STUDENTS' CONCEPTUAL UNDERSTANDING OF CALCULUS By ZAHRA GOOYA B.S, Boston State College (University of Massachusetts-Boston), 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics and Science Education) (Faculty of Education) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1988 ® Zahra Gooya, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, 1 agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by . the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MfZJZD The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The purpose of t h i s study was conceptual understanding and function. (a) modification of sessions. being of two As a way to i d e n t i f y the nature of students' concepts of calculus namely, derivative of c o l l e c t i n g data two Piagetean clinical methods were employed: interview; and, Whenever the students seemed to be confused discussed, the researcher provided (b) tutorial about the issues instructions through the t u t o r i a l sessions. The analysis of data was done by developing individual p r o f i l e s and by response categories. It was found that the interview methodology was effective in revealing some aspects of students' concept students were to derivative. which definition. definition lead to It was adequate s k i l l calculus. have little meaningful understanding the also development evident from of the an appropriate study of that concept students had instructors of in using algorithm to solve problems. results of the It was would study would suggested that contribute to be useful to the introducing a concept by students' confusions Yet i f a concept i s presented by means of meaningful had The A number of students held proper concept images of function should The found images. better opportunity to develop them to form concept d e f i n i t i o n s . that more challenging exercises be solving s i t u a t i o n s . their The and i t s formal difficulties. examples, students concept images. Thus leading researcher strongly recommended posed to the students in problem- i i i T A B L E O F CONTENTS Page Abstract Table of Contents ii i i i L i s t of Figures vii Acknowledgement vi i i CHAPTER ONE 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Background of the Study D e f i n i t i o n of Terms General Statement of the Problem S p e c i f i c Statement of the Problem Overview of Methods of Study Limitation of the Study J u s t i f i c a t i o n of the Study 1 2 4 4 5 6 7 CHAPTER TWO 2.1 2.2 2.3 2.4 2.5 Introduction Literature in the Broad Problem Area 2.2.1 Related causes to students' f a i l u r e The a b i l i t y of instructors to teach calculus Rote versus meaningful learning in calculus Students' high school background Instructors awareness of students' backgrounds Textbooks Students' conceptual a b i l i t y 2.2.2 Suggestions to overcome students' failure The S p e c i f i c Problem Area Vinner's Conceptual Model A B r i e f Summary of Constructivism 2.5.1 Constructivist Perspective of Knowledge i 2.5.2 Constructivism and Mathematics 2.5.3 Teaching and Learning from a Constructivist Perspective 9 9 9 10 10 11 11 12 12 13 15 17 18 19 21 23 iv Page CHAPTER THREE 3.1 3.2 3.3 3.4 3.5 The Rationale f o r Using the C l i n i c a l Interview The Subjects The Interview Procedure Method of Analysis S p e c i f i c Interview Questions . 25 27 28 28 29 Introduction Individual Students' Profiles 4.2.1 Jenifer's P r o f i l e 4.2.2 Richard's P r o f i l e 4.2.3 Brian's P r o f i l e . . . 4.2.4 Magan's P r o f i l e 4.2.5 Jason's P r o f i l e 4.2.6 Joe's P r o f i l e 4.2.7 Owen's P r o f i l e 4.2.8 Kathy's P r o f i l e 4.2.9 Gary's P r o f i l e 4.2.10 Barbara's P r o f i l e 4.2.11 Nick's P r o f i l e 4.2.12 Ted's P r o f i l e Response Category 4.3.1 Categories of responses to the questions of derivative Categories of responses to the d e f i n i t i o n of derivative 32 33 33 37 41 46 48 51 54 61 68 73 81 86 89 CHAPTER FOUR 4.1 4.2 4.3 Category I Category II Category III Category IV Definition of derivativetextbook d e f i n i t i o n Derivative as rate of change-velocity Derivative as a "slope"... Derivative as a rule of differentiation Categories of responses to the concept of slope of the tangent l i n e Category I Category II Slope of the tangent l i n e as derivative of function Slope of the tangent l i n e as the l i m i t of slopes of other secants Categories of responses to the question of d i f f e r e n t i a b i l i t y 90 90 90 91 91 92 92 92 93 V Page Category I Category II Category III 4.3.2 93 There i s more than one tangent l i n e 93 Function i s not d i f f e r e n t i a b l e at sharp points 94 Categories of responses to the questions of function Category I Category II Category III Category IV 4.3.3 The right l i m i t and l e f t 1 imit are not equal Some elements of the formal d e f i n i t i o n of a function Function as a r e l a t i o n between two variables Function as an algebraic term an equation Idiosyncratic responses Summary of Results 94 94 95 95 96 97 CHAPTER FIVE 5.1 5.2 5.3 5.4 5.5 Summary of the Study Method 5.2.1 Tutorial Session Conclusions of the Study 5.3.1 The nature of the students' conceptual understanding The nature of the students' conceptual understanding of derivative The nature of the students' conceptual understanding of the slope of the tangent l i n e The nature of the students' conceptual understanding of d i f f e r e n t i a b i l i t y The nature of the students' conceptual understanding of function 5.3.2 A system f o r categorizing students' understanding Discussion and Implications - S i g n i f i c a n t Issues Recommendations f o r Further Research 101 101 102 102 103 103 103 104 104 104 105 109 REFERENCES Ill APPENDIX A 115 APPENDIX B 116 APPENDIX C 117 vi Page APPENDIX D 118 APPENDIX E 119 APPENDIX F 146 vii LIST OF FIGURES Page Figure 3.1 The graph of Parabola 30 Figure 3.2 The graph of a non-smooth function 31 viii ACKNOWLEDGEMENT I wish to express my sincere gratefulness to the chairman of my thesis committee, Dr. David R o b i t a i l l e f o r his advice and encouragement which he has offered throughout the duration of this study. I am e s p e c i a l l y indebted to Dr. Gaalen Erickson f o r his patient understanding and continued help and assistance during the conducting of t h i s study and writing of t h i s t h e s i s . I would l i k e to express my thanks to Dr. James S h r r i l l f o r agreeing to serve in my committee. A l s o , I would l i k e t h i s study. to thank those students who were involved in It i s true that without t h e i r p a r t i c i p a t i o n , the conducting of the present study would not have been possible. Finally, encouragement sincerely I am during grateful my indebted to my to formative my mother years and for her support thereafter. I also and am husband, Bijan, for his everlasting patience, maintenance and understanding, and to my children Sahar and A l i f o r t h e i r cooperation, kindness and tolerance during my years of study. 1 CHAPTER ONE 1.1 Background of the Study "The teaching of calculus i s in a state of disarray and near c r i s i s at most American Colleges and U n i v e r s i t i e s . The p r i n c i p l e evidence being a f a i l u r e rate of about 50%" (Rodi, 1985, p. 1 ) . The calculus course has the greatest drop-out rate at the University of B r i t i s h Columbia (U.B.C.) among f i r s t year students and yet i s required f o r the greatest number of students. year "At U.B.C, only about 55% of students who enroll mathematics (Walsh, p. 30). problems relating successfully complete There has l a t e l y both been to the teaching Mathematics in f i r s t 100 and 101" serious discussions about the of calculus courses to t r y and determine some of the factors which may have led to t h i s f a i l u r e rate. Many researchers have t r i e d to i d e n t i f y the possible causes, because as Stein (1985) says: "Before we propose the medicine, we had better agree on diagnosis..." (p. 3 ) . Once these possible causes suggestions may be made to reduce the f a i l u r e r a t e . given are: inexperienced instructors are known, then Among the reasons (Maures, 1985; Lax, 1985; Davis, 1985), the compactness of the calculus course (Davis, 1985; Epp, 1985), student's poor high school background (Renz, 1985; Rodi, 1985), the lack of awareness of the students' needs and expectations (Ash, 1985; Epp, 1985), poor textbooks (Stein, 1985; Kenelly, 1985; Rodi, 1985; Stevenson, 1985), and conceptual d i f f i c u l t i e s in students' understanding Orton, 1980, 1983; Vinner, 1982, 1983, 1986). (Epp, 1985; The l i t e r a t u r e regarding these possible causes w i l l be reviewed in Chapter 2. Some suggestions researchers. f o r improvement have also been given by various Many authors proposed that d i f f e r e n t calculus courses can be developed to meet the special students' needs (Stein, 1985; Epp, 1985; 2 Rodi, 1985). concepts Other authors are the focus discussed of instruction teaching methods where: (Lax, 1985; Davis, key 1985), an i n t u i t i v e approach i s used to introduce calculus (Renz, 1985; Ash, 1985), a variety of open-ended problems are presented to students (Stein, 1985), application of mathematical 1985; Lax, 1985), and the emphasis i s on the development of conceptual not just purely mechanical Since concepts, calculus rote performance in understanding The concepts are the focus of instruction (Rodi, writer's understanding (Epp, 1985). involves memorization solving the nature position a broad which 'type' understanding leads problems to does of the relationships i s that students a of type not interrelated of mechanical lend itself between these experience failure to concepts. in calculus because they have d i f f i c u l t i e s in understanding both the concepts and the relationships between the key concepts of the calculus course. It does not mean that other factors are not involved in the students' f a i l u r e . Perhaps the most important concepts. Vinner difficulties using one i s the students' understanding (1981,1983,1986) a model has f o r cognitive examined processes notions of "concept image" and "concept d e f i n i t i o n " . upon this model to investigate the nature some that of of the these involves the The study w i l l draw of students' conceptual d i f f i c u l t i e s of selected concepts of c a l c u l u s . 1.2 D e f i n i t i o n of Terms In the following, some of the terms that are being used in this study w i l l be presented very b r i e f l y . The mathematical d e f i n i t i o n s , are based on the Adam's Calculus book which i s the text f o r Math 100 and 101 at U.B.C. 3 Concept Image According to Vinner associated with the (1983), concept image i s a set of properties concept together with a mental picture of that concept. Concept D e f i n i t i o n Concept definition, as Vinner (1983) explains, i s verbal d e f i n i t i o n of a concept in a non-circular an accurate way. Tutorial Session Tutorial session, as opposed to c l i n i c a l interview (Piaget, 1929), is a situation in which an interviewer intervenes and acts as a teacher. Cobb and Steffe researcher to (1983) explore indicated the that teaching student's activity construction of helps the mathematical knowledge. Tangent Line (Adams, 1986, p. 52) Suppose that the function f i s continuous at x = x Q . x limit f ( p + ) h-0 h n If x f ( p ) = m e x i s t s , then the straight l i n e having slope m and passing through the point P=[x Q ,f(x Q )], that i s , the l i n e with equation y = f ( x Q ) + m (x - x „ ) , i s tangent to the graph of y=f(x) at P. If l i m i t h-0 f x + ( p n f x ) " ( p ) = * (or - w ) , then the v e r t i c a l straight l i n e h x = x0 i s tangent to y=f(x) at P. If lim h-0 f ( x + 0 n e x i s t in in any way other than by being either » or y = f(x) has no tangent l i n e at P. f ) - ( h x 0 ) f a i l s to then the graph of 4 Derivative (Adams, 1986, p. 56) If l i m i t f(x + h) - f(x) = f'(x) exists as a f i n i t e real number, we h-0 h say that the function f i s d i f f e r e n t i a b l e at x. In t h i s case the number f'(x) i s c a l l e d the derivative of f at x, and the function f is called the derivative of f . Function (Adams, 1986, p. 14) A function f i s a rule that assigns to each real number x in some set D(f) (called the domain of f) a unique real number f(x) c a l l e d the value of f at x. 1.3 General Statement of the Problem From a c o n s t r u c t i v i s t view, what a student "knows" i s dependent in some important ways upon how he has come to know i t (Erickson, 1987). is very important for a teacher to know the student's It primitive conceptions, so as to better understand what errors and misunderstandings may follow, sophisticated how these conceptions may change into wider and more ones, through which s i t u a t i o n s , which explanations, which steps to pursue in the instructional setting (Vergnaud, 1982). The aim of t h i s study was to reveal some of the d i f f i c u l t i e s students were struggling with to understand and function. that the concepts of derivative In so doing, t h i s study also hopes to make a contribution to the teaching of the c a l c u l u s . 1.4 S p e c i f i c Statements of the Problem The purpose of t h i s study was to examine some of the conceptual d i f f i c u l t i e s in calculus that students experience in f i r s t year calculus 5 courses. were Problems related to the concepts of "function" and "derivative" discussed in interviews with students in order to address the following research questions: What i s the nature of the students' conceptual understanding of the d e r i v a t i v e , slope of a tangent l i n e , how derivative and slope are r e l a t e d , d i f f e r e n t i a b i l i t y , and function? Can a category system be developed f o r these understandings which might provide some insight into the nature of the d i f f i c u l t i e s experienced by students learning these concepts. 1.5 Overview of Methods of Study The interviews with volunteer first year students at U.B.C. who were taking Math 100 during the f i r s t term. The study was based on individual interviews were tape-recorded. The transcripts of the interviews along with the students' written work were used f o r the further analyses. The tutorial sessions were used whenever i t seemed apparent to the researcher that the students were confused and in need of c l a r i f i c a t i o n . The interview was based upon broad Chapter 3. The questions were questions that w i l l aimed to explore be presented in the students' d i f f i c u l t i e s in understanding the concepts of derivative and function. A s t r i c t standardized protocol was not used because the interviewer's f i r s t priority was to have the opportunity to study the process of a dynamic passage from one state of knowledge to another. The interviewer did not pre-structure the direction of inquiry. The students had enough time f o r their ideas responses and to express their toward a question. The students' responses to questions about each one of the concepts were then categorized to f a c i l i t a t e were interviewed. the analyzing procedures. Sixteen Students However, only the l a s t 12 students' interviews' were used i n the a n a l y s i s . As a f i r s t step of data reduction, individual 6 profiles were made f o r every student and then the p r o f i l e s were summarized according to defined categories of responses. The study. f i r s t four students interviewed provided the data f o r a p i l o t The p i l o t study gave the investigator the opportunity to revise the questions and to improve her interviewing technique as w e l l . 1.7 Limitations of the Study The solicited subjects by sections. being public Therefore interviewed announcements were a l l volunteers. in calculus the sample was not a random subjects were f i r s t year U.B.C. students. 80 minute classes interview combined with They were and sample. tutorial All the The study was a one shot 60 to the t u t o r i a l sessions. Therefore, there was not enough time f o r some of the students to r e f l e c t on their actions and t h i s might be a potential shortcoming of the interview. The students' weaknesses in algebra, t h e i r even simple analysis. calculations, e t c . were lack of a b i l i t y not taken into account understanding which d i f f i c u l t i e s in the concepts of derivative and function. Also the investigator acknowledges that meaning of the interviewees' verbal indicates, different i n the This r e s t r i c t i o n was due to the purpose of the study focused upon an examination of the students' conceptual (1985) to perform the "different curiosities people about the area she i s constructing the and written responses. with different As Jones perspectives and of investigation will inevitably f i n d d i f f e r e n t categories with which to structure and make sense of the data" (p. 59). analyzing Every the same data investigator could and might arrive use d i f f e r e n t at d i f f e r e n t methods f o r findings. The 7 investigator t r i e d to avoid systematic bias although she believes everyone views the world through one's own 1.6 that perspective. J u s t i f i c a t i o n of the Study Two chosen concepts for fundamental this in of calculus, study because learning needs any not calculus mathematics, but also these two anybody who namely level derivative only and they later concepts and are on in are quite function very basic more crucial were and advanced f o r almost of mathematics in higher education such as engineering, economics, e t c . Most college and university students have conceptual in understanding the concepts of calculus (Lochhead, 1983; difficulties Davis, 1985; Epp, 1985; Zorn, 1985), yet very l i t t l e research has been done into this important issue. "In Tall some a series of papers (Vinner and Hershkowitz, 1980; Vinner, 1983; and Vinner, 1981; Dreyfus and Vinner, 1982) mathematical notions there d e f i n i t i o n and the concept image. concept images definitions. which These are are conflicts claimed that for between the concept Namely, p a r t i c u l a r individuals develop inconsistent concept i t was images with are the quite mathematical common and concept widespread" (Vinner, 1986, p. 2 ) . He then poses the following question: "Is t h i s phenomena a result of vbad pedagogy' or i s i t inherent to the concept? In other words: Is there a way to teach these concepts, so that such images w i l l not be formed or these images are unavoidable and they w i l l be formed no matter how the concept i s taught?" (Vinner, 1986, p. 2 ) . 8 The function present and study has derivative addressed t h i s question for the in terms of a c o n s t r u c t i v i s t view of researcher, as a model builder. notion of teacher- 9 CHAPTER TWO REVIEW OF LITERATURE 2.1 Introduction The four major categories into which the l i t e r a t u r e in this chapter w i l l be reviewed are as follows: 2.2 The broad problem area of d i f f i c u l t i e s in teaching-learning calculus 2.3 The more s p e c i f i c problem area of students' d i f f i c u l t i e s in learning the concepts of c a l c u l u s . 2.4 Vinner's conceptual model. 2.5 A b r i e f summary on constructivism as a theoretical framework of t h i s study. 2.2 Literature in the broad problem area This section i s divided into two 2.2.1 Related causes to student f a i l u r e . 2.2.2 Suggestions to overcome students' f a i l u r e . 2.2.1 Related causes to students' f a i l u r e In December 1985, a conference was in high school and among them, a few high subsections: school and college. held on the teaching of calculus Many eminent instructors of calculus and researchers, spoke up about the status of calculus in college. The conference proposal teaching of calculus i s in a state of disarray and stated that "the near c r i s i s . . . [with a] f a i l u r e rate of nearly half at many colleges and u n i v e r s i t i e s " (Stein, 10 1985). Many researchers c r i t i c i z e d universities responsible and colleges. They the way suggested that calculus was several factors for the current state of teaching in c a l c u l u s . taught in which are These factors are: The a b i l i t y of instructors to teach c a l c u l u s . Rote versus meaningful learning in c a l c u l u s . Students' high school background. Instructors' awareness of students' backgrounds. Textbooks. Students' conceptual ability. The a b i l i t y of instructors to teach calculus Maures (1985) complains that calculus i s taught by anyone including the least energetic member of the graduate student. Lax and a large yet it i s taught backgrounds; mathematics to and mathematics department or (1985) states that calculus department. number of is taught David students by (1985) i s a big with anyone who has also diverse is the even by enterprise needs available same a idea and in a about unqualified i n s t r u c t o r s . Rote versus meaningful learning in calculus Davis (1985), says that calculus courses have been presented as a series of notions, routine problems, and a few simple a p p l i c a t i o n s . way to of presenting calculus does not understand the techniques and concepts but formulas. He This provide an opportunity for students i t focuses on continues to say learning that how "...The to use the difficulty comes mainly from the rapid pace of moving through the material, and from 11 an attempt to cover a large number of d e t a i l s without much focus on main key ideas" relating (p. 10, to the emphasis content choice, w i l l we understanding of less? his). Epp of calculus and settle for s u p e r f i c i a l (1985) says: proposes "...If knowledge of a question i t comes to a a l o t or deeper Perhaps less i s more" (p. 18). Students' high school background Some researchers view the students' poor high school background as a potential cause for f a i l u r e . Renz (1985) says that many students have been taking less mathematics in high school. Rodi (1985) complains about "a generation of c e r t a i n l y unsophisticated, and probably even i l l i t e r a t e , high school graduates" (p. 2 ) . He says that "These students not i n t e l l e c t u a l l y ready for i t . and trigonometry" simply are They do not have the s k i l l s in algebra (p. 4). Instructors awareness of students' backgrounds Epp (1985) talks about an professors and t h e i r students. intellectual She gulf between mathematics says that professors reactions vary from, ignoring t h i s gulf and teaching as though the students were mature enough to primarily understand on computations. calculus skills Ash courses a l l the to enable formal students (1985) claims are in proofs, to that engineering, to perform while the those certain most of calculus whose foci the is are mechanical students in presented by instructors "...as i f the entire audience were planning a career in pure mathematics" (p. 2 ) . students' needs and She says that because of backgrounds, "...formal lack of mathematical awareness of language which 12 was intended to prevent misunderstanding had precisely the opposite e f f e c t " (p. 3 ) . Textbooks A great number of researchers talk about poor textbooks which contribute to students' f a i l u r e . But i t seems that they appropriate suggestions to solve the textbook problem. "why do not may have Steen (1985) asks do calculus books weigh so much" (p. 4), and says that the economics of publishing causes encyclopedic textbooks. "Authors have less t h e i r books w i l l Kenelly Stein (1985) proposes that choice, for i f they omit someone's f a v o r i t e not be adopted and soon w i l l (1985), Rodi (1985), Stevenson be out of p r i n t " (1985), and Masurer topic, (p. 3 ) . (1985) have also presented the same idea about the poor quality of textbooks. Students' conceptual Hadas Rin by examining students do ability (1983) has studied students' d i f f i c u l t i e s with their not spontaneous know the "known" theorems to new definitions situations. students do not understand of written questions. and they Davis equations and students they have taken calculus. able to apply Lochhead (1983) wrote that able to read "...seem to variable or of function" (p. 2 ) . students' conceptual that the proofs and do not have a clear conception are not also not found (1985) believes that many the d i f f i c u l t i e s that must be overcome. many college are She calculus lack or write any algebraic defined notion of Epp (1985) says that the state of most knowledge of mathematics Zorn well simple i s "abysmal" after they (1985) stated that "Instead of learning to 13 create, v e r i f y , and analyze algorithms, calculus students learn mainly how to perform them" (p. 3 ) . 2.2.2 Suggestions to overcome students' failure Many researchers have t r i e d to not only t a l k about the causes of students' f a i l u r e , but also have t r i e d to give productive suggestions in order to deal with students' f a i l u r e in c a l c u l u s . Stein discrete students, students (1985) suggests mathematics that would "Such a course for c a l c u l u s . if a be quarter available to first year and course Epp thus prepare to make them include additional work Rodi poses the question be more consistent to make normal expectation for entering students in a thorough pre-calculus course before attempting heady and adult s t u f f of calculus?" and (1985) suggests thoroughly, that "key so that d i f f i c u l t i e s students are able to see how (p. 11). are (1985) i s talking about the p o s s i b i l i t y to increase students' l o g i c a l maturity" (p. 10). Davis of university It could, i n c i d e n t a l l y , weed out those who "...to modify pre-calculus courses "would i t not semester could help develop maturity not ready to go on" (p. 4). of: or the (p. 5, emphasis h i s ) . ideas are presented very c a r e f u l l y are c l e a r l y perceived and these d i f f i c u l t i e s are met so that and dealt with" Lax (1985) says that ...for a concept, when presented properly, can be absorbed a whole, while an algorithm remains a sequence of steps. is only after a concept has been understood, and made part one's t h i n k i n g , that we turn to the intriguing task devising e f f i c i e n t algorithms (p. 6 ) . as It of of Renz (1985) says that "calculus has been, i s , and w i l l continue to be a basic computational and conceptual tool for students studying the 14 hard sciences and engineering" (p. 6 ) . Then he suggests i n t u i t i v e foundation and careful imaginative thinking w i l l that more be needed f o r introductory c a l c u l u s , in order to make i t understandable f o r students. Ash (1985) c a l l s f o r a change in mathematicians' s t y l e of teaching. She says: Mathematics does not come into existence f u l l y developed with theorems and proofs. It arises from imagination and intuition aided by physical and geometric reasoning. Students should be taught in a style that r e f l e c t s the creation of mathematics and not in style that would s a t i s f y a professional mathematician tidying things up years a f t e r the fact. It i s more important to learn how to formulate and solve interesting problems than to learn the techniques of writing formal proofs (p. 4). Stein (1985) suggests t h e i r own and to express that students t h e i r thoughts, should to think on and t h i s chance can be given them by providing open-ended problems with ".. .