1 on question 6 of the questionnaire. 3. Students should use dynamic language in their responses to questions 1, 7, and 8. Dynamic language is considered to be language that emphasizes the use of motion in understanding the limit concept. Such language might include key phrases such as “moves towards,” “approaches,” or “gets closer to.” 4. Students should provide written descriptions that demonstrate an ability to express themselves in a clear manner. From these four criteria, students were invited to participate in the interview portion of the study in the following manner: Questionnaires were collected from all students indicating interest in participating in the interviews. Questionnaires that failed to meet #1 above were removed from consideration. The remaining questionnaires were analyzed and those that did not meet #2 or #3 above were removed from consideration. The remaining questionnaires were analyzed along both #3 and #4 above, and students were judged as to how strongly they met each of the criteria. Students who were judged to have used strong dynamic language in their responses as described in #3 above and who were judged to demonstrate a strong ability to express their thinking as describes in #4 above were invited to participate in the interviews. In total nine students were invited to participate in the interviews. Seven of the nine students agreed to participate in the interviews and all seven completed interview process. 42 Data Collection Recalling that the goal of this study is to create a description of student behavior, it was decided that a qualitative study using taskbased interviews would be the best method for data collection. Thomas (1998) described various interview strategies used in qualitative studies. The interviews in this study contained questions of two types: general questions discussing the meaning of the limit concept and taskbased questions centered on specific limit problems. The general questions can be described as loose questions in the sense that their goal is to “reveal the variable ways respondents interpret a general question” (Thomas, 1998, p. 129). Taskbased questions can be described as responseguided questions in the sense that they “consist of the interviewer beginning with a prepared question, then spontaneously creating followup queries relating to the interviewee‟s answer to the opening question” (p. 132). In the case of this study, the initial questions take the form of a mathematics problem and followup questions are asked to clarify meaning about students‟ responses while solving the mathematical problem. Data from the interviews took three forms: written work in response to mathematical limit problems, verbal responses to questions throughout the interview, and observation of student behavior throughout the interview (important behaviors might include pointing at a graph or the use of hand gestures). In order to capture both verbal and observational data throughout the course of the interviews, it was decided that videotaping would be a primary source of data collection. Each video segment was stored on DVD disks and viewed only by those involved in overseeing the study. 43 In order to analyze students‟ prior understanding of the limit concept, the first interview session (see Appendix B) focused solely on the students‟ understanding of limits in a traditional introductory calculus course. Participants for this study were carefully chosen so that they possessed a personal model of limit that involves some element of dynamic imagery. Question 1 and question 2 of the first interview session were designed to evaluate the strength and nature of this dynamic imagery in each student‟s personal model of limit. Question 1: Review your answers to the questionnaire given earlier. Would you like to change any of your answers? Are there any answers that you would like to clarify? Question 2: When describing a function as “approaching” or “getting close to” a point, this idea would best be explained as: a) Evaluating a function at different numbers over time with those numbers successively getting closer to the point in question. b) Mentally envisioning a point on a graph moving closer and closer to the limit point. Question 1 reviewed with each student his/her responses to the questionnaire, providing an opportunity for students to explain their responses in detail. Question 2 was designed to evaluate the dynamic nature of each student‟s conception of limit by providing them with two options, one which involves examining a function at various points over time and a second which involves mentally envisioning a point in motion along a graph. Williams (1991) would refer to those students who select (a) as having a dynamicpractical model of limit and those selecting (b) as having a dynamictheoretical model of limit. The remainder of the first interview session was designed to observe students‟ behavior on traditional introductory calculus limit problems in order to 44 determine how the students‟ personal model of limit is put into practice on actual problems (see Appendix B). This portion of the interview involved both symbolic and graphical limit problems from introductory calculus, and a calculator was made available to students for use with these problems. The remaining interview sessions were designed to provide students with the opportunity to encounter multivariable limits in four different settings: traditional symbolic manipulation, symbolic manipulation involving polar coordinates, threedimensional graphing, and contour graphing. These were designed as four separate treatments (see Appendices C and D). Each treatment introduces students to the multivariable setting in question, asks students to describe how to determine whether a multivariable limit exists or not in that setting, asks students to explain why they believe their method should work, and observes students using this method on problems from this setting. It was decided that for purposes of time, these four treatments would take place over the course of two interview sessions, with each session containing a symbolic and graphing portion. The first session contained traditional symbolic manipulation and threedimensional graphing while the second session contained symbolic manipulation involving polar coordinates and contour graphing. In order to allow the students to interact with a larger number of multivariable functions, a computer was used to experience graphs during the three dimensional graphing portion of the interviews and computer generated graphs were printed out and provided during the contour graphing portion of the interview. The mathematical software package Maple 11© was used to create all the graphs used in this study. 45 In addition to these four treatments, students were asked several interview questions following each multivariable limit experience designed to help them reflect on their experiences with multivariable limits. This portion of the interviews was created to provide students with an opportunity to discuss their overall understanding of the limit concept and the connection between single variable and multivariable limits. Researcher’s Perspective In qualitative research it is necessary that the researcher be actively involved in the setting of the research study, and as a consequence the collection and interpretation of the data will be effected by the role the researcher plays. Holliday (2002) explains, “The presence of the researcher in the research setting is unavoidable and must be treated as a resource” (p. 173). Because of this fact, the perspective that I bring with me into the study should be assessed in order to put the data from the interviews in context. For this reason, I will spend the remainder of this section discussing my beliefs and expectations prior to the collection of data for the study. In this study I took the position that students would enter the study with an initial personal model of the limit concept. I use the word model much in the same sense as Williams (1991) to be a collection of cognitive structures which has an internal meaning to the student and carries with it some predictive qualities. I use the phrase personal model in contrast to the phrase theoretical model which captures a hypothetical conceptualization of the limit concept. Throughout the study I expected students to be able to coherently express their beliefs about their personal model of limit and use the model to determine the truthfulness of related mathematical statements. I also expected the students to be able to use this model to make sense of related mathematical ideas; in 46 particular, I expect the students to try to use their personal model of limit to make sense of limit problems in a multivariable setting. I anticipated that the students‟ models of limit would be relatively static, until confronted with a discrepant event, in the sense of Nussbaum and Novick (1982). At the same time, I expected students involved in the interviews to be in the early developmental stages in terms of their personal model of multivariable limits. I expected this notion of multivariable limits to be less coherent and less consistent than their personal model of single variable limits. However, I anticipated that students would use their single variable limit model to interpret multivariable limits, providing a basis for their actions and statements in the new context. It is worthwhile to note that I use the term model to mean something very similar to Dubinsky‟s (1991) use of the word schema. The primary difference is my emphasis on the predictive qualities of a students‟ model of limit, while Dubinsky emphasizes the collection of actions, processes and objects which are contained in a limit schema. However, both these notions are more specific than Tall and Vinner‟s (1981) notion of a student‟s concept image. Both a model and a schema are intended to be coherent in the sense that students are, to some degree, aware of these structures and able to use them in productive ways. Meanwhile, the term concept image is the collection of all cognitive structures connected with the concept. This concept image may not be coherent and a student may have little awareness of it or ability to use it productively. Many past studies have looked at students‟ abilities to use the imagery of motion to understand the limit concept (Monk, 1992; Thompson, 1994b; Carlson et al, 2002). Williams (1991) found that 30 percent of the students in his study contained what he 47 called a dynamictheoretical view of limit, while 80% of the students believed that a dynamictheoretical definition of limit was true. This dynamictheoretical view of limit is marked by a student‟s use of motion to understand the limit concept. Students involved in this study were carefully chosen to have a personal model of limit similar to the dynamictheoretical model of limit and I expected this aspect of their thinking to influence the way they conceptualize multivariable limits. In a multivariable setting, the idea of motion easily assists showing that a limit does not exist, since a limit such as 2 2 2 2 ( , ) (0,0) lim x y x y x y can be shown to not exist by observing that when