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 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 c l i n i c a l 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 l i t t l e 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 s k i l l 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 i t s formal definition would contribute to students' confusions and d i f f i c u l t i e s . Yet i f 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. i i i T A B L E OF CONTENTS Page Abstract i i Table of Contents i i i List of Figures v i i Acknowledgement vi i i CHAPTER ONE 1.1 Background of the Study 1 1.2 Definition of Terms 2 1.3 General Statement of the Problem 4 1.4 Specific Statement of the Problem 4 1.5 Overview of Methods of Study 5 1.6 Limitation of the Study 6 1.7 Justification of the Study 7 CHAPTER TWO 2.1 Introduction 9 2.2 Literature in the Broad Problem Area 9 2.2.1 Related causes to students' failure 9 The abil i t y of instructors to teach calculus 10 Rote versus meaningful learning in calculus 10 Students' high school background 11 Instructors awareness of students' backgrounds 11 Textbooks 12 Students' conceptual abil i t y 12 2.2.2 Suggestions to overcome students' failure 13 2.3 The Specific Problem Area 15 2.4 Vinner's Conceptual Model 17 2.5 A Brief Summary of Constructivism 18 2.5.1 Constructivist Perspective of Knowledge i 19 2.5.2 Constructivism and Mathematics 21 2.5.3 Teaching and Learning from a Constructivist Perspective 23 i v Page CHAPTER THREE 3.1 The Rationale for Using the Clinical Interview . 25 3.2 The Subjects 27 3.3 The Interview Procedure 28 3.4 Method of Analysis 28 3.5 Specific Interview Questions 29 CHAPTER FOUR 4.1 Introduction 32 4.2 Individual Students' Profiles 33 4.2.1 Jenifer's Profile 33 4.2.2 Richard's Profile 37 4.2.3 Brian's Profile... 41 4.2.4 Magan's Profile 46 4.2.5 Jason's Profile 48 4.2.6 Joe's Profile 51 4.2.7 Owen's Profile 54 4.2.8 Kathy's Profile 61 4.2.9 Gary's Profile 68 4.2.10 Barbara's Profile 73 4.2.11 Nick's Profile 81 4.2.12 Ted's Profile 86 4.3 Response Category 89 4.3.1 Categories of responses to the questions of derivative 90 Categories of responses to the definition of derivative 90 Category I Definition of derivative-textbook definition 90 Category II Derivative as rate of change-velocity Category III Derivative as a "slope"... 91 Category IV Derivative as a rule of differentiation 91 Categories of responses to the concept of slope of the tangent line 92 Category I Slope of the tangent line as derivative of function 92 Category II Slope of the tangent line as the limit of slopes of other secants 92 Categories of responses to the question of differentiability 93 V Page Category I The right limit and l e f t 1 imit are not equal 93 Category II There is more than one tangent line 93 Category III Function is not differentiable at sharp points 94 4.3.2 Categories of responses to the questions of function 94 Category I Some elements of the formal definition of a function 94 Category II Function as a relation between two variables 95 Category III Function as an algebraic term an equation 95 Category IV Idiosyncratic responses 96 4.3.3 Summary of Results 97 CHAPTER FIVE 5.1 Summary of the Study 101 5.2 Method 101 5.2.1 Tutorial Session 102 5.3 Conclusions of the Study 102 5.3.1 The nature of the students' conceptual understanding 103 The nature of the students' conceptual understanding of derivative 103 The nature of the students' conceptual understanding of the slope of the tangent line 103 The nature of the students' conceptual understanding of differentiability 104 The nature of the students' conceptual understanding of function 104 5.3.2 A system for categorizing students' understanding 104 5.4 Discussion and Implications - Significant Issues 105 5.5 Recommendations for Further Research 109 REFERENCES I l l APPENDIX A 115 APPENDIX B 116 APPENDIX C 117 v i Page APPENDIX D 118 APPENDIX E 119 APPENDIX F 146 v i i LIST OF FIGURES Page Figure 3.1 The graph of Parabola 30 Figure 3.2 The graph of a non-smooth function 31 v i i i ACKNOWLEDGEMENT I wish to express my sincere gratefulness to the chairman of my thesis committee, Dr. David Robitaille for his advice and encouragement which he has offered throughout the duration of this study. I am especially indebted to Dr. Gaalen Erickson for his patient understanding and continued help and assistance during the conducting of this study and writing of this thesis. I would like to express my thanks to Dr. James Sh r r i l l for agreeing to serve in my committee. Also, I would like to thank those students who were involved in this study. It is true that without their participation, the conducting of the present study would not have been possible. Finally, I am grateful to my mother for her support and encouragement during my formative years and thereafter. I also am sincerely indebted to my husband, Bijan, for his everlasting patience, maintenance and understanding, and to my children Sahar and Ali for their cooperation, kindness and tolerance during my years of study. 1 CHAPTER ONE 1.1 Background of the Study "The teaching of calculus is in a state of disarray and near c r i s i s at most American Colleges and Universities. The principle evidence being a failure rate of about 50%" (Rodi, 1985, p. 1). The calculus course has the greatest drop-out rate at the University of British Columbia (U.B.C.) among f i r s t year students and yet is required for the greatest number of students. "At U.B.C, only about 55% of students who enroll in f i r s t year mathematics successfully complete both Mathematics 100 and 101" (Walsh, p. 30). There has lately been serious discussions about the problems relating to the teaching of calculus courses to try and determine some of the factors which may have led to this failure rate. Many researchers have tried to identify 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 are known, then suggestions may be made to reduce the failure rate. Among the reasons given are: inexperienced instructors (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 (Epp, 1985; Orton, 1980, 1983; Vinner, 1982, 1983, 1986). The literature regarding these possible causes will be reviewed in Chapter 2. Some suggestions for improvement have also been given by various researchers. Many authors proposed that different calculus courses can be developed to meet the special students' needs (Stein, 1985; Epp, 1985; 2 Rodi, 1985). Other authors discussed teaching methods where: key concepts are the focus of instruction (Lax, 1985; Davis, 1985), an intuitive approach is used to introduce calculus (Renz, 1985; Ash, 1985), a variety of open-ended problems are presented to students (Stein, 1985), application of mathematical concepts are the focus of instruction (Rodi, 1985; Lax, 1985), and the emphasis is on the development of conceptual not just purely mechanical understanding (Epp, 1985). Since calculus involves a broad understanding of interrelated concepts, rote memorization which leads to a type of mechanical performance in solving 'type' problems does not lend i t s e l f to understanding the nature of the relationships between these concepts. The writer's position is that students experience failure 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' failure. Perhaps the most important one is the students' understanding of the concepts. Vinner (1981,1983,1986) has examined some of these d i f f i c u l t i e s using a model for cognitive processes that involves the notions of "concept image" and "concept definition". The study will draw upon this model to investigate the nature of students' conceptual d i f f i c u l t i e s of selected concepts of calculus. 1.2 Definition of Terms In the following, some of the terms that are being used in this study will be presented very b r i e f l y . The mathematical definitions, are based on the Adam's Calculus book which is the text for Math 100 and 101 at U.B.C. 3 Concept Image According to Vinner (1983), concept image is a set of properties associated with the concept together with a mental picture of that concept. Concept Definition Concept definition, as Vinner (1983) explains, is an accurate verbal definition of a concept in a non-circular 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 (1983) indicated that teaching activity helps the researcher to explore the student's construction of mathematical knowledge. Tangent Line (Adams, 1986, p. 52) Suppose that the function f is continuous at x = xQ. If limit f (xp + n) f (xp ) = m exists, then the straight line having slope m h-0 h and passing through the point P=[xQ,f(xQ)], that i s , the line with equation y = f(xQ) + m (x - x„), is tangent to the graph of y=f(x) at P. If limit f(xp + n) " f(xp ) = * (or - w ) , then the vertical straight line h-0 h x = x0 is tangent to y=f(x) at P. If lim f(x0 + n) - f(x0) f a i l s to h-0 h exist in in any way other than by being either » or then the graph of y = f(x) has no tangent line at P. 4 Derivative (Adams, 1986, p. 56) If limit 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 is differentiable at x. In this case the number f'(x) is called the derivative of f at x, and the function f is called the derivative of f . Function (Adams, 1986, p. 14) A function f is 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) called the value of f at x. 1.3 General Statement of the Problem From a constructivist view, what a student "knows" is dependent in some important ways upon how he has come to know i t (Erickson, 1987). It is very important for a teacher to know the student's primitive conceptions, so as to better understand what errors and misunderstandings may follow, how these conceptions may change into wider and more sophisticated ones, through which situations, which explanations, which steps to pursue in the instructional setting (Vergnaud, 1982). The aim of this study was to reveal some of the d i f f i c u l t i e s that students were struggling with to understand the concepts of derivative and function. In so doing, this study also hopes to make a contribution to the teaching of the calculus. 1.4 Specific Statements of the Problem The purpose of this 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. Problems related to the concepts of "function" and "derivative" were discussed in interviews with students in order to address the following research questions: What is the nature of the students' conceptual understanding of the derivative, slope of a tangent l i n e , how derivative and slope are related, 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 for 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 study was based on individual interviews with volunteer f i r s t year students at U.B.C. who were taking Math 100 during the f i r s t term. The interviews were tape-recorded. The transcripts of the interviews along with the students' written work were used for 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 questions that will be presented in Chapter 3. The questions were aimed to explore the students' d i f f i c u l t i e s in understanding the concepts of derivative and function. A st 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 for their responses and to express their ideas 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 the analyzing procedures. Sixteen Students were interviewed. However, only the last 12 students' interviews' were used in the analysis. As a f i r s t step of data reduction, individual 6 profiles were made for every student and then the profiles were summarized according to defined categories of responses. The f i r s t four students interviewed provided the data for a pilot study. The pilot study gave the investigator the opportunity to revise the questions and to improve her interviewing technique as well. 1.7 Limitations of the Study The subjects being interviewed were a l l volunteers. They were solicited by public announcements in calculus classes and tutorial sections. Therefore the sample was not a random sample. All the subjects were f i r s t year U.B.C. students. The study was a one shot 60 to 80 minute interview combined with the tutorial sessions. Therefore, there was not enough time for some of the students to reflect on their actions and this might be a potential shortcoming of the interview. The students' weaknesses in algebra, their lack of ab i l i t y to perform the even simple calculations, etc. were not taken into account in the analysis. This restriction was due to the purpose of the study which focused upon an examination of the students' conceptual d i f f i c u l t i e s in understanding the concepts of derivative and function. Also the investigator acknowledges that she is constructing the meaning of the interviewees' verbal and written responses. As Jones (1985) indicates, "different people with different perspectives and different curiosities about the area of investigation will inevitably find different categories with which to structure and make sense of the data" (p. 59). Every investigator could use different methods for analyzing the same data and might arrive at different findings. The 7 investigator tried to avoid systematic bias although she believes that everyone views the world through one's own perspective. 1.6 Justification of the Study Two concepts of calculus, namely derivative and function were chosen for this study because not only they are very basic and fundamental in learning calculus and later on in more advanced mathematics, but also these two concepts are quite crucial for almost anybody who needs any level of mathematics in higher education such as engineering, economics, etc. Most college and university students have conceptual d i f f i c u l t i e s in understanding the concepts of calculus (Lochhead, 1983; Davis, 1985; Epp, 1985; Zorn, 1985), yet very l i t t l e research has been done into this important issue. "In a series of papers (Vinner and Hershkowitz, 1980; Vinner, 1983; Tall and Vinner, 1981; Dreyfus and Vinner, 1982) i t was claimed that for some mathematical notions there are conflicts between the concept definition and the concept image. Namely, particular individuals develop concept images which are inconsistent with the mathematical concept definitions. These concept images are quite common and widespread" (Vinner, 1986, p. 2). He then poses the following question: "Is this phenomena a result of vbad pedagogy' or is i t inherent to the concept? In other words: Is there a way to teach these concepts, so that such images will not be formed or these images are unavoidable and they will be formed no matter how the concept is taught?" (Vinner, 1986, p. 2). 