-.problem-sol ving' courses be able variations and introducing to compensate f o r the narrowness of our mission" (p. 10). Rodi (1985) says that "Applications are a c r i t i c a l part of teaching and learning calculus precisely because they constitute one of the best places both to understanding" expose (p. 8 ) . and to reinforce Then he warns that intuitive i f research conceptual mathematicians teach c a l c u l u s , they may pay less attention to application and pay more attention to more abstract g e n e r a l i z a t i o n . Lax (1985) seems to have d i f f e r e n t ideas from Rodi. He says that the teaching of calculus should be entrusted to those who a c t i v e l y use i t in t h e i r own research. application Although he sees the same role f o r the use of in calculus when he says: "Teaching of calculus i s the natural vehicle f o r introducing applications, and that applications give 1 5 the proper shape to calculus: i s used" (p. 1). They show how, and to what end, calculus Rodi (1985) believes that applications do not give the proper shape to calculus i f they w i l l not enrich students' i n t u i t i o n and will not allow them to see each successive stage of generalization. He says that by applications should be carefully selected i n s t r u c t o r s , since applications "...are as powerful and expressed a tool in revealing i t ' s message as metaphor i s to the poet" (p. 10). Epp states that "...the primary aim of calculus instruction should be development understanding" of (p. conceptual 9). But as she opposed provides to some purely mechanical suggestions to help students and among these suggestions i s one that would appear to lead to mechanical understanding: Make your students memorize precisely-worded d e f i n i t i o n s and perhaps theorem statement a l s o . Memorization i s greatly underrated as a Pedagogical t o o l . At the l e a s t , memorization of a d e f i n i t i o n forces students to read i t c a r e f u l l y , at best, i t encourages them to understand i t (since i t i s easier to memorize something i n t e l l i g i b l e than gibberish) (p. 12). This writer did not find any study to support t h i s suggestion. 2.3 The S p e c i f i c Problem Area Calculus i s not a course f o r the e l i t e any more. There i s a great demand f o r c a l c u l u s , and yet so l i t t l e research has been done into the study calculus. of students' d i f f i c u l t i e s with A l l the suggestions mentioned in the previous section represented the teachers' perspective but there were no suggestions directly involving students. Unfortunately, only a few studies have been done in t h i s large area and many more studies are d i f f i c u l t i e s in calculus. needed to clarify the nature of students' 16 The concepts of derivative and function are two fundamental areas upon which students' understanding of many other concepts of calculus are based. Few researchers have studied the nature of student in these two p a r t i c u l a r areas. Among those are Orton difficulties (1980,1983), Monk (1987), Vinner (1983) and Markovits et a l . (1986). In 1983, Orton conducted understanding of d i f f e r e n t i a t i o n . 22 years o l d . concerned study a study to investigate the students' He interviewed 110 students aged 16 to He found out that one of the most d i f f i c u l t questions was with the understanding generally r e l a t i v e l y easy. found Orton that of d i f f e r e n t i a t i o n . the application Students of his of d i f f e r e n t i a t i o n believes that students' errors was in dealing with calculus are mostly conceptual. Markovits et a l . (1986) conducted a study with some 400 9th grade students to investigate the students' understanding of some components of the concept of function. The study was designed in a way to give the researchers the opportunity to identify the students' d i f f i c u l t i e s and the probable causes of these d i f f i c u l t i e s . The investigators were not, as they s a i d , "interested in the students' overall success but rather in the types of d i f f i c u l t y they encountered" (p. 24). Among t h e i r findings were that three types of functions caused d i f f i c u l t y namely, the constant function, a function defined piecewise and a function represented by a discrete set of points. Vinner (1979) also found that students had d i f f i c u l t y with piecewise functions. Monk (1987) involved 628 f i r s t year university students in a study which had as i t ' s objective the investigation of students' understanding of the concept of function. His findings showed that the students were r e l a t i v e l y good in answering those questions which did not ask f o r t h e i r 1 7 understanding but rather more of t h e i r s k i l l , yet they d i d very poorly in responding to those questions that asked f o r t h e i r understanding of the concept of function. 2.4 Vinner's Conceptual Model Vinner "Function" has examined (1983), students' understanding and "Tangent Line" (1982), of "Limit" by using (1986), his model of cognitive processes that involves the notions of concept d e f i n i t i o n and concept image. Concept d e f i n i t i o n , as he explains (1981), i s an accurate verbal d e f i n i t i o n of a concept in a non-circular way. According to him, concept image i s a set of properties associated with the concept together with a mental picture of that concept. Vinner (1975) used an example to i l l u s t r a t e what he means by a mental picture of a concept. He says that i f C i s a concept and i f P i s a person, then "P's mental image of C w i l l be defined as the set of a l l pictures that have ever been associated with C i n P's mind, namely the set of a l l pictures of objects denoted by C in P's mind" (p. 339). images by Vinner (1983,1986) claims that the developed concept different individuals are inconsistent with the concept d e f i n i t i o n s that are usually i n a c t i v e . For every concept, Vinner (1981) presents a model by assuming the existence of two d i f f e r e n t c e l l s in the cognitive structure, one c e l l f o r the concept d e f i n i t i o n ( s ) , and the other one f o r the concept Depending on how a concept interactions will occur is first between image(s). introduced to students, d i f f e r e n t the two mentioned cells. The interactions vary from the construction of a d e f i n i t i o n based on one's own experience with the concept which of his concept i s a description image to introducing the concept d e f i n i t i o n p r i o r to the existence of any 1 8 concept images. For Vinner (1982, 1983, 1986) the important task i s to recognize the students' concept images and t r y to reveal them whenever i t i s possible. This revealing w i l l help teachers to not only acquire some understanding of t h e i r students' mental a c t i v i t i e s but also t r y to find the probable causes for formation of students' wrong concept images. Fifteen Grade 12 students were involved in a study of "The notion of limit" conducted by Vinner introduced to students formation of mathematical wrong ability, in a way or (1986). that was inadequate as Vinner The concept says, were concept supposedly of Limit was to prevent the images. The unquestionable. students' Yet he arrived at a conclusion that the formation of certain concept images are probably unavoidable. He introducing new mathematical means of concept d e f i n i t i o n . suggests a reconsideration of the way of concepts which i s usually presented by the He also found that the students' concept images play a crucial role in construction of t h e i r mathematics yet the textbooks on are based the concept supported by his study of function (1982). definitions. His finding was (1980) and study of the tangent line Vinner then suggested that presenting a concept by i t ' s formal d e f i n i t i o n may not be the best way of doing that. 2.5 A B r i e f Summary of Constructivism This section i s divided into three subsection as follows: 2.5.1 Constructivist perspective of knowledge 2.5.2 Constructivism and Mathematics 2.5.3 Teaching and Learning from a Constructive Perspective 19 2.5.1 Constructivist Perspective of Knowledge Constructivism as an epistemology knowing (VonGlasersfeld, 1984 the what and cited in Erickson, 1987, Constructivism i s "...a general way variety of phenomena. concerns how of emphasis i s h i s ) . of interpreting and making sense of a It constitutes a framework within which to address situations of complexity, uniqueness, and uncertainty that Schon (1985) c a l l s 'messes' and to transform them into p o t e n t i a l l y solvable problems. Thus, l i k e any epistemology, constructivism influences both the posed and the c r i t e r i a question for what counts as an adequate solution" (Cobb; Wood; Yackel, 1988, p. 2). As stated by Vergnaud (1987), Piaget theorist of constructivism in his time. to refer cognitive to Piaget. The development has constructivist psychology essential of way senses, but first research been and Piaget major (Magoon, 1977). knowing the real the most systematic It seems therefore, appropriate of the was world and his influence "According i s not foremost through our toward a into more to Piaget, the directly through our actions. action has to be understood in the following way: inquiry In t h i s context, a l l behavior by which we bring about a change in the world around us or by which we change our own situation in r e l a t i o n to the world. In other words, i t i s behavior that changes the knower-known relationship" ( S i n c l a i r , 1987, p. 28). The knower should draw upon his existing knowledge and also should be able to extend his knowledge base i f he wants to make sense of his (Erickson, 1987; p. ( K i l p a t r i c k , 1987). "new 22) which Sinclair knowledge i s constructed i s the only reality that he experience can know (1987) pointed out that in Piaget's view, from the changes or transformations subject introduces in the knower-known relationship" (p. 28). the 20 The major assumption of constructivism, as Magoon (1977) s t a t e s , i s that the subjects being studied must be considered as knowing beings and their knowledge actions are has important interpreted. consequences The K i l p a t r i c k (1987), involves two view, as behavior or described by principles: Knowledge i s actively constructed by the cognizing not passively received from the environment. 2. Coming to know i s an adaptive process that organizes one's experiential world; i t does not discover an independent, preexisting world outside the mind of the knower (p. 7). argues that the constructivism, as VonGlaserfeld Davis and Mason c a l l i t , from constructivism for constructivist their 1. Kilpatrick calls for how second calls separates trivial i t , or simple constructivism, as i t , or empiricist-oriented constructivism, as Cobb radical constructivism rejects metaphysical objective truth principle subjects should be (1987). realism and stopped. He says claims Kilpatrick that radical that the summarizes search radical constructivism as: ...an epistemology that makes a l l knowing active and a l l knowledge subjective. Following modern physical sciences in i t s rejection to the p o s s i b i l i t y of coming to know ultimate r e a l i t y , i t treats the cognizing subject as the organizer of his or her own experience and the constructor of his or her own r e a l i t y . It views coming to know as a process in which, rather than taking in information, the cognizing subject through t r i a l and error constructs a viable model of the world (p. 10). The consequences of a radical i d e n t i f i e d by VonGlasersfeld as: constructivist position have been (1983, in press, c i t e d in K i l p a t r i c k , 1987) 2 1 ...(a) teaching (using procedures that aim at generating understanding) becomes sharply distinguished from t r a i n i n g (using procedures that aim at r e p e t i t i v e behavior); (b) processes inferred as inside the student's head become more interesting than overt behavior; (c) l i n g u i s t i c communication becomes a process for guiding a student's l e a r n i n g , not a process f o r transforming knowledge; (d) students' deviations from the teacher's expectations become means for understanding t h e i r e f f o r t s to understand; and (e) teaching interviews become attempts not only to i n f e r congnitive structures but also to modify them. 2.5.2 The Constructivism and Mathematics fundamental question of whether mathematics i s discovered invented i s sometimes viewed as a choice between either the position or the constructivist position. Plato believes or platonist that "The concepts of mathematics are independent of experience and have a r e a l i t y of t h e i r own. 1972, p. They are discovered, not 43). independently (1928, c i t e d As of stated by Thorn, "The thought, as in Kline 1985) invented or fashioned" mathematical Platonic ideas" (Kline, entities exist 696). Hardy (1971, p. believes that "...mathematical r e a l i t y lies outside us, that our function i s to discover or observe i t , and that the theorems which we prove, and which describe grandiloquently 'creations,' are simply our notes of our observations" means that mathematicians do nothing but as our (p. 205). to discover the This concepts and t h e i r properties. The other view sees mathematics as a product of human thought. "Herman Hankel, Richard Dekekind, and Karl Weierstrass a l l believed that mathematics i s a human creation" (Kline, 1985). Brouwer's idea gave thought in mathematics. (1980)" was rise to constructivism as His position as described by that the natural numbers are given to us a new Davis by school and of Hersh a fundamental 3 22 i n t u i t i o n , which i s the starting point for a l l mathematics. that a l l mathematics numbers" (p. 334). should be based constructively He demanded on the natural Bishop says that Brouwer and his followers "were much more successful in t h e i r c r i t i c i s m of c l a s s i c a l mathematics than in t h e i r efforts to replace i t with something better" (p. ix). Furthermore, Bishop claims that a s a t i s f a c t o r y alternative exists although Brouwer did not convince the others that there i s an a l t e r n a t i v e . Bishop, within a constructive framework, develops a large portion of abstract analysis in order to give numerical meaning to c l a s s i c a l abstract analysis since he think that " c l a s s i c a l mathematics i s d e f i c i e n t in numerical meaning" (p. ix). "The c o n s t r u c t i v i s t s regard as genuine mathematics only what can obtained by a f i n i t e construction" (Davis & Hersh, 1980, believes that "when a man show how is not on Bishop proves a positive integer to e x i s t , he to find i t " (1967, p. 2). emphasis p. 320). foundational As was be should stated by Goodman (1983), "The questions, but on the hard work of finding constructive versions and constructive proofs of actual theorem" (p. 61). the In c o n s t r u c t i v i s t ' s view, the purpose of proof i s to theorem, "to make the theorem obvious, so that f u l l y revealed, with nothing hidden" (Goodman, 1983, the clarify phenomenon i s p. 63). From a c o n s t r u c t i v i s t ' s view, mathematical objects as Cobb (1987), says are: operations reality. "phenomenological rather than correlates elements Regardless of the of a of systems of mind-independent conceptual mathematical interpretation adopted, i t would seem that viable models of learning or problem solving in mathematics must account for the experience of mathematical objects" more concerned with the relation between (pp. 9-10). the Kilpatrick is constructivism and the 23 practice of mathematics and he consequently expresses his concern for school mathematics and what should be taught. For Piaget, the increasing evolution of mathematical comprehensiveness constructivism, meaning and rigor i s cumulative. He structure since then in sees i s towards his the view of process of moving mathematics toward increasing o b j e c t i v i t y . However, other options seem to be available to answer the question of how does mathematics come into being (Wheeler, 1987). Wheeler seems to be in favour of a combination of both as a better answer. 2.5.3 The Teaching and Learning From a Constructivist Perspective activity of exploring children's construction of mathematical knowledge, as Cobb and Steffe (1983) believe, must involve teaching (p. 33). They make a distinction between the constructivist and nonconstructivist teacher by the emphasis they place on the a c t i v i t y of modelling children's r e a l i t i e s . through the overt behavior The c o n s t r u c t i v i s t teacher aims to see in contrast to the behaviorist teacher that attempts to see in the overt behavior. By seeing through t h i s behavior, the reconstruct teacher would help students to their mathematical learning contexts. Teachers solved only by using mere d e f i n i t i o n s ; i f they 1982, p. 33). is an 1988). is should r e a l i z e that the problem of teaching w i l l c o n f l i c t with "...students' conceptions situations they fail not be can change to handle" (Vergnaud, Learning, which i s stimulating and relevant to the student indispensable part of the constructivist program (Cobb et a l . In the c o n s t r u c t i v i s t s ' view, the core of mathematical the problem solving process (Cobb, 1987, also Cobb, 1986; learning Confrey, 24 1987; Thompson, 1985; VonGlasersfeld, 1983 cited in Cobb et a l . 1988). In t h i s s i t u a t i o n , the students t r y to reach t h e i r goals by constructing t h e i r solutions to problems which have a r i s e d . The focus of constructivism in r e l a t i o n to mathematics education i s exclusively realities on of the active individual processes students of construction of (Cobb et a l . 1988). mathematical The student's process of gaining and constructing knowledge i s more important than the structure of the student's knowledge i t s e l f (Erickson, 1987). Pines and West (1986), in referring to Vygotsky (1962), introduces two kinds knowledge. of knowledge, namely spontaneous knowledge The f i r s t i s the knowledge that children obtain and formal spontaneously from t h e i r surroundings, while the second i s the knowledge that children gain through between the formal schooling. learner's current "Learning understanding gleened" (Pine & West, 1986, p. 587). is and always the an interaction new information 25 CHAPTER 3 METHODS This chapter consists of the following sections: 3.1 3.1 The Rationale for Using the C l i n i c a l 3.2 The Subjects 3.3 The Interview Procedure 3.4 Method of Analysis 3.5 S p e c i f i c Interview Questions The Rationale for Using the C l i n i c a l Research into mathematical Interview thinking, claimed, has three basic aims which Interview are: as Ginsburg (1981) has the discovery of cognitive processes; the i d e n t i f i c a t i o n of cognitive processes; and the evaluation of competence. Depending on the research purpose, d i f f e r e n t methods can be used (Ginsburg, 1981). learning mathematics mathematical numerous is knowledge. the interviews (Ginsburg, will help 1981). (Magoon, investigate tests This to recognize students' cognitive processes. beings" intellectual may underlying be i n e f f e c t i v e researcher some 1981, p. 652). thinks the processes are or at least that of the complexities Clinical that these constructing mathematical concepts. clinical of the The depth interviews knowing will help to beings use when "In order to understand why persons act as they do, we need to understand actions. process The students being studied are "knowing some of the processes give to t h e i r in teaching and Since the underlying cognitive and complex, standard inefficient What i s important clinical the meaning and significance they [ s i c ] interview i s one way, not the 26 only way and ways-of doing she must also often used most appropriately in conjunction with so" (Jones, 1985, p. 46). act as a students' understanding. opportunity to not questions, but also only teacher in This researcher believed that order to explore Acting as a teacher gave the see what the to understand how other students do and they why the nature of researcher the in order did to answer i t (Cobb and S t e f f e , 1983). The interviewees were asked to do a series of questions related to the concepts of "function" and "derivative". what they were doing while they They were asked to describe answered these questions. investigator's purpose for conducting these interviews was The to attempt to find the nature of some of the d i f f i c u l t i e s that students are struggling with in c a l c u l u s . these d i f f i c u l t i e s . Probing questions were used to explore the nature of In the t u t o r i a l sessions, the investigator acted as a teacher while using probing questions. As Cobb and Steffe have stated: The actions of a l l teachers are guided, at least i m p l i c i t l y , by t h e i r understanding of t h e i r students' mathematical r e a l i t i e s as well as by t h e i r own mathematical knowledge. The teacher's mathematical knowledge plays a crucial role in their decisions concerning what knowledge could be constructed by the students in the immediate future. Through reflecting on their interactions with students, they formulate, at least i m p l i c i t l y , models of t h e i r students' mathematical knowledge (p. 85). This investigator had broad questions that the interview was upon, but the f i r s t p r i o r i t y was based to have the opportunity to study the process of a dynamic passage from one state of knowledge to another. For t h i s reason, a s t r i c t standardized protocol was suggests that interviewers should inquiry with t h e i r own not not used. prestructure the Jones (1985) direction of frame of reference in ways that give l i t t l e time 27 and space f o r t h e i r respondents than half of the questions to elaborate. were covered, In some cases not more however, the students opportunities to express t h e i r ideas and even to sometimes express had their feelings toward a question. According to Pine et a l . (1978), conduction of a p i l o t study i s a necessity p r i o r to every research study that uses c l i n i c a l i t s data c o l l e c t i o n methodology (cited in Aguirre, 1981). students were interviewed f o r a p i l o t study. interview as Four volunteer This gave the investigator the opportunity to revise the questions and her interviewing technique as well. The Subjects 3.2 Sixteen volunteer students participated all in the study. They were f i r s t year U.B.C. students and were s o l i c i t e d by public announcement in Math 101 classes and t h e i r t u t o r i a l students were asked to read sections (see Appendix A ) . a consent form prior Appendix B f o r a copy of the consent form). their real names would not be used in the to The interviews (see They also were told that study for the sake of confidentiality. Only twelve First year of the university students' interviews were used students were chosen because f o r analysis. they had been introduced to the concepts of function and derivative in the f i r s t term of the same year. The The sample consisted of four females and eight males. students' marks for Math 100 varied from 47% to 100%. This gave researcher the opportunity to investigate the conceptual d i f f i c u l t i e s of students of d i f f e r e n t mathematical backgrounds. students being studied along with the small Given the nature of the size of the sample, no 28 attempt was made to generalize the findings of t h i s study. of t h i s More studies nature would be needed to provide supporting evidence f o r the findings of t h i s study. 3.3 The Interview Procedure The data f o r t h i s study were collected during the months of March and A p r i l , 1987. The investigator conducted the interviews during a time convenient to the student, in a quiet o f f i c e located in the Department of Mathematics at U.B.C. Each interview lasted included the t u t o r i a l sessions). 60 to 80 minutes A l l the interviews were tape recorded. There were two sets of questions, one on function derivative. (which and the other on Students were free to start with any of the two sets that they wanted t o . They also were asked to explain t h e i r work. The f i r s t few minutes of each interview was mostly informal conversation until the student seemed to feel comfortable. The students were t o l d that they could terminate the interview whenever they l i k e d , although t h i s d i d not happen. On the contrary, some of them were discussion. eager to continue the So t h e i r wishes were granted, yet they were not recorded since there was not more than one 90-minute casette tape available f o r every interview. At l a s t i t i s worth mentioning that students being interviewed were extremely cooperative. 3.4 Method of Analysis The investigator was looking f o r patterns in the data. The data were q u a l i t a t i v e and provided by transcripts of the most relevant parts of the interviews and the students' written work. Immediately a f t e r each 29 interview, the interviewer made notes to facilitate further data analysis. Jones' (1985) advice least twice was followed. interview while based The f i r s t time provided the second s p e c i f i c questions. reduction, to l i s t e n to the tapes of each interview at on allowed a sense of the whole f o r a more d e t a i l e d analysis of Individual p r o f i l e s were made as a f i r s t the transcripts of the relevant parts step data of the interviews, students' written work and the interviewer's notes. Then the students' of responses reduction. for final were categorized as the second step data Later on the students' responses were "coded" into categories analysis. Vinner's model for cognitive processes was used f o r data a n a l y s i s . 3.5 S p e c i f i c Interview Questions The followings are the s p e c i f i c interview questions. of two sets of questions, the f i r s t set i s questions They consist related to the concept of derivative and the second set i s the questions on function. 1) 2) Derivative 2 a) Find the derivative of f(x) = x + 1 at x = 1. b) Sketch the graph of f(x) = x + 1. 2 The diagram shows the graph of the above function and a fixed point P on the curve (Parabola). Lines, PQ are drawn from P to points Q on the Parabola and are extended in both d i r e c t i o n s . Such lines across a Parabola are called secants, and some examples are shown in diagram. a) How many d i f f e r e n t secants could be drawn in addition to the ones already in the diagram? b) As Q gets closer and closer to P, what happens to the secant? 30 c) Find the slope of PQ1 Find the slope of PQ2 d) Find the slope of L at Point P = (1,2) Figure 3.1 The Graph of Parabola \ yy // \ \ \\ 0 / / ! P(l,2) 3) Give the d e f i n i t i o n of d e r i v a t i v e . 4) What's the relation between this curve and i t s derivative? 5) Compute the derivative of above function (f(x) = x by using the d e f i n i t i o n of d e r i v a t i v e . 6) (a) 7) compute the derivative of f(x) = compute d_ dx [1 + 1/(7 - 5x) 1/2 2 + 1) at x = 0 ] 0 x+1 •x+1 0 x<-l -l<x<0 0<x<l x>l at: a) x = -2 -1 b) c) -1/2 d) 1/3 e) 1 f) x 10 31 Figure 3.2 The Graph of a Non-Smooth Function l i i Function: x 2 + y 2 1) Consider the equation equation. = 1. 2) Determine whether the above equation represents a function y = f(x) or not. 3) I f not, determine a domain and a range such that the above equation i s a function. 4) What i s the r e l a t i o n between the domain and the range of a function and i t s inverse? and l / f ( x ) . 5) Find the inverse of f(x) in part (3) i f i t i s i n v e r t i b l e . your answer. 6) What i s the difference between Cos (x) Sec (x)? 7) I f f(x) = 7 x _1 (a) fog(x) (b) gof(x) (c) gog(x) (d) fof(x) Sketch the graph of this Justify _1 (Cos (x) = arc Cos(x)) and and g(x) = x + 1, compute one of the following. 32 CHAPTER FOUR RESULTS 4.1 Introduction The objective of this study was to provide p a r t i a l answers to the following s p e c i f i c research questions. 1. What i s the nature of the students' understandings of d e r i v a t i v e , slope of a tangent l i n e , how derivative and slope are r e l a t e d , d i f f e r e n t i a b i l i t y and function? 