moving along the yaxis the limit tends to 1, while when moving along the xaxis, the limit tends to 1. Since these two directional limits are unequal, the limit does not exist. This should be somewhat familiar to students; since, in the twodimensional case you can show a limit does not exist by showing that lim f (x) lim f (x) x a x a . However, unlike in the twodimensional case, the multivariable limit is not solved by simply showing that the two directional limits are equal. In fact, in the threedimensional case there are uncountably many directional limits, following any possible path to the point (a,b), that must all be equal for lim ( , ) ( , ) ( , ) f x y x y a b to exist. Therefore, the imagery of motion towards a point is insufficient to show that a multivariable limit exists. Because this use of motion is insufficient to completely understand the multivariable limit problem, I expected students to encounter a cognitive obstacle from their application of motion into multivariable limits. From there, I anticipated an effort 48 on the part of the students to reconstruct their understanding of the limit concept to allow for a complete understanding of multivariable limits. The nature of this anticipated reconstruction process is one of the primary focuses in this study. These expectations color the way I interacted with students during the interviews. As I engaged in responseguided questioning, they affected the types of questions I asked and the manner in which these questioned were presented. I do not believe my role in the interviews should be perceived as a negative aspect of the study design; rather, I believe that my expectations enabled me to guide the interviews towards a line of discourse that would be profitable for answering the study‟s research questions. 49 CHAPTER IV DATA ANALYSIS In this chapter I will describe how the collected data was analyzed. I will begin with a description of the questionnaire results which show how the seven selected interview participants compare to the entire calculus III student population. Then I will describe how the interview data was analyzed qualitatively. Finally, I will describe how this analysis of the transcripts along with an analysis of formal mathematics led to the development of three models of limit: neighborhood, dynamic, and topological. These three models will be used to code the interview data and shape the results provided in chapter V. Questionnaire Results During the first week of the fall semester, the study questionnaire was distributed in all five sections of calculus III offered by the university. All willing students completed the questionnaire at this time and a total of 208 students returned their responses. The results on questions one through five are given in Tables 1 and 2 below. Strongly Somewhat Somewhat Strongly Likert Average Agree Agree Neither Disagree Disagree (4 = strongly agree) Statement 1 105 68 3 22 10 3.13 Statement 2 46 97 19 33 13 2.62 Statement 3 63 93 28 14 5 2.96 Statement 4 84 51 20 30 23 2.69 Statement 5 32 66 35 37 37 2.09 Table 1: Questionnaire Results (Cumulative) 50 Strongly Somewhat Somewhat Strongly Agree Agree Neither Disagree Disagree Statement 1 50% 33% 1% 11% 5% Statement 2 22% 47% 9% 16% 6% Statement 3 31% 46% 14% 7% 2% Statement 4 40% 25% 10% 14% 11% Statement 5 15% 32% 17% 18% 18% Table 2: Questionnaire Results (Percentage) As opposed to a Likert Scale analysis, the result can also be viewed as binomial data, as in tables 3 and 4 below. Agree Disagree Statement 1 173 32 Statement 2 143 46 Statement 3 156 19 Statement 4 135 53 Statement 5 98 74 Table 3: Questionnaire Results (Binomial) Agree Disagree Statement 1 83% 15% Statement 2 69% 22% Statement 3 77% 9% Statement 4 65% 25% Statement 5 47% 36% Table 4: Questionnaire Results (Binomial Percentage) It can be observed from the tables above that the respondents have a strong tendency towards agreeing with the statements as presented. This corresponds with the finding from Williams (1991) that students are often capable of believing several models of limit simultaneously. This result, however, should be treated carefully due to the known phenomenon of acquiescence bias that students tend to agree with statements as presented. Due to this known fact, students were asked which model best described the way they understood the limit concept and the results are presented in tables 5 and 6 below. 51 Statement: 1 2 3 4 5 None 78 15 37 58 5 8 Table 5: Question #6 Results (Cumulative) Statement: 1 2 3 4 5 None 39% 7% 18% 29% 2% 4% Table 6: Question #6 Results (Percentage) From the 208 students who completed the study questionnaire, 36 agreed to take part in the interview portion of the study. Their responses follow in tables 7 and 8. Strongly Somewhat Somewhat Strongly Likert Average Agree Agree Neither Disagree Disagree (4 = strongly agree) Statement 1 20 12 1 3 0 3.36 Statement 2 7 15 6 7 1 2.56 Statement 3 10 17 3 2 3 2.83 Statement 4 13 7 5 6 5 2.47 Statement 5 7 11 3 6 9 2.03 Table 7: Questionnaire Results, Interview Volunteers Statement: 1 2 3 4 5 None 14 4 6 9 0 1 Table 8: Question #6 Results, Interview Volunteers These responses are closely aligned with the responses of the student population as a whole, as tables 9 and 10 show. Agree Disagree 36 Volunteers 208 Students 36 Volunteers 208 Students Statement 1 89% 83% 8% 15% Statement 2 63% 69% 23% 22% Statement 3 84% 77% 16% 9% Statement 4 65% 65% 35% 25% Statement 5 67% 47% 56% 36% Table 9: Questionnaire Results, Volunteer Comparison Statement 1 2 3 4 5 None 36 Volunteers 42% 12% 18% 27% 0% 3% 208 Students 39% 7% 18% 29% 2% 4% Table 