8 The present study has addressed this question for the notion of function and derivative in terms of a constructivist view of teacher-researcher, as a model builder. 9 CHAPTER TWO REVIEW OF LITERATURE 2.1 Introduction The four major categories into which the literature in this chapter will 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 specific problem area of students' d i f f i c u l t i e s in learning the concepts of calculus. 2.4 Vinner's conceptual model. 2.5 A brief summary on constructivism as a theoretical framework of this study. 2.2 Literature in the broad problem area This section is divided into two subsections: 2.2.1 Related causes to student fai l u r e . 2.2.2 Suggestions to overcome students' fa i l u r e . 2.2.1 Related causes to students' failure In December 1985, a conference was held on the teaching of calculus in high school and college. Many eminent instructors of calculus and among them, a few researchers, spoke up about the status of calculus in high school and college. The conference proposal stated that "the teaching of calculus is in a state of disarray and near c r i s i s . . . [with a] failure rate of nearly half at many colleges and universities" (Stein, 10 1985). Many researchers cr i t i c i z e d the way that calculus was taught in universities and colleges. They suggested several factors which are responsible for the current state of teaching in calculus. These factors are: 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 . The a b i l i t y of instructors to teach calculus Maures (1985) complains that calculus is taught by anyone including the least energetic member of the mathematics department or even by a graduate student. Lax (1985) states that calculus is a big enterprise and is taught to a large number of students with diverse needs and backgrounds; and yet i t is taught by anyone who is available in a mathematics department. David (1985) has also the same idea about unqualified instructors. 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 applications. This way of presenting calculus does not provide an opportunity for students to understand the concepts but i t focuses on learning how to use the techniques and formulas. He continues to say that "...The d i f f i c u l t y comes mainly from the rapid pace of moving through the material, and from 11 an attempt to cover a large number of details without much focus on main key ideas" (p. 10, emphasis his). Epp (1985) proposes a question relating to the content of calculus and says: "...If i t comes to a choice, will we settle for superficial knowledge of a lot or deeper understanding of less? Perhaps less is more" (p. 18). Students' high school background Some researchers view the students' poor high school background as a potential cause for failure. Renz (1985) says that many students have been taking less mathematics in high school. Rodi (1985) complains about "a generation of certainly unsophisticated, and probably even i l l i t e r a t e , high school graduates" (p. 2). He says that "These students simply are not intellectually ready for i t . They do not have the s k i l l s in algebra and trigonometry" (p. 4). Instructors awareness of students' backgrounds Epp (1985) talks about an intellectual gulf between mathematics professors and their students. She says that professors reactions vary from, ignoring this gulf and teaching as though the students were mature enough to understand a l l the formal proofs, to those whose foci are primarily on s k i l l s to enable students to perform certain mechanical computations. Ash (1985) claims that while most of the students in calculus courses are in engineering, the calculus is presented by instructors "...as i f the entire audience were planning a career in pure mathematics" (p. 2). She says that because of lack of awareness of students' needs and backgrounds, "...formal mathematical language which 12 was intended to prevent misunderstanding had precisely the opposite effect" (p. 3). Textbooks A great number of researchers talk about poor textbooks which may contribute to students' failure. But i t seems that they do not have appropriate suggestions to solve the textbook problem. Steen (1985) asks "why do calculus books weigh so much" (p. 4), and says that the economics of publishing causes encyclopedic textbooks. Stein (1985) proposes that "Authors have less choice, for i f they omit someone's favorite topic, their books will not be adopted and soon will be out of print" (p. 3). Kenelly (1985), Rodi (1985), Stevenson (1985), and Masurer (1985) have also presented the same idea about the poor quality of textbooks. Students' conceptual a b i l i t y Hadas Rin (1983) has studied students' d i f f i c u l t i e s with calculus by examining their spontaneous written questions. She found that students do not know the definitions and they are not able to apply "known" theorems to new situations. Davis (1985) believes that many students do not understand the proofs and do not have a clear conception of the d i f f i c u l t i e s that must be overcome. Lochhead (1983) wrote that many college students are not able to read or write simple algebraic equations and they also "...seem to lack any well defined notion of variable or of function" (p. 2). Epp (1985) says that the state of most students' conceptual knowledge of mathematics is "abysmal" after they have taken calculus. Zorn (1985) stated that "Instead of learning to 13 create, verify, and analyze algorithms, calculus students learn mainly how to perform them" (p. 3). 2.2.2 Suggestions to overcome students' failure Many researchers have tried to not only talk about the causes of students' f a i l u r e , but also have tried to give productive suggestions in order to deal with students' failure in calculus. Stein (1985) suggests that i f a quarter or semester course of discrete mathematics would be available to f i r s t year university students, "Such a course could help develop maturity and thus prepare students for calculus. It could, incidentally, weed out those who are not ready to go on" (p. 4). Epp (1985) is talking about the possibility "...to modify pre-calculus courses to make them include additional work to increase students' logical maturity" (p. 10). Rodi poses the question of: "would i t not be more consistent to make normal expectation for entering students in a thorough pre-calculus course before attempting the heady and adult stuff of calculus?" (p. 5, emphasis h i s ) . Davis (1985) suggests that "key ideas are presented very carefully and thoroughly, so that d i f f i c u l t i e s are clearly perceived and so that students are able to see how these d i f f i c u l t i e s are met and dealt with" (p. 11). Lax (1985) says that ...for a concept, when presented properly, can be absorbed as a whole, while an algorithm remains a sequence of steps. It is only after a concept has been understood, and made part of one's thinking, that we turn to the intriguing task of devising efficient algorithms (p. 6). Renz (1985) says that "calculus has been, i s , and will continue to be a basic computational and conceptual tool for students studying the 14 hard sciences and engineering" (p. 6). Then he suggests that more intuitive foundation and careful imaginative thinking will be needed for introductory calculus, in order to make i t understandable for students. Ash (1985) calls for a change in mathematicians' style 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 reflects the creation of mathematics and not in style that would satisfy a professional mathematician tidying things up years after the fact. It is 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 that students should be able to think on their own and to express their thoughts, and this chance can be given them by providing open-ended problems with variations and introducing ".. .-.problem-sol ving' courses to compensate for 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 expose and to reinforce intuitive conceptual understanding" (p. 8). Then he warns that i f research mathematicians teach calculus, they may pay less attention to application and pay more attention to more abstract generalization. Lax (1985) seems to have different ideas from Rodi. He says that the teaching of calculus should be entrusted to those who actively use i t in their own research. Although he sees the same role for the use of application in calculus when he says: "Teaching of calculus is the natural vehicle for introducing applications, and that applications give 1 5 the proper shape to calculus: They show how, and to what end, calculus is used" (p. 1). Rodi (1985) believes that applications do not give the proper shape to calculus i f they will not enrich students' intuition and will not allow them to see each successive stage of generalization. He says that applications should be carefully selected and expressed by instructors, since applications "...are as powerful a tool in revealing i t ' s message as metaphor is to the poet" (p. 10). Epp states that "...the primary aim of calculus instruction should be development of conceptual as opposed to purely mechanical understanding" (p. 9). But she provides some suggestions to help students and among these suggestions is one that would appear to lead to mechanical understanding: Make your students memorize precisely-worded definitions and perhaps theorem statement also. Memorization is greatly underrated as a Pedagogical tool. At the least, memorization of a definition forces students to read i t carefully, at best, i t encourages them to understand i t (since i t is 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 this suggestion. 2.3 The Specific Problem Area Calculus is not a course for the e l i t e any more. There is a great demand for calculus, and yet so l i t t l e research has been done into the study of students' d i f f i c u l t i e s with calculus. All 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 this large area and many more studies are needed to c l a r i f y the nature of students' d i f f i c u l t i e s in calculus. 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 d i f f i c u l t i e s in these two particular areas. Among those are Orton (1980,1983), Monk (1987), Vinner (1983) and Markovits et a l . (1986). In 1983, Orton conducted a study to investigate the students' understanding of differentiation. He interviewed 110 students aged 16 to 22 years old. He found out that one of the most d i f f i c u l t questions was concerned with the understanding of differentiation. Students of his study generally found that the application of differentiation was relatively easy. Orton believes that students' errors 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 said, "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 their 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 relatively good in answering those questions which did not ask for their 1 7 understanding but rather more of their s k i l l , yet they did very poorly in responding to those questions that asked for their understanding of the concept of function. 2.4 Vinner's Conceptual Model Vinner has examined students' understanding of "Limit" (1986), "Function" (1983), and "Tangent Line" (1982), by using his model of cognitive processes that involves the notions of concept definition and concept image. Concept definition, as he explains (1981), is an accurate verbal definition of a concept in a non-circular way. According to him, concept image is a set of properties associated with the concept together with a mental picture of that concept. Vinner (1975) used an example to illustrate what he means by a mental picture of a concept. He says that i f C is a concept and i f P is a person, then "P's mental image of C will be defined as the set of a l l pictures that have ever been associated with C in P's mind, namely the set of all pictures of objects denoted by C in P's mind" (p. 339). Vinner (1983,1986) claims that the developed concept images by different individuals are inconsistent with the concept definitions that are usually inactive. For every concept, Vinner (1981) presents a model by assuming the existence of two different cells in the cognitive structure, one cell for the concept definition(s), and the other one for the concept image(s). Depending on how a concept is f i r s t introduced to students, different interactions will occur between the two mentioned c e l l s . The interactions vary from the construction of a definition based on one's own experience with the concept which is a description of his concept image to introducing the concept definition prior to the existence of any 1 8 concept images. For Vinner (1982, 1983, 1986) the important task is to recognize the students' concept images and try to reveal them whenever i t is possible. This revealing will help teachers to not only acquire some understanding of their students' mental activities but also try 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 (1986). The concept of Limit was introduced to students in a way that was supposedly to prevent the formation of wrong or inadequate concept images. The students' mathematical a b i l i t y , as Vinner says, were unquestionable. Yet he arrived at a conclusion that the formation of certain concept images are probably unavoidable. He suggests a reconsideration of the way of introducing new mathematical concepts which is usually presented by the means of concept definition. He also found that the students' concept images play a crucial role in construction of their mathematics yet the textbooks are based on the concept definitions. His finding was supported by his study of function (1980) and study of the tangent line (1982). Vinner then suggested that presenting a concept by i t ' s formal definition may not be the best way of doing that. 2.5 A Brief Summary of Constructivism This section is 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 concerns the what and how of knowing (VonGlasersfeld, 1984 cited in Erickson, 1987, emphasis is his). Constructivism is "...a general way of interpreting and making sense of a variety of phenomena. It constitutes a framework within which to address situations of complexity, uniqueness, and uncertainty that Schon (1985) calls 'messes' and to transform them into potentially solvable problems. Thus, like any epistemology, constructivism influences both the question posed and the c r i t e r i a for what counts as an adequate solution" (Cobb; Wood; Yackel, 1988, p. 2). As stated by Vergnaud (1987), Piaget was the most systematic theorist of constructivism in his time. It seems therefore, appropriate to refer to Piaget. The research of Piaget and his inquiry into cognitive development has been the major influence toward a more constructivist psychology (Magoon, 1977). "According to Piaget, the essential way of knowing the real world is not directly through our senses, but f i r s t and foremost through our actions. In this context, action has to be understood in the following way: 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 relation to the world. In other words, i t is behavior that changes the knower-known relationship" (Sinclair, 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 experience (Erickson, 1987; p. 22) which is the only reality that he can know (Kilpatrick, 1987). Sinclair (1987) pointed out that in Piaget's view, "new knowledge is constructed from the changes or transformations the subject introduces in the knower-known relationship" (p. 28). 