2. To develop a category system f o r these understandings which might provide some insight into the nature of the d i f f i c u l t i e s experienced by students learning these concepts. One method of addressing these questions has been to investigate i f there are c o n f l i c t s between the concept d e f i n i t i o n concept images of the concepts of derivative and the students' and function. Questions were chosen in such a way so as to give the subjects an opportunity to reveal t h e i r concept images and also to help the researcher to see the possible conflicts between students' concept definitions and their concept images. One way of investigating these issues, in the researcher's view, was to act as a teacher while conducting the interviews. Cobb and Steffe personal (1983), relationships "By acting with contexts within which they As stated by as teachers, and by forming children, we help learn mathematics" them close reconstruct the (P. 85). The " t u t o r i a l sessions" are the result of the researcher's teaching action whenever i t was appropriate. Because i t i s important and This constructing, excerpts to see the process of students' thinking of each student's interview were investigator believed that the students' individual analyzed. profiles will 33 help the readers in two ways: sessions" and students' understanding. To secondly facilitate the to students' help data questions on derivative and from the f i r s t to show the nature of the them to analysis, capture the students' essence responses function have been categorized and interviews have been quoted. "tutorial At the end of the to the excerpts of this chapter, a summary of results i s provided. 4.2 Individual Students' P r o f i l e s This section deals with the Individual Students' Profiles. the relevant parts of the students' interviews have been analyzed. effort has been made to i l l u s t r a t e the process Only An of students' knowledge construction and the role of the t u t o r i a l sessions in helping students to do t h a t . Only one example of a t u t o r i a l been discussed. Informal language was session for a given concept has used by the researcher to create a more f r i e n d l y environment for the interview. Some of the p r o f i l e s are lengthier than the others due to the fact that some of the students were better able to a r t i c u l a t e t h e i r ideas than others. any s p e c i f i c order. It should be mentioned that pseudo names have been used for the sake of the students' c o n f i d e n t i a l i t y . interviewer in the p r o f i l e s . 4.2.1 " I " represents the Twelve p r o f i l e s are presented below. Jenifer's P r o f i l e J e n i f e r got 77% in math 100. she P r o f i l e s are not in She wanted to major in sociology but fear of mathematics seemed to affect her d e c i s i o n . She said: "I heard that you have to take math 200, oh, I don't know, I have not r e a l l y decided y e t " . 34 For finding vertical the slope of the secant displacement l i n e , she t r i e d (dy) and horizontal displacement trouble finding the y-component of the point Q. to find the (dx). She had Her attention was drawn to the y-component of a few points with x = 1, x = 2, e t c . I expected her to possibly see a pattern f o r finding the y values of the point Q. Instead she said: Jenifer: Ya, but this i s not, but this l i n e i2 s not a parabola [she meant the secant PQ]. This [f(x) = x + 1] i s the equation of parabola, you can't use t h i s . She did not r e a l i z e that the points P and Q are on the parabola. Her main function. problem was the lack of understanding of the concept of J e n i f e r did not understand that the height (y) of any point (including Q with the length of x + h) on the parabola was 2 y = f(x) = x + 1. t h i s part. In a t u t o r i a l session, an e f f o r t was made to c l a r i f y This matter was discussed again during the interview. Later on she was asked: I: Can you give the d e f i n i t i o n of derivative? Jenifer: No, I can't, I can do i t . I: o.k., do i t please. Jenifer: f(x) = x + 1, f'(x) = 2x. I: No, not by using the formula, do you know how you got this formula? Jenifer: No, I don't, understand i t . [Emphasis i s mine.] 2 my teacher d i d the proof, but I didn't But I can do i t . [Emphasis i s mine.] There was a discussion about the slope of a tangent l i n e and the d e f i n i t i o n of d e r i v a t i v e . The question was raised again. This time she said: "In terms of l i m i t ? Oh yes, we did [emphasis i s mine] the d e f i n i t i o n of d e r i v a t i v e , i t ' s from l i m i t . . . , Oh, I don't know". Her p r i o r knowledge was "I remember" or "I don't inadequate. remember". She mostly used the phrases: She mathematics, as she s a i d , not understanding i t . guessing without thinking. was performing (doing) She was advised to avoid She then was asked to see i f there were any r e l a t i o n between the derivative of a function and the slope of the l i n e tangent to i t . Jenifer: The derivative i s the slope of the tangent l i n e . I: The derivative of the function at what point? She was not sure. and d i f f e r e n t curve. She Jenifer: session, the point P was tangent lines were drawn at the d i f f e r e n t then was those points. In a t u t o r i a l asked to find the derivative changed points on the of the function at She f i n a l l y defined the derivative c o r r e c t l y and said: You know, when you do i t , you don't think about t h i s . i t so mechanical. On the question of d i f f e r e n t i a b i l i t y that f(x) = 0 at x = -1 as w e l l . You do (#7, b ) , she skipped the fact When she was told of her wrong answer, she said: Jenifer: Oh ya, you have to do l i m i t or something. I: What do you mean? Jenifer: O.K., as x approaches from one side, approaches to 1 and then x approaches from the other side, approaches to 0. Doesn't exist...no! I: Why doesn't i t exist? Jenifer: Because i t approaches to two d i f f e r e n t numbers. [The derivative.] 36 Tutoring helped her to improve her concept image of derivative and the concept of d i f f e r e n t i a b i l i t y . Although she understood the concept of derivative with d i f f i c u l t y , the acquired knowledge helped the question on d i f f e r e n t i a b i l i t y with no d i f f i c u l t y . her to answer Not many of the interviewees were comfortable with t h i s question. 2 On the section on function, she said that the c i r c l e (x + y 2 = 1) was a function: Jenifer: Because i t has a certain number of points. I: Do you think that each set of points represent a function? Jenifer: Ya. I: What i s the d e f i n i t i o n of function? Jenifer: I don't know. I: How can we say that t h i s [x + y What i s function? Jenifer: What i s the function? 2 2 =1] represents a function? Oh, I don't know. These kinds of answers were widespread among the students. sometimes did not have any j u s t i f i c a t i o n for t h e i r answers. They The t u t o r i a l helped her understand the function in a very concrete manner. Then the next question was: I: Can we r e s t r i c t the domain and the range of t h i s equation in order to have a function? How can we change t h i s c i r c l e to a function? Jenifer: Make i t . a 1 ine. I: Can we change a c i r c l e to a l i n e . Jenifer: No, we can't, but that's the only way. Her [Emphasis i s mine.] suggestion to change a c i r c l e to a l i n e was very She did not understand the concept of function. interesting. She d i d not believe that 37 the domain of the equation could be r e s t r i c t e d , while the equation remained unchanged. circumference of She the rather aimed circle. That changing the c i r c l e to a l i n e . last said that, positive y's". "we Her can last to conserve seemed to be the length why she itself of the suggested After explanation and discussion, she at restrict response i t either to the negative showed her progress in y's or understanding the concept. 4.2.2 Richard's P r o f i l e Richard was e l e c t i v e courses He said: goal". a commerce major. because he He took Math 100 and Math 101 said that he l i k e d mathematics very much*. "I'd l i k e to teach high school mathematics, that's actually His reason for majoring in opportunity in the future job market. got 100%. as commerce was to have a better He did very well in Math 100. Richard introduced himself as an eager student who hard to understand the concepts in c a l c u l u s . my He strived so He said that he volunteered to p a r t i c i p a t e in the interview to learn something from i t . A minimum amount of time was compared to interviews. the amount of Mostly time spent on tutoring in t h i s interview, that was spent a hint or suggestion was attention (not his memory). on tutoring sufficient in other to d i r e c t his For finding the slope of the l i n e PQ, (#2, C) he said: Richard: If I know the value of the point [he meant Q]. I: You can get i t , you have the f i r s t coordinate of Q. *For commerce majors, math 140 and math 141 are requirements less theoretical and are mostly based on the applications. which are 38 Richard: Oh, I don't know the y value for that, r i g h t ? . . . You mean I can find i t by myself and everything?! [surprisingly.] I: Yes, f i r s t write the x value for Q. Richard: O.K., Q^h + 1, (h + l ) + 1 ) . 2 He then continued d i f f i c u l t y at a l l . and got the slope of the secant l i n e with no 2 He then d i f f e r e n t i a t e d the f(x)=x +l to get the slope of the tangent l i n e . 2 Richard: f(x) = y = x + 1 f'(x) = dy/dx = 2x I: What I'd l i k e you to do i s to follow the same procedure that you did f o r finding the slopes of the other secants, which means PQ's, in order to get the slope of the tangent l i n e L. After derivative". a few minutes he s a i d , "I can give you the d e f i n i t i o n of He was advised to forget the derivative f o r the time being and to get the slope of the tangent l i n e as i f i t were any other secant. He responded as follows: Richard: The l i n e i s tangent to the graph, but there i s only one point, I can't use the same thing here. [Emphasis mine.] I: What happens to this between P and Q.] Richard: h tends to zero, right? I: Then how can we find the slope? Richard: Use the same point over again?! distance? [Pointing to the distance Limit i s one of the most problematic concepts f o r students. a l l experienced some kind of d i f f i c u l t i e s Although investigating concept of 1imit was in understanding the students' d i f f i c u l t i e s not the objective of They t h i s concept. in understanding the study, the the above 39 questioning was brought up i n t e n t i o n a l l y . the concept of l i m i t was one of the missing students from understanding saw that graph). the point The lack of understanding of chains which prevented other concepts such as d e r i v a t i v e . Q got closer and closer Richard to P (by looking at the He could see that the, distance between Q and P ( i . e . h) became shorter and shorter. He was then stuck at this point. what to do and asked "use the same point over again?!" He did not know Richard knew that he could not do that, but there was nothing else that he could think o f . He was eager to take a fresh look and t r y to answer the same question again: I: What i s the slope of this tangent l i n e at t h i s point? P(l,2).] Richard: I t ' s 2 [he used the d i f f e r e n t i a t i o n answer i . e . f'(x) = 2x and f ' ( l ) = 2]. I: What i s the slope of the tangent l i n e [ i . e . , not this p a r t i c u l a r case]. Richard: Sorry, Oh no, I don't actually know. formula [at to get this at any given point When the question of the slope of the tangent l i n e was r a i s e d , his f i r s t response was to give the d e f i n i t i o n of d e r i v a t i v e . thing again. between these tangent l i n e . He did the same This researcher was wondering i f he could see any r e l a t i o n two concepts, i . e . , the derivative In a t u t o r i a l and the slope of the session, an e f f o r t was made to help him understand what would happen to h as Q got closer and closer to P. session helped him to get the slope of the tangent l i n e . This Then he was asked to compare what he got as a slope of the tangent l i n e with what he gave as the d e f i n i t i o n of d e r i v a t i v e . f'(x) = l i m i t f(x+h)-f(x). h+0 h He gave the d e f i n i t i o n as: His concept image related these two concepts 40 together, since he offered the d e f i n i t i o n of derivative to express the slope of the tangent l i n e . he used the d e f i n i t i o n the tangent l i n e . derivative was question He related these two concepts in a way of derivative as a "method" to find the slope of The tutorial session helped him the slope of the tangent l i n e . (#5) was that to understand that His response to the next more l i k e the appreciation of the t u t o r i a l session, since he knew what he was doing. 2 I: f(x) = x + 1, find the d e f i n i t i o n of d e r i v a t i v e . Richard: I think I have the answer, because i t ' s gonna be the tangent (he meant x = 0 ) , i t would be the horizontal l i n e and the slope of that l i n e would be zero, [ i . e . , f'(0) = 0.] His correct. first x = 0 by using response to the question on d i f f e r e n t i a b i l i t y was The investigator's attention to the function "there f'(x) at i s no...it's intervention and i t s graph singular point. helped him to pay (in #7), and he then said There i s no tangent line the not more that to i t . There i s no suitable tangent l i n e to i t " . He had an adequate concept image and a correct concept for the concept of function. of the related session, the concepts inverse definition He had d i f f i c u l t i e s in understanding some to the function concept was of discussed. function. He then In a tutorial restricted domain and the range of semi-circle such that to make i t an the invertible function. At the end of the toward mathematics. interview, He was the concepts of c a l c u l u s . he liked to express his feelings aware of his d i f f i c u l t i e s in understanding He said that his 100% mark in Math 100 did not guarantee his understandings of the concepts. He said: 41 Richard: I went through half of the year and I didn't know that the dy/dx i s actually changing of y over changing of x. So for example, i f a question comes up that asks you to v i s u a l i z e the problem, r i g h t ? ! Something that i s not straight out of the textbook, i t would be d i f f i c u l t i f you didn't know that t h i s i s what i t meant because you don't get these d i f f i c u l t questions that asking for r e a l l y understanding [Emphasis i s mine.] His comment was quite i n t e r e s t i n g . He said that "you these d i f f i c u l t questions that ask for real understanding". don't get The fact of the matter was that these questions were selected from the very beginning of the textbook. The first part of his claim was not true that "you don't get these d i f f i c u l t questions" because these questions were not the difficult ones compared to for example what they got for t h e i r homework or t h e i r exams. Yet the second part of his claim was questions were not could f u l f i l l asked for the real the course, and understanding true that since the those students the instructor's expectation as w e l l , by doing them well i . e . , getting a good mark! 4.2.3 Brian's P r o f i l e Brian was got 70% a science major. in Math 100. Brian had He hoped to get into engineering. a vague and mixed up concept image of derivative. Although, he at l a s t was he did not have a clear understanding difficulties were deeply rooted He able to answer the questions, yet of the discussed concepts. in his understanding of the basic His and crucial concepts of c a l c u l u s , namely the concepts of function and 1imit. He had an inadequate sense of g e n e r a l i z a b i l i t y . accept their difficulty existence. He, He had to see things to l i k e most of the other to find the y-component of the point. interviewees, had He could not see the 42 difference between a point and i t s coordinates. He responded as following in order to find the slope of the secant l i n e : Brian: So i t would be r i s e over run...Q - 2. I: Q i s a point, right? Brian: Yes, r i g h t , so i t would be...but we don't know what the Q i s ! The coordinates of a point were discussed saying that the length of a point was point, i . e . , f(x) was in a t u t o r i a l the x-component and the height of a the y-component. I: The height of Q i s : f ( x 0 + h) which i s equal t o . . . Brian: I can't see that. The told how tutorial session session went on with more concrete examples. the value of f(x) was changing of f(x) was dependent on dependent on the value the changing of x. find the f(x) when x = 1, x = 2, x = 3 and x = x0 f ( 2 ) , and f ( 3 ) . "The But for f ( x Q + h) he said: of x, He + h. He i . e . the was asked to He got f ( l ) , function...right. mean...but we are told to use this function [f(x) = x 2 was You + 1]. His response revealed an important fact that he did not understand the concept of function. As long as x had numerical to get f(x) without any d i f f i c u l t i e s . doing t h i s sort of problems. changed. So he was variable. For him, anything, because x stuck. i t was was a value, he was able Because he learnt the mechanism of For f(x + h), the syntax of the problem was Brian hard could to variable, not possibly see conceptualize and since x that was an the x x as a could be independent v a r i a b l e , so f(x) would be the subject to change because of x. It was 43 d i f f i c u l t to see x as just a symbol not a value. A f t e r the t u t o r i a l he said: Brian: So the slope would just be, OK, so i t would be... [in the beginning he said that the slope i s r i s e a n r j that's run why he was trying to get the y-component of the p o i n t ] . I: You said the slope i s r i s e run Brian: But...wouldn't i t be...the slope would be the derivative of it?!! His concept image of slope was formed of two d i s c r e t e elements, namely slope as the derivative and slope as the r i s e . For him there was run no difference between the derivative as slope of the tangent l i n e and a slope of secant l i n e . (#2, d) he said: He was In order to find the slope of the tangent l i n e "Wouldn't be the derivative of the function". reluctant to proceed with the followed for finding the slope of the secant. l i m i t , yet tutorial his understanding session, an of e f f o r t was the same concept was The derivative. f(x) = x 2 + 1 questionable. at x The = 0 he that In how point of the tangency and got the slope of the tangent l i n e at x = 1. to give the d e f i n i t i o n of d e r i v a t i v e . that Although he mentioned the slope of the tangent l i n e were discussed as w e l l . of procedure made to describe to him secant l i n e s became the tangent l i n e . derivative as A f t e r the t u t o r i a l a the the he Yet i t took awhile for him next question was by He correctly found f'(x) at x = 0. using the This was to find the definition of a good evidence to show that he did not just agree with the interviewer in the t u t o r i a l session, but the t u t o r i a l derivative. helped him to understand the d e f i n i t i o n of the 4 4 His response to the question of d i f f e r e n t i a b i l i t y (#7) was the same as many other interviewees. He f i r s t said that f'(x) at x = -1 i s 1. The interviewer stopped him to draw his attention to the fact that f'(x) = 0 at x = -1 as w e l l . He said: Brian: OK, i t would be 0 a l s o . I: The derivative of function i s 1 at x = -1 and again the f'(x) = 0 at x = -1. How can we interpret this? There was asked him no response to the above question except s i l e n c e . i f he could r e c a l l the d e f i n i t i o n of d e r i v a t i v e . questions and I With probing interviewer's intervening, he understood that there could be i n f i n i t e tangent lines for f(x) passing through the point x = -1. I: Can we decide which one [f'(x) derivative of f(x) at x = -1. Brian: No, we I: Then, what do you suggest? There was he was told = 0 or f'(x) = 1] i s the can't. What can you say about i t ? no answer to t h i s l a s t question. about d i f f e r e n t i a b i l i t y . In a t u t o r i a l session, Differentiability's property was only t o l d by the aid of the geometric representation, not in a rigorous language. He had d i f f i c u l t y understanding understanding the concept of the concept of f u n c t i o n . function, caused answering the questions of derivative as w e l l . The him frustration I: Is i t [the c i r c l e ] a function or not? Brian: Is y a function of x? Yes, sure. in section on function started with an intention to see whether or not the c i r c l e i s a function. Lack of (x 2 + y 2 = 1) 45 2 2 I: Why? Brian: [Silence].. I: What i s the d e f i n i t i o n of a function? Brian: I don't know. I: Why you said that this i s a function? Brian: Because I could draw a c i r c l e , that's a l l . I: Is c i r c l e a function? Brian: Yes, sure, i t has two...I don't know, of course, you don't want the d e f i n i t i o n of the equation. For him, and Why i s x + y =l a function? function was nothing few numbers with examples were used but a few symbols (such as x and y) an equality s i g n . to describe In a t u t o r i a l the concept session, many of function. The l a t t e r question was looking for the possible r e s t r i c t i o n that could be imposed to the domain and range of the equation 2 2 (x + y = l ) , in order to make i t a function. I: How can we determine the domain and the range equation in order to have a function. Brian: You want to make i t into a function? I: Yes, because we saw that tutorial]. Brian: You could expand i t just l i k e that I: We are not allowed to do that. i t ' s not a function He t r i e d and he could not succeed. the time frame) to help time to finish interesting. the him understand. interview. But of this [after the The interviewer t r i e d He unfortunately his last (within had no more response was very He said that the c i r c l e could be expanded to make i t as a function (refer to Jenifer's p r o f i l e for the discussion on t h i s matter). 46 4.2.4 Megan's P r o f i l e Megan was majoring in biology. She got 50% in Math 100. questions on derivative to start with. She chose When she wanted to get the slope of the secant l i n e , she, l i k e most of the other students, had d i f f i c u l t y to find the y-component of the point. Probing questions helped get the y-component of Q and the slope of the secant as w e l l . not know how to find the slope of the tangent l i n e . matter was concept not adequate image f o r the enough to help coordinates a She did Her knowledge of the her to establish of her to point. She the proper had trouble understanding the concepts of l i m i t , tangent l i n e and point of tangency. In a t u t o r i a l session, the discussions aimed to help her understand that the slope of the tangent l i n e would be the l i m i t of the slopes of the secant l i n e s . I: We are looking f o r the slope of the tangent l i n e at P [1,2]. Megan: So to finding the slope of the tangent derivative of the equation. The to get the derivative in order to find the slope of the tangent line? She had no The t u t o r i a l liked to know that session continued why get the she wanted answer. interviewer l i n e , we f o r another 20 minutes. Her response after the t u t o r i a l was as follows: Megan: The derivative i s the slope of tangent l i n e at point P. I: Derivative at which x? Megan: 1. Her response was an evidence to show that the t u t o r i a l to understand the concept of derivative to some degree. helped her 47 In answering the question same d i f f i c u l t y that most of the other students had. almost the same. of the reasons (#7), of d i f f e r e n t i a b i l i t y The responses were She did not know the constant function. that she was not able she had the to see that This was one whether f(x) was d i f f e r e n t i a t e at point x=-l or not. I: You said f(x) = 0, then what i s f'(x)? Megan: f'(x)?!...There i s no x. A tutorial constant session aimed to help her understand function and concept of d i f f e r e n t i a b i l i t y . derivative was discussed by r e f e r r i n g d e f i n i t i o n of d e r i v a t i v e . the concept of The existence of to the e a r l i e r discussion on the Although after the t u t o r i a l , she thought that she understood the discussed concepts, yet there was not enough evidence to prove that the new knowledge was constructed by her. The function. interview continued She said that x 2 by + y 2 her responses to the questions of = 1 was a function f o r the following reason: Megan: Because.. .they are related?!! seem to c a r e ] . I don't know [and she did not I: There i s a r e l a t i o n between x and y, so x equation. But why i s i t a function? Megan: The equation i s a function. I: Why do we name them two d i f f e r e n t things same, why do we say equation and function? 2 + y 2 = 1 i s an i f they are the She did not have any answer f o r i t . She had a limited knowledge about function understanding. in general. Tutoring helped her to acquire some Enough to enable her to answer some of the questions. But she had no idea of what the inverse function was. 1 4.2.5 48 Jason's P r o f i l e Jason was a science major. dentistry. as He aimed to continue his study in Jason got 90% in Math 100. He did not at a l l use such words guessing in the interview. He was quietly thinking unless he c e r t a i n l y had a point to make. He l i k e d to start with the questions of d e r i v a t i v e . wondering how he would find the slope of the secant. on in silence f o r about 10 minutes. The He was The interview went interviewer necessarily intervened and asked: I: Do you have any idea that how to find the slope of the line? Jason: It's r i s e over run. I: OK, find r i s e and run. Jason: slope = Q - 2 h I: What do you mean by Q? Jason: It's distance. I: Distance from where? Jason: Distance from p to Q. He had a misconception point itself. He could about the y-component of a point and the see the distance component of P, because i t had a numerical 3.1). from the centre to the y- value which was 2 (see Figure But he was not able to find the distance from 0 to y(Q) since there was no numerical value for i t . I: What i s Q? Jason: Q i s a point. I: What are the coordinates of Q? 49 Jason: Oh, ya... I: Can you find the coordinates of Q? Jason: No, I haven't [have] a height here. After a b i t of thinking he wrote: 2 Jason: Q = (X0 + h) + 1 I: Why did you write: Q = (x 0 + h ) + 1 . Q i t s e l f i s a point. Every point i s indicated by i t ' s length and i t ' s height. What you got here i s the height of Q, i t ' s not Q. Q i s a point. 