10: Question #6 Results, Volunteer Comparison 52 As the above tables show, with a few slight variations the two groups responded to the questionnaire in a similar fashion. The only exception was found in the response to question 5 which was found to be significantly different (p < .05). Since the emphasis of this study is on students with a dynamic understanding of limit, this difference in the students‟ opinions of the approximations model of limit was deemed to be insignificant in light of the study‟s goals. Therefore, this difference was not further explored in this study. From these 36 volunteers 9 were contacted to take part in the individual interview sessions, and of those 9, 7 participated in the interviews. All participants who began the interview process completed all 3 interviews. A total of three males and four females took part in the interview portion of the study. These seven interview participants will be given the pseudonyms Mike, Jessica, Jennifer, Amanda, Josh, Ashley, and Chris for the remainder of this study. These seven students were chosen using a method of purposeful selection so that their questionnaires showed a tendency towards understanding the limit using dynamic imagery. These seven students‟ responses to the questionnaire are shown in tables 11 and 12. Strongly Somewhat Somewhat Strongly Agree Agree Neither Disagree Disagree Statement 1 5 2 0 0 0 Statement 2 2 3 0 1 1 Statement 3 1 3 1 0 2 Statement 4 2 2 1 1 1 Statement 5 1 1 0 2 3 Table 11: Questionnaire Results, Interview Participants Model: 1 2 3 4 5 None 6 0 1 0 0 0 Table 12: Question #6 Results, Interview Participants 53 The above tables demonstrate the clear preference for participants who agree with statement 1 on the questionnaire. As shown by the entire class results, these seven students represent a sizeable portion of the class. From table 4, 83% of the students agree with statement 1 and from table 6, 38% of the students believe statement one best describes how they understand the concept of limit. Both of these numbers were the highest recorded for any of the five statements. Qualitative Data Analysis Qualitative data was analyzed using a grounded theory approach to data analysis. Maxwell (1996) describes grounded theory when he says, “The theory is grounded in the actual data collected, in contrast to a theory that is developed conceptually and then simply tested against empirical data” (p. 33). The fact that little previous research had been recorded on multivariable limits requires that a grounded theory approach be employed. This process of theory development led to the realization that the initial theoretical models of limit created for the study questionnaire were inadequate to describe students‟ interactions with the multivariable limit concept. This led to the creation of three new theoretical models of limit which were observed as part of students‟ descriptions of multivariable limits. These models grew out of the observation that students tend to conceptualize limits using either a) a sense of „closeness,‟ b) a dynamic process, or c) an examination of external features. In addition these models of limit were closely tied to the formal mathematics of the limit concept. 54 Investigating these three models led to a coding scheme for examining how students understand multivariable limits. This coding scheme was used to analyze the data and provide a description of student behavior throughout the interviews. In the remainder of this chapter, the development of the three models of limit will be described, resulting in a coding scheme which shall be used to bring clarity to the interview data. Definitions of Limit in Formal Mathematics In the effort to describe how students understand the multivariable limit concept, it is worthwhile to consider the formal mathematics behind the concept of limit. In this section, we will discuss how the concept of limit is developed formally and what cognitive structures might be necessary to understand this formal development. In formal mathematics, there are two ways to define the concept of limit in ℝn. The traditional definition requires the use of universal and existential quantifiers. Formal Limit Definition: If there exists a value L such that for every positive number, ε > 0, there exists a value, δ > 0 such that f (x) L whenever 0 x a , then we say that the limit of the function, f, as x approaches a is L, and we write f x L x a lim ( ) . In contrast to this formal definition there is a definition based on sequences. Sequential Limit Definition: If for every sequence (x : x a) n i with x a n n lim( ) we have that f x L n n lim ( ) , then we say that the limit of the function, f, as x approaches a is L, and we write f x L x a lim ( ) . 55 For a function, f: ℝn →ℝm, these two definitions can be proven to be equivalent. However, in practice, using each definition involves inherently different cognitive processes. To compare the two definitions, I will describe the possible cognitive processes necessary to develop an understanding of each definition. A possible description of the cognitive structures required for understanding the formal limit definition is given below: 1. Mentally construct a neighborhood of values around the point L. 2. Mentally construct a corresponding neighborhood of values around the point a. a. Coordinate these two constructions such that the neighborhood around a is mapped into the neighborhood around L. 3. Construct a process of reducing the size of the neighborhoods around L while maintaining corresponding, coordinated neighborhoods around a. Note that coordinating these two neighborhoods requires reverse thinking as described by Roh Hah (2005). In contrast to the formal definition, a possible description of the cognitive structures required for understanding limits using the sequential definition of limit is given below: 1. Construct a schema for evaluating the limit using a single sequence. a. This involves mentally constructing a sequence, (xn), that approaches the point a. b. Then constructing the resulting sequence, ( f (xn) ). c. Finally, evaluating the result of this infinite sequence, the point L. 56 2. Mentally construct an infinite process of evaluating the limit of these sequences successively. 