20 The major assumption of constructivism, as Magoon (1977) states, is that the subjects being studied must be considered as knowing beings and their knowledge has important consequences for how their behavior or actions are interpreted. The constructivist view, as described by Kilpatrick (1987), involves two principles: 1. Knowledge is actively constructed by the cognizing subjects not passively received from the environment. 2. Coming to know is an adaptive process that organizes one's experiential world; i t does not discover an independent, pre-existing world outside the mind of the knower (p. 7). Kilpatrick argues that the second principle separates t r i v i a l constructivism, as VonGlaserfeld calls i t , or simple constructivism, as Davis and Mason call i t , or empiricist-oriented constructivism, as Cobb calls i t , from radical constructivism (1987). He says that radical constructivism rejects metaphysical realism and claims that the search for objective truth should be stopped. Kilpatrick summarizes radical constructivism as: ...an epistemology that makes a l l knowing active and a l l knowledge subjective. Following modern physical sciences in its rejection to the possibility 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 constructivist position have been identified by VonGlasersfeld (1983, in press, cited in Kilpatrick, 1987) as: 2 1 ...(a) teaching (using procedures that aim at generating understanding) becomes sharply distinguished from training (using procedures that aim at repetitive behavior); (b) processes inferred as inside the student's head become more interesting than overt behavior; (c) linguistic communication becomes a process for guiding a student's learning, not a process for transforming knowledge; (d) students' deviations from the teacher's expectations become means for understanding their efforts to understand; and (e) teaching interviews become attempts not only to infer congnitive structures but also to modify them. 2.5.2 Constructivism and Mathematics The fundamental question of whether mathematics is discovered or invented is sometimes viewed as a choice between either the platonist position or the constructivist position. Plato believes that "The concepts of mathematics are independent of experience and have a reality of their own. They are discovered, not invented or fashioned" (Kline, 1972, p. 43). As stated by Thorn, "The mathematical entities exist independently of thought, as Platonic ideas" (1971, p. 696). Hardy (1928, cited in Kline 1985) believes that "...mathematical reality l i e s outside us, that our function is to discover or observe i t , and that the theorems which we prove, and which describe grandiloquently as our 'creations,' are simply our notes of our observations" (p. 205). This means that mathematicians do nothing but to discover the concepts and their 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 is a human creation" (Kline, 1985). Brouwer's idea gave rise to constructivism as a new school of thought in mathematics. His position as described by Davis and Hersh (1980)" was that the natural numbers are given to us by a fundamental 3 22 intuition, which is the starting point for a l l mathematics. He demanded that a l l mathematics should be based constructively on the natural numbers" (p. 334). Bishop says that Brouwer and his followers "were much more successful in their criticism of classical mathematics than in their efforts to replace i t with something better" (p. i x ) . Furthermore, Bishop claims that a satisfactory alternative exists although Brouwer did not convince the others that there is an alternative. Bishop, within a constructive framework, develops a large portion of abstract analysis in order to give numerical meaning to classical abstract analysis since he think that "classical mathematics is deficient in numerical meaning" (p. i x ) . "The constructivists regard as genuine mathematics only what can be obtained by a f i n i t e construction" (Davis & Hersh, 1980, p. 320). Bishop believes that "when a man proves a positive integer to exist, he should show how to find i t " (1967, p. 2). As was stated by Goodman (1983), "The emphasis is not on foundational questions, but on the hard work of finding constructive versions and constructive proofs of actual theorem" (p. 61). In constructivist's view, the purpose of proof is to c l a r i f y the theorem, "to make the theorem obvious, so that the phenomenon is f u l l y revealed, with nothing hidden" (Goodman, 1983, p. 63). From a constructivist's view, mathematical objects as Cobb (1987), says are: "phenomenological correlates of systems of conceptual operations rather than elements of a mind-independent mathematical re a l i t y . Regardless of the interpretation adopted, i t would seem that viable models of learning or problem solving in mathematics must account for the experience of mathematical objects" (pp. 9-10). Kilpatrick is more concerned with the relation between the constructivism and the 23 practice of mathematics and he consequently expresses his concern for school mathematics and what should be taught. For Piaget, the evolution of mathematical structure is towards increasing comprehensiveness and rigor since in his view of constructivism, meaning is cumulative. He then sees the process of moving mathematics toward increasing objectivity. 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 Teaching and Learning From a Constructivist Perspective The 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 activity of modelling children's r e a l i t i e s . The constructivist teacher aims to see through the overt behavior in contrast to the behaviorist teacher that attempts to see in the overt behavior. By seeing through this behavior, the teacher would help students to reconstruct their mathematical learning contexts. Teachers should realize that the problem of teaching will not be solved by using mere definitions; "...students' conceptions can change only i f they conflict with situations they f a i l to handle" (Vergnaud, 1982, p. 33). Learning, which is stimulating and relevant to the student is an indispensable part of the constructivist program (Cobb et a l . 1988). In the constructivists' view, the core of mathematical learning is the problem solving process (Cobb, 1987, also Cobb, 1986; Confrey, 24 1987; Thompson, 1985; VonGlasersfeld, 1983 cited in Cobb et a l . 1988). In this situation, the students try to reach their goals by constructing their solutions to problems which have arised. The focus of constructivism in relation to mathematics education is exclusively on the active processes of construction of mathematical reali t i e s of individual students (Cobb et a l . 1988). The student's process of gaining and constructing knowledge is 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 of knowledge, namely spontaneous knowledge and formal knowledge. The f i r s t is the knowledge that children obtain spontaneously from their surroundings, while the second is the knowledge that children gain through formal schooling. "Learning is always an interaction between the learner's current understanding and the new information gleened" (Pine & West, 1986, p. 587). 25 CHAPTER 3 METHODS This chapter consists of the following sections: 3.1 The Rationale for Using the Clinical Interview 3.2 The Subjects 3.3 The Interview Procedure 3.4 Method of Analysis 3.5 Specific Interview Questions 3.1 The Rationale for Using the Clinical Interview Research into mathematical thinking, as Ginsburg (1981) has claimed, has three basic aims which are: the discovery of cognitive processes; the identification of cognitive processes; and the evaluation of competence. Depending on the research purpose, different c l i n i c a l methods can be used (Ginsburg, 1981). What is important in teaching and learning mathematics is the intellectual process underlying the mathematical knowledge. Since the underlying cognitive processes are numerous and complex, standard tests may be ineffective or at least inefficient (Ginsburg, 1981). This researcher thinks that c l i n i c a l interviews will help to recognize some of the complexities of the students' cognitive processes. The students being studied are "knowing beings" (Magoon, 1981, p. 652). Clinical interviews will help to investigate some of the processes that these knowing beings use when constructing mathematical concepts. "In order to understand why persons act as they do, we need to understand the meaning and significance they give to their actions. The depth [sic] interview is one way, not the 26 only way and often used most appropriately in conjunction with other ways-of doing so" (Jones, 1985, p. 46). This researcher believed that she must also act as a teacher in order to explore the nature of students' understanding. Acting as a teacher gave the researcher the opportunity to not only see what the students do in order to answer questions, but also to understand how and why they did i t (Cobb and Steffe, 1983). The interviewees were asked to do a series of questions related to the concepts of "function" and "derivative". They were asked to describe what they were doing while they answered these questions. The investigator's purpose for conducting these interviews was 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 calculus. Probing questions were used to explore the nature of these d i f f i c u l t i e s . In the tutorial 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 implicitly, by their understanding of their students' mathematical rea l i t i e s as well as by their 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 implicitly, models of their students' mathematical knowledge (p. 85). This investigator had broad questions that the interview was based upon, but the 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. For this reason, a s t r i c t standardized protocol was not used. Jones (1985) suggests that interviewers should not prestructure the direction of inquiry with their own frame of reference in ways that give l i t t l e time 27 and space for their respondents to elaborate. In some cases not more than half of the questions were covered, however, the students had opportunities to express their ideas and even to sometimes express their feelings toward a question. According to Pine et a l . (1978), conduction of a pilot study is a necessity prior to every research study that uses c l i n i c a l interview as i t s data collection methodology (cited in Aguirre, 1981). Four volunteer students were interviewed for a pilot study. This gave the investigator the opportunity to revise the questions and her interviewing technique as well. 3.2 The Subjects Sixteen volunteer students participated in the study. They were al l f i r s t year U.B.C. students and were solicited by public announcement in Math 101 classes and their tutorial sections (see Appendix A). The students were asked to read a consent form prior to interviews (see Appendix B for a copy of the consent form). They also were told that their real names would not be used in the study for the sake of confidentiality. Only twelve of the students' interviews were used for analysis. First year university students were chosen because they had been introduced to the concepts of function and derivative in the f i r s t term of the same year. The sample consisted of four females and eight males. The 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 different mathematical backgrounds. Given the nature of the students being studied along with the small size of the sample, no 28 attempt was made to generalize the findings of this study. More studies of this nature would be needed to provide supporting evidence for the findings of this study. 3.3 The Interview Procedure The data for this 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 office located in the Department of Mathematics at U.B.C. Each interview lasted 60 to 80 minutes (which included the tutorial sessions). All the interviews were tape recorded. There were two sets of questions, one on function and the other on derivative. Students were free to start with any of the two sets that they wanted to. They also were asked to explain their 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 told that they could terminate the interview whenever they liked, although this did not happen. On the contrary, some of them were eager to continue the discussion. So their wishes were granted, yet they were not recorded since there was not more than one 90-minute casette tape available for every interview. At last i t is worth mentioning that students being interviewed were extremely cooperative. 3 .4 Method of Analysis The investigator was looking for patterns in the data. The data were qualitative and provided by transcripts of the most relevant parts of the interviews and the students' written work. Immediately after each 29 interview, the interviewer made notes to f a c i l i t a t e further data analysis. Jones' (1985) advice to listen to the tapes of each interview at least twice was followed. The f i r s t time provided a sense of the whole interview while the second allowed for a more detailed analysis of specific questions. Individual profiles were made as a f i r s t step data reduction, based on the transcripts of the relevant parts of the interviews, students' written work and the interviewer's notes. Then the students' responses were categorized as the second step of data reduction. Later on the students' responses were "coded" into categories for final analysis. Vinner's model for cognitive processes was used for data analysis. 3.5 Specific Interview Questions The followings are the specific interview questions. They consist of two sets of questions, the f i r s t set is questions related to the concept of derivative and the second set is the questions on function. 