2 He f i n a l l y that. got the slope of the PQ by spending a few minutes on Later on he was asked to find the slope of the tangent l i n e at p. He wrote: 2 Jason: f(x) = x + 1,, f'(x) = 2x, f ' ( l ) = 2 = slope L [tangent l i n e to the curve at point P ] , I: Why did you take the derivative of f(x) at x = 1? Jason: Why did I do i t ? The derivative i s slope of l i n e ? ! ! I: How are they related? Jason: I'm not sure... I: Will you find the slope of L in the same way that you got the slope of the other secants. It took interviewer's 11 minutes f o r him to get the answer without intervention. He did no mistake in doing any of the that. The questions followed by: I: What i s the derivative? Jason: Slope. I: Slope of what? derivative? What i s the definition of 50 Jason: Slope of the l i n e at a point given. I: Slope of which line? Jason: f(x). After a few minutes of thinking, he said: Jason: Derivative...is the slope of the tangent l i n e f(x) at a point x. I: Derivative at which point? Jason: at any point [Emphasis i s h i s ] . The following t u t o r i a l session, aimed to show that to the curve the value of derivative was changed by changing the tangent l i n e s to the curve i . e . , by changing the point of tangency. geometric representations. This was done by the means of the After the t u t o r i n g , h i s response was the unique one among a l l the interviewees. Jason: The equation of derivative i s the same at a l l point, isn't i t ? ! Putting d i f f e r e n t numbers for x. This understanding was rare among the students. That the derivative function f'(x) was not changed but the value of derivative was subject to change by changing the points of tangencies. He got the f ' (x) derivative. at x = 0 by using the d e f i n i t i o n of the He also sketched the graph showing the tangent l i n e to the curve at x = 0. The correct answer to the above question showed that he seemed to be confident in understanding the concept of d e r i v a t i v e . the section on d i f f e r e n t i a b i l i t y , although towards solving the problem (as he said: he had a d i f f e r e n t v For approach can I do i t by inspection'), y e t , he was experiencing the same d i f f i c u l t y , i . e . , facing two values f o r 51 f ' (x) at x = -1. With an exclusive discussion and the probing questions, he f i n a l l y answered the l a s t question of t h i s part: I: Can a function have two d i f f e r e n t tangent l i n e s at one [and s t i l l have derivative at that p o i n t ] . Jason: No, i t doesn't have derivative at that point then. The next set of questions was concept image of function. set, on function. In response to the Jason had first an accurate question of this he said: 2 Jason: 2 It's [x + y = 1] not a function, because i f we perpendicular l i n e [he meant v e r t i c a l l i n e ] , there values of y for one x. For him, f " 1 and 1/f were the same. 1 inverse function and reciprocal of a function, was with some of the have already other been tutorial discussed. sessions with He then draw a i s two He said that 1/f = f " . t u t o r i a l session, in which the researcher aimed to c l a r i f y the of point other The concepts, the same in nature interviewees confidently answered that a l l the related questions of the function. Joe's P r o f i l e 4.2.6 Joe was interview. a quiet person. At least 1/3 was mostly thinking. answer. He got 90% He hardly talked or even wrote during the of the tape was by s i l e n t moments. He He did not speak unless he thought that he had the in Algebra 12. He filled had no calculus at high except a very basic introductory at the end of Grade 12. Math 100. I: Interviewer t o l d him that: It's very good that you got 100%. school He got 100% in 52 Joe: Even though I got 100%, s t i l l there are l o t s of anxiety to understand. When the test came up, I knew how to do everything but when i t ' s going to problems, there are problems that I don't know how to do. [Emphasis i s mine.] The Way he expressed himself difference between performing calculus. showed that f o r him there calculus and understanding As described by him, he was w i l l i n g was a the concepts of to r e a l l y understand the concepts but he did not get enough help from the class or the t e x t s . The interview continued by talking about d e r i v a t i v e . His response for finding the slope of the secant was: Joe: I'm not r e a l l y sure how to find i t . I: Give i t a t r y , what i s the slope? Joe: I know i t ' s r i s e over run, but I don't see what the r i s e i s . He found the r i s e by himself but he, l i k e many others, had a hard time to substitute the value f o r f ( x 0 + h). from lack of understanding His d i f f i c u l t i e s stemmed the concept of function. He d i d not know the 2 behavior of f(x) = x + 1 (The behavior of f(x) has been discussed in one of the other p r o f i l e s ) . errors t i l l slope In case of f ( x + h ) , he d i d many t r i a l s and he got the right answer. of the tangent understanding line (at P ) . He did not know how to find the He d i d not have an adequate of the concept of l i m i t in order to help him to understand the slope of the tangent l i n e . In a t u t o r i a l session, i t was explained to him that the slope of the tangent l i n e was the l i m i t of the slopes of the secants when h approached to zero. correctly. He defined the derivative almost His concept image of derivate led him e a s i l y to the accurate concept d e f i n i t i o n of d e r i v a t i v e . 53 Joe: The slope of the tangent l i n e at point on the curve i s the derivative of function. I: At where? Joe: At that point. In response to the question of d i f f e r e n t i a b i l i t y , he thought that f(x) (#7) was undefined at greater than 2 since f(x) = 0. The interview with the appropriate interviewer's intervention went on until he said: Joe: at x = -1 i s undefined [f(x) i s undefined]. I: Why Joe: At that point f'(x) i s between 1 and 0, but undefined at that point because you can't plug t h i s number into f ( x ) . I: What do you mean by: Joe: Since your formula. i s i t undefined? "you can't plug t h i s number into f(x)?" f(x) = 1, and x = -1, you can't put into this His reason f o r the derivative of function to be undefined at x=-l was f(x) interesting. i s constant He did not know the concept of constant function which for a l l x belongs to domain of function. partly the reason for his l a s t response as: t h i s [f(x) = 1 ] " . This was "you can't put [x = -1] into His main problem was his d i f f i c u l t i e s in understanding the concept of function. He gave differentiable. a list of incorrect conditions for a function Probing questions were posed to c l a r i f y After the discussion he said: the "Oh, this i s a smooth curve. to be problem. I remember t h i s part, I wasn't quite sure how to do that, I taught myself how to do i t and that one worked (Emphasis i s mine)". These honest words must be t h r i l l i n g curriculum makers. Solving and answering f o r calculus instructors problems has and become a game. 54 Some of the students play i t smart and some of them do not know the rules of the game. He had a r e l a t i v e l y good understanding of function. No special incident happened while he was answering the questions of t h i s part. Owen's P r o f i l e 4.2.7 Owen's major was science. he decided caused me He wanted to continue in engineering but not to "...because Math i s one of my d i f f i c u l t to take through i t second time Math 101". subjects, He had an introductory to calculus at the end of Grade 12, but as was stated by him, i t d i d not help him to be more prepared f o r university calculus. "When I came here l a s t year, i t was just like...Oh r e a l l y , what's going on"? Owen was f i r s t in a crowded calculus class and he f a i l e d that term. For the second time he changed to a smaller c l a s s . be more relaxed in small classes". understand anything and he only As he said: "you can He insisted on saying that he did not repeated his instructor's words. He could not see himself as being a student who understands the concepts. For finding the slope of the secant he asked: Owen: Can I do y = m + b? I: How do you find the slope? Owen: From the tangent, the d e r i v a t i v e . I: What i s the r e l a t i o n between the derivative and the tangent l i n e , why do you want to take the derivative? Owen: The tangent i s the derivative at that point. He had an unclear knowledge of the discussed concepts p r i o r to the interview. these He had heard a l l the notions before, but he did not know how notions or concepts were r e l a t e d . He could not v i s u a l i z e the 55 difference between the slope of the secant l i n e , the slope of the tangent l i n e and thing. derivative. He dealt with them as i f they a l l were the same Probing questions were posed to f i n d out more about his primitive knowledge of these concepts. The interview went on until asked to answer the question of the slope of the secant Owen: Oh, Oh, i t would be: Qr =Qi' p x0+h-x0 What do you mean by Qj - P? Owen: Q i s a point on the curve minus P. I: What i s xQ + h - xQ? Owen: That's the run. I: OK, what about the rise? got "run" because he could again line: p h I: He he was see them (see Figure 3.1) but couldn't see the y-component of the points (because he had to f i r s t them). In a t u t o r i a l the point. did he find session, I talked to him about the coordinates of For y-component of the point he used the point i t s e l f . not know c l e a r l y how a point was indicated on the plane. He It i s not reasonable to expect a student to understand derivative while he needs to know many basic concepts prior to t h a t . Owen: So, Qx-P would be: x After the t u t o r i a l he said: f( n)'f(l) does i t make sense to you? h I: I want i t to make sense to you, does i t ? Owen: Ya, i t makes sense, because i t ' s the amount of height. Interviewer was just about to say something when he said: 56 Owen: OK, OK, I got i t , f(x 0 +h)-f(x 0 ) = f(x 0 +h)-f(x Q ) x0+h-x0 His last response was more l i k e h a jump. I think h i s response showed the evidence that he had, in his mind, the discrete pieces of the concept either by understanding those pieces or by remembering them. He was then asked to find the slope of the tangent l i n e at P. Owen: h i s zero, then f x +l-x +l ...I 2 2 just plugged in h=0, because 0 there isn't any distance along x axes, so i t would be x2+l-x2+l , but i t ' s not r i g h t . 0 I: As we move along the curve, what w i l l happen to the secants? Owen: Secants w i l l become the tangent, i t ' s just the d e r i v a t i v e , so that would be 2. The interviewer asked him i f he could consider any given function other than f(x) = x 2 + 1 and say what w i l l happen i f Q moves toward the fixed point P, he answered: Owen: The l i m i t . . . h-*» I: Why » ? Owen: Oh, I just s a i d , the l i m i t f(x n +h)-f(x n ) . h+0 (x0+h)-x0 Then he went on and got the slope of the tangent l i n e . r i g h t solution did not seem to be an evidence of his understanding concept. But his of the For example, when he was asked that why he took the l i m i t as h approaches to » , he s a i d , "Oh...I just said i t " . But in investigator's opinion, based on her observations, Owen did not have clear of the concept of l i m i t , approaching to something, e t c . understanding He usually had 57 seen the notion of l i m i t which was accompanied by h approaching to either o or 0 and he naturally t r i e d them both to see which one worked better. He d i d not see h approached to 0 from the graph. slope he was of the tangent line, asked After he found the to give the d e f i n i t i o n of derivative. Owen: It's the tangent l i n e at a certain point of that graph, such as the derivative of equation i s defined. I: OK, what i s the derivative of function at that point? Owen: At that point! I: What i s the slope of the tangent l i n e at t h i s point [at P]? Owen: 2. There...[no answer], For him, the concepts of secant, tangent l i n e , slope and derivative were a l l mixed up, although he sometimes used them quite c o r r e c t l y . He knew that these concepts were somehow r e l a t e d , but he wasn't sure how. In another t u t o r i a l session, an e f f o r t was made; with the aid of graphs; to c l a r i f y these concepts as much as i t was possible After the t u t o r i a l , he defined derivative as: (time constrain). "The derivative i s at a certain point defined that i t approaches as the same as the l i m i t of the secant l i n e as changing value of x approaches to zero." tutoring The process of and questioning went on and on a couple of times, until he seemed to understand this s p e c i f i c case of d e r i v a t i v e . His answer to the next question showed his lack of g e n e r a l i z a b i l i t y along with his uncertainty. of f(x) = x 2 The question was to compute the derivative + 1 at x = 0 (the f'(x) at x = 1 was discussed in d e t a i l ) . He f i r s t got i t quite r i g h t , but he then changed his mind and said: no, i t ' s not r i g h t " . "no, He t r i e d again and got the same answer (f'(x) = 0) 58 and again he repeated that: " i t i s not r i g h t " . time and again f'(x) = 0 was his f i n d i n g . He t r i e d for the t h i r d He said: Owen: I: It's wrong. Why i s t h i s wrong? Owen: Because the d e r i v a t i v e , the slope should be 2 [he pointed to the derivative at x = 1 ] . What he did was i n t e r e s t i n g . in his understanding to believe that His d i f f i c u l t i e s were deeply of the concept of d e r i v a t i v e . the derivative d i f f e r e n t points. had d i f f e r e n t rooted For him, i t was hard numerical values at He did not know the reason that he got two d i f f e r e n t answers f o r the f'(x) was that he was computing the derivative of the function at two d i f f e r e n t points. He simply thought that there must be something wrong with his computation. He was even asked to "look back" (Polya, 1945) and see i f h i s procedure correct. (and computation as well) was He "looked back" and he d i d not find any mistakes, yet he was not s a t i s f i e d with h i s findings. The interviewer suggested to him that both, he and her could go through the whole procedure again to get the f'(x) at x = 0 (by using the d e f i n i t i o n of d e r i v a t i v e ) . We d i d i t and when we f i n i s h e d , he said: Owen: It's 0. I: Then why do you think anything wrong. Owen: Because f o r what I've been taught, derivative...Oh, that's r i g h t ! This i s r i g h t . Because f'(x) = 2x and i f x = 0 then f'(x) = 0 . I .just wasn't thinking. [Emphasis i s mine.] I: Because at x = 1, the tangent l i n e i s [see Figure 3.1], but at x = 0. the tangent l i n e i s the horizontal l i n e (I sketched the graph). Owen: Right, r i g h t , 1 .just wasn't thinking. that i t ' s wrong? You didn't do [Emphasis i s mine.] 59 Twice he said that he was not thinking but the investigator thinks that h i s error was rather "structural" (Orton, 1983). He had a vague concept image of the related concepts to the concept of d e r i v a t i v e . interviewer believes that he at l a s t accepted The that f'(x) = 0 at x = 0 because he plugged the x = 0 to the formula and he was s a t i s f i e d with the result. But i t d i d not necessarily prove that he got i t right because he understood the concept thoroughly. tutoring by means of graphs. That was the reason f o r further That helped him to understand the concept better. For him, the function which was d i f f e r e n t i a b i l i t y , needed more explanation. used in the question of His response to question 7 (after the function i t s e l f was explained to him) was the common one. Owen: At x = -1, f(x) = x + 1, f'(x) = 1. I: What about t h i s : f(x) = 0 x<-l. (x<-l and -l<x<o). Owen: The f(x) says that: f'(x) = 0 at x = -1 and f'(x) = 1 at x = -1, so i t ' s gonna be one of the two. I: Can you decide which interrupted by him]. Owen: You can't say that because i t should be both. I: What? Owen: It should be both; i t can be 0, i t can be 1. The interview went one? on by asking x = -1 i s in both intervals Can you say that...[I him whether was the function was d i f f e r e n t i a b l e at x = -1 or not. Owen: Probably, because i t ' s not a smooth curve. together. Is that correct??! Two curves chain I: Why because i t ' s not smooth, i t ' s not d i f f e r e n t i a b l e ? Owen: Oh, because two d i f f e r e n t l i n e s cross each other. 60 His responses showed that d i f f e r e n t i a b i l i t y but he was possible that with this he remembered not sure why he was something about saying those. It i s much understanding of any concept, a student could get a s a t i s f a c t o r y mark f o r his course, e s p e c i a l l y those who experts in getting good marks. are But i_s there any understanding behind the marks i s a question that should be answered. The students' responses could be more or l e s s , a part of the answer. He did mentioned not the know "smooth d i f f e r e n t i a b i l i t y was of graphs and the concept curve". of d i f f e r e n t i a b i l i t y , In a short tutorial although session, he the discussed in a very simple way, mainly by the aid referring to the discussion at the beginning of the interview (on the d e f i n i t i o n of d e r i v a t i v e ) . In order to answer the questions of function, he c e r t a i n l y hard to remember a l l he was succeed. told by his i n s t r u c t o r s . His concept image of function was But tried he did not the same as f o r the r e l a t i o n and equation, although he did not have proper concept image of those two concepts as w e l l . vice versa. Four times he changed his answers from NO to YES He f i n a l l y gave up and said: and "I don't know how to describe it". His main idea for f to be a function was: defines points concepts of the y along the x-axis" and " i t defines y" or " i t so on. inverse function, l / f ( x ) , e t c . He He did not was know the stuck, but he seemed to be anxious to know something about function in general (domain, range, inverse...). investigator, within That was why he asked me to tutor him. The her time frame, t r i e d to use the various examples and sketched d i f f e r e n t graphs to explain the concept of function to him. Tutoring took about 15 minutes. Unfortunately he had no more time after 61 the tutoring to see i f the t u t o r i a l session on function helped him to understand the concept or not. 4.2.8 Kathy's P r o f i l e Kathy's although she major had f a i l e d Math 100. Kathy: was biology. to write the She was doing supplementary well in Math exam in August since 101, she She gave following reason f o r her f a i l u r e in Math 101: The reason i s , because I hate math. I never did my homework, and I usually just cram at l a s t minute and you can't do that for c a l c u l u s , calculus i s quite d i f f e r e n t from Algebra. Kathy had taken Honor Algebra 12 and Geometry. of the subjects. She got 'A' in both She decided to start the interview with the concept of function. x 2 + y 2 I: Determine whether the equation: not. = 1 i s a function or Kathy: You mean i f I solve i t for x? or i f I solve i t f o r y? It i s a function, you mean i f you solve i t , what would be the function of x or what could be the function of y,? And she then continued: Kathy: Is t h i s a function?? Oh, Oh, I see, isn't that you draw a horizontal l i n e through i t ? ! That's not a function, because you can draw a l i n e through i t and i t intersects both points...or i t i s a v e r t i c a l l i n e ? ! I: You decide. Kathy: There i s another d e f i n i t i o n , i f you draw a horizontal l i n e through i t . There i s a word f o r i t , I don't know, I have forgotten. Her conception of the function was t r i e d to remember that. limited to a l i n e . She only It seemed to me that she was taught the concept 62 of the function in such understanding of vertical, then she the a hurry concept. said: that no When she "Yes, and time was spent on the was that line told i t intersects the the y-axis real was at two understand the points, so i t ' s not a function". It i s a wishful thinking to expect students to concept by only giving them the d e f i n i t i o n of the concept (Vinner, i . e . , before constructing any meaning into the concept. 1983) Students need many concrete examples in order to build up adequate concept images. If the concept image were established by the means of the f a m i l i a r examples and geometric representations, they then, would be ready to understand the d e f i n i t i o n of the concept. Otherwise an abstract d e f i n i t i o n could be forgotten and a l l that might be l e f t i s a vague memory. Kathy expressed t h i s in her words as: Kathy: I don't remember doing those in September, you see, I don't have any feeling for math. Everything that I learn i t ' s kind of goes in there and stays there, sometimes, you see, I got confused with what i s the function already, l i k e with that definition. I know that i t ' s a business of drawing a l i n e . [Emphasis i s mine.] After the tutoring the researcher asked her that: I: Don't you think that i t would are two y values for one function, instead of keeping have to draw a v e r t i c a l l i n e can't have two y values. Kathy: Ya, o.k., that makes sense. learned i t . Next, she was _1 be x, in or better to then you your mind horizontal say that i f there couldn't have a that whether you line. Say one x I agree with that, because I asked to respond on the question of inverse function ( f ) and reciprocal of function ( l / f ( x ) ) . She c e r t a i n l y had heard about the d e f i n i t i o n of inverse function, yet she did not understand i t . Her 63 knowledge of the matter was l i k e more of remembering few f a c t s , without even t r u s t i n g on those memories. Kathy: Isn't that a function and of each other? I: What do you mean by mirror image? Kathy: That i s . . . that the value of y for one of them i s the value of x for the other one and vice versa. And l / f ( x ) i s that you just inverted, i s not the inverse. It i s the inverse, but not inverse of the function. She tried different to things. make her But in i t ' s inverse are the mirror image point that answering the the l/f(x) next _1 difference between Cos (x) = arc Cos(x) and and f"* are two is the could not see question(what sec x ) , she them as two d i f f e r e n t things. Kathy: Sec i s 1/Cos x, arc Cos x?... [With doubt.] Could that be also 1/Cos I: You Kathy: I don't know, I guess so, I think so, because that would be 1 Cos" which i s 1/Cos*1 because i f you have anything to negative one [e.g. ( x ) " ] , I think i t just goes one1 over that 1 x to the one [she meant: ( x ) " = (1/x)]. Cos" would be 1/Cos and sec i s 1/Cos, that i s the same t h i n g . . .There must be some d i f f e r e n c e . Of course. They give you definition l i k e that, and I never ran across i t , but 1 would think, they would be the same thing, although I'm wrong. [Emphasis i s mine.] decide. In a t u t o r i a l session, the notions of 1/f and f " well as t h e i r related concepts. f" 1 was introduced function in action of the composition of two After the t u t o r i a l she in September... confused" said: "Oh, I'm 1 were discussed as as the functions, and inverse of function in action of the m u l t i p l i c a t i o n help me x?! of two inverse of 1/f as functions. learning something...That would I memorized a l l these things, that's why (Emphasis i s mine.). the I'm 64 The questions f(x) = x points 2 + 1. of derivative the sketching Kathy had a hard time to do t h a t . on the axes. without started with She also t r i e d any successions. Kathy to sketch had a peculiar graph of She t r i e d many d i f f e r e n t so many curves way of sketching the curves. Although sketching the graph of the curve was not i n the p a r t i c u l a r interest of the interview, yet i t was revealing in analyzing conceptual d i f f i c u l t i e s in calculus. not f a i r to put a l l the blame on her. Kathy was a bright student. It was She got A in both Honor Algebra 12 and Geometry in high school, and she s t i l l procedure to sketch a simple graph. Kathy's did not have any reasonable She d i d i t by t r i a l and e r r o r . By locating d i f f e r e n t points of the graph and j o i n them together to obtain the desired curve. well. The growing concern successfully 1986). it's She survived escaping i n high facts and i n university as i s that "How did so many students manage t o . . . their teachers' recognition" (Gorodetsky et a l . , She had right to "hate calculus" basic school since she d i d not understand and yet she had to work hard to pass the course. Following passage was the way that Kathy t r i e d to sketch the graph of 2 y = x + 1. Kathy: Would i t centre at (1,1) or at (1,0)? I: You can test i t . Kathy: Oh, t h i s one: test i t . I: Really? Then how do you sketch the graph? you follow? Kathy: I don't r e a l l y follow one...I just make sure that I knew i t before the t e s t , set of rules! [Emphasis i s mine.] (0,1), Oh, Gee! [She meant the curve]. I never knew that you can What procedure do 65 I: If you don't know the procedures, i t ' s hard to keep a l l the shapes in your mind, this function has t h i s graph, the other function has d i f f e r e n t one...you know what I mean? But i f you t r y to understand how to do i t , then i t i s much e a s i e r . Kathy: I have a r u l e , i f y i s on 2 this side [ i . e . the l e f t side of equality sign, e.g. y = x + 1]. i t w i l l open up, I think that's the way i t i s , i f x = y' + 1 then i t opens on the sides, because i t looks l i k e an x, or may be i t i s opposite! Second question asked for the slope of the trouble locating a point on the plane. was secant PQ. She She did not know that any indicated on the plane by i t s coordinates. had point She was wondering how to get the y-component of point Q. Kathy: The Q also l i e s on the parabola. find the two points x and y of Q. Can I use the parabola to She immediately added that: Kathy: I can't find the values. I don't know the between x and y, that comes from the parabola. In a t u t o r i a l session, diagrams were used to show that what would be the y-component of Q. square i t and there was add one To show that the role of f was to take every x, to i t . one y which was doing secant. that. I also concluded that for every point x, the y-component of that x. get the slope of the secant PQ. was She then t r i e d to Nothing s i g n i f i c a n t happened while It took 20 minutes for her She did i t correctly and said: to get the slope of she the " i t ' s l i k e p u l l i n g teeth!" She moved on to get the slope of the tangent l i n e . vague understanding relationship Kathy had a of the concepts of l i m i t and d e r i v a t i v e . Kathy: I wouldn't know how to find the slope of i t , a l l I know...What I heard i s to find the derivative of i t . I: O.K., find the derivative of i t . 66 Kathy: I d i d already, i t i s 2. I: No, I mean by using the d e f i n i t i o n of d e r i v a t i v e , not using a formula. Can you give the d e f i n i t i o n of derivative? Kathy: I can give i t , but I don't know how to relate i t to finding the derivative of i t . I: Can you write i t down, then we can t r y to relate i t . Kathy: No, no, I wouldn't be able to do i t . I: Do you know what i s the d e f i n i t i o n of derivative? Kathy: Yes, isn't t h a t . . . l i k e the slope of the line?...but I don't know, I don't know how to find the slope of the l i n e without using the formula. It was interesting that her concept image of the slope of the secant l i n e , the slope of the tangent l i n e and derivative were t i e d up together although i t seemed that she was not aware of i t . She f i r s t said that she could find the slope of the tangent l i n e by finding d e r i v a t i v e . Later on she said that derivative i s the slope of the tangent l i n e . But she said that she couldn't see any r e l a t i o n between the derivative of a function and the slope of the tangent l i n e . Previously, she said that "I don't know how to find the slope of the l i n e without using the formula". She was asked to give that formula. I: What i s the formula? Kathy: Well, there i s some formula, I can't remember i t , i t had h at the bottom...limit f(h+x Q )-f(x Q ) h I: O.K., you already did i t . Kathy: Oh, that's t r u e , i t looks f a m i l i a r . . . Oh, I remember for exam we had to know how to do that without using chain rule or some rule to finding i t . 