3. Capture this infinite process into a coherent understanding of the limit of all possible sequences simultaneously. Capturing infinite processes is an inherently difficult task as described in chapter IV, “Infinite Processes.” Just as Nunez and his colleagues (1999) found that the concept of continuity has two cognitively different conceptualizations, the two notions of limit based on the two formal definitions above appear to be built on inherently different conceptualizations. The formal definition must first be grounded on a notion of „closeness‟ that can be used to create mathematical neighborhoods. This conceptualization is static and relies heavily on the conceptualization of the real number system. On the other hand, the sequential definition is built upon a process of examining points along a sequence. The conceptualization of this process is significantly different from that of mathematical neighborhoods in that it is both dynamic in nature and infinite. Although students rarely understand the limit concept using one of these formal definitions, the two definitions do provide a blueprint for different cognitive structures that can be used to develop an understanding of the limit concept. The idea of „closeness‟ used in the formal definition will form the basis for the neighborhood model of limit, and the dynamic process used in the sequential definition will form a basis for the dynamic model of limit. In the following sections, I will describe these models along with a third model, the topographical model, which is not based in formal mathematics. 57 The Neighborhood Model of Limit It is well documented that students struggle to understand the formal definition of limit. This is, in part, due to the need for “reverse thinking” (Roh Hah, 2005). A description of limit using reverse thinking was observed in only one student throughout the course of the study. This discussion occurred in response to question 5 on the questionnaire which posed an approximation view of the limit concept. Excerpt 1 JESSICA: Yes, because if you get closer to it, like, if you like, x, well you can make it as close to 2 as you want. I mean, yeah, it could take you like years to figure it out but you could get as close as you wanted to depending on what the number that you put in. Jessica‟s use of the phrases “you can make it as close to 2 as you want” and “you could get as close as you wanted to depending on what the number that you put in” both show some evidence of reverse thinking. They show that Jessica has an awareness that goes beyond simple closeness to an understanding that this closeness can be controlled. Oehrtman (2003) has argued that a metaphor of approximation can be used to help students better understand the definition of limit, and Jessica‟s response seems to support this notion. However, this was the only incident of her using language that demonstrated the use of reverse thinking, and it seems unlikely that this notion plays a vital role in her understanding of limits. Statement 3 on the questionnaire was designed to illustrate the formal definition of limit; however, no students used reverse thinking when responding to this statement. Amanda shows her confusion towards the problem in the excerpt below 58 Excerpt 2 AMANDA: Alright, number three, „A limit is a number that a function can be made arbitrarily close to by taking values sufficiently close to a certain point.‟ This one I read over and over several times and I was like, „Huh?‟ I know it‟s given other people problems. INTERVIEWER: Yeah. AMANDA: See, the arbitrarily close to, isn‟t that like, you can make a function close to certain point? Like, to me it‟s, it looks like you‟re choosing where the function is going. But, I don‟t really know. See, I don‟t think you can really choose where you‟re… where you‟re limit is. But, maybe I interpreted that wrong? Amanda takes the notion of reverse thinking and interprets it as “choosing where the function is going.” To her, the dynamic nature of the limit concept requires that the independent variable be considered first, and the dependent variable is analyzed as a result. This perspective shuts out the ability to apply reverse thinking to the limit concept. I argue that “reverse thinking” is only part of the cognitive structure needed to develop an understanding of the formal definition of limit. It is also important for students to develop a sense of „closeness‟ that can lead to the construction of mathematical neighborhoods. In Nunez‟s (1999) description of continuity, he points out that a key metaphor for CauchyWeierstrauss continuity is “limit IS preservation of closeness near a point” (p. 58). Using this metaphor, dynamic language such as the words “approaching” or “tending to” lose their meaning and are replaced by the static elements of a neighborhood consisting of values near the limit point. This idea of “closeness” will be the basis for the neighborhood model of limit discussed