1) Derivative a) Find the derivative of f(x) = x2 + 1 at x = 1. b) Sketch the graph of f(x) = x2 + 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 directions. Such lines across a Parabola are called secants, and some examples are shown in diagram. a) How many different 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 definition of derivative. 4) What's the relation between this curve and its derivative? 5) Compute the derivative of above function (f(x) = x2 + 1) at x = 0 by using the definition of derivative. 6) (a) compute d_ [1 + 1/(7 - 5x)1/2] dx 7) compute the derivative of f(x) = 0 x+1 •x+1 0 x<-l -l<x<0 0<x<l x>l at: a) x = b) c) d) e) f) x -2 -1 -1/2 1/3 1 10 31 Figure 3.2 The Graph of a Non-Smooth Function l i i Function: 1) Consider the equation x2 + y2 = 1. Sketch the graph of this equation. 2) Determine whether the above equation represents a function y = f(x) or not. 3) If not, determine a domain and a range such that the above equation is a function. 4) What is the relation 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 is invertible. Justify your answer. 6) What is the difference between Cos_1(x) (Cos_1(x) = arc Cos(x)) and Sec (x)? 7) If f(x) = 7 x and g(x) = x + 1, compute one of the following. (a) fog(x) (b) gof(x) (c) gog(x) (d) fof(x) 32 CHAPTER FOUR RESULTS 4.1 Introduction The objective of this study was to provide partial answers to the following specific research questions. 1. What is the nature of the students' understandings of derivative, slope of a tangent l i n e , how derivative and slope are related, dif f e r e n t i a b i l i t y and function? 2. To develop a category system for 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 conflicts between the concept definition and the students' concept images of the concepts of derivative and function. Questions were chosen in such a way so as to give the subjects an opportunity to reveal their 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. As stated by Cobb and Steffe (1983), "By acting as teachers, and by forming close personal relationships with children, we help them reconstruct the contexts within which they learn mathematics" (P. 85). The "tutorial sessions" are the result of the researcher's teaching action whenever i t was appropriate. Because i t is important to see the process of students' thinking and constructing, excerpts of each student's interview were analyzed. This investigator believed that the students' individual profiles will 33 help the readers in two ways: f i r s t to show the nature of the "tutorial sessions" and secondly to help them to capture the essence of the students' understanding. To f a c i l i t a t e the data analysis, students' responses to the questions on derivative and function have been categorized and excerpts from the students' interviews have been quoted. At the end of this chapter, a summary of results is provided. 4.2 Individual Students' Profiles This section deals with the Individual Students' Profiles. Only the relevant parts of the students' interviews have been analyzed. An effort has been made to illustrate the process of students' knowledge construction and the role of the tutorial sessions in helping students to do that. Only one example of a tutorial session for a given concept has been discussed. Informal language was used by the researcher to create a more friendly environment for the interview. Some of the profiles are lengthier than the others due to the fact that some of the students were better able to articulate their ideas than others. Profiles are not in any specific order. It should be mentioned that pseudo names have been used for the sake of the students' confidentiality. "I" represents the interviewer in the profiles. Twelve profiles are presented below. 4.2.1 Jenifer's Profile Jenifer got 77% in math 100. She wanted to major in sociology but she fear of mathematics seemed to affect her decision. She said: "I heard that you have to take math 200, oh, I don't know, I have not really decided yet". 34 For finding the slope of the secant l i n e , she tried to find the vertical displacement (dy) and horizontal displacement (dx). She had trouble finding the y-component of the point Q. Her attention was drawn to the y-component of a few points with x = 1, x = 2, etc. I expected her to possibly see a pattern for finding the y values of the point Q. Instead she said: Jenifer: Ya, but this is not, but this line is not a parabola [she meant the secant PQ]. This [f(x) = x2 + 1] is the equation of parabola, you can't use t h i s . She did not realize that the points P and Q are on the parabola. Her main problem was the lack of understanding of the concept of function. Jenifer did not understand that the height (y) of any point (including Q with the length of x + h) on the parabola was y = f(x) = x2 + 1. In a tutorial session, an effort was made to cl a r i f y this part. This matter was discussed again during the interview. Later on she was asked: I: Can you give the definition of derivative? Jenifer: No, I can't, I can do i t . [Emphasis is mine.] I: o.k., do i t please. Jenifer: f(x) = x2 + 1, f'(x) = 2x. I: No, not by using the formula, do you know how you got this formula? Jenifer: No, I don't, my teacher did the proof, but I didn't understand i t . But I can do i t . [Emphasis is mine.] There was a discussion about the slope of a tangent line and the definition of derivative. The question was raised again. This time she said: "In terms of limit? Oh yes, we did [emphasis is mine] the definition of derivative, i t ' s from l i m i t . . . , Oh, I don't know". Her prior knowledge was inadequate. She mostly used the phrases: "I remember" or "I don't remember". She was performing (doing) mathematics, as she said, not understanding i t . She was advised to avoid guessing without thinking. She then was asked to see i f there were any relation between the derivative of a function and the slope of the line tangent to i t . Jenifer: The derivative is the slope of the tangent l i n e . I: The derivative of the function at what point? She was not sure. In a tutorial session, the point P was changed and different tangent lines were drawn at the different points on the curve. She then was asked to find the derivative of the function at those points. She f i n a l l y defined the derivative correctly and said: Jenifer: You know, when you do i t , you don't think about t h i s . You do On the question of differentiability (#7, b), she skipped the fact that f(x) = 0 at x = -1 as well. When she was told of her wrong answer, she said: Jenifer: Oh ya, you have to do limit or something. 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 t so mechanical. I: What do you mean? I: Why doesn't i t exist? [The derivative.] Jenifer: Because i t approaches to two different numbers. 36 Tutoring helped her to improve her concept image of derivative and the concept of di 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 her to answer the question on differentiability with no d i f f i c u l t y . Not many of the interviewees were comfortable with this question. On the section on function, she said that the c i r c l e (x2 + y2 = 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 is the definition of function? Jenifer: I don't know. I: How can we say that this [x2 + y2 =1] represents a function? What is function? Jenifer: What is the function? Oh, I don't know. These kinds of answers were widespread among the students. They sometimes did not have any justification for their answers. The tutorial helped her understand the function in a very concrete manner. Then the next question was: I: Can we restrict the domain and the range of this equation in order to have a function? How can we change this c i r c l e to a function? Jenifer: Make it. a 1 ine. [Emphasis is mine.] I: Can we change a ci r c l e to a l i n e . Jenifer: No, we can't, but that's the only way. Her suggestion to change a ci r c l e to a line was very interesting. She did not understand the concept of function. She did not believe that 3 7 the domain of the equation could be restricted, while the equation i t s e l f remained unchanged. She rather aimed to conserve the length of the circumference of the c i r c l e . That seemed to be why she suggested changing the c i r c l e to a l i n e . After explanation and discussion, she at last said that, "we can restrict i t either to the negative y's or positive y's". Her last response showed her progress in understanding the concept. 4.2.2 Richard's Profile Richard was a commerce major. He took Math 100 and Math 101 as elective courses because he said that he liked mathematics very much*. He said: "I'd like to teach high school mathematics, that's actually my goal". His reason for majoring in commerce was to have a better opportunity in the future job market. He did very well in Math 100. He got 100%. Richard introduced himself as an eager student who strived so hard to understand the concepts in calculus. He said that he volunteered to participate in the interview to learn something from i t . A minimum amount of time was spent on tutoring in this interview, compared to the amount of time that was spent on tutoring in other interviews. Mostly a hint or suggestion was sufficient to direct his attention (not his memory). For finding the slope of the line 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 which are less theoretical and are mostly based on the applications. 38 Richard: Oh, I don't know the y value for that, right?... 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 )2 + 1). He then continued and got the slope of the secant line with no d i f f i c u l t y at a l l . He then differentiated the f(x)=x2+l to get the slope of the tangent l i n e . Richard: f(x) = y = x2 + 1 f'(x) = dy/dx = 2x I: What I'd like you to do is to follow the same procedure that you did for finding the slopes of the other secants, which means PQ's, in order to get the slope of the tangent line L. After a few minutes he said, "I can give you the definition of derivative". He was advised to forget the derivative for the time being and to get the slope of the tangent line as i f i t were any other secant. He responded as follows: Richard: The line is tangent to the graph, but there is only one point, I can't use the same thing here. [Emphasis mine.] I: What happens to this distance? [Pointing to the distance 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?! Limit is one of the most problematic concepts for students. They all experienced some kind of d i f f i c u l t i e s in understanding this concept. Although investigating the students' d i f f i c u l t i e s in understanding the concept of 1imit was not the objective of the study, the above 39 questioning was brought up intentionally. The lack of understanding of the concept of limit was one of the missing chains which prevented students from understanding other concepts such as derivative. Richard saw that the point Q got closer and closer to P (by looking at the graph). He could see that the, distance between Q and P ( i . e . h) became shorter and shorter. He was then stuck at this point. He did not know what to do and asked "use the same point over again?!" Richard knew that he could not do that, but there was nothing else that he could think of. He was eager to take a fresh look and try to answer the same question again: I: What is the slope of this tangent line at this point? [at P(l,2).] Richard: It's 2 [he used the differentiation formula to get this answer i.e. f'(x) = 2x and f ' ( l ) = 2]. I: What is the slope of the tangent line at any given point [ i . e . , not this particular case]. Richard: Sorry, Oh no, I don't actually know. When the question of the slope of the tangent line was raised, his f i r s t response was to give the definition of derivative. He did the same thing again. This researcher was wondering i f he could see any relation between these two concepts, i.e., the derivative and the slope of the tangent l i n e . In a tutorial session, an effort was made to help him understand what would happen to h as Q got closer and closer to P. This session helped him to get the slope of the tangent l i n e . Then he was asked to compare what he got as a slope of the tangent line with what he gave as the definition of derivative. He gave the definition as: f'(x) = limit f(x+h)-f(x). His concept image related these two concepts h+0 h 40 together, since he offered the definition of derivative to express the slope of the tangent l i n e . He related these two concepts in a way that he used the definition of derivative as a "method" to find the slope of the tangent l i n e . The tutorial session helped him to understand that derivative was the slope of the tangent l i n e . His response to the next question (#5) was more like the appreciation of the tutorial session, since he knew what he was doing. I: f(x) = x2 + 1, find the f'(x) at x = 0 by using the definition of derivative. 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 line and the slope of that line would be zero, [ i . e . , f'(0) = 0.] His f i r s t response to the question on di f f e r e n t i a b i l i t y was not correct. The investigator's intervention helped him to pay more attention to the function and its graph (in #7), and he then said that "there is no...it's singular point. There is no tangent line to i t . There is no suitable tangent line to i t " . He had an adequate concept image and a correct concept definition for the concept of function. He had d i f f i c u l t i e s in understanding some of the related concepts to the concept of function. In a tutorial session, the inverse function was discussed. He then restricted the domain and the range of semi-circle such that to make i t an invertible function. At the end of the interview, he liked to express his feelings toward mathematics. He was aware of his d i f f i c u l t i e s in understanding the concepts of calculus. 