67 Her understandings was mostly l i k e disjoint pieces of memories. Sometimes they were easy to r e c a l l and sometimes they were not. I: What happens to h? Kathy: It gets smaller and smaller, i t approaches to i n f i n i t y ? ! I: I n f i n i t y i s when i t ' s getting bigger and bigger. Kathy: So minus i n f i n i t y ? ! [no trust in her words]. Oh no, i t approaches to 0 and that's the formula: l i m i t (x+h)-f(x) w h+0 h Her response to every responded f i r s t single question and thought l a t e r . was amazing. When she was t o l d She usually that h was not approaching to the i n f i n i t y , she just changed i t to minus i n f i n i t y . One might think that t h e o r e t i c a l l y , she was right i f h was greater than zero (h>0) well. was not the only condition and h could be less"than zero (h<0) as But i t was not the case. was - » . She said i t , because the opposite of + » THis type of responses was heard a l l along the interview. It was true that she got 47% i n Math 100, but she was smart and had a good memory indeed. One of the reasons f o r her misunderstanding researcher's view, her naive attitude towards mathematics. convince was, in She t r i e d to her audience that she hated mathematics and she would never be able to understand mathematics. which caused course many her great other Her negative attitude acted as a barrier difficulty factors were intention to discuss a l l of them. in understanding involved indeed mathematics. which there Of was no 68 Gary's P r o f i l e 4.2.9 Gary was interested majoring into either pharmacy or psychology. He had f a i l e d Math 100 twice and he got 86% the t h i r d time. reasons that he had said: "It was class with for his f a i l u r e was the size of the classroom. a big c l a s s , I didn't show up to the c l a s s , now 20 people. I can stop and Among the I'm interrupt the class...so He in a I just stop the c l a s s " . Gary's opinion about the classroom size was expressed by many other students as w e l l . the students' The main objective of this study was to reveal some of conceptual understanding of calculus, yet it was interesting to come across the other understanding barriers that students were dealing with (from t h e i r point of view). Gary l i k e d graph of x 2 + y 2 to start with questions on function. He sketched the = 1 and he then said: 2 Gary: y=± /1-x i s a function. I: Why i s i t a function? function? Gary: I don't know...[laughter]. I: What do you think a function i s ? Gary: Function i s just an equation which she would take some number and put i t through. It's just l i k e . . . I don't know, i t ' s l i k e the button of your c a l c u l a t o r , you put something in there, you punch that button and use another number which for all...which a l l numbers put i n , give you a related output number. I: You used a good analogy. You said that as you punch the button of the c a l c u l a t o r , you'll get something as an output. When you punch the button, you get one outcome, but look at here. For this one, as you said f o r example, you punch this 2 one, then you w i l l get two outcomes: y=+Jl-x and y=-7l-x . You solved i t f o r y, why 2 is i t a 69 Gary: Oh, O.K., I 2know what you mean. I forgot about that v e r t i c a l 2 l i n e ! /1-x or y=-/l-x , two d i f f e r e n t functions put together. He used a good example for function. to It i s not f a r from the truth say that Gary was the only one who dared to argue about his views. Also to think of the non-textbook examples to better understanding of the concepts. His concept image of function helped him to be led to a proper concept d e f i n i t i o n . His He evidently did understand the concept of function. further responses supported t h i s claim. Following i s h i s responses to question #4: I: What i s the difference between the inverse of the function and l / f ( x ) ? Gary: I don't know. I: Don't you want to think about i t ? Gary: It's just whatever f(x) i s , l/f(x) i s one over that and the inverse i s the...I guess what you c a l l opposite of i t ? ! [Emphasis i s mine.] I: What do you mean by that? Gary: Like..., I cannot be sure about inverse. I: You could t r y . Gary: J I: What do you mean? Gary: ...[silent] His Declaring 2 and ( ) ? ! [square root and the square] Can you be more p r e c i s e . immediate responses to many questions were the same in nature. "I don't know" was the very common and widespread response among students, yet i t was irresponsible not a thoughtful one. He needed a push in order to reveal his concept images. to him in the f i r s t place. Everything seemed blurry But he came along very n i c e l y . Many students 70 did not have the habit of thinking about the questions. They usually learned the algorithm of doing the problems, although sometimes there was no understanding behind i t . He had a vague but a promising concept image of the inverse of a function. His concept image allowed him to see the function and i t ' s inverse as opposite of each other. example of J and ( ) 2 His interesting stemmed from his concept images. He had no answer to the question of how one function can be the opposite of the other. I: How are the domain and the range of the function related to the domain and the range of i t ' s inverse function? Gary: I don't know, I forgot. I: Can you think about i t ? Gary: I think i t was...the domain o f . . . [ s i l e n t ] . I: The domain of what? Gary: It's way back, I forgot. I: You said that the domain of...,you wanted to say something. Gary: It's supposed to be switched with some...it was equal to the range of something e l s e , but I forget which. I: We are talking about function and i t ' s inverse and you that [I was interrupted by him] Gary: Aha, I remember. It was, the domain of the function i s the range of the inverse of the function. The unwilling above questionings to express t h e i r self-confidence and Go on. could thinking self-respect prove a point that students was rotated 9 0 ° . other one were because they did not have enough toward themselves. His answers used, to show him how the domain and the range were switched. rote learning of the concept. said were He had a I flipped the page on the space while i t He saw how the domain of one became the range of the (function and i t ' s inverse) since the Y-axis and X-axis were 71 switched. I described that the function and i t ' s inverse were the mirror image of each other with respect to the l i n e Y = X (the text's language). After the t u t o r i a l , he answered the rest of the questions c o r r e c t l y . was He surprised to hear that the invertable function was defined by him as: "there i s one y for every x and one x for every y". Gary's answer made i t clear that his concept image had necessary components in order to lead him to an accurate concept d e f i n i t i o n . barrier was his uncertainty of his understanding. The His concept images were disorganized and he was not aware of them. After f i n i s h i n g on function, he continued with the interesting to see that how many students had d i f f i c u l t y in finding the coordinates of a point. Gary got questions the questions on d e r i v a t i v e . It was "(Q-P/h)" for slope of the secant PQ. He was asked why and he then, x 2 2 corrected himself and got i t right ( ( o + h ) + 1 - (x Q + 1) h = h + 2). I: Now, can you find the slope of the tangent l i n e at t h i s point [P(l,2)]? Gary: f(x) = x + 1, f'(x) = 2x at x = 1 the derivative i s 2. I: What i s the r e l a t i o n between the derivative and t h i s slope? Why d i d you use the derivative to find the slope of the tangent line? Gary: The application of derivative i s that i t defines the slope of the tangent l i n e to curve. 2 I was anxious to find out how he would define the slope of the tangent l i n e as an application of d e r i v a t i v e . help. Probing questions did not He was asked to forget about the derivative for the time being and yet t r y to find the slope of the tangent l i n e using the same procedure as he d i d to find structured. the slope of the secant l i n e . The interview was not So mutual discussion was in favor to c l a r i f y the matter. 72 After the tutoring and explaining the slope of the tangent l i n e as the l i m i t of slopes of the secant l i n e s (as Q gets very close to P), he said: Gary: Ya, but I usually don't think could, but I don't. of that term, I guess you I: How do you think? Gary: I just think as a road and t h i s has sort of a stationary point and I think of car road that goes, in vision car touches that point [P], and when i t gets the point [pointing P] i t stops here and we have a tangent l i n e . He was the only one who t r i e d to make himself to understand using the concrete examples, the tutoring helped him to state the d e f i n i t i o n of the d e r i v a t i v e as: "The slope of a tangent l i n e at a given point, i s the derivative of the function at that point." Gary: He then said: I was aware of t h i s since I took the course three times, but i t ' s much e a s i e r . . . i f you just use i t . It i s much e a s i e r , much quicker. Gary had the same d i f f i c u l t y differentiability conclude that in answering as most of the others had. "Smooth derivative at cosps". curves the question of the The t u t o r i a l helped him to are d i f f e r e n t i a b l e and there i s no At the end of the interview, the interviewer said: I: I am anxious to know i f you think that you f i n a l l y understood the concepts [after taking Math 100 three times] or d i d you find the way to get a good mark [he got 86% in Math 100 after the t h i r d time]. Gary: Ya, more of the l a t t e r , i t was just to find out what to do in each s i t u a t i o n ! I think that was most obvious in 'errorestimate tangent approximations', I didn't l e a r n , I couldn't understand from the l e c t u r e s , I saw the examples, I saw what they d i d and I just did i t that way and i t worked. And he then expressed his ideas about the teaching derivative. of concept of 73 Gary: I think more concrete d e f i n i t i o n of derivative and to that the derivative i s speed i s helping to understand better. I think we shouldn't bother t h i s much with actual derivation or where they come from, and how to them. More the a p p l i c a t i o n . [Emphasis i s mine.] His l a s t comment was about i t deeply. said: And may see it the get important for teachers-researchers to think motivate them to do "In f i r s t two terms, I was something about i t . He r e a l l y f r u s t r a t e d , I didn't know what the function was, I didn't know why they are doing those things [Emphasis i s mine.]". Gary's complaint was as w e l l . of expressed in d i f f e r e n t ways by other students Some of the students were aware of t h e i r d i f f i c u l t i e s and some them were not. But they a l l shared the same pain which was their d i f f i c u l t i e s in understanding the concepts of function and d e r i v a t i v e . to 4.2.10 Barbara's P r o f i l e Barbara was a physics major. express herself and she was She got 75% in Math 100. successful in doing started to answer the questions, she changed f(x) = x and said: "I c a l l i t y. the secant P Q , she said: I l i k e y". 2 it. would be the difference between these two points. liked Before + 1 to y = x In order to get "Y of Q would be equal... She the 2 she + 1 slope of Just wild guessing, many guessings are going on here" (Emphasis i s mine.). She started with the word guessing right t h i s word was used in her interview more than slope of P Q without a major d i f f i c u l t y . from the beginning 15 times. She was She and got the one of the exceptions that got the slope of the tangent l i n e and the d e f i n i t i o n of derivative correctly. concept. Yet she s t i l l had some d i f f i c u l t i e s in understanding of the She was asked to investigate the r e l a t i o n between the curve and 74 its derivative. Her uncertainty could be witnessed by her intention to change her responses from time to time. Barbara: Well, the derivative w i l l give you so l i k e a tangent and on the curve at a point you pick. So the slope of that point as h gets very small. I: What do you mean by the slope of the point? Barbara: The tangent sort l i k e to the curve. I: Does a point have a slope? Barbara: Not r e a l l y , but the slope of the curve at that point. I: Does a curve have slope [What she said was r i g h t . I just asked to see that i f she was trying everything that she knew or i f there was any understanding behind i t ] . Barbara: I guess not. I: Then the slope of what? Barbara: Slope of the function, not what the d e r i v a t i v e . Her answers showed that one of her problems was accurate (not talking about vigorous) language. interview she had no trouble getting the questions tangency. revealed that she had trouble But lack of In the beginning slope of the consequently the slope of the tangent l i n e . the of the secant and l a t e r on, the understanding the an then probing concept of Same questions about tangency y e t , in d i f f e r e n t contexts were posed on her. The intention was to find out whether she could see r e l a t i o n between the curve and the l i n e L tangent to i t . Barbara: How are they [the curve and l i n e L] related?! at one point only. I: What does that mean? Barbara: I don't know what I mean! [Emphasis i s mine.] I: You said that i t touches i t at one c a l l t h i s line? She the said: It touches i t point, then what do we 75 Barbara: In What do we c a l l the line? a tutorial session, I don't know. the graph was sketched to explain the concept of tangent l i n e and tangency. I: Let's start from the beginning. As Q gets closer and closer to P, what happens to secant? You said i t by yourself [And she was r i g h t ] . Barbara: The secant i s disappeared. [emphasis i s mine]. I: What do you mean by disappears? Barbara: Because they are so close at two points. [She d i d not pay attention to the d e f i n i t i o n of secant l i n e ] . I: We said that these secant l i n e s are extended from both d i r e c t i o n s , so when P and Q are getting very c l o s e , i t doesn't mean that the secant PQ disappears, the l i n e i s there. The Q i s just moved towards P, so i t means that i t i s the same l i n e which has rotated around P. It just hits the curve at one point. What we c a l l t h i s l i n e ? Barbara: I c a l l i t the tangent. i t ' s really disappeared! The above questions served the writer's purpose, since she believed that the students' prior knowledge were playing t h e i r further construction of mathematical significant knowledge. roles in In the beginning, she did not think of the tangent l i n e as a concept, but she remembered how to find i t s slope by rote. Later on the probing questions put her into a position that she had to think about what she was saying. she was frustrated and confused. degree, that she even said: Then She did not believe in her words to the "I don't know what I mean". It was hard to understand the concept of derivative without knowing the concepts of tangency, l i m i t , and function. She gave the d e f i n i t i o n of derivative in response to the "slope of the tangent l i n e " , as 76 f'(x) = l i m f(x+h)-f(x) h^O h I: What i s the f ( x ) ? Barbara: It i s d e r i v a t i v e . I: What i s derivative? Barbara: What i s i t ? . . . Do you want me to write the d e f i n i t i o n ? I: No, you gave me the d e f i n i t i o n as: f ( x ) = l i m f(x+h)-f(x) , h-0 h What i s t h i s f'(x)? You said that derivative i s a slope, slope of what? Barbara: Derivative i s the slope curve. I: The derivative of function at what point? Barbara: Any. point on the curve, you just [Emphasis i s mine.] I: You say any. point along the curve. You got the slope of the tangent l i n e at what point? [Emphasis i s mine.] Barbara: At point P there. I: Which you named i t the f ( x ) at what point? at any point [Emphasis i s mine.] Barbara: This would be true f o r any point [Emphasis i s mine.] of the tangent to a point pick i t and plug tangent to the curve. on the curve, In a t u t o r i a l in that w i l l be i . e . , changing the lines session, I changed the point P(l,2) and drew d i f f e r e n t tangent l i n e s to the curve. to it Is i t the f'(x) She d i d not see the derivative as a dynamic process changed by changing the point on the The writer's purpose was show her that the derivative of function at any point was not the slope of line tangent to curve at a fixed point. Geometric representation helped to establish the fact that derivative has d i f f e r e n t numerical values at d i f f e r e n t points. 77 Barbara: But the tangent changes, going around. I: Then what happens to the derivative when the tangent changes? Barbara: Changes with again. I: Is i t true that the derivative at any point i s the slope of the tangent l i n e at a fixed point [emphasis i s mine]. Barbara: That i s the s p e c i f i c of any. which i s the general term, that part of any group [emphasis i s mine], I: Means what? Barbara: The derivative of a function at a point x i s the slope of the tangent l i n e at that point ( x ) . I: This i s correct! It was the shape of the curve... ask me the question interesting questions helped to see that how the t u t o r i a l s her to improve her concept image until vigorous) concept d e f i n i t i o n was reached. and probing the proper (not In the next questions, she got the derivative of f(x) at x=0 by using the d e f i n i t i o n of derivative with no trouble. The question of d i f f e r e n t i a b i l i t y caused the same trouble as f o r most of the other interviewees, the same misunderstandings and the same misconceptions. Her immediate response to the derivative of f(x) at x = -1 (question #7) was: and she wanted "f'(x) = 1". to continue. She was confident in her answer But she was questions concerning the f'(x) at x = -1. stopped to answer a few It was easy to see that f(x) = x + 1 at x = -1 but she did not notice that f(x) = 0 at x = -1 as well. Her attention was drawn to f(x) = 0 at x<-l but she said: Barbara: Her function. Actually i t ' s nothing to plug i n , that's the real t h i n g . answer was quite revealing. She could not imagine that She d i d not know the constant the value of the function was 78 constant at a l l points. One concept image of function. two d i f f e r e n t way of her causing problems was It was the inadequate hard for her to see that there were of writing a function at one point (x = -1). Barbara: Oh boy, what a question! Something strange coming... Could i t be two d i f f e r e n t answers?! I: What do you think? Barbara: I'm not gonna argue with that, i t ' s just there. I: How do you interpret this? Barbara: ...[silent]. In a t u t o r i a l was session, the derivative as a slope of tangent l i n e explained again. question was This was repeated done by sketching d i f f e r e n t graphs. The again. I: What does i t mean when the function has two d i f f e r e n t values for the derivative at one point? Barbara: Then there i s two tangent at that point. The tutorial went on discussing about r e l a t i o n with the slope of the tangent l i n e . the derivative She was and i t ' s showed that there were so many tangent lines passing through the point x = -1. After a l l I asked: -1]? I: Is this function d i f f e r e n t i a b l e at t h i s point [x = Barbara: Yes, something e x i s t s , so much exists...so much e x i s t s . [Emphasis i s hers by changing her tonation.] Barbara did not know that the existence of the derivative means the existence slopes). of limit (even she defined the derivative as the limit of And the existence of the l i m i t requires that the l e f t l i m i t and right l i m i t should be equal, i . e . there should be only one tangent l i n e 79 passing through the point in order f o r function to be d i f f e r e n t i a b l e at that point. I: What i s the condition for a function to be d i f f e r e n t i a b l e ? Barbara: I guess at point x, there should be only one answer f o r the derivative [Emphasis i s mine.] I: What i s that answer for f'(x) at x = -1? Barbara: What i s that answer?!! that point. I: So you are saying that there must be one tangent l i n e at that point, but there i s not one tangent l i n e at that point. There are so many, so i s this function d i f f e r e n t i a b l e at x=-l or not. Barbara: That's weird, doesn't reply that, i t should be one, but this guy has more than one. I: Then what do you conclude? Barbara: Is not differentiable...[no trust in her words]. i s the answer. Her doubt was n a t u r a l . differentiable because We c a l l Her eyes she saw that passing through x = -1. the slope of the tangent at agreed that It i s , here function was not there were so many tangent But numbers were convincing reason lines f o r her to believe that the derivative did exist since she got the numerical value for i t . I: Why i s i t d i f f e r e n t i a b l e and why i s i t not? Barbara: I guess i t i s , because...[silent for few minutes]. Barbara: What does d i f f e r e n t i a b i l i t y mean any way?! I: You just told me! Barbara: Oh, about the tangent business?! though, oh, what comprehensive! We have so many tangents 80 She was one of the exceptions that l e t her concept images to be revealed by themselves. Barbara: I guess i t i s , isn't i t ? Oh, I have to take i t overnight and get back to you. I've never seen anything l i k e t h i s , i t i s good s t u f f , yes or no?!..no I guess. I: Why no? Barbara: I think, i f i t ' s yes, because i t existed, l i k e you calculate it... I: We calculated and we came up with two d i f f e r e n t Barbara: Oh, my l o r d . . . , and i t seems to l i k e a range of slopes of that tangent l i n e . I: What do you mean by the range of slopes? Barbara: O.K. I say no, there tangent l i n e s . The session. If i t ' s no, why not? I f i t ' s yes, why yes? isn't, because answers. there are too many existence of derivative was discussed with her in a t u t o r i a l By showing and proving to her that the right l i m i t l i m i t of f(x) at x = -1 were not equal text at U.B.C). and l e f t (The same discussion in Math 100 She saw that the function was d i f f e r e n t i a b l e a l l along the l i n e f(x) = 0 and a l l along the l i n e f(x) = x + 1 as w e l l , but not at point x = -1 although x = -1 belonged to both l i n e s . and the smoothness of a function were brought The sharp points up by her. She f i n a l l y said: Barbara: Consider at the pick, I d i f f e r e n t i a b l e at x = - 1 ] . guess not [the f ( x ) i s not Barbara concluded t h i s section by her following comment: Barbara: ** This That's neat. I never knew anything l i k e t h a t , nobody told us anything about that. It's pretty neat.** question was very similar to those in math 100 text at U.B.C. 81 Obviously a l l the discussed questions were the same (in context) as those that she had in her Math 100 c l a s s . that how fast students could But i t was interesting to see forget mathematics i f there were not the reasonable understanding of concepts. Once she said: Barbara: Oh, Oh, I remember doing that. That was a long time ago, few months ago, i t ' s ancient history [Emphasis i s mine.] Barbara: We should have done t h i s [interview] at the end of December. There were some kind of memory [interview was conducted at March]. The probing questions and the interviewer's intervention at the appropriate times, allowed her to answer the questions on the sections of function. She said that the c i r c l e was not a function. drew a v e r t i c a l two points. l i n e to the c i r c l e . The l i n e was h i t t i n g the c i r c l e at Her logic was quite r i g h t . answered the question But the point was that she by remembering the v e r t i c a l about the function and i t ' s properties. l i n e not by thinking Her concept image of function was accompanied with the notion of " v e r t i c a l l i n e " . interesting. She said: As a reason she Her l a s t comment was "I got 75% in Math 100, i t ' s good, but that doesn't r e a l l y r e f l e c t like...you know. It was good to see that she was aware of her weaknesses which i s a good s t a r t to understand the meanings and concepts. 4.2.11 Nick's P r o f i l e Nick was in f i r s t year science. first class). honor c l a s s . He d i d well He did not have calculus at high He was thinking computer science. He said that of majoring he l i k e d in Math 100 (he got school, yet he was in in either biochemistry mathematics. or He was a b i t 82 nervous in the beginning things t i l l of the interview. We talked about d i f f e r e n t the ice melted between us and he f e l t comfortable. interview started. The interviewer asked every single interviewee to say the things that they were w r i t i n g . t h i s idea and kept quiet. than t a l k . Then the Some of them d i d not seem to l i k e Nick was one of those who mostly wrote rather At least 1/3 of the tape was f i l l e d with the s i l e n t moments. He d i d not l i k e to be interrupted while he was working on problems. His wish was respected unless both Nick and the interviewer f e l t that the tutoring was necessary. The interviewees' d i f f i c u l t i e s most others finding had trouble finding much a l i k e . the y-component the slope of the secant "run", but he said: were very line, Nick like of the point. For he was looking "Here the trouble comes. f o r " r i s e " and I don't have the value (she meant y - d i f f e r e n c e . ] " . Before he started to find the slope of the secant PQj, he said that he could d i f f e r e n t i a t e the f'(x) to get the slope of the tangent line. But he had d i f f i c u l t y to understand that the slope of the tangent l i n e at a fixed point P was the l i m i t of the slopes of the secants PQ while Q got closer and closer to P. explain that the slope approaches to 0. major d i f f i c u l t y after line approached session, an e f f o r t was made to to i t s l i m i t when the distance h If he understood the concept of l i m i t , he would have no i n finding the slope of the tangent l i n e . the t u t o r i a l tangent In a t u t o r i a l proved without much this claim when he found of a problem. He substituting f o r f(x+h) because he had d i f f i c u l t y only His answer the slope of the had trouble in in understanding the behavior of function. Then the question of the derivative was r a i s e d . I: What i s the r e l a t i o n between t h i s curve and i t ' s derivative? 