throughout this study. The phrase “neighborhood thinking” will refer to the 59 use of „closeness‟ by a student to describe his/her conceptualizations of the limit concept. In practical terms, neighborhood thinking can manifest itself in both graphical and symbolic settings. In a graphical sense, students “look around” the function to analyze the behavior of its graph near the limit point to create arguments about the value of the limit. In a symbolic way, students may calculate values very near the limit point to draw conclusions about the limit itself. Mathematicians use inequalities to create symbolic arguments involving „closeness‟; however, there was no evidence that students associated the use of inequalities with neighborhood thinking during this study. The Dynamic Model of Limit There is very little chance that students involved in this study had previously encountered the sequential definition of limit. Even though the study of infinite sequences was developed the prior semester, there is little in the curriculum to connect the ideas of sequences with those of finding traditional limits in calculus one. However, the related infinite process is a concept that is encountered by almost every student in introductory calculus. This process tends to manifest itself in two ways, symbolically using a process of evaluating the function at successive points and visually using a metaphor of motion. The symbolic process of analyzing limits is generally introduced early in an introductory calculus course. This process is often conceptualized in the form of a table of values in which the independent variable gets successively closer to the limit point and the corresponding dependent variable is analyzed. This process is intended to be both dynamic and infinite in nature, with the independent variable getting closer and closer to the limit point with each iteration of the process. In action, however, this is never treated 60 as an infinite process, but rather the process is ended after some finite number of steps and a conclusion is drawn about the limit of the function. In this way, as noted by Tall and his colleagues (2000) the concept of limit comes to be understood as both a process of approaching a point and the conclusion of that infinite process, while the process itself can never be carried out to its conclusion. Visual representations of this dynamic motion have been referred to by others as the „fictive motion metaphor‟ (Talmy, 1988; Lakoff and Nunez, 2000). Using this metaphor, the static curve of a graph actually represents the results of motion. Colloquial language supports the use of static objects embodying motion. For example, the phrase “this road goes to the lake” describes a static object, a road, as conveying a sense of motion. The road itself is not in motion, but motion is visualized on top of this structure. In a similar way, students may visualize motion in the static graph of a function. In the case of a single variable limit, this motion takes place from either the right or the left, creating the right and left hand limits. This visualization is different from the process of analyzing a function using a sequence of points in the fact that the visualization does not carry the same infinite nature as the process. Instead, the fluid motion from either side can be conceptualized as a single action and the conclusion can be visualized without using an infinite process. Similar to the symbolic process, the term „limit‟ can refer to both the visualized motion towards the limit point and the conclusion of that motion. Although the two manifestations above are conceptualized in significantly different ways, they both describe a dynamic process, either symbolic or visual, which results in the value of the limit. This process forms the basis of the dynamic model of limit discussed throughout this study. The phrase “dynamic thinking” will refer to 61 students‟ use of a dynamic process, either symbolically or visually, which is intended to result in the limit value. The Topographical Model of Limit A student in calculus I can be quite successful solving most limit problems encountered by only looking at the surface features of a function and never developing a sense of limit related to that of either the formal definition or the sequential definition of limit. This way of thinking will be referred to as the topographical model of limit. It begins with an ability to recognize and classify discontinuities in a function, and through this classification procedure, it is possible to know the resulting limit value. A possible description of the cognitive structures required for developing topographical thinking in calculus I is given below: 1. The function must be viewed as an object with inherent characteristics and properties. 2. Points must be classified as either continuous or discontinuous. If it is discontinuous, then the type of discontinuity (removeable, jump, infinite) must be classified. 3. The result of the limit must be deduced from the classification process. a. A continuous function‟s limit is given by evaluating the function at the point. b. A removable discontinuity must be „removed‟ before the limit can be evaluated. That is, using a Gestalttype viewpoint, a point must be identified which can fill in the hole on the function. c. A jump discontinuity has no limit. 