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 is actually changing of y over changing of x. So for example, i f a question comes up that asks you to visualize the problem, right?! Something that is 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 this is what i t meant because you don't get these d i f f i c u l t questions that asking for really understanding [Emphasis is mine.] His comment was quite interesting. He said that "you don't get these d i f f i c u l t questions that ask for real understanding". The fact of the matter was that these questions were selected from the very beginning of the textbook. The f i r s t 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 d i f f i c u l t ones compared to for example what they got for their homework or their exams. Yet the second part of his claim was true that those questions were not asked for the real understanding since the students could f u l f i l l the course, and the instructor's expectation as well, by doing them well i.e., getting a good mark! 4.2.3 Brian's Profile Brian was a science major. He hoped to get into engineering. He got 70% in Math 100. Brian had a vague and mixed up concept image of derivative. Although, he at last was able to answer the questions, yet he did not have a clear understanding of the discussed concepts. His d i f f i c u l t i e s were deeply rooted in his understanding of the basic and crucial concepts of calculus, namely the concepts of function and 1imit. He had an inadequate sense of generalizability. He had to see things to accept their existence. He, like most of the other interviewees, had d i f f i c u l t y to find the y-component of the point. 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 line: Brian: So i t would be rise over run...Q - 2. I: Q is a point, right? Brian: Yes, right, so i t would be...but we don't know what the Q i s ! The coordinates of a point were discussed in a tutorial session saying that the length of a point was the x-component and the height of a point, i.e., f(x) was the y-component. I: The height of Q i s : f ( x0 + h) which is equal to... Brian: I can't see that. The tutorial session went on with more concrete examples. He was told how the value of f(x) was dependent on the value of x, i.e. the changing of f(x) was dependent on the changing of x. He was asked to find the f(x) when x = 1, x = 2, x = 3 and x = x0 + h. He got f ( l ) , f(2), and f(3). But for f ( xQ + h) he said: "The function...right. You mean...but we are told to use this function [f(x) = x2 + 1]. His response revealed an important fact that he did not understand the concept of function. As long as x had numerical value, he was able to get f(x) without any d i f f i c u l t i e s . Because he learnt the mechanism of doing this sort of problems. For f(x + h), the syntax of the problem was changed. So he was stuck. Brian could not possibly see the x as a variable. For him, i t was hard to conceptualize that x could be anything, because x was a variable, and since x was an independent variable, 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. After the tutorial he said: Brian: So the slope would just be, OK, so i t would be... [in the beginning he said that the slope is rise a n rj that's run why he was trying to get the y-component of the point]. I: You said the slope is rise run Brian: But...wouldn't i t be...the slope would be the derivative of i t ? ! ! His concept image of slope was formed of two discrete 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 line and as a slope of secant l i n e . In order to find the slope of the tangent line (#2, d) he said: "Wouldn't be the derivative of the function". He was reluctant to proceed with the same procedure that he followed for finding the slope of the secant. Although he mentioned the l i m i t , yet his understanding of the concept was questionable. In a tutorial session, an effort was made to describe to him that how the secant lines became the tangent l i n e . The point of the tangency and the slope of the tangent line were discussed as well. After the tutorial he got the slope of the tangent line at x = 1. Yet i t took awhile for him to give the definition of derivative. The next question was to find the derivative of f(x) = x2 + 1 at x = 0 by using the definition of derivative. He correctly found f'(x) at x = 0. This was a good evidence to show that he did not just agree with the interviewer in the tutorial session, but the tutorial helped him to understand the definition of the derivative. 4 4 His response to the question of dif 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 is 1. The interviewer stopped him to draw his attention to the fact that f'(x) = 0 at x = -1 as well. He said: Brian: OK, i t would be 0 also. I: The derivative of function is 1 at x = -1 and again the f'(x) = 0 at x = -1. How can we interpret this? There was no response to the above question except silence. I asked him i f he could recall the definition of derivative. With probing questions and interviewer's intervening, he understood that there could be in 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) = 0 or f'(x) = 1] is the derivative of f(x) at x = -1. Brian: No, we can't. I: Then, what do you suggest? What can you say about it? There was no answer to this last question. In a tutorial session, he was told about d i f f e r e n t i a b i l i t y . Differentiability's property was only told 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 the concept of function. Lack of understanding the concept of function, caused him frustration in answering the questions of derivative as well. The section on function started with an intention to see whether or not the c i r c l e (x2 + y2 = 1) is a function. I: Is i t [the circle] a function or not? Brian: Is y a function of x? Yes, sure. 45 I: Why? Why is x2 + y2=l a function? Brian: [Silence].. I: What is the definition of a function? Brian: I don't know. I: Why you said that this is 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 definition of the equation. For him, function was nothing but a few symbols (such as x and y) and few numbers with an equality sign. In a tutorial session, many examples were used to describe the concept of function. The latter question was looking for the possible restriction that could be imposed to the domain and range of the equation (x2 + y2=l), in order to make i t a function. I: How can we determine the domain and the range of this equation in order to have a function. Brian: You want to make i t into a function? I: Yes, because we saw that i t ' s not a function [after the tu t o r i a l ] . Brian: You could expand i t just like that I: We are not allowed to do that. He tried and he could not succeed. The interviewer tried (within the time frame) to help him understand. He unfortunately had no more time to finish the interview. But his last response was very interesting. He said that the circl e could be expanded to make i t as a function (refer to Jenifer's profile for the discussion on this matter). 46 4.2.4 Megan's Profile Megan was majoring in biology. She got 50% in Math 100. She chose questions on derivative to start with. When she wanted to get the slope of the secant l i n e , she, like 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 her to get the y-component of Q and the slope of the secant as well. She did not know how to find the slope of the tangent l i n e . Her knowledge of the matter was not adequate enough to help her to establish the proper concept image for the coordinates of a point. She had trouble understanding the concepts of l i m i t , tangent line and point of tangency. In a tutorial session, the discussions aimed to help her understand that the slope of the tangent line would be the limit of the slopes of the secant lines. I: We are looking for the slope of the tangent line at P [1,2]. Megan: So to finding the slope of the tangent l i n e , we get the derivative of the equation. The interviewer liked to know that why she wanted to get the derivative in order to find the slope of the tangent line? She had no answer. The tutorial session continued for another 20 minutes. Her response after the tutorial was as follows: Megan: The derivative is the slope of tangent line at point P. I: Derivative at which x? Megan: 1. Her response was an evidence to show that the tutorial helped her to understand the concept of derivative to some degree. 47 In answering the question of differentiability (#7), she had the same d i f f i c u l t y that most of the other students had. The responses were almost the same. She did not know the constant function. This was one of the reasons that she was not able to see that 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 is f'(x)? Megan: f'(x)?!...There is no x. A tutorial session aimed to help her understand the concept of constant function and concept of d i f f e r e n t i a b i l i t y . The existence of derivative was discussed by referring to the earlier discussion on the definition of derivative. 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 interview continued by her responses to the questions of function. She said that x2 + y2 = 1 was a function for the following reason: Megan: Because.. .they are related?!! I don't know [and she did not seem to care]. I: There is a relation between x and y, so x2 + y2 = 1 is an equation. But why is i t a function? Megan: The equation is a function. I: Why do we name them two different things i f they are the same, why do we say equation and function? She did not have any answer for i t . She had a limited knowledge about function in general. Tutoring helped her to acquire some understanding. Enough to enable her to answer some of the questions. But she had no idea of what the inverse function was. 1 48 4.2.5 Jason's Profile Jason was a science major. He aimed to continue his study in dentistry. Jason got 90% in Math 100. He did not at a l l use such words as guessing in the interview. He was quietly thinking unless he certainly had a point to make. He liked to start with the questions of derivative. He was wondering how he would find the slope of the secant. The interview went on in silence for about 10 minutes. The interviewer necessarily intervened and asked: I: Do you have any idea that how to find the slope of the line? Jason: It's rise over run. I: OK, find rise 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 about the y-component of a point and the point i t s e l f . He could see the distance from the centre to the y-component of P, because i t had a numerical value which was 2 (see Figure 3.1). 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 is Q? Jason: Q is 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 bit of thinking he wrote: Jason: Q = (X0 + h)2 + 1 I: Why did you write: Q = (x0 + h)2 +1. Q i t s e l f is a point. Every point is indicated by i t ' s length and i t ' s height. What you got here is the height of Q, i t ' s not Q. Q is a point. He f i n a l l y got the slope of the PQ by spending a few minutes on that. Later on he was asked to find the slope of the tangent line at p. He wrote: Jason: f(x) = x2 + 1,, f'(x) = 2x, f ' ( l ) = 2 = slope L [tangent line to the curve at point P], I: Why did you take the derivative of f(x) at x = 1? Jason: Why did I do it? The derivative is slope of line?!! 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 11 minutes for him to get the answer without any of the interviewer's intervention. He did no mistake in doing that. The questions followed by: I: What is the derivative? What is the definition of derivative? Jason: Slope. I: Slope of what? 50 Jason: Slope of the line 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 line to the curve f(x) at a point x. I: Derivative at which point? Jason: at any point [Emphasis is h i s ] . The following tutorial session, aimed to show that the value of derivative was changed by changing the tangent lines to the curve i.e., by changing the point of tangency. This was done by the means of the geometric representations. After the tutoring, his response was the unique one among a l l the interviewees. Jason: The equation of derivative is the same at a l l point, isn't i t ? ! Putting different 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) at x = 0 by using the definition of the derivative. He also sketched the graph showing the tangent line 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 derivative. For the section on di f f e r e n t i a b i l i t y , although he had a different approach towards solving the problem (as he said: vcan I do i t by inspection'), yet, he was experiencing the same d i f f i c u l t y , i.e., facing two values for 51 f' (x) at x = -1. With an exclusive discussion and the probing questions, he f i n a l l y answered the last question of this part: I: Can a function have two different tangent lines at one point [and s t i l l have derivative at that point]. Jason: No, i t doesn't have derivative at that point then. The next set of questions was on function. Jason had an accurate concept image of function. In response to the f i r s t question of this set, he said: Jason: It's [x2 + y2 = 1] not a function, because i f we draw a perpendicular line [he meant vertical l i n e ] , there is two values of y for one x. For him, f"1 and 1/f were the same. He said that 1/f = f"1. The tutorial session, in which the researcher aimed to c l a r i f y the concepts, of inverse function and reciprocal of a function, was the same in nature with some of the other tutorial sessions with other interviewees that have already been discussed. He then confidently answered a l l the related questions of the function. 4.2.6 Joe's Profile Joe was a quiet person. He hardly talked or even wrote during the interview. At least 1/3 of the tape was f i l l e d by silent moments. He was mostly thinking. He did not speak unless he thought that he had the answer. He got 90% in Algebra 12. He had no calculus at high school except a very basic introductory at the end of Grade 12. He got 100% in Math 100. Interviewer told him that: I: It's very good that you got 100%. 52 Joe: Even though I got 100%, s t i l l there are lots 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 is mine.] The Way he expressed himself showed that for him there was a difference between performing calculus and understanding the concepts of calculus. As described by him, he was willing to really understand the concepts but he did not get enough help from the class or the texts. The interview continued by talking about derivative. His response for finding the slope of the secant was: Joe: I'm not really sure how to find i t . I: Give i t a try, what is the slope? Joe: I know i t ' s rise over run, but I don't see what the rise i s . He found the rise by himself but he, like many others, had a hard time to substitute the value for f( x0 + h). His d i f f i c u l t i e s stemmed from lack of understanding the concept of function. He did not know the behavior of f(x) = x2 + 1 (The behavior of f(x) has been discussed in one of the other profiles). In case of f(x + h), he did many t r i a l s and errors t i l l he got the right answer. He did not know how to find the slope of the tangent line (at P). He did not have an adequate understanding of the concept of limit in order to help him to understand the slope of the tangent l i n e . In a tutorial session, i t was explained to him that the slope of the tangent line was the limit of the slopes of the secants when h approached to zero. He defined the derivative almost correctly. His concept image of derivate led him easily to the accurate concept definition of derivative. 