83 Nick: The derivative i s the slope of the tangent l i n e to a point on the curve, i f i t ' s defined in that point. I: The derivative of function at what point? Can d e f i n i t i o n of derivative from what you s a i d . Nick: The derivative of a function at the point tangent to i t ' s secant l i n e i s the slope of the secant l i n e . His vague answer needed to be clarified. revealed some aspects of his concept images. you Probing give the questions His concept images made him to believe that the value of the derivative of a function w i l l not be changed at various points. Nick: Shouldn't be one mine.] derivative for one He knew that the derivative was he did not pay tangent l i n e s function! the slope of the tangent l i n e , enough attention to the at d i f f e r e n t points on fact that there geometric but were d i f f e r e n t the curve so there were d i f f e r e n t values of derivative associated to each of those points. sessions, the [Emphasis i s representation was used and In two tutorial d i f f e r e n t tangent l i n e s were drawn to explain that the derivative of function had d i f f e r e n t values at d i f f e r e n t points of tangencies. Nick: He again said: Shouldn't be one derivative for t h i s function?! mine.] He did not believe in his words. of derivative as a function and further tutoring and mutual [Emphasis i s He had a mixed up concept images derivative as a numerical value. discussion he at last said After that: "My d e f i n i t i o n would be wrong". His conclusion was helped him a reasonable evidence to prove that the tutoring to develop his concept image toward acquiring of the correct 84 concept d e f i n i t i o n . that the derivative points. definition He then believed i n (not just of the function had d i f f e r e n t He got the derivative of f(x) = x of d e r i v a t i v e . Because 2 accepted) the fact values at d i f f e r e n t + 1 at x = 0 using the of his lack of understanding of function, he again had a hard time to substitute f o r f(x+h). For the question of d i f f e r e n t i a b i l i t y (#7) he was wondering whether to use the d e f i n i t i o n . He f e l t derivative since he understood it. confident to use the d e f i n i t i o n of He then asked i f he could answer the question by using the graph. Nick: Oh, I can do i t from the graph. I: That's why I sketched the graph, you can v i s u a l i z e i t . Nick: At x = 0 i s 0[f'(x) = 0 ] , at x = -1 i s 0 again [f'(x) = 0 ] . I: Look at your function again [see Figure 3.1]. The function i s 0 i f x<-l and here [pointing the function and i t ' s graph] f(x) i s x+1 i f -l<x<0. Nick: So anyway, f(x) i s 0, we substitute x into i t . I: What i s the f'(x) at x = -1. Nick: At (a) f'(x) = 0 (at x = 0 ) , and at (b) i s 1 (at x = -1), [He referred to the parts (a) and (b) of question #7]. I: Then what i s the f'(x) at x = -1. Nick: Should we change i t a l i t t l e function.] I: We don't want to change i t . Nick: I don't know. I: Can you see i t from the curve? Nick: No response. bit?! [He meant changing the We want to keep i t t h i s way. 85 He was stuck. asked f o r t u t o r i n g . Silence was his answer to the question. Later on he In a tutoring session, I explained to him that the derivative of function at x = -1 had two d i f f e r e n t values. I related t h i s to the d e f i n i t i o n of derivative and how the l i m i t f(x+h)-f(x) w h+0 h a s varied i f the approach to the point x = -1 were from l e f t or from r i g h t . I: Then does f(x) have derivative at this point? Nick: I don't think there i s a d e r i v a t i v e . I: Why? Nick: Adams*** said that i t i s a sharp point, because there i s no tangent l i n e . I: Actually you can draw so many tangent l i n e s , because by tangent l i n e you mean secant passing through one point on the curve. Nick: I guess I know i t now. I: What i s the condition for a function to have derivative? Nick: Not be sharp, just one tangent l i n e , Adams s a i d , continuous, smooth interval [Professor Adams i s teaching Math 100 and math 101 at U.B.C. His calculus book i s the text f o r Math 100 and 101 at U.B.C.]. He referred to Professor Adams many times. self-confidence. He d i d not have enough I think that was partly the reason that he referred to Adams as a superior, to put him in-charge of his words. Also to make his audience to believe in his words, because Adams said so! (He referred to him whether he himself was right or wrong). For the questions of function, he said that: definition of function i s . Is i t f o r only one x? correspondence or that doesn't come to i t ? "I forgot what the Is i t one to one I would say i t i s " . Professor Adams i s teaching calculus at U.B.C. 86 He t r i e d to relate his concept image to d e f i n i t i o n of function. He was saying whatever he heard about the function without them. questions did not help and he asked f o r t u t o r i n g . The simple examples were used to explain the concept of the function. Also the Probing understanding textbook d e f i n i t i o n described. and geometric representation of function were His answer to the next question (inverse of the function) proved that he did not only agree with whatever was t o l d in the t u t o r i n g , but he rather understood them. Because he could r e s t r i c t the domain and the range of the c i r c l e in order to have a function. He also said that the semi-circle was not i n v e r t i b l e since there would be two y's f o r one x. 4.2.12 Ted's P r o f i l e Ted was a science major. again. He got 60% in Math 100 but he took i t He said that "I r e a l l y didn't understand i t much, so I took i t again in summer and I got B, about 70%". Ted in his words, was an average Algebra 12). eager at high school (B in He did not know many of the interview questions, yet he was to understand derivative. student them. He chose He did not get stuck t i l l slope of the secant l i n e . to begin with the questions on he was faced with the question of He then said: Ted: The slope i s the... I think, the derivative of the function. I: Without using the d e r i v a t i v e . Ted: O.K., Ay Ax I: What do you mean by Ay and Ax? Ted: Changing of r i s e over the changing of run. 87 I: What i s r i s e and what i s run for PQX? Ted: Rise i s Qx - P, I say i t ' s 2 u n i t s . I: No, Q i s an arbitrary point. What Ted said was rather from 2 (Y(P): point Q). interesting. He measured the distance The y-component of point P) to Y(Q) (The y-component of He d i d not notice that Q, was an arbitrary point. each space as one u n i t . He assumed He measured the y-differences according diagram which was approximately two centimeter to his and said that "I say i t ' s 2 units". The response lack of g e n e r a l i z a b i l i t y was not the only could single be seen incident of in h i s answer. this sort. His This investigator witnessed many other cases that students' responses were the same with t h i s one in nature. He also had d i f f i c u l t y in understanding the coordinates of point and to indicate the point on the plane. He said that " r i s e i s Qi - P" which means, he substituted a point i t s e l f for ycomponent of point. other related In a t u t o r i a l session, the coordinates matters of t h i s problem were of point and discussed. A f t e r the t u t o r i a l , he had no d i f f i c u l t y to find the slope of the tangent l i n e as: 2 2 l i m i t (x 0 +h) +l)-(x 0 +l) h^O h rather w e l l . that t u t o r i a l = l i m i t 2 + h = 2. Ted explained t h i s part h+0 He j u s t i f i e d whatever he did and i t was quite convincing helped him to f u l f i l l his understanding gaps. The next question was to define the d e r i v a t i v e . He said: Ted: Slope of tangent l i n e i s the derivative of function f ( x ) . •I: O.K. Then can you define the derivative of the function? 88 Ted: It's the, the derivative i s the point...no, the derivative i s the...Oh, i s the...I guess another d e f i n i t i o n i s the tangent to the slope of the l i n e at point x. He was and a b i t confused. tangent were r e l a t e d . His attention was and said: He knew that the words d e r i v a t i v e , slope His concept image connected them drawn to what he was saying. together. Ted noticed his mistake "O.K., you got a function f ( x ) , at a point x, the derivative i s going to be the slope of the tangent l i n e that touches the graph of the function f(x) at that point". His concept correct image answer was towards an surprising. adequate The concept He had was definition. session helped him to reach the goal which was derivative by goal to lead The his tutorial the correct d e f i n i t i o n of him. an interesting response to question #7. f'(x) at x = -1 was 0 and 1. He I asked for his reasons. said that the He said that: "Well, I can see i t and i t i s 0 (he pointed the Figure 3.1)". His answer was interesting. He could look at the graph and see that the derivative of f(x) = 0 was 0 at x = -1 (f'(x) = 0) since f(x) = 0 was a l i n e with a slope of zero. Yet he used formula to get the derivative of f(x) = x + 1 at x = -1, as f ( x ) = 1. He got two values f o r derivative of function at the same point (x = -1). know what he should do. correct. different He did not He believed that the both of his answers were Yet he could not decide which one can be representative of the value of the derivative at x = -1. I: How can you interpret this? The function has two d i f f e r e n t derivative at one point, at x = -1 you got f'(x) = 0 and f'(x) = 1. 89 Ted: It's discontinuous at this point...Oh no, forgetting what they taught in calculus book, i s a corner here [Pointing x = -1 in Figure 3.1]. I asked him at the corner. the reason that why His answer was a function was a correct one. not d i f f e r e n t i a b l e That there was no suitable tangent l i n e to the function at x = -1. Interview continued as we moved to the questions on function. gave correct answers to the questions on functions. x 2 + y 2 = values. He 1 i s not a function, "because for one He said that: value of x i s two had a good concept image of inverse function and He showed, geometrically, function to be i n v e r t i b l e . the necessary reciprocal of condition for eager to know, even though the time was to record up and there was He no more tape on. Response Category This section i s divided into two subsections: 4.3.1 Categories of responses to the question of d e r i v a t i v e . 4.3.2 Categories of responses to the concept of function. Each category w i l l be defined quoted whenever i t i s required. questions and the interviewer's t h e i r concept images. all a He asked for more tutoring about the related concepts to the concept of function, (such as one-to-one f u n c t i o n ) . 4.3 y So i t can't be a function". function. was He the students had and the students' responses w i l l It should be mentioned that be probing intervention helped students to improve For example, in response to question #7, difficulties to understand the d i f f e r e n t i a l i t y yet t h e i r l a t e r answers were mostly c o r r e c t . almost concept of 90 Categories of responses t o the question of derivative 4.3.1 Response categories on concept of derivative concepts are discussed in t h i s section. and i t s related Following sections define these categories: Categories of responses to the d e f i n i t i o n of d e r i v a t i v e . Categories of responses to the concept of slope of the tangent 1 ine. Categories of responses to the question of "How the slope of a tangent l i n e and derivative are related." Categories of responses to the question of d i f f e r e n t i a b i l i t y . Categories of responses to the Question of derivative Four main categories of students' responses to the question of derivative have been i d e n t i f i e d . Category I D e f i n i t i o n of derivative - textbook d e f i n i t i o n Two of the students gave the textbook d e f i n i t i o n of derivative as: f'(x)=limit f(x-fh)-f(x), h+0 h T r a d i t i o n a l l y , using t h i s d e f i n i t i o n to get the derivative of a function i s one of the questions on the f i n a l exam at U.B.C. (The copies of f i n a l exams in Math 100 and Math 101 are available from the Department of Mathematics in U.B.C). this definition, derivative. had difficulties understanding Their understanding was rote. a tool to express the d e r i v a t i v e . Those students who gave the concept of For them t h i s d e f i n i t i o n was This concept d e f i n i t i o n d i d not lead them to acquire the proper concept image. 91 Category II Derivative as rate of change - v e l o c i t y Only one of the student derivative. This student referred to "rate of change" to define had a physical interpretation of d e r i v a t i v e . In fact his concept image could lead him to the concept d e f i n i t i o n i f they were developed. Category III Derivative as a "slope" This category consists of those responses in which derivative and "slope" had the same place in the students' minds. One of them defined derivative c o r r e c t l y in terms of slope of the tangent l i n e : "The slope of a tangent l i n e at a given derivative of the function at that point". point, i s the While some others' responses were not accurate, for example: "The derivative i s the slope of tangent l i n e at point P". Two of the students l i n e at a point given". simply said that, "derivative i s slope of the Seven students' responses have f a l l e n into this category. Category IV Derivative as a rule of d i f f e r e n t i a t i o n Students computation. derivative. know how to use derivative One of the students as a gave the formula tool to do their as d e f i n i t i o n of The following quote i s self-explanatory: "No, I don't [defined d e r i v a t i v e ] , I can do i t , 2 f(x) = x + 1, f'(x) = 2x. My teacher did the proof, but I didn't understand i t , but I can do i t " [Emphasis i s mine.] 92 And l a t e r on she said: "in terms of l i m i t ? ! Oh, Ya, we d i d the d e f i n i t i o n of d e r i v a t i v e . . . i t ' s from l i m i t . I don't know... you know, when you do i t , you don't think about t h i s , you do i t so mechanical [Eemphasis i s mine.] Categories of responses to the concept of slope of the tangent l i n e Two main categories were distinguished based on the students' responses to the questions of slope of the tangent l i n e . Category I Slope of the tangent l i n e as derivative of function Almost half of the students used the formula to get the derivative which they then c a l l that the slope of the tangent l i n e . 2 "OK, f(x) = y = x + 1, f'(x) = dy/dx = 2x... There i s only one point, I can't use the same thing here, but I can give you the d e f i n i t i o n of d e r i v a t i v e . This i s giving you a right answer". It i s interesting to know that the above quote belongs to one of the students that gave the textbook d e f i n i t i o n of d e r i v a t i v e . I f he understood that d e f i n i t i o n , he would not say that "I can't use the same thing here". He defined derivative as: "f'(x) = l i m i t f(x+h)-f(x)» but h-0 h he was not sure that why h approached to 0, or what was the role of l i m i t in t h i s formula. Category II Slope of the tangent line as the l i m i t of slopes of other secants Many students had d i f f i c u l t y in understanding the concept tangent l i n e . Only two of them responded c o r r e c t l y . "I guess h approaches to 0, then i f we take the l i m i t f(x+h)-f(x) while h-*0 h of 93 we can get the slope of tangent l i n e " . Three others responded correctly after the t u t o r i a l session. Some of the students had an idea that the slope of the tangent l i n e should be the l i m i t of the slopes of other secants. lack of understanding Their stumbling block was the the concept of l i m i t and tangency. Categories of responses to the question of d i f f e r e n t i a b i l i t y The following three major categories present the students' responses to the question #7 and why f(x) i s not d i f f e r e n t i a b l e . Category I The right l i m i t and l e f t l i m i t are not equal Only one of the students' responses f a l l s into t h i s category. Oh, Ya, you have to do l i m i t or something... as x approaches from one side approaches to 1 and then x approaches from the other side, approaches to 0, [Derivative] doesn't e x i s t , no?! Because approaches to two d i f f e r e n t numbers. The above quote shows that he understood the concept of derivative because a function i s d i f f e r e n t i a b l e i f derivative of function- exists at that point. By d e f i n i t i o n of d e r i v a t i v e , t h i s existence i s equivalent to the existence of l i m i t f(x+h)-f(x) t h+0 h He knew that f o r existence of l i m i t , the right l i m i t and l e f t l i m i t should be equal. Category II There i s more than one tangent l i n e "No, there i s n ' t , because there are too many tangent l i n e s " . "No, it doesn't have derivative at that then?!...because i t has two d i f f e r e n t tangent l i n e s point". point at one 94 The responses of t h i s category are not the same as those in category I, since in here they are talking about the existence of tangent line and category I talks about the existence of d e r i v a t i v e . Yet the right answer obtains from both categories. Category III Function i s not d i f f e r e n t i a b l e at sharp points "It i s a sharp point, because there i s no tangent l i n e " . "Probably, because i t ' s not a smooth curve. together". Two curves chain "It i s a sharp point, because there i s no tangent l i n e " . These responses stem from a practical and concrete understanding of the concept of d i f f e r e n t i a b i l i t y . It i s more convenient f o r students to look at the graph of function to see that whether i t i s smooth or i t i s sharp at some points. 4.3.2 Categories of responses to the questions of function Students' responses to the question of function i s categorized into four major ones. Category I Some elements of the formal d e f i n i t i o n of a function "It's not a function, because f o r one value of x i s two y values, so i t can't be a function". "It's not a function, because f o r every x there i s two y, perpendicular 1ine". "It's not a function, because i f we draw l i n e , there i s two value of y f o r one x". a perpendicular "Yes, and i t intersects the y axes at two points, so i t ' s not a function". 95 2 2 "71-x or y i - x , two d i f f e r e n t functions put together". "Something defines the y values, certain values f o r certain given y's. You are giving value x, and function defines the value of y". The purpose of the interview has been to investigate the student's understanding that these of function by means of an example. students have some proper The responses show concept images of function which could lead to t h e i r developing of an appropriate concept d e f i n i t i o n ; yet the above responses in themselves contain only some of the elements of the concept d e f i n i t i o n of function. Category II Function as a r e l a t i o n between two variables \ This variables. category represents a function as a r e l a t i o n between two In fact function i s a r e l a t i o n between x and y such that f o r each x there i s only one y. The following quotes show that the concept images of these students see a function as a r e l a t i o n without having the r e s t r i c t i o n that for each x there i s only one y. "Yes, sure, because I could draw a c i r c l e , that's a l l " . "Because...they are related?!! a function". Category III I don't know, the equation i s Function as an algebraic term, an equation The concept images of those students whose responses f a l l category, view the function only as an algebraic function. have studied the transcendental such as exponential function). into this Although they functions ( i . e . , non-algebraic function function, trigonometric function and logarithmic 96 "You mean i f I solve i t for x? or i f I solve i f for y? i s a function, you mean i f you solve i t , what would be function of x or what could be the function of y". "Function i s just an equation". Category IV Idiosyncratic responses This category consists of a variety of responses. quote: It the "I know that following i t ' s a business of drawing l i n e " , shows that concept image of function was that whether the l i n e was drawing a v e r t i c a l The line r e l a t i o n i s a function her limited to a 1ine without even remembering vertical or horizontal to the graph. Although i s a good test to check that whether or not ( i f vertical l i n e hits the graph of r e l a t i o n a at two points, i t shows that a r e l a t i o n i s not a f u n c t i o n ) . The following i s an example of another i d i o s y n c r a t i c responses to the question of function: Owen: "No, 2 i t doesn't, y = /1-x , y 1 2 = Ml-x )~ % (-2x) = ) 2 3 (1-x ) * I: Why did you take derivative? Owen: "I don't know, just i n s t i n c t " . "Yes, i t i s a function, because i t has points?!!" a certain number of His response (as he called i t i n s t i n c t ) i s reminiscent of a quote from Kalmykova who was interviewing "When I cannot arrive at subtract, multiply, or answer" (Kalmykova, 1975, the divide p.2). a pupil. answer to the the numbers He said that (the problem, I begin until I obtain pupil) to the add, right 97 The question of inverse function and reciprocal of function caused students great deal of d i f f i c u l t i e s . that f 1 Five of the interviewees thought and 1/f. are the same while four of them said "I don't know". Two of the students d i d not have extra time to f i n i s h t h i s part. one of the students responded c o r r e c t l y . Only Except f o r two students who did not f i n i s h t h i s part, and one student who had trouble with symbols, the rest of the students had no d i f f i c u l t y to answer the questions on composition of function. Summary of Results 4.3.3 Calculus courses are mostly designed in a way to cover many topics in a limited time. In calculus classes there i s not enough time f o r mutual discussion between the instructor and the students. Many concepts are considered to be known by the students p r i o r to t h e i r enrollment in Math 100, such as finding the coordinates of a point. For example, students were given points P(1,2) and an arbitrary point Q on the parabola and were asked to get the slope of PQ. Five out of twelve students whose marks in Math 100 ranged from 60% to 90%, wrote "Q - 2" or "Q - P" f o r the difference in y (the r i s e ) , while finding the difference in x (the run) caused them no d i f f i c u l t y since i t was a matter of a simple tutorial of Q. subtraction: (x + h) - x = h. sessions helped them to understand Probing questions and how to find the y-component Later on a l l of them got the slope of PQ c o r r e c t l y . Another d i f f i c u l t y f o r many students was seeing the tangent l i n e as the l i m i t of other secants and seeing i t s slope as the l i m i t of the slope of other secants. When Q moved along the graph and got closer and closer to p, one of the students s a i d , "The secant has disappeared, i t ' s r e a l l y 98 disappeared". In his discussion tangent l i n e , Orton of students' misconceptions of the (1983) also found that the students sometimes viewed the tangent as the disappearance of the secant. As a f i r s t response, the subjects of this study a l l d i f f e r e n t i a t e d f(x) to get the slope of the tangent l i n e . the same procedure lines. that they They were advised to follow did f o r getting the slope of the secant Only a few of them, then, got the slope of the tangent l i n e as the l i m i t of other secants. In order to get the slope of the l i n e tangent to f(x) = x P(l,2), a l l the students offered to either derivative or d i f f e r e n t i a t e f(x) = x 2 give 2 + 1 at the d e f i n i t i o n of + 1, yet they mostly d i d not know the r e l a t i o n between derivative of function and slope of l i n e tangent to it. Many of the students viewed the derivative as a useful many applications. tool Three out of twelve students d i d not believe that the derivative of a function at d i f f e r e n t points has d i f f e r e n t values. of the students were given the graph of f(x) = x tangent f(x) with 2 Some + 1 which had many l i n e s drawn on i t . They had trouble finding the derivative of = x 2 + 1 at d i f f e r e n t points of tangency. One of them "shouldn't [there] be one derivative f o r one function"?! said: It was hard f o r them to conceptualize that the function of derivative remained the same while i t s numerical values were changed by changing the point of tangency. Students' d i f f i c u l t i e s t h e i r understanding component of Q on in understanding of d e r i v a t i v e . the parabola function, hindered them in They had d i f f i c u l t y (f(x) = x 2 + finding the y- 1) with arbitrary coordinates, since a majority of them did not know the behavior of this function. In the question on d i f f e r e n t i a b i l i t y , (#7), the function 99 caused them more d i f f i c u l t y since i t was not defined under one correspondence (see question #7, Chapter t h r e e ) . rule of A l s o , i t was hard for them to understand the concept of function. The the graph of function (#7) question correctly. understanding those some of the students subjects of had difficulty in the harder. students computation of the derivative had difficulty in doing straightforward (#6). The equation of a c i r c l e (x 2 + y 2 = 1) was were asked to determine that whether or not function. The question. Three of them drew v e r t i c a l not represent who to answer the concepts of tangency and l i m i t , grasping on the concept of d i f f e r e n t i a b i l i t y was None For helped given and the subjects the circle immediate responses of seven students lines and a function since the l i n e represented was "yes" to a the said that c i r c l e does hits the graph at two points. Only two of them explained that since there are more than one y for every x, the c i r c l e i s therefore not a function. After the t u t o r i a l most of the subjects r e s t r i c t e d the domain and the range of x ( c i r c l e with radius 1) in order to have a function. the upper semi-circle as t h e i r desired function. Two same nature were given said: "make i t [ c i r c l e ] a l i n e , that's the only way" "you thought that could as expand long as i t [circle] the length just of like between x and y should be 2 + y conserved. 2 2 = 1 They mostly chose One of the students and the other that" _ . circumference of They circle conserved, they were allowed to make any changes to the graph. not understand the behavior of x + y interesting answers of the said: for this question. sessions, 2 one both was They did = 1 which means that t h i s r e l a t i o n 1 0 0 Later research questions drew students' attention to the d e f i n i t i o n of inverse inverse function. of the They were asked to check that whether or not semi-circle was that the function and of them said that i t s inverse are mirror the domain switched with each other. that function and a function. its and the Those who inverse Although what they said was A number of subjects the function explain Three must be used the term "mirror images", said were "mirror true, using image" of each t h i s term does not students understood the concept of invertable function. to show and said image of each other. range of the t h e i r responses by using other. imply that They were asked the graph of s e m i - c i r c l e . They gave incomplete explanations as to what they meant when they used the term "mirror image". restricted the After the t u t o r i a l and discussion, many of them domain and the range of semi-circle and determined q u a r t e r - c i r c l e to be a function whose inverse i s also a function. they did i t , this writer invertable function as one told them which has that they had now defined to be a one-to-one function. the After the They were very surprised by this r e s u l t . For most of the interviewees, the difference between the inverse of 1 a function f" ) and reciprocal of a function of them could not distinguish between 1/f and dealt with said that f 1/f -1 and f" 1 as 1 i s l i k e x" i f they were the (1/f) was 1 f" . clear. Most Their concept images same. which can be written as not 1/x. One of the students 101 CHAPTER FIVE CONCLUSIONS, EDUCATIONAL IMPLICATION AND RECOMMENDATIONS FOR FURTHER RESEARCH 5.1 Summary of the Study The main objectives of this study has been: A. To identify the nature of the students' conceptual understanding of the concepts of d e r i v a t i v e , function and t h e i r other related concepts. B. To develop a category system f o r those understandings which might provide some insight into the nature of the d i f f i c u l t i e s experienced by students learning these concepts. Twelve f i r s t study. year* university students were interviewed i n t h i s The collected data were analyzed and the individual p r o f i l e s were produced f o r every interviewee. Students' responses to the questions on derivative and function were then categorized to enable the researcher to look at the degree of students' progress in acquiring the proper concept images that may lead them to concept d e f i n i t i o n . 5.2 Method The purpose of present study has been to investigate the nature of students' understanding of the concepts other related concepts. The researcher's aim has been to choose a method of collecting of function, d e r i v a t i v e , and data that enables her to see the students' processes in constructing t h e i r mathematical knowledge. What students d i d in solving the problems discussed during the interview was not the investigator's only concern, but she also was interested in how they d i d i t and why. *Some of them were in second year but they were a l l taking Math 101 at the time that interview was conducted. 