62 d. An infinite discontinuity must be examined to determine if a sign change takes place. The term “function” in stage 1 is used in a loose sense. It is quite possible for students to treat external elements of the function, such as an equation or a graph, as the object being encountered. Even though for some students these external elements are not necessarily connected to the concept of function, they may still be used in a productive manner for solving most calculus I limit problems. However, it should be noted that this method does not cover every possible function that a student could encounter. For example, the function x f (x) sin 1 cannot be classified under the above system at x = 0. This means of conceptualizing the limit concept, by itself, potentially weakens the students‟ ability to understand important limiting situations in calculus. For example, Carlsen, Oehrtman, and Thompson (2007) argue that a topographical understanding of limits does not provide an understanding of the limiting processes necessary to understand differentiation and integration. It can also be observed that this conceptualization may lead to difficulties in understanding limits at infinity, which require an inherently different classification system in order to be understood topographically. It is also important to observe that the topographical understanding of limit directly contradicts with the limit concept as introduced in formal mathematics. In formal mathematics, the notion of continuity is defined using the definition of limit as its basis. The topographical understanding of limit described here requires an understanding 63 of continuity prior to the concept of limit, which is opposite that of the formal mathematics. Throughout this study, the term “topographical thinking” will refer to the use of external characteristics of the function to make decisions about the value of the limit. These external characteristics tend to be visual in the form of a graph; however, it is also possible for students to use the external characteristics of a symbolic expression, for example the value of a function at a single point, to draw conclusions about the limit. This type of topographical thinking has several characteristics that distinguish it from both the neighborhood and dynamic model. Topographical thinking is static in the sense that it refers to the external features of a function as opposed to dynamic thinking which envisions motion involved in a function, and topographical thinking also places an emphasis on classifying functions based on external characteristics. This classification process need not be well defined; rather it can be based on loose feelings about the external characteristics. However, it is different from dynamic and neighborhood thinking in the fact that it aims to classify and not analyze the features of the function. Textbook Treatment of the Limit Concept Students involved in this study used the textbook Calculus by James Stewart (2003). This textbook introduces the concept of limit with the following definition. 64 Definition 1: We write f x L x a lim ( ) and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. (p. 71) Immediately following this definition, the text states that “roughly speaking, this says that the values of f (x) get closer and closer to the number L as x gets closer to the number a (from either side of a) but x a ” (p. 71). These two statements seem to be an attempt to connect students with the two formal definitions of limit discussed previously. The words “arbitrarily” and “sufficiently” appear to be an attempt to capture the notion of „closeness‟ used in the formal definition of limit. The words “closer and closer” appear to be an attempt to capture the dynamic nature of the sequential definition. Students using this textbook are introduced to both the neighborhood and the dynamic model of limit. The implication from the written textual material is that the neighborhood model of limit should be considered the „official‟ understanding of limit, while the dynamic model is described as a “rough” description of the neighborhood model. There is no mention of understanding the limit using conceptualizations connected to the topographical model, and there is no implication that the two models of limit described are, in any way, cognitively different. 65 Coding Scheme The three models of limit described were used to create a coding scheme for analyzing the qualitative interview data. Interview statements were analyzed, and when applicable, categorized into one of the three categories in Table 13. Table 13: Three Models of Limit The analysis of the transcripts frequently revealed language relfelcting cognitive structures inherent in the three models of limit. Applicable text was sorted into „instances‟ of language supporting one of the three categories. An „instance‟ is understood to be a statement describing one complete thought. Statements were divided into two or more instances when the interviewee appeared to switch the focus of his/her description from one thought to another. For example in Ashley‟s statement below while referring to a multivariable contour graph. Her statement was considered to have two separate instances. Excerpt 3 ASHLEY: I‟m trying to think back to the pattern idea I had with those, so there‟s a very distinct movements and this one seems to have a more distinct movement. Let‟s see… This one probably exists, just because again it has that upward sloping of values and similar line patterns. I would say this one doesn‟t exist. Category Description Dynamic Uses motion to develop meaning in a function. Neighborhood Uses a sense of „closeness‟ to develop meaning in a function. Topographical Uses the shape of a graph to develop meaning in a function. 