53 Joe: The slope of the tangent line at point on the curve is 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 is undefined [f(x) is undefined]. I: Why is i t undefined? Joe: At that point f'(x) is between 1 and 0, but undefined at that point because you can't plug this number into f ( x ) . I: What do you mean by: "you can't plug this number into f(x)?" Joe: Since your f(x) = 1, and x = -1, you can't put into this formula. His reason for the derivative of function to be undefined at x=-l was interesting. He did not know the concept of constant function which f(x) is constant for a l l x belongs to domain of function. This was partly the reason for his last response as: "you can't put [x = -1] into this [f(x) = 1]". 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 a l i s t of incorrect conditions for a function to be differentiable. Probing questions were posed to c l a r i f y the problem. After the discussion he said: "Oh, this is a smooth curve. I remember this part, I wasn't quite sure how to do that, I taught myself how to do i t and that one worked (Emphasis is mine)". These honest words must be t h r i l l i n g for calculus instructors and curriculum makers. Solving and answering problems has 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 relatively good understanding of function. No special incident happened while he was answering the questions of this part. 4.2.7 Owen's Profile Owen's major was science. He wanted to continue in engineering but he decided not to "...because Math is one of my d i f f i c u l t subjects, caused me to take i t second time through Math 101". He had an introductory to calculus at the end of Grade 12, but as was stated by him, i t did not help him to be more prepared for university calculus. "When I came here last year, i t was just like...Oh really, what's going on"? Owen was f i r s t in a crowded calculus class and he failed that term. For the second time he changed to a smaller class. As he said: "you can be more relaxed in small classes". He insisted on saying that he did not understand anything and he only 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 derivative. I: What is the relation between the derivative and the tangent l i n e , why do you want to take the derivative? Owen: The tangent is the derivative at that point. He had an unclear knowledge of the discussed concepts prior to the interview. He had heard all the notions before, but he did not know how these notions or concepts were related. He could not visualize the 55 difference between the slope of the secant l i n e , the slope of the tangent line and derivative. He dealt with them as i f they a l l were the same thing. Probing questions were posed to find out more about his primitive knowledge of these concepts. The interview went on until he was again asked to answer the question of the slope of the secant line: Owen: Oh, Oh, i t would be: Q r p = Q i ' p x0+h-x0 h I: What do you mean by Qj - P? Owen: Q is a point on the curve minus P. I: What is xQ + h - xQ? Owen: That's the run. I: OK, what about the rise? He got "run" because he could see them (see Figure 3.1) but he couldn't see the y-component of the points (because he had to f i r s t find them). In a tutorial session, I talked to him about the coordinates of the point. For y-component of the point he used the point i t s e l f . He did not know clearly how a point was indicated on the plane. It is not reasonable to expect a student to understand derivative while he needs to know many basic concepts prior to that. After the tutorial he said: Owen: So, Qx-P would be: f (xn ) ' f ( l ) does i t make sense to you? h I: I want i t to make sense to you, does it? 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(x0+h)-f(x0) = f(x0+h)-f(xQ) x0+h-x0 h His last response was more like a jump. I think his 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 line at P. Owen: h is zero, then f x2+l-x2+l ...I 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 right. 0 I: As we move along the curve, what will happen to the secants? Owen: Secants will become the tangent, i t ' s just the derivative, so that would be 2. The interviewer asked him i f he could consider any given function other than f(x) = x2 + 1 and say what will happen i f Q moves toward the fixed point P, he answered: Owen: The li m i t . . . h-*» I: Why »? Owen: Oh, I just said, the limit f(xn+h)-f(xn) . h+0 (x0+h)-x0 Then he went on and got the slope of the tangent l i n e . But his right solution did not seem to be an evidence of his understanding of the concept. For example, when he was asked that why he took the limit as h approaches to », he said, "Oh...I just said i t " . But in investigator's opinion, based on her observations, Owen did not have clear understanding of the concept of l i m i t , approaching to something, etc. He usually had 57 seen the notion of limit which was accompanied by h approaching to either o or 0 and he naturally tried them both to see which one worked better. He did not see h approached to 0 from the graph. After he found the slope of the tangent l i n e , he was asked to give the definition of derivative. Owen: It's the tangent line at a certain point of that graph, such as the derivative of equation is defined. I: OK, what is the derivative of function at that point? Owen: At that point! There...[no answer], I: What is the slope of the tangent line at this point [at P]? Owen: 2. 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 correctly. He knew that these concepts were somehow related, but he wasn't sure how. In another tutorial session, an effort was made; with the aid of graphs; to c l a r i f y these concepts as much as i t was possible (time constrain). After the t u t o r i a l , he defined derivative as: "The derivative is at a certain point defined that i t approaches as the same as the limit of the secant line as changing value of x approaches to zero." The process of tutoring and questioning went on and on a couple of times, until he seemed to understand this specific case of derivative. His answer to the next question showed his lack of generalizability along with his uncertainty. The question was to compute the derivative of f(x) = x2 + 1 at x = 0 (the f'(x) at x = 1 was discussed in de t a i l ) . He f i r s t got i t quite right, but he then changed his mind and said: "no, no, i t ' s not right". He tried again and got the same answer (f'(x) = 0) 58 and again he repeated that: " i t is not right". He tried for the third time and again f'(x) = 0 was his finding. He said: Owen: It's wrong. I: Why is this wrong? Owen: Because the derivative, the slope should be 2 [he pointed to the derivative at x = 1]. What he did was interesting. His d i f f i c u l t i e s were deeply rooted in his understanding of the concept of derivative. For him, i t was hard to believe that the derivative had different numerical values at different points. He did not know the reason that he got two different answers for the f'(x) was that he was computing the derivative of the function at two different 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 his procedure (and computation as well) was correct. He "looked back" and he did not find any mistakes, yet he was not satisfied with his 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 definition of derivative). We did i t and when we finished, he said: Owen: It's 0. I: Then why do you think that i t ' s wrong? You didn't do anything wrong. Owen: Because for what I've been taught, derivative...Oh, that's right! This is right. Because f'(x) = 2x and i f x = 0 then f'(x) =0. I .just wasn't thinking. [Emphasis is mine.] I: Because at x = 1, the tangent line is [see Figure 3.1], but at x = 0. the tangent line is the horizontal line (I sketched the graph). Owen: Right, right, 1 .just wasn't thinking. [Emphasis is mine.] 59 Twice he said that he was not thinking but the investigator thinks that his error was rather "structural" (Orton, 1983). He had a vague concept image of the related concepts to the concept of derivative. The interviewer believes that he at last accepted that f'(x) = 0 at x = 0 because he plugged the x = 0 to the formula and he was satisfied with the result. But i t did not necessarily prove that he got i t right because he understood the concept thoroughly. That was the reason for further tutoring by means of graphs. That helped him to understand the concept better. For him, the function which was used in the question of di f f e r e n t i a b i l i t y , needed more explanation. 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 this: f(x) = 0 x<-l. x = -1 is in both intervals (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 one? Can you say that...[I was 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 on by asking him whether the function was differentiable at x = -1 or not. Owen: Probably, because i t ' s not a smooth curve. Two curves chain together. Is that correct??! I: Why because i t ' s not smooth, i t ' s not differentiable? Owen: Oh, because two different lines cross each other. 60 His responses showed that he remembered something about dif f e r e n t i a b i l i t y but he was not sure why he was saying those. It is possible that with this much understanding of any concept, a student could get a satisfactory mark for his course, especially those who are experts in getting good marks. But i_s there any understanding behind the marks is a question that should be answered. The students' responses could be more or less, a part of the answer. He did not know the concept of d i f f e r e n t i a b i l i t y , although he mentioned the "smooth curve". In a short tutorial session, the di f f e r e n t i a b i l i t y was discussed in a very simple way, mainly by the aid of graphs and referring to the discussion at the beginning of the interview (on the definition of derivative). In order to answer the questions of function, he certainly tried hard to remember a l l he was told by his instructors. But he did not succeed. His concept image of function was the same as for the relation and equation, although he did not have proper concept image of those two concepts as well. Four times he changed his answers from NO to YES and vice versa. He f i n a l l y gave up and said: "I don't know how to describe i t " . His main idea for f to be a function was: " i t defines y" or " i t defines points y along the x-axis" and so on. He did not know the concepts of the inverse function, l / f ( x ) , etc. He was stuck, but he seemed to be anxious to know something about function in general (domain, range, inverse...). That was why he asked me to tutor him. The investigator, within her time frame, tried to use the various examples and sketched different 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 tutorial session on function helped him to understand the concept or not. 4.2.8 Kathy's Profile Kathy's major was biology. She was doing well in Math 101, although she had to write the supplementary exam in August since she failed Math 100. She gave following reason for her failure in Math 101: Kathy: The reason i s , because I hate math. I never did my homework, and I usually just cram at last minute and you can't do that for calculus, calculus is quite different from Algebra. Kathy had taken Honor Algebra 12 and Geometry. She got 'A' in both of the subjects. She decided to start the interview with the concept of function. I: Determine whether the equation: x2 + y2 = 1 is a function or not. Kathy: You mean i f I solve i t for x? or i f I solve i t for y? It is 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 this a function?? Oh, Oh, I see, isn't that you draw a horizontal line through i t ? ! That's not a function, because you can draw a line through i t and i t intersects both points...or i t is a vertical line?! I: You decide. Kathy: There is another definition, i f you draw a horizontal line through i t . There is a word for i t , I don't know, I have forgotten. Her conception of the function was limited to a l i n e . She only tried to remember that. It seemed to me that she was taught the concept 62 of the function in such a hurry that no time was spent on the real understanding of the concept. When she was told that the line was v e r t i c a l , she then said: "Yes, and i t intersects the y-axis at two points, so i t ' s not a function". It is a wishful thinking to expect students to understand the concept by only giving them the definition of the concept (Vinner, 1983) i.e., before constructing any meaning into the concept. Students need many concrete examples in order to build up adequate concept images. If the concept image were established by the means of the familiar examples and geometric representations, they then, would be ready to understand the definition of the concept. Otherwise an abstract definition could be forgotten and a l l that might be l e f t is a vague memory. Kathy expressed this 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 is the function already, like with that definition. I know that i t ' s a business of drawing a l i n e . [Emphasis is mine.] After the tutoring the researcher asked her that: I: Don't you think that i t would be better to say that i f there are two y values for one x, then you couldn't have a function, instead of keeping in your mind that whether you have to draw a vertical line or horizontal l i n e . Say one x can't have two y values. Kathy: Ya, o.k., that makes sense. I agree with that, because I learned i t . Next, she was asked to respond on the question of inverse function ( f_ 1) and reciprocal of function ( l / f ( x ) ) . She certainly had heard about the definition of inverse function, yet she did not understand i t . Her 63 knowledge of the matter was like more of remembering few facts, without even trusting on those memories. Kathy: Isn't that a function and i t ' s inverse are the mirror image of each other? I: What do you mean by mirror image? Kathy: That i s . . . that the value of y for one of them is the value of x for the other one and vice versa. And l/f(x) is that you just inverted, is not the inverse. It is the inverse, but not inverse of the function. She tried to make her point that the l/f(x) and f"* are two different things. But in answering the next question(what is the difference between Cos_ 1(x) = arc Cos(x) and sec x), she could not see them as two different things. Kathy: Sec is 1/Cos x, arc Cos x?... Could that be also 1/Cos x?! [With doubt.] I: You decide. Kathy: I don't know, I guess so, I think so, because that would be Cos"1 which is 1/Cos* because i f you have anything to negative one [e.g. ( x ) "1] , I think i t just goes one over that x to the one [she meant: (x)"1 = (1/x)]. Cos"1 would be 1/Cos and sec is 1/Cos, that is the same thing.. .There must be some difference. Of course. They give you definition like that, and I never ran across i t , but 1 would think, they would be the same thing, although I'm wrong. [Emphasis is mine.] In a tutorial session, the notions of 1/f and f "1 were discussed as well as their related concepts. f "1 was introduced as the inverse of function in action of the composition of two functions, and 1/f as the inverse of function in action of the multiplication of two functions. After the tutorial she said: "Oh, I'm learning something...That would help me in September... I memorized a l l these things, that's why I'm confused" (Emphasis is mine.). 