102 Incorporating some instruction along with the conventional aspects of a clinical The interview, provided tutorial sessions and the opportunity to address probing questions helped this concern. students to improve t h e i r concept images. 5.3.2 Tutorial Session This students' writer formation calculus. The teacher and effective believes of tutorial provided concept teaching images has sessions, in which the the an particularly some instructions f o r revealing derivative. that f o r the students' concept essential in an role area in like researcher acted as students, seemed to images of function a be and Some of the students preferred to quietly write the answers to the questions, or more often, t h e i r immediate responses questions were simply "I don't know". While some aspects of students' concept images were revealed by some students discussed. appeared to be probing discussed. questions, in other quite confused In such a s i t u a t i o n , the t u t o r i a l enable the students to better understand These sessions were judged to about the specific instances issues being sessions were provided to the mathematical concepts to be useful being i f they helped the students to acquire the adequate concept images. 5.3 Conclusions of the Study A number of tentative They are presented research questions. in two conclusions are different offered in t h i s section. sections in the same order of the 103 5.3.1 The nature of the students' conceptual The following conclusions are summary statements understanding obtained from the data analysis presented in chapter 4. The nature of the students' conceptual understanding of derivative 1) The students had very l i t t l e concept of derivative as was meaningful understanding evident in terms of the types of of the responses given in the interview s e t t i n g . 2) The students, except one, did not have a physical interpretation of d e r i v a t i v e . 3) For some students, the algorithm of d i f f e r e n t i a t i n g the function became the d e f i n i t i o n of d e r i v a t i v e . 4) Some of the students did not believe that the derivative of a function at d i f f e r e n t points has d i f f e r e n t values. In general terms, then, t h e i r concept images were based upon the notion of derivative as a rule which assigns a number to each even though derivative i s a rule which assigns a new function 1 function f" to each function f . The nature of the students' conceptual understanding of the slope of the tangent l i n e 1) Almost half of the students believed that there was between the slope of the tangent l i n e and d e r i v a t i v e . not know that the slope of the tangent line i s the a relation Although most did derivative of a function at point of tangency. 2) Some of the students viewed the tangent as the disappearance the secant l i n e . of 104 The nature of the students' conceptual understanding of d i f f e r e n t i a b i l i t y 1) The students' d i f f i c u l t i e s in understanding differentiability was caused by t h e i r lack of understanding of the concepts of tangency and l i m i t , also by t h e i r lack of knowledge of the properties of a function and, in p a r t i c u l a r , the constant function. 2) Those who said that a function was points and that the smooth function was not d i f f e r e n t i a b l e at sharp d i f f e r e n t i a b l e , appreciated the geometric representation since t h e i r geometric intuition helped them to do so. The nature of the students' conceptual understanding of function 1) A which number of students should lead to the held proper development concept of an images of function appropriate concept definition. 2) two Few variables of the students, understood function as a r e l a t i o n between (without having the r e s t r i c t i o n that f o r each x there i s only one y ) . 3) For some students, a function was only considered to be an algebraic function. 4) For most students there appeared to be a c o n f l i c t between the students' concept images and the concept d e f i n i t i o n s of inverse function -1 ( f ) and reciprocal of function A system for categorizing students' understanding 5.3.2 1) The (1/f). students' understanding of the concepts of derivative and function were varied in nature which enabled the researcher to categorize their responses. The criterion used to categorize the students' 105 understanding concept was their definition. degree These of categories instructional point of view for two (a) they provide understanding students closeness the of appear the to responses to the useful from an be reasons: researcher with some insight into the conceptual d i f f i c u l t i e s experienced by the in the interview setting and hence enhanced the f r u i t f u l n e s s of the t u t o r i a l sessions; (b) they appear to make some i n t u i t i v e sense from a mathematics point of view since they structure student responses from more primitive conceptions to more sophisticated ones. 5.4 Discussion and Implications - S i g n i f i c a n t Issues The concept of limit was quite hard for students to grasp. Furthermore, they need to have an adequate concept image of l i m i t , that i s an i n t u i t i v e understanding, to enable them to understand This writer believes that l i m i t should not be presented definition. The abstract d e f i n i t i o n derivative. by i t s concept in the absence of students' proper concept images w i l l not be of great help to students. Math 100 applications. to is concerned primarily with the derivative and its It enables students to do a l l sorts of d i f f e r e n t i a t i o n and mechanically substitute different values function to get d i f f e r e n t d e r i v a t i v e s . of x in the derivative None of the students f a i l e d to answer question #6 where they had to use the chain-rule to compute the derivative. to applying The question differentiation aimed rules. check The confident working with the d e r i v a t i v e . useful tool to do a l l sorts of things. on the subjects To students' seemed to skill be in quite them, the derivative was a 106 In the Math usually taught 100 c l a s s , the concept in a short period of time. of function and l i m i t are About two weeks of class i s devoted to the teaching of l i m i t , function and a l l t h e i r related concepts (see Appendix C ) . them, students derivative. Function and l i m i t are p i l l a r s of c a l c u l u s . will In the not be able interview to section understand on such Without concepts derivative, the as students experienced great d i f f i c u l t i e s because of t h e i r lack of understanding of function. One of the educational implications of t h i s study is for instructors to spend more time on the teaching of function and l i m i t in calculus c l a s s e s . Furthermore, they should use meaningful teaching the concepts. This study shows that students unable to generalize t h e i r understanding examples f o r are frequently since only few routine examples are used by i n s t r u c t o r s . Instructors usually only lecture and mark the f i n a l exams. Beyond doing the r e p e t i t i v e calculus problems, and getting good marks, students need to be confronted with more challenging and interesting This cannot instructors' be done without direct in the problem-solving involvement process. of In t h i s problems. students and process, the instructors should be focusing on students' d i f f i c u l t i e s and looking f o r t h e i r causes. direct their Being aware of the nature of students' d i f f i c u l t i e s , may attention to the p o s s i b i l i t y of a l t e r i n g their teaching method. It i s important f o r a teacher to r e a l i z e the students' b e l i e f s and how they might affect t h e i r further understanding. (1968) says: prior Ausubel 107 If I had to reduce a l l of educational psychology to just one p r i n c i p l e , I would say t h i s : the most important single factor influencing learning i s what the learner already knows. Ascertain t h i s and teach him accordingly (p. v i ) . The information of students' background and t h e i r p r i o r knowledge of concepts should guide an instructor in choosing the way present those concepts to the students. in high school in a d i f f e r e n t way is hard for definition students of university. to function The formal see and in which to For instance, function i s taught than i t i s presented in u n i v e r s i t y . It a the connection one between that is the high presented to school them d e f i n i t i o n of function i s not easy for in students to understand since for the most part t h e i r concept images of function are not adequate. Concepts are usually presented by t h e i r formal d e f i n i t i o n s . (1983) said that i t i s wishful formed by means of the 295). This expectation that students Vinner thinking for "the concept image [to concept d e f i n i t i o n i s an idealistic and under i t ' s control" be] (P. view since many studies show usually forget the abstract d e f i n i t i o n s and, since they do not have a well established concept image, a l l that remains for them i s the technique and some procedural knowledge so that they can do some computational work. For a majority of students, d e f i n i t i o n s in calculus are to t h e i r i n t u i t i o n . her, the An example of this i s the student who inverse of a function 1 ( f ) and reciprocal (1/f) are the different names for them, i t means that they are contradiction same even though she the will clearly believed that prevent understanding of these concepts. her from unrelated said that, for of a function since there not the obtaining are two same. This any real 108 There are c o n f l i c t s definitions. An example of t h i s i s a student who was asked to inspect 2 that whether or not x + y (x 2 + y 2 between students' concept images and concept 2 = 1 was a function. = 1) i s a function without giving She said that the c i r c l e any reason for i t . The interviewer suggested to her to do the " v e r t i c a l l i n e t e s t " which i s used in calculus classes to check whether a graph represents a function. She f a i l e d to do the t e s t . circle (x 2 + y 2 The researcher did the test to show her that the = 1) i s not a function since the l i n e intersect the c i r c l e at two points, which means there are two y values f o r one x. Her response was quite i n t e r e s t i n g . x >1). Her response She drew the v e r t i c a l l i n e further (with revealed two aspects of her concept image of function: 1. She ignored the fact that the projection of the v e r t i c a l on the x-axis should be in the domain of function. line In other words, f(x) can be defined f o r x in the domain of function. 2. She did not know that t h i s lines not some l i n e s . " l i n e test" must be true f o r a l l In other words f o r every domain there should be one y such that y = f (x). x in the It i s not enough to show that there are some x and there are some y such that y = f ( x ) . Her concept image of function lead her to view a function as: a rule such that f o r some x in domain (or outside of domain) there i s one and only one y in range. One of the stumbling blocks of students of calculus i s t h e i r lack of self-confidence in dealing with mathematics. derivative of f(x) = x 2 A student computed the + 1 (#5) at x = 0 by using the d e f i n i t i o n of 109 derivative. in He did i t correctly but he crossed out his work even though, question #2, he correctly d i f f e r e n t i a t e f(x) = x answer should be 2. 2 used the + 1 at x = 1. definition of He said he was derivative to wrong because the He went through the whole thing again and again (see Appendix D for the copy of his work) without knowing that the value of derivative varies at d i f f e r e n t points. Many students complained about the class s i z e . there is Students little mutual passively take remain unaddressed. students and discussion their notes between and In large classes, instructor their and questions students. and concerns Small classes could provide the opportunity to both teachers to have better communications. An alternative model would be a mixture of large classes and t u t o r i a l sessions. researcher's opinion, the t u t o r i a l In this sessions w i l l be more helpful i f they are offered by the same instructor of the large c l a s s . 5.5 Recommendations for Further Research More research should be done to investigate the nature of students' d i f f i c u l t i e s in c a l c u l u s . The following are recommendations for further study which aim to identify students' understanding 1. None of the students high school. The been controversial few years. give some of this of c a l c u l u s . subject had calculus in existence of calculus in high school, has in B r i t i s h Columbia (B.C.) within the last Many studies should be conducted in t h i s area to insight into the question as calculus should be taught in high school in 2. taken to whether or not B.C. The p o s s i b i l i t y of having a pre-calculus course in high school should be examined. A pre-calculus course could prepare 110 students fewer concepts, therefore instructor has more time to elaborate on questions and f o r calculus. Since i t presents student has more time to understand them by the means of problem-solving. A majority of students had a weak background in geometry which hindered them in their development of concept images. Research into the nature of students' d i f f i c u l t i e s in geometry and also the status of geometry in high school i s a necessity. Research i s needed to investigate the teaching of function in high school. Ill REFERENCES Adams, Robert A. (1983). Single Variable Calculus. Wesley Publishers Limited. New York: Addison- Ash, C a r o l , Ash, Robert, & VanValkenburg, M.E. (1985). A Sensible Approach to Calculus. Selected Papers on the Teaching of Calculus. Aquirre, J . (1981). Students' Preconceptions of Quantities. Unpublished Doctoral D i s s e r t a t i o n . B r i t i s h Columbia, Vancouver. Ausabel, D.P. (1968). 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(1985, Nov.). Calculus as a General Education Selected Papers on the Teaching of Calculus. Kilpatrick, J . (1987). What Constructivism might be in Mathematics Education. In J . Bergeron, N. Herscovics, & C. Kieran (Eds.). Proceeding of the Eleventh International Conference. Psychology of Mathematics. 1, (pp. 3-27). Montreal. K l i n e , Morris. (1972). Mathematical Thought from Times. New York: Oxford University Press. Ancient to Modern K l i n e , Morris. (1985). Mathematics and the Search f o r Knowledge. York: Oxford University Press. Lax, Peter D. (1985, D e c ) . On the Teaching of Calculus. Papers on the Teaching of Calculus. New Selected Lockhead, Jack. (1983). The Mathematical needs of Students in the Physical Sciences. Future of College Mathematics. Edited by Anthony Ralston and Gail S. Young, pp. 55-70. New York: SpringerVerlag. Magoon, A.J. (1985, F a l l ) . Constructivist Approaches in Educational Research. Review of Educational Research. 47(4), pp. 651-693. 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Concept D e f i n i t i o n , Concept Image and the Notion of Function. The Journal of Mathematical Education in Science and Technology. 14: 293-305. Vinner, Shlomo. (1986, A p r i l ) . The Notion of Limit - Some Unavoidable Misconception Stages. Paper Presented at the AERA Conference: San Francisco. Walsh, C. (1988). Calculus in the Secondary School. Vector: Journal of the B r i t i s h Columbia Association of Mathematics Teachers. 29(3), (pp. 25-31). Wheeler, D. (1987). The World of Mathematics: Dream, Myth or Reality? In J . Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceeding of the Eleventh International Conference. Psychology of Mathematics. 1, (pp. 55-66). Montreal. Zorn, Paul. (1985, D e c ) . Computer symbolic manipulation in elementary c a l c u l u s . Selected Papers on the Teaching of Calculus. 1 1 6 APPENDIX B Oral Consent Form (To be read or given to students p r i o r to interview) This project i s not connected to your mathematics class in any way. You are not required to respond to a l l of the questions and i f f o r any reason you feel uncomfortable you can withdraw from the interview at any time without any consequences. 1 1 7 APPENDIX C M A T H 100 — S e p t e m b e r - D e c e m b e r , 198% COURSE OUTLINE Sections Topic 1 DIFFERENTIATION a functions, d o m a i n s , ranges, graphs, compositions, odd and even symmetry, inverses limits ( i n t u i t i v e - t-6 not necessary), properties of l i m i t s , one sided l i m i t s , infinite l i m i t s , l i m i t s at i n f i n i t y continuity, intermediate value theorem (statement) tangents, n o r m a l s , derivatives and differentials differentiation rules i n t e r p r e t a t i o n of the derivative, rate of change, velocity and acceleration, marginals higher order derivatives, differential equations, initial-value problems implicit differentiation the mean-value theorem (omit proof), increasing and decreasing functions antiderivatives and indefinite integrals b c d e f g h i j II ELEMENTARY FUNCTIONS a trig functions and their derivatives, projectiles, simple h a r m o n i c m o t i o n inverse t r i g functions and their derivatives n a t u r a l l o g a r i t h m and exponential functions general exponentials and logarithms, l o g a r i t h m i c differentiation exponential g r o w t h and decay, logistic growth b c d e Hrs 1.2 2 1.3-1.4 2 1.5 2.1-2.2 2.3-2.4 2.5 1 2 3 1 2.6 2.7 2.8 2.9 1.5 1.5 2 1 3.1-3.2 3.3 3.4-3.5 3.6 3.7 4 2 3 1.5 1.5 '""! III a b c d e f 8 APPLICATIONS OF DIFFERENTIATION local and absolute extreme values, c r i t i c a l and singular points, first derivative test concavity and inflections, second derivative test asymptotes, f o r m a l curve sketching optimization problems related rates p r o b l e m s tangent line a p p r o x i m a t i o n , error estimate, Newton's m e t h o d i n d e t e r m i n a t e forms, l'Hopital's rules 4.1 1 4.2 4.3 4.4 4.5 4.6 4.7 1 2 3 2 3 2 43 6 49 Total Tests and leeway A p p r o x i m a t e n u m b e r of class hours O P T I O N A L T O P I C S (if t i m e permits — not examinable) hyperbolic f u n c t i o n s and their inverses parametric curves vector v e l o c i t y and acceleration in the plane 3.8 5.1-5.2 5.3 • T e x t sections refer to SINGLE-VARIABLE CALCULUS, Revised Edition, R. A. A d a m s , Addison-Wesley Canada, 1986. • Calculators, while useful for some parts of the course, w i l l not be required for the final examination, • T e r m marks for the course s h o u l d be based a l l or m o s t l y on in-class tests, and should be scaled w i t h i n sections (when the course is finished) so that the final m a r k s for a section have the same median or mean as final exam marks for that section. MATX 1215 L o c a l 3782 1 1 8 APPENDIX D Written Work From One of the Students' Profi1es A1 - • 4 7- - .L • ^ r.±l \ - rti/ ^ - ** 1 .^ - -. . "n y \ 1 V - X" • •-J3 ' ; •• — - ~ g 1 1 9 APPENDIX E Exemplary Transcript From A Student Interview I: Interviewer Barbara: Subject I: Before you s t a r t , I l i k e you to read this consent form. I: How did you do in Math 100? Barbara: Like marks? but... I: What did you get? Barbara: I got 75% for the term. But that r e a l l y doesn't r e f l e c t l i k e . . . you know, so... I: What about this term? Barbara: This term? I think [Math] 101 i s a b i t harder, because you know, in [Math] 100, you j u s t . . . l i k e chain rule does a chain r u l e , power rule i s power r u l e , you just gonna do i t . That's not so hard, because there i s something s p e c i f i c , but in [Math] 101, there are so many methods, you can integrate some functions that they give you, so i f your mind having remember what to do or manipulate i t , then, oh w e l l , you w i l l make i t . I: What i s your major? Barbara: I don't know. I think I ' l l gonna go to Physics though. I: So you w i l l have lots of Math... Barbara: Ya, I'm taking lots of Math at the same time, because I don't want to go into Biology. I don't l i k e Biology and I don't r e a l l y l i k e Commerce because I'm not r e a l l y into the money, so...Math i s l i k e . . . I don't r e a l l y l i k e i t that much, but You know i s good, Analyst Comments 1 2 0 i t doesn't bother me, but i t ' s more l i k e a t o o l , you know, when you need i t , for...1 ike Physics. I: You didn't have calculus in high school, d i d you? Barbara: Ya, actually I d i d about a month of i t . My teacher wanted to introduce us to these s t u f f . I: Was i t in Algebra 12. Barbara: Ya, just the r i c h Algebra or something, but he only gave us a month and gave us the t e s t . Because we did d i f f e r e n t i a t i n g and integrating and i t was too much f o r us, too much. I have forgotten a l o t of i t . I: O.K., Now, I l i k e to discuss the concept of function and derivative with you. It's up to you to start with which one. Do you want to start with function or derivative? Barbara: [Laughing] What i s the other choices? I: Leave i t and go [Laughing]. Barbara: Is i t [Laughing]?! I: I was just kidding. with d e r i v a t i v e . Barbara: Do you want me to do i t out loud or on paper? I: It's up to you. Barbara: I think I need a paper. do i t mentally. Question 1: Barbara: You may start It's hard to a) Find the2 derivative of f(x) = x + 1 at x = 1 b) Sketch the graph of 2 f(x) = x + 1 I love sketching the graphs! I'm good on that. She started to read the question out loud. 121 I: Do you l i k e to go through a l l of the questions and then come back and discuss each one of them or do you l i k e to f i n i s h one question and then move to the other one. Barbara: When you say discuss i t , do you mean my d i f f i c u l t i e s and... I: No, I just l i k e to see how you are doing i t . Barbara: To see how I: Ya, then i f you have any questions, we can discuss i t . Barbara: The questions that I have trouble with?! I: Ya. Barbara: O.K., you want me to read t h i s [question2 1], O.K., so...I guess, f(x) = x + 1, f'(x) = 2x at x = 1, f ' ( l ) = 2. I l i k e that [laughing]. O.K., sketch the graph of i t . I: Yes. Barbara: O.K., I c a l l i t y, because I like y 2 [She wrote y = f(x) = x + 1] and t h i s i s a parabola [She sketched the graph c o r r e c t l y ] . I: Right on! I do i t ! She moved to question 2: 2) The diagram shows the graph of the above function and a fixed point P on the curve (Parabola). Lines, PQ are drawn from P to point Q's on the Parabola and are extended in both d i r e c t i o n s . Such lines across a Parabola are called secants, and some examples are shown in diagram. a) How many d i f f e r e n t secants could be drawn in addition to the ones already in the diagram? [See Figure 3.1] b) As Q gets closer and closer to P, what happens to the secant? 1 2 2 e) Find the slope of Find the slope of PQ1 PQ2 d) Find the slope of L at point P = (1,2). She f i r s t read the question. Barbara: Lines PQ's? I: Yes, PQj, PQ2, PQ3... Barbara: What do you mean [ l i n e extended from] both directions? I: I mean i t ' s not bounded, i t ' s not l i m i t e d , you know, you can extend as [I was interrupted]. Barbara: Oh, Just a l i n e with arrows. I: It doesn't have two arrows... Barbara: Oh, O.K., I: Yes. Barbara: Such lines across a parabola are c a l l e d secant [reading a part of the question again], O.K., that's true, ya...How many d i f f e r e n t secants could be drawn in addition to the ones already in the diagram? Through point P you mean? not a segment, a l i n e . Yes. Barbara: How many?! I guess...a l o t [Emphasis i s hers], i n f i n i t e numbers I guess, because you can just...you know, angle i t or something. Do you want me to write that down? I: No, i t ' s o.k., because I have i t [I am recording i t ] . Barbara: As Q gets closer and closer to P, what happens to the secant? Oh, 1 2 3 becomes more l i k e a tangent [laughter]. That's what I think any way. O.K., that's c o r r e c t . Barbara: Find the slope of...here to here?! [She showed PQ: on the graph]. Yes. Barbara: O.K., what i s t h i s point [Pointing P ] . This i s point P = (1,2). Barbara: That's (1,2)? O.K., O.K., so this minus t h i s . What i s the height of t h i s [Pointing Q J . The length of this [QJ i s x0 + h. Barbara: This distance? [she showed x Q + h on the graph, but she was not sure]. I: Ya, the whole thing. Barbara: O.K., I just redraw i t . And you want the slope of this [PQJ? The copy of her work i s included, (see Appendix F) Yes. I: Oh, i t ' s that the function? 2 meant f(x) = x + 1 Barbara: Ya, the function i s f(x) = x + 1. The diagram shows the graph of the 2 above function which i s f(x) = x + 1. [The figure was in front of her], I: Barbara: [She 2 O.K., so I guess y at Qj would equal...Just wild guessing now...I guess y of Q;. I have no idea...well, I guess i t w i l l be difference between these two points I guess. Many guessings are going on here [laughter]. That's [slope of PQ,] Ay = +1-2 Ax 1 + h -1 2 What do you mean by Qx + 1? be 1 2 4 2 Barbara: What i s t h i s [Q: + 1] for...?! I: Qj i s the point right? Then what do you mean by + 1? Barbara: Oh, Ya, you're r i g h t , you have to put x in eh...O.K., so that's 2 (1 + h) + 1, y a , that's more l i k e h something that we were doing. That's i t . I: This i s [(1 + h) + 1] height of Q r Barbara: Ya, y a . I: Then what would be t h i s distance [the amount of r i s e ] . See what did you get 2f o r i t [she got (1 + h) + 1). Barbara: Oh, I forgot 2 -2, O.K., so: Ay = (1 + h) + 1 - 2 2 Ax h I also make l o t s of middle e r r o r s . . . [laughter]. Do you want me to finish it? I: Yes, please. Barbara: Slope?...Last term memories are coming back. We take the l i m i t I guess. I: What i s t h i s [I showed her what she got as 2 r i s e = (1 + h) + 1 - 2] run h Barbara: Oh, that's the slope. I: O.K., then... Barbara: That's i t ? ! I: Are you looking f o r something e l s e . Barbara: I don't know! 125 I: Because 2 you told me that this i s [(1 + h + 1 - 2 ] slope. You just h f i n i s h i t up. Barbara: Oh, Oh, I know what are trying me to do, 2 2 l + 2h + h - 1 = 2 + h?! I: O.K., that's f i n e . the slope of PQ2. Barbara: [Laughter] No! I: PQ2 i s another secant. Barbara: Are they a l l about the same?! Usually?! I: What do you mean that they are a l l the same? Barbara: What would be the x part of t h i s part? I: Just another h. xQ plus another h. We can name i t h p h 2 , whatever, just another h. Barbara: Oh, d i f f e r e n t h?! I: We named this distance xQ + h, so when we move down, you know, then t h i s distance w i l l be changed. So t h i s i s not the same h, i t would be another one. Barbara: Ya, so I guess slope of l i n e [PQ 2 ], w i l l be 2 + h r I call i t h r I: O.K., then what w i l l be the difference between 2 + h and 2 + h Barbara: The h part. I: What happened to h part. Barbara: It gets smaller. I: O.K. Now f i n d the slope of L at P(l,2). What would be No! She understood t h i s part rather well. 126 Barbara: Find the slope of L?! O.K... f'(x) = 2x and f ( l ) = 2. I: No, you already got that. Barbara: Oh, Oh, you want me to actually do i t ! O.K., I guess h i s approaching zero, so i t would be one of these things again, so i t would be 2 + h. We take l i m i t as h approaching zero, because i t ' s too close l i k e point [Q] coming down down down and then get 2. [Emphasis i s hers.] I: O.K. Barbara: So I pointed up there. I: O.K., Now, can you give the d e f i n i t i o n of d e r i v a t i v e . Barbara: Oh, O.K., what I do... *f'(x) = f ( x 0 + h) - f ( x ) (x + h) - h It's not x Q , because you said f'(x) Barbara: Ya, O.K., O.K., inconsistency! That was 1 so there i s only x, so f(x + h) - f(x) f'(x) = (x + h) - x f(x + h) - 1 I: Remember what did you do here! Barbara: Oh, Ya, where am I?! I: You just simplify i t . It was 2 (1 + h ) + 1 - 2, you just add them up h together and got that. I: I l i k e you to give me the d e f i n i t i o n of derivative in general. Barbara: Oh, in general, i t i s f ( x + h) - f ( x ) (x + h) - x I guess which i s r e a l l y h at the bottom. *I intentionally did not say that her d e f i n i t i o n of derivative was wrong. I did not want to confuse her by the notion of l i m i t at t h i s stage. 