66 The first instance is her description of how motion plays a role in her understanding of multivariable limits. The phrase “there‟s a very distinct movements and this one seems to have a more distinct movement” was judged to be an instance of dynamic thinking since the statement emphasized the “movement” of the graph. However, it was judged that in the second part of her statement she switched her focus from describing her use of motion in general to describing her thought process on one particular problem. So it was judged that the statement, “because again it has that upward sloping of values and similar line patterns” constituted a second instance which was classified as topographical thinking because its central focus is that of exterior features of the contour graph. In some cases, however students used a very long statement to describe a single thought. For example, Chris made the following statement when reflecting on multivariable limits. Excerpt 4 CHRIS: It depends which direction you‟re looking at as to where the limit is coming from, where it‟s going, what kind of thing, how is moving or approaching, using those two words that are confusing. But, yeah, in the first two, from a different direction, from a different x or a different y direction it changes like where it is or where it‟s moving to. But on this one, on the third one, from every direction it‟s the same. This statement was judged to be a single instance since Chris maintained focus on describing his thought about which direction the limit was coming from throughout his statement. Even though he mentioned three separate problems, it was judged that he was not switching his focus from one problem to the next, but rather he was using the three problems as examples to illustrate his thinking. For these reasons, the above text was 67 considered one instance of dynamic thinking since the focus of the entire passage was on the use of motion from different directions on the graph. Statements which were not easily associated with one of the three primary categories were not recorded as an instance. Josh‟s statement below is an example of a noninstance. Excerpt 5 JOSH: Ok, First one (limit describes how a function moves as you approach a given point) the way my calc I teacher […] described it was just like, it was the behavior of the function. That you may have a discontinuous function that has a point above but if you look at the limit your analyzing the behavior on each side, you‟re not just evaluating a value at that given point. So that‟s why I thought this one was the most accurate description of it because it kind of described the behavior. Josh‟s statement above had elements of all three coding categories in it. He used motion, closeness, and the shape of the graph to describe his thinking; however, the main emphasis of the description surrounded the word “behavior” which was used ambiguously in this statement. Since the statement did not contain a strong description of any of the three categories, it was considered a noninstance and was not recorded as part of the coding scheme. An important issue when coding the interview results is that of conventional mathematics language usage. In calculus, common language about limits involves words such as “approaches” or “goes towards.” Although these words carry with them a connotation of dynamic motion, the meaning a student may give to them can vary significantly. For this reason, the use of common mathematical language alone is not considered sufficient evidence that a student is using dynamic reasoning. Below is an example of Amanda‟s description of a multivariable limit 68 Excerpt 6 AMANDA: Ok, so a limit from a line is kind of like how the line approaches the point. So, say zero, if the line 2x is how it approaches zero that just, you just have to worry about the x and y variables, same with in a curve, like a parabola or something, same thing, you just have to worry about the x and y coordinates and you can easily see just the x and y coordinates on a graph. But in multivariable, you have several different variables that are approaching the same point, and so you can‟t exactly see how that happens on a graph, easily, so you have to take several slices and look at those curves at the slices and put it all together and analyze it that way. In her description, Amanda frequently used the word “approaches,” but it is not clear that Amanda was actually evoking a sense of motion to describe the multivariable limit. Instead, it is quite plausible that Amanda was using the word “approaches” as similar to the word “behavior” which does not necessarily connote dynamic motion. In general, a conscious effort was made during the coding process to err on the side of exclusion rather than inclusion. In that sense, a statement such as Excerpt 4 above might actually refer to dynamic motion, but was excluded because reasonable doubt of its appropriateness remained. Upon completion of the initial coding process, all coded text was reexamined to determine accuracy in coding. Instances from different interviews and involving different students were compared to determine that the coding scheme was implemented in a consistent basis. Any discrepancies in the coded instances were addressed and changed as necessary. In all, 283 total instances were recorded with the majority describing dynamic thinking. Table 14 below shows the number of recorded instances for each coded category. 69 Coded Category Instances (Percentage) Dynamic Thinking 160 (56.5%) Neighborhood Evaluation 41 (14.5%) Topographical Examination 82 (29.0%) Total 283 Instances Table 14: Instances for Each Cod 



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