64 The questions of derivative started with the sketching graph of f(x) = x2 + 1. Kathy had a hard time to do that. She tried so many points on the axes. She also tried to sketch many different curves without any successions. Kathy had a peculiar way of sketching the curves. Although sketching the graph of the curve was not in the particular interest of the interview, yet i t was revealing in analyzing Kathy's conceptual d i f f i c u l t i e s in calculus. Kathy was a bright student. It was not f a i r to put a l l the blame on her. She got A in both Honor Algebra 12 and Geometry in high school, and she s t i l l did not have any reasonable procedure to sketch a simple graph. She did i t by t r i a l and error. By locating different points of the graph and join them together to obtain the desired curve. She survived in high school and in university as well. The growing concern is that "How did so many students manage to... successfully escaping their teachers' recognition" (Gorodetsky et a l . , 1986). She had right to "hate calculus" since she did not understand i t ' s basic facts and yet she had to work hard to pass the course. Following passage was the way that Kathy tried to sketch the graph of y = x2 + 1. Kathy: Would i t centre at (1,1) or at (1,0)? [She meant the curve]. I: You can test i t . Kathy: Oh, this one: (0,1), Oh, Gee! I never knew that you can test i t . I: Really? Then how do you sketch the graph? What procedure do you follow? Kathy: I don't really follow one...I just make sure that I knew i t before the test, set of rules! [Emphasis is mine.] 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 this graph, the other function has different one...you know what I mean? But i f you try to understand how to do i t , then i t is much easier. Kathy: I have a rule, i f y is on this side [ i . e . the l e f t side of equality sign, e.g. y = x2 + 1]. i t will 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 like an x, or may be i t is opposite! Second question asked for the slope of the secant PQ. She had trouble locating a point on the plane. She did not know that any point was indicated on the plane by its coordinates. She was wondering how to get the y-component of point Q. Kathy: The Q also li e s on the parabola. Can I use the parabola to find the two points x and y of Q. She immediately added that: Kathy: I can't find the values. I don't know the relationship between x and y, that comes from the parabola. In a tutorial session, diagrams were used to show that what would be the y-component of Q. To show that the role of f was to take every x, square i t and add one to i t . I also concluded that for every point x, there was one y which was the y-component of that x. She then tried to get the slope of the secant PQ. Nothing significant happened while she was doing that. It took 20 minutes for her to get the slope of the secant. She did i t correctly and said: " i t ' s like pulling teeth!" She moved on to get the slope of the tangent l i n e . Kathy had a vague understanding of the concepts of limit and derivative. Kathy: I wouldn't know how to find the slope of i t , a l l I know...What I heard is to find the derivative of i t . I: O.K., find the derivative of i t . 66 Kathy: I did already, i t is 2. I: No, I mean by using the definition of derivative, not using a formula. Can you give the definition 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 try to relate i t . Kathy: No, no, I wouldn't be able to do i t . I: Do you know what is the definition of derivative? Kathy: Yes, isn't that...like the slope of the line?...but I don't know, I don't know how to find the slope of the line 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 line and derivative were tied 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 line by finding derivative. Later on she said that derivative is the slope of the tangent l i n e . But she said that she couldn't see any relation 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 line without using the formula". She was asked to give that formula. I: What is the formula? Kathy: Well, there is some formula, I can't remember i t , i t had h at the bottom...limit f(h+xQ)-f(xQ) h I: O.K., you already did i t . Kathy: Oh, that's true, i t looks familiar... 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 like disjoint pieces of memories. Sometimes they were easy to recall and sometimes they were not. I: What happens to h? Kathy: It gets smaller and smaller, i t approaches to infinity?! I: Infinity is when i t ' s getting bigger and bigger. Kathy: So minus infinity?! [no trust in her words]. Oh no, i t approaches to 0 and that's the formula: limit (x+h)-f(x)w h+0 h Her response to every single question was amazing. She usually responded f i r s t and thought later. When she was told 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 theoretically, she was right i f h was greater than zero (h>0) was not the only condition and h could be less"than zero (h<0) as well. But i t was not the case. She said i t , because the opposite of +» was -». THis type of responses was heard a l l along the interview. It was true that she got 47% in Math 100, but she was smart and had a good memory indeed. One of the reasons for her misunderstanding was, in researcher's view, her naive attitude towards mathematics. She tried to convince her audience that she hated mathematics and she would never be able to understand mathematics. Her negative attitude acted as a barrier which caused her great d i f f i c u l t y in understanding mathematics. Of course many other factors were involved indeed which there was no intention to discuss a l l of them. 68 4.2.9 Gary's Profile Gary was interested majoring into either pharmacy or psychology. He had failed Math 100 twice and he got 86% the third time. Among the reasons that he had for his failure was the size of the classroom. He said: "It was a big class, I didn't show up to the class, now I'm in a class with 20 people. I can stop and interrupt the class...so I just stop the class". Gary's opinion about the classroom size was expressed by many other students as well. The main objective of this study was to reveal some of the students' conceptual understanding of calculus, yet i t was interesting to come across the other understanding barriers that students were dealing with (from their point of view). Gary liked to start with questions on function. He sketched the graph of x2 + y2 = 1 and he then said: Gary: y=± /1-x2 is a function. I: Why is i t a function? You solved i t for y, why is i t a function? Gary: I don't know...[laughter]. I: What do you think a function is? Gary: Function is just an equation which she would take some number and put i t through. It's just like...I don't know, i t ' s like the button of your calculator, you put something in there, you punch that button and use another number which for all...which all 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 calculator, 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 for example, you punch this one, then you will get two outcomes: y=+Jl-x2 and y=-7l-x2. 69 Gary: Oh, O.K., I know what you mean. I forgot about that vertical line! /1-x2 or y=-/l-x2, two different functions put together. He used a good example for function. It is not far from the truth to 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 definition. He evidently did understand the concept of function. His further responses supported this claim. Following is his responses to question #4: I: What is 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 it? Gary: It's just whatever f(x) i s , l/f(x) is one over that and the inverse is the...I guess what you call opposite of i t ? ! [Emphasis is mine.] I: What do you mean by that? Gary: Like..., I cannot be sure about inverse. I: You could try. Gary: J and ( )2?! [square root and the square] I: What do you mean? Can you be more precise. Gary: ...[silent] His immediate responses to many questions were the same in nature. Declaring "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. Everything seemed blurry to him in the f i r s t place. But he came along very nicely. 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. His interesting example of J and ( )2 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 it? Gary: I think i t was...the domain o f . . . [ s i l e n t ] . I: The domain of what? Go on. 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 else, but I forget which. I: We are talking about function and i t ' s inverse and you said that [I was interrupted by him] Gary: Aha, I remember. It was, the domain of the function is the range of the inverse of the function. The above questionings could prove a point that students were unwilling to express their thinking because they did not have enough self-confidence and self-respect toward themselves. His answers were used, to show him how the domain and the range were switched. He had a rote learning of the concept. I flipped the page on the space while i t was rotated 9 0 ° . He saw how the domain of one became the range of the other one (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 line Y = X (the text's language). After the tu t o r i a l , he answered the rest of the questions correctly. He was surprised to hear that the invertable function was defined by him as: "there is 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 definition. The barrier was his uncertainty of his understanding. His concept images were disorganized and he was not aware of them. After finishing the questions on function, he continued with the questions on derivative. It was 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 "(Q-P/h)" for slope of the secant PQ. He was asked why and he then, corrected himself and got i t right ( (xo + h)2 + 1 - (xQ2 + 1) = h + 2). h I: Now, can you find the slope of the tangent line at this point [P(l,2)]? Gary: f(x) = x2 + 1, f'(x) = 2x at x = 1 the derivative is 2. I: What is the relation between the derivative and this slope? Why did you use the derivative to find the slope of the tangent line? Gary: The application of derivative is that i t defines the slope of the tangent line to curve. I was anxious to find out how he would define the slope of the tangent line as an application of derivative. Probing questions did not help. He was asked to forget about the derivative for the time being and yet try to find the slope of the tangent line using the same procedure as he did to find the slope of the secant l i n e . The interview was not structured. 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 line as the limit of slopes of the secant lines (as Q gets very close to P), he said: Gary: Ya, but I usually don't think of that term, I guess you could, but I don't. I: How do you think? Gary: I just think as a road and this 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 tried to make himself to understand using the concrete examples, the tutoring helped him to state the definition of the derivative as: "The slope of a tangent line at a given point, is the derivative of the function at that point." He then said: Gary: I was aware of this since I took the course three times, but it ' s much easier...if you just use i t . It is much easier, much quicker. Gary had the same d i f f i c u l t y in answering the question of the dif f e r e n t i a b i l i t y as most of the others had. The tutorial helped him to conclude that "Smooth curves are differentiable and there is no derivative at cosps". 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 did you find the way to get a good mark [he got 86% in Math 100 after the third time]. Gary: Ya, more of the latter, i t was just to find out what to do in each situation! I think that was most obvious in 'error-estimate tangent approximations', I didn't learn, I couldn't understand from the lectures, I saw the examples, I saw what they did and I just did i t that way and i t worked. And he then expressed his ideas about the teaching of concept of derivative. 73 Gary: I think more concrete definition of derivative and to see that the derivative is speed is helping to understand i t better. I think we shouldn't bother this much with the actual derivation or where they come from, and how to get them. More the application. [Emphasis is mine.] His last comment was important for teachers-researchers to think about i t deeply. And may motivate them to do something about i t . He said: "In f i r s t two terms, I was really frustrated, I didn't know what the function was, I didn't know why they are doing those things [Emphasis is mine.]". Gary's complaint was expressed in different ways by other students as well. Some of the students were aware of their d i f f i c u l t i e s and some of them were not. But they all 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 derivative. 4.2.10 Barbara's Profile Barbara was a physics major. She got 75% in Math 100. She liked to express herself and she was successful in doing i t . Before she started to answer the questions, she changed f(x) = x2 + 1 to y = x2 + 1 and said: "I call i t y. I like y". In order to get the slope of the secant P Q , she said: "Y of Q would be equal... Just wild guessing, would be the difference between these two points. many guessings are going on here" (Emphasis is mine.). She started with the word guessing right from the beginning and this word was used in her interview more than 15 times. She got the slope of P Q without a major d i f f i c u l t y . She was one of the exceptions that got the slope of the tangent line and the definition of derivative correctly. 