127 I: What i s the difference between whatever you got here and...[I was interrupted]. Barbara: Oh, f'(x) = lim f(x + h) - f ( x ) . h-0 h I: O.K. Now can you t e l l me that what i s the r e l a t i o n between t h i s curve and i t ' s derivative? Barbara: You mean the parabola?! I: Yes. Barbara: And i t ' s derivative!...Well, the derivative w i l l give you so l i k e a tangent and on the curve at a point you pick. So the slope of that point as h gets very small. I: What do you mean by the slope of the point. Barbara: The tangent sort l i k e to the curve. I: You mean slope of a point? Barbara: Yes. I: Does a point have a slope. Barbara: Not r e a l l y , but...the slope of the curve at that point. I: Does a curve have a slope? Barbara: I guess not. I: Then the slope of what? Barbara: Slope of the function, not what the derivative. I: Just look at your graph again. Barbara: Ya. I: You have the idea, you just have to put i t in precise form. The slope of what? Look, this derivative i s equal to the slope of what? [Laughter.] By definition, slope of the curve at point P i s the slope of the tangent l i n e at the point. Her answer was correct. I just asked t h i s question to see that i f she was trying everything that she knew or i f there was any understanding 1 2 8 Barbara: Oh, between Q and P?! points on the curve?! They are two I: It wasn't the d e r i v a t i v e . That was just a secant. How did you here [I was interrupted]. Barbara: Just shrunk i t ! I: Here, how did you get the slope of that. You said you have to take this l i m i t . Barbara: Ya, the l i m i t of h approaches to zero. I: O.K., then what i s L, what's the position of t h i s l i n e to the curve. Barbara: I r e a l l y don't know [laughter]! I: You know! Barbara: I do?! I: Ya, what's the position of this l i n e to this curve [I showed her the tangent l i n e ] . Barbara: I see the l i n e . I: How t h i s l i n e and this curve are related? Barbara: How are they related?! at one point only. I: O.K., Barbara: I don't know what I mean [laughter]! I: You said i t touches i t at one point, then what we c a l l this line? Barbara: What we c a l l the l i n e ? ! don't know! I: Let's go back to this part. You said as Q gets closer and closer to point P, what happens to the secant? Barbara: What happens to the secant?! You think I know?! behind i t . By this l i n e l i n e of reasoning she seemed to get confused. I should be more explicit. The term position might be misleading here. It touches i t what does that mean? Oh...I Tutorial Session 1 2 9 I: You said i t by y o u r s e l f . As Q gets closer and closer to P, then what happens to the secant? Barbara: The secant i s disappeared, does disappear, i t r e a l l y isn't here. I: What do you mean by disappear? Barbara: Because they are so close at two points. I: O.K., we said that these l i n e s are extended from both d i r e c t i o n s . So i t doesn't mean that i t disappears. The secant ya, PQ w i l l be disappeared, but the l i n e i s there. P i s fixed and we are just moving Q towards P, so i t means that i t i s the same l i n e which has rotated around P. It just hits the curve at one point. Barbara: Ya. I: Then what we c a l l this line? Barbara: I c a l l i t the tangent, but... I: O.K., back to here again. You found the slope of the tangent l i n e at this point [P = (1,2)]. You got the slope of L [tangent l i n e ] by taking the l i m i t of a l l these slopes. You said that i t approaches to i t s l i m i t . Then you gave the d e f i n i t i o n of d e r i v a t i v e . Barbara: That! [Pointing f'(x) = l i m f(x + h) - f ( x ) ] . h-0 h I: Ya, how d i d you get this d e f i n i t i o n ? Barbara: How d i d I get the d e f i n i t i o n ? ! I: Yes. Barbara: I just did t h i s , sort o f , only more general term. [She meant f(x + h) - f [ x ) h h 1 3 0 I: O.K., then what i s the r e l a t i o n between t h i s curve and i t ' s derivative? Look at here that what did you get f o r the slope of the tangent l i n e . Do you see any kind of r e l a t i o n . Barbara: I should! I: How did you come to the idea that this l i m i t [lim f ( x + h) - f ( x ) ] i s f ( x ) . h+0 h Barbara: It looks to me...I always c a l l i t the slope sort of that l i n e that P and Q l i k e about almost very c l o s e . I: O.K., then c a l l i t again... I: O.K., you are saying that the derivative...you c a l l that the derivative i t slope of what? Barbara: The slope of the secant which I guess i s almost l i k e a tangent at a point. I: O.K., This derivative wasn't the slope of PQ1} PQ2,...you didn't name the slopes of any of these [secants] as f ' ( x ) . You just got the slope of L [tangent l i n e ] as the l i m i t of a l l those slopes and you called i t f ' ( x ) . Now you can see that what the f'(x) i s . Because we just got i t , then what i s i t ? Barbara: Derivative [laughter]. I: What i s t h i s derivative? Barbara: What i s i t [Emphasis i s hers]?! Do you want the d e f i n i t i o n ? ! I: [Laughter]... No, you already gave the d e f i n i t i o n as f'(x) = lim f(x + h) - f ( x ) ] h-0 h which i s quite r i g h t . Then what i s t h i s derivative? You said that t h i s derivative i s sort of slope. Slope of what? 1 3 1 Barbara: Slope of the tangent?! on the curve! To a point I: O.K. Barbara: [Laughter] O.K., that eased l o t s of pain. I: O.K., this i s the f'(x) at what point? Barbara: Any point along the curve. pick i t and plug i t i n . I: You say any point along the curve. But at what point d i d you get the slope of the tangent line? Barbara: At point P there. I: Which you named i t f'(x) at what point? Is i t the f'(x) at any point? Barbara: This would be true f o r any point. I: You said that f'(x) i s the slope of the tangent l i n e at any point. Let's change the point P = (1,2) [Point of tangency] to P, = (0,1). Then what would be the derivative function at this point? [I sketched the graph]. Yes. You just Tutorial Session of Barbara: Oh! I: You said that at any point. But at t h i s point [P = (0,1)] you got 0 f o r derivative and at P = (1,2) you got 2 for d e r i v a t i v e . x Barbara: Ya, i t ' s the slope of the tangent at the point. But the tangent changes, going around. I: Then what happens to derivative when the tangent changes? Barbara: Changes with the shape of curve?!...[Laughter]...ask me the question again. the 1 3 2 I: O.K., you said that this [ f ( x ) = l i m f(x+h)-f(x)] i s the h-0 h derivative of function at any point. But we said that t h i s derivative at length of any point i s equal to the slope of the tangent l i n e at that point. Barbara: Ya, at that point. I: O.K., what i s that any point and what i s t h i s s p e c i f i c point? How can you relate these two points together? At any point and at that point [Emphasis i s mine]. You said any point and that point. How are these two related? Barbara: That i s the s p e c i f i c of any which i s the general term. That part of that group [Emphasis i s hers]. I: Which means?! Barbara: The derivative of any! I: Can you restate again? The derivative of a function at...what? Barbara: Point x. I: Is equal to the slope of... Barbara: The tangent at point x. I: Now, you got i t . Before you said that "derivative of function at any point i s equal to the slope of the tangent l i n e at that point". I just wanted to be sure that i f these two points are the same point or not... which are the same. Barbara: Ya, I got i t . I: Because for example here [pointing the graph] we got the derivative of function at point p = (1,2) which i s equal to the slope of the tangent l i n e at point p = (1,2). If we change This i s a very common mistake to say that "derivative of function at any point is the slope of the tangent l i n e at that point". "The derivative at the xcomponent of 1 3 3 any point". I did not pick on t h i s since they realized the difference. Just the wording was not accurate. the point, we can see that the value of derivative w i l l be changed when the point of tangency i s changed. Barbara: O.K., you wanted a word [laughter]. I: If we don't put i t in precise form, we don't understand i t . Now you saw that by changing the points of tangencies, we got d i f f e r e n t values for d e r i v a t i v e . Barbara: Oh, what a struggle any way... I: Do you know that where have a l l d i f f e r e n t i a t i o n formulas come from? Like chain rule e t c . Barbara: Ya, that's the whole bunch of these stuffs. I: We derive a l l these formulas from t h i s d e f i n i t i o n of derivative [pointing f'(x) = lim f(x + h) - f ( x ) ] , h-0 h I: Now, compute the derivative of above 2 function [f(x) = x + 1] at x = 0 by using the d e f i n i t i o n of d e r i v a t i v e . Barbara: lim f(x + h) - f(x) = h-0 h lim ((x + h) h-0 2 + 1) h the 2 (x + 1) That's r e a l l y a d e f i n i t i o n ? ! I: Why you doubt i t ? Barbara: I don't know, I have mental elapses right now [laughter]. I: Just look at the curve again and check t h i s d e f i n i t i o n . See i f i t makes sense to you. Barbara: That's the y part and that's the h part [pointing the graph]. I mean the x part. I: O.K. This axis indicates the y values which i s f ( x ) . Then you want to find this [f(x + h) - f ( x ) ] . 1 3 4 2 2 lim (x + 2xh + h + 1) - x h-0 2x + h 2 -1) = Then f'(x) = 2x at x = (0) = 0. We should have done t h i s [interview] at the end of December. There was some kind of memory. Right, you check i t by the graph too. This derivative i s the slope of the tangent l i n e at x = 0 which obviously is... A f l a t angle. And i t s slope i s 0. Ya. Now we can move to next question. Please compute d/dx (1 + 1/77 - 5x) 50 Oh, the derivative of that? Not using that way [she meant using the d e f i n i t i o n of derivative]?! Oh, No, No. Using the formula. =50 (1 + 1/77 - 5x) 49 d/dx (1/77 - 5x) What i s this formula called? The one that I'm using i t ? Ya. Chain r u l e , O.K., Woh, what a struggle to remember that s t u f f . But you are using i t again, in Math 101. Ya, Ya, we are in series now, power s e r i e s , woh! O.K., I have to d i f f e r e n t i a t e this as w e l l , so d/dx (1 + 1/77 - 5x) = (7 - 5x)then d/dx (1 + 1/77 - 5x) 50 = 1/2 " 2/2 1 3 5 49 Thanks. That's f i n e . Now compute the derivative of f(x) = fO x<-l - x+1 -l<x<l -x+1 0<x<l 0 x>l at: a) b) c) d) e) f) x=-2 x=-l x=-l/2 x=l/3 x=l x=10 [See Figure 3.2 f o r the diagram]. Barbara: a) Oh, O.K., O.K., when x = -2, that's where x<-l so f(x) = 0?! Therefore the f'(x) = 0 . at b) x = -1 s t i l l i s the same thing, x<-l again and f(x) = 0. c) I: [I did not l e t her to continue] what about here [I showed her that f(x) = x + 1 at x = -1 as w e l l ] . Barbara: Oh, Jee, two points! i s f'(x) = 1. I: O.K., so what i s the derivative of function at x = -1? Barbara: ...[silent] I: You said 0 and 1 right? Barbara: Actually there i s nothing that's the real thing. I: Barbara: The derivative to plug i n , Why there i s nothing to plug in? The derivative of t h i s i s 1. for any x? Is that I: We want to find the f'(x) at x = -1, but we know that at x = -1, f(x) = x + 1 as well and f'(x) = 1. Then how you can interpret this? Barbara: How we can interpret t h i s ? ! 136 I: Yes. Look at the function again. f(x) = 0 at x = -1, but f(x) = x + 1 at x = -1 as w e l l . Barbara: Oh...what a question! Something strange i s coming. Could i t be two d i f f e r e n t answers?! I: You t e l l Barbara: Because there i s two function f o r . . . I: There i s not two functions... Barbara: One function. I: This function i s defined this Barbara: Ya, i t ' s defined this way. Two d e f i n i t i o n s of x = -1 and i f you derive i t for each of the ways, you define i t , then you get two separate answers. I'm not gonna argue with that, i t ' s just there. I: How do you interpret this? Barbara: How do I interpret t h i s ? ! I: Ya, that the function has two d i f f e r e n t derivatives at one point. Remember we said that the derivative of a function at a given point i s the slope of the tangent l i n e at that point, right? Barbara: Ya. I: me. way. O.K. Barbara: Then there are two tangents at that point! I: Can i t be? Can a function have two tangents at one point. Barbara: In your graph i t looks l i k e i t sort of [see Figure 3.2]. I: Well, you said there are two tangent l i n e s . I'm arguing that there are 1 3 7 more than two you said two but I l i k e to have this one and this one [I drew many tangent l i n e s to the curve such that they a l l passed through x = - 1 ] . Then what i s the derivative of function at that point? Barbara: I guess from 0 to 1. Such a t h i n g . So I guess derivative i s from 0 to 1 l i k e the range goes from 0 to 1. I: What does that mean, derivative? Barbara: I don't know, - I I guess. I: No, I mean you said that derivative at a length of a given point, i s the derivative of function at that point. Barbara: Oh, Oh, What a fun! I: When you say that derivative i s from 0 to 1, you mean that the slope of the tangent l i n e i s between 0 and 1. Barbara: Passing through that point. I: I showed you more tangent through that the tangents 0 and 1. Barbara: I think these may be the highest tangent and lowest tangent, l i k e highest slope and lowest slope. I: Have you remembered something l i k e this? Barbara: What? I'm not used to having seen two answers, because we don't do very many of these things... I: O.K., forget i t for now. Is this function d i f f e r e n t i a b l e at x = -1. Barbara: Yes, something e x i s t s , so much exists [laughter]. What i s the that we can have many lines which they pass point, they're not only with the slopes of Her response was quite interesting. She i s defining derivative as a range of slopes. In advanced mathematics when derivative does not exit, a subderivative is defined as a set of points l i k e an i n t e r v a l . For example f'(x 0 ) i s value of subderivative and in this case i s [0,1]. I f f is differentiable at point Pxn, then f i x . ) = {c}. She t r i e d to complete her definition l a t e r on. I was not aware of this definition, when I conducted the interview. 1 3 8 I: What i s the condition f o r a function to be d i f f e r e n t i a b l e ? In what condition the function i s differentiable? Barbara: Oh...I guess at point x, there should be only one answer f o r the derivative [Emphasis i s hers]. I: What i s that answer for f'(x) at x = -1. Barbara: What i s that answer?! We c a l l the slope of the tangent at that point. I: So you are saying that there must be one tangent l i n e at that point. But there i s not a suitable tangent l i n e at that point. There are so many, so i s t h i s function d i f f e r e n t i a b l e at x = -1 or not. Barbara: That's weird! I l i k e that. Doesn't reply that, i t should be one, but this guy has more than one. I: O.K. then, what do you conclude? Barbara: Is not d i f f e r e n t i a b l e [no trust on her words]?!...Yes i t i s , here i s the answer. Why i t i s [ d i f f e r e n t i a b l e ] and why i t i s not. Barbara: Oh, I guess i t i s , because... What does the d i f f e r e n t i a b i l i t y mean any way [laughter]! I: You just told me [laughter]. Barbara: Oh, about the tangent business?! We have so many tangents though! What a comprehensive! I: O.K. [laughter], you decide [that whether f(x) i s d i f f e r e n t i a b l e at x = -1 or n o t ] . Barbara: I guess i t i s , isn't i t ? ! Oh, I have to take i t overnight and get back to you. I've never seen anything l i k e t h i s . It i s good s t u f f . Yes or No?! No I guess. She l i k e d challenging questions. 1 3 9 I: Then why no? I f i t ' s no, why not. If i t ' s yes, why yes. Barbara: I think, i f i t ' s yes, because...it existed l i k e you calculate i t . I: O.K., but we calculated and we came up with two d i f f e r e n t answers. Barbara: Oh my Lord... and...it seems to be l i k e a range of slopes of that tangent l i n e . I: What do you mean by "the range of slopes"? Barbara: O.K., O.K., I say no, there i s n ' t , because there are too many tangent l i n e s [laughter]. I: O.K., the f(x) i s not d i f f e r e n t i a b l e at x = -1, because when we move from l e f t to right which means from -» to -1, the slope [of the tangent l i n e ] would be 0, right? Barbara: Ya, Ya, r i g h t . I: But when we move from right to l e f t , the slope [of the tangent l i n e ] would be 1, which i s the slope of this l i n e [pointing on the graph] including x = -1. Then t h i s function i s not d i f f e r e n t i a b l e at this point. Because the l i m i t from l e f t i s not equal to the l i m i t from r i g h t . Barbara: Oh, Oh, that... I: Because the derivative i s the l i m i t of t h i s slope. When that distance [ i t was showed on the graph] approaches to 0. But from here, [right l i m i t ] we get two d i f f e r e n t l i m i t s . Which mean two d i f f e r e n t values f o r l i m i t at t h i s point. So the function i s d i f f e r e n t i a b l e a l l along t h i s l i n e [pointing l i n e f(x) = 0] and a l l along that l i n e [pointing l i n e f(x) = x + 1], but function i s not d i f f e r e n t i a b l e at this point. She i s s t i l l thinking about her definition of subderivative. Tutorial Session 1 4 0 Barbara: Ya, because the l e f t and right l i m i t aren't equal. I: O.K. Barbara: I remember that, I had so much trouble with that. Because they gave us 1 ike...I don't know. I: Look at the graph again, x = -1 belongs to both parts. Can you conclude anything from i t that why t h i s function was not different i a b l e at x = -1. Barbara: Because, .didn't you t e l l me?! The l e f t and right l i m i t s are not equal. I: An in a very simple way, those functions are d i f f e r e n t i a b l e that are smooth. Smooth functions are d i f f e r e n t i a b l e . Remember the 2 parabola [f(x) = x + 1] we could draw tangent l i n e at any point on the curve. Consider t h i s one. This i s d i f f e r e n t i a b l e at a l l the points of the tangencies [I sketched two smooth curves]. But consider t h i s [a graph with a pick] Is t h i s function d i f f e r e n t i a b l e at that point [pointing the p i c k ] . Barbara: Oh, at the pick?! I: Yes. Barbara: I guess not, because t h i s side i s l i k e t h i s and t h i s side i s l i k e t h i s . These two are straight l i n e s . [So function i s not d i f f e r e n t i a b l e at the p i c k ] . That's r i g h t . When a graph of a function i s sharp at some points, so function i s not d i f f e r e n t i a b l e at picks. And the same thing here again. For example y = |x| i s not d i f f e r entiable at x - 0. When the function i s smooth, then i t i s d i f f e r e n t i a b l e like lines. Barbara: This i s not continuous. 141 I: No, i t i s continuous but i t ' s sharp at that point [pointing the graph], by continuous we mean that when you start from one point, you...[I was interrupted]. Barbara: Oh, Oh, you don't stop. We didn't stop here [at the pick] but...this i s a sharp point. To be d i f f e r e n t i a b l e , function must be smooth, without a sharpness. And we didn't stop here [at the pick] but...this i s a sharp point. To be d i f f e r e n t i a b l e , function must be smooth, without a sharpness. Barbara: O.K. I: So f a r so good! that. Barbara: That's neat! I never knew something l i k e that. Nobody told us anything about that. That's pretty neat! I: We can skip the other parts of the question [Parts c to f ] . Because for other parts we have the same reasoning and same l o g i c . Because I'm interested on your responses on function and we don't have more than 10 minutes. Barbara: Oh, Oh, I have physics [class] right now. But who cares, I ' l l stay. I: I won't take more than 5-6 minutes of your time. Barbara: "Consider the equation x + y = 1. Sketch the graph of t h i s equation! O.K., t h i s i s a c i r c l e with radius 1 [She sketched the graph]. Is that i t ? ! Any questions on 2 2 Yes, that's f i n e . Barbara: "Determine whether the above equation represents a function y 2 = f(x) or 2 not". Oh...strange! y = 1 - x so y = ± 71 remember. 7. Is i t +?! I can't 1 4 2 Ya. Barbara: Ya, better be! Is i s a function or not...[silent]. I: Can you say i t from the graph! Barbara: No! I: Why? Barbara: Because...for certain point of x there are two y's, y value. In function, f o r a point x, we have to have only one y value. It's not a function. That's r i g h t . Barbara: "Determine..." didn't I answer that [question 2 on functioning]. Yes, you d i d . Barbara: O.K., " I f not, determine a domain and a range such that the above equation i s a function". O.K., y...Just the positive values, y > 0, so r e s t r i c t to only positive y's. I: Then what i s the domain and the range of t h i s function [I meant upper semi-circle]. Barbara: Domain i s . . , •1 to 1 and range i s from 0 to 1. Right Barbara: O.K., "What i s the r e l a t i o n between the domain and the range of a function and i t s inverse and i t s reciprocal [ l / f ( x ) ] " ? I guess you can say either you f l i p the variable around in an equation, f o r example y = x + 1, i t ' s inverse w i l l be x = y + 1. I: O.K., now what's the difference between f(x) and l/f(x)? She started to read the next question, but she stopped. 1 4 3 The difference between t h i s 2 [y = 71 - x which i s upper semic i r c l e ] and l/f(x)? Barbara: The inverse i s when you switch x's for y's and y's f o r x's. But l/f(x) is the r e c i p r o c a l , and that's not necessarily equal, so...what am I going to conclude [laughter]?! I: [I repeated question 4 again]. Barbara: Oh, they are not the same, are they?! I: Look at the graph. This i s the domain and range of your function [D = {-1, 1} and R = {0,1}], now what would be the domain and the range of i t s inverse function? Barbara: Can I redraw i t ? [She c o r r e c t l y did i t . ] So was not a function any more i f we invert i t . I: Good. Barbara: O.K., O.K., then no domain! no range! I: O.K., how we can r e s t r i c t the domain or the range of the function 2 [y = y i - x ] in order to have an invertable function. Barbara: Say i t again. I: You said that i f we f l i p i t over, i t i s not function any more. Now what can we do to the domain and the range of the semi-circle 2 [y = y i - x ] to make i t an invertable function. Barbara: O.K., Let's cut o f f another quarter off and then you just have a quarter of a c i r c l e . Then we f l i p i t over, we get one x, one y. So I r e s t r i c t the domain to just positive x's or just negative x's. 1 4 4 I: Now you just defined the invertable function. It i s a function that for every x [I was interrupted by h e r ] . Barbara: There i s only one y and... I: Remember you r e s t r i c t e d again. Barbara: Then...you mean a function i s inverse, better work out both ways?! I: You said that i f we convert the semi-circle, i t won't be a function because then for every x we could have two d i f f e r e n t y's and you r e s t r i c t e d to quarter of the c i r c l e . Then i t means that before you cut o f f semi-circle you had two x's for one y. But after you cut o f f one quarter of i t , then we have what? Barbara: One x for one I: O.K., and one y for one x [She got t h i s from the beginning]. We c a l l t h i s [pointing quarter of c i r c l e ] one-to-one function such that there i s one y for every x and there i s one x for every y. I: Now what i s the domain and range of inverse function. Barbara: Oh, [0,1] and [0,1]. I: Let's see what i s i t in general. You said that x values go for y values and y values go to x values. Then what would be the domain and the range of inverse function with respect to function. If domain of f(x) i s D and range of f(x) i s R, then what would be the domain and range of inverse function? Barbara: y. [laughter]...backwards! I: What do you mean by backwards? Barbara: Oh, switch them around. I: O.K., which means... 1 4 5 Barbara: Ohm...What does that mean?! I'm not very good at words [laughter]. I: You said backwards. What do you mean by backwards? Barbara: I guess domain and range... D[domain] becomes R[range] and Range becomes Domain. I: That's f a i r enough. What about l/f(x) i . e . the relation between the D and R of f(x) and l / f ( x ) ? Barbara: 1/D, 1/R?!...D and R?! I: Do you have any r e s t r i c t i o n f o r i t ? Barbara: The bottom [f(x)] better not be 0. I: Better not be or should not be 0. Barbara: The f(x) should not be 0. I: O.K., that's f i n e . Now compute one of the following please [composing of functions]. Barbara: "If f(x) = /x and g(x) = x + 1, compute one of the following: a) fog(x), b) gof(x), c) gog(x), d) f o f ( x ) " . Oh I remember doing that. That was a long time ago, few months or so. I ' l l t r y one...woh . . . i t ' s ancient history now. O.K., a) f(g(x)) = !x~ + 1. I: Thank you very much f o r your time. Barbara: You're welcome. 1 4 6 APPENDIX F Written Work from the Exemplary Student Interview f(^V-y* H 3) '~a*'*-. T5T -2X AT / / / )r r - 1 J V \ 'i \ V \ v <\* — — W • SC. 1 1. r v 1;-z ^ 1 \*\\ o \_SN-. /A _ t . < — —) A- _ 1. a / W , *«,».! it*. W / 1 4 7 1 4 8 149 1 5 0 SPECIAL COLLECTIONS PHOTOCOPY REQUEST •. Fill in one form per item. ',* Please pay when ordering. * Photocopies will not be held after 3 months. • * Photocopying will be done by staff only. NAME TELEPHONE DATE CALL NO. 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Students' conceptual understanding of calculus Gooya, Zahra 1988
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Title | Students' conceptual understanding of calculus |
Creator |
Gooya, Zahra |
Publisher | University of British Columbia |
Date Issued | 1988 |
Description | The purpose of this study was to identify the nature of students' conceptual understanding of two concepts of calculus namely, derivative and function. As a way of collecting data two methods were employed: (a) modification of Piagetean clinical interview; and, (b) tutorial sessions. Whenever the students seemed to be confused about the issues being discussed, the researcher provided instructions through the tutorial sessions. The analysis of data was done by developing individual profiles and by response categories. It was found that the interview methodology was effective in revealing some aspects of students' concept images. The students were found to have little meaningful understanding of derivative. A number of students held proper concept images of function which should lead to the development of an appropriate concept definition. It was also evident from the study that students had adequate skill in using algorithm to solve problems. The results of the study would be useful to the instructors of calculus. It was suggested that introducing a concept by its formal definition would contribute to students' confusions and difficulties. Yet if a concept is presented by means of meaningful examples, students had better opportunity to develop their concept images. Thus leading them to form concept definitions. The researcher strongly recommended that more challenging exercises be posed to the students in problem-solving situations. |
Subject |
Calculus University of British Columbia -- Students Mathematics -- Study and teaching (Higher) -- British Columbia -- Vancouver |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0097722 |
URI | http://hdl.handle.net/2429/28056 |
Degree |
Master of Arts - MA |
Program |
Mathematics Education |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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