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 concept. She was asked to investigate the relation between the curve and 74 i t s derivative. Her uncertainty could be witnessed by her intention to change her responses from time to time. Barbara: Well, the derivative will give you so like 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 like to the curve. I: Does a point have a slope? Barbara: Not really, but the slope of the curve at that point. I: Does a curve have slope [What she said was right. 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 derivative. Her answers showed that one of her problems was the lack of an accurate (not talking about vigorous) language. In the beginning of the interview she had no trouble getting the slope of the secant and then consequently the slope of the tangent l i n e . But later on, the probing questions revealed that she had trouble understanding the concept of tangency. Same questions about tangency yet, in different contexts were posed on her. The intention was to find out whether she could see the relation between the curve and the line L tangent to i t . She said: Barbara: How are they [the curve and line L] related?! It touches i t at one point only. I: What does that mean? Barbara: I don't know what I mean! [Emphasis is mine.] I: You said that i t touches i t at one point, then what do we call this line? 75 Barbara: What do we call the line? I don't know. In a tutorial session, the graph was sketched to explain the concept of tangent line 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 right]. Barbara: The secant is disappeared. i t ' s really disappeared! [emphasis is mine]. I: What do you mean by disappears? Barbara: Because they are so close at two points. [She did not pay attention to the definition of secant l i n e ] . I: We said that these secant lines are extended from both directions, so when P and Q are getting very close, i t doesn't mean that the secant PQ disappears, the line is there. The Q is just moved towards P, so i t means that i t is the same line which has rotated around P. It just hits the curve at one point. What we call this line? Barbara: I call i t the tangent. The above questions served the writer's purpose, since she believed that the students' prior knowledge were playing significant roles in their further construction of mathematical knowledge. In the beginning, she did not think of the tangent line 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. Then she was frustrated and confused. She did not believe in her words to the degree, that she even said: "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 definition of derivative in response to the "slope of the tangent l i n e " , as 76 f'(x) = lim f(x+h)-f(x) h^ O h I: What is the f ( x ) ? Barbara: It is derivative. I: What is derivative? Barbara: What is i t ? . . . Do you want me to write the definition? I: No, you gave me the definition as: f ( x ) = lim f(x+h)-f(x) , h-0 h What is this f'(x)? You said that derivative is a slope, slope of what? Barbara: Derivative is the slope of the tangent to a point on the curve. I: The derivative of function at what point? Barbara: Any. point on the curve, you just pick i t and plug i t in [Emphasis is mine.] I: You say any. point along the curve. You got the slope of the tangent line at what point? [Emphasis is mine.] Barbara: At point P there. I: Which you named i t the f ( x ) at what point? Is i t the f'(x) at any point [Emphasis is mine.] Barbara: This would be true for any point [Emphasis is mine.] She did not see the derivative as a dynamic process that will be changed by changing the point on the curve, i.e., changing the lines tangent to the curve. In a tutorial session, I changed the point P(l,2) and drew different tangent lines to the curve. The writer's purpose was to 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 different numerical values at different points. 77 Barbara: But the tangent changes, going around. I: Then what happens to the derivative when the tangent changes? Barbara: Changes with the shape of the curve... ask me the question again. I: Is i t true that the derivative at any point is the slope of the tangent line at a fixed point [emphasis is mine]. Barbara: That is the specific of any. which is the general term, that part of any group [emphasis is mine], I: Means what? Barbara: The derivative of a function at a point x is the slope of the tangent line at that point (x). I: This is correct! It was interesting to see that how the tutorials and probing questions helped her to improve her concept image until the proper (not vigorous) concept definition was reached. In the next questions, she got the derivative of f(x) at x=0 by using the definition of derivative with no trouble. The question of differentiability caused the same trouble as for 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: "f'(x) = 1". She was confident in her answer and she wanted to continue. But she was stopped to answer a few questions concerning the f'(x) at x = -1. 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: Actually i t ' s nothing to plug i n , that's the real thing. Her answer was quite revealing. She did not know the constant function. She could not imagine that the value of the function was 78 constant at all points. One of her causing problems was the inadequate concept image of function. It was hard for her to see that there were two different way of writing a function at one point (x = -1). Barbara: Oh boy, what a question! Something strange coming... Could i t be two different 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: . . . [ s i l e n t ] . In a tutorial session, the derivative as a slope of tangent line was explained again. This was done by sketching different graphs. The question was repeated again. I: What does i t mean when the function has two different values for the derivative at one point? Barbara: Then there is two tangent at that point. The tutorial went on discussing about the derivative and i t ' s relation with the slope of the tangent l i n e . She was showed that there were so many tangent lines passing through the point x = -1. After all I asked: I: Is this function differentiable at this point [x = -1]? Barbara: Yes, something exists, so much exists...so much exists. [Emphasis is hers by changing her tonation.] Barbara did not know that the existence of the derivative means the existence of limit (even she defined the derivative as the limit of slopes). And the existence of the limit requires that the l e f t limit and right limit should be equal, i.e. there should be only one tangent line 7 9 passing through the point in order for function to be differentiable at that point. I: What is the condition for a function to be differentiable? Barbara: I guess at point x, there should be only one answer for the derivative [Emphasis is mine.] I: What is that answer for f'(x) at x = -1? Barbara: What is that answer?!! We call the slope of the tangent at that point. I: So you are saying that there must be one tangent line at that point, but there is not one tangent line at that point. There are so many, so is this function differentiable 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]. It i s , here is the answer. Her doubt was natural. Her eyes agreed that function was not differentiable because she saw that there were so many tangent lines passing through x = -1. But numbers were convincing reason for her to believe that the derivative did exist since she got the numerical value for i t . I: Why is i t differentiable and why is i t not? Barbara: I guess i t i s , because...[silent for few minutes]. Barbara: What does differentiability mean any way?! I: You just told me! Barbara: Oh, about the tangent business?! We have so many tangents though, oh, what comprehensive! 80 She was one of the exceptions that let her concept images to be revealed by themselves. Barbara: I guess i t i s , isn't it? Oh, I have to take i t overnight and get back to you. I've never seen anything like t h i s , i t is good stuff, yes or no?!..no I guess. I: Why no? If i t ' s no, why not? If i t ' s yes, why yes? Barbara: I think, i f i t ' s yes, because i t existed, like you calculate i t . . . I: We calculated and we came up with two different answers. Barbara: Oh, my lord..., and i t seems to like 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 isn't, because there are too many tangent lines. The existence of derivative was discussed with her in a tutorial session. By showing and proving to her that the right limit and le f t limit of f(x) at x = -1 were not equal (The same discussion in Math 100 text at U.B.C). She saw that the function was differentiable a l l along the line f(x) = 0 and a l l along the line f(x) = x + 1 as well, but not at point x = -1 although x = -1 belonged to both lines. The sharp points and the smoothness of a function were brought up by her. She fi n a l l y said: Barbara: Consider at the pick, I guess not [the f(x) is not differentiable at x = -1]. Barbara concluded this section by her following comment: Barbara: That's neat. I never knew anything like that, nobody told us anything about that. It's pretty neat.** ** This 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 class. But i t was interesting to see that how fast students could 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 is mine.] Barbara: We should have done this [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 circl e was not a function. As a reason she drew a vertical line to the c i r c l e . The line was hitting the ci r c l e at two points. Her logic was quite right. But the point was that she answered the question by remembering the vertical line not by thinking about the function and i t ' s properties. Her concept image of function was accompanied with the notion of "vertical l i n e " . Her last comment was interesting. She said: "I got 75% in Math 100, i t ' s good, but that doesn't really reflect like...you know. It was good to see that she was aware of her weaknesses which is a good start to understand the meanings and concepts. 4.2.11 Nick's Profile Nick was in f i r s t year science. He did well in Math 100 (he got f i r s t class). He did not have calculus at high school, yet he was in honor class. He was thinking of majoring in either biochemistry or computer science. He said that he liked mathematics. He was a bit 82 nervous in the beginning of the interview. We talked about different things t i l l the ice melted between us and he f e l t comfortable. Then the interview started. The interviewer asked every single interviewee to say the things that they were writing. Some of them did not seem to like this idea and kept quiet. Nick was one of those who mostly wrote rather than talk. At least 1/3 of the tape was f i l l e d with the silent moments. He did not like 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 were very much alike. Nick like most others had trouble finding the y-component of the point. For finding the slope of the secant l i n e , he was looking for "rise" and "run", but he said: "Here the trouble comes. I don't have the value (she meant y-difference.]". Before he started to find the slope of the secant PQj, he said that he could differentiate the f'(x) to get the slope of the tangent l i n e . But he had d i f f i c u l t y to understand that the slope of the tangent line at a fixed point P was the limit of the slopes of the secants PQ while Q got closer and closer to P. In a tutorial session, an effort was made to explain that the slope approached to its limit when the distance h approaches to 0. If he understood the concept of l i m i t , he would have no major d i f f i c u l t y in finding the slope of the tangent l i n e . His answer after the tutorial proved this claim when he found the slope of the tangent line without much of a problem. He only had trouble in substituting for f(x+h) because he had d i f f i c u l t y in understanding the behavior of function. Then the question of the derivative was raised. I: What is the relation between this curve and i t ' s derivative? 83 Nick: The derivative is the slope of the tangent line to a point on the curve, i f i t ' s defined in that point. I: The derivative of function at what point? Can you give the definition of derivative from what you said. Nick: The derivative of a function at the point tangent to i t ' s secant line is the slope of the secant l i n e . His vague answer needed to be c l a r i f i e d . Probing questions revealed some aspects of his concept images. His concept images made him to believe that the value of the derivative of a function will not be changed at various points. Nick: Shouldn't be one derivative for one function! [Emphasis is mine.] He knew that the derivative was the slope of the tangent l i n e , but he did not pay enough attention to the fact that there were different tangent lines at different points on the curve so there were different values of derivative associated to each of those points. In two tutorial sessions, the geometric representation was used and different tangent lines were drawn to explain that the derivative of function had different values at different points of tangencies. He again said: Nick: Shouldn't be one derivative for this function?! [Emphasis is mine.] He did not believe in his words. He had a mixed up concept images of derivative as a function and derivative as a numerical value. After further tutoring and mutual discussion he at last said that: "My definition would be wrong". His conclusion was a reasonable evidence to prove that the tutoring helped him to develop his concept image toward acquiring of the correct 84 concept definition. He then believed in (not just accepted) the fact that the derivative of the function had different values at different points. He got the derivative of f(x) = x2 + 1 at x = 0 using the definition of derivative. Because of his lack of understanding of function, he again had a hard time to substitute for f(x+h). For the question of differentiability (#7) he was wondering whether to use the definition. He f e l t confident to use the definition of derivative since he understood i t . 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 visualize i t . Nick: At x = 0 is 0[f'(x) = 0], at x = -1 is 0 again [f'(x) = 0]. I: Look at your function again [see Figure 3.1]. The function is 0 i f x<-l and here [pointing the function and i t ' s graph] f(x) is x+1 i f -l<x<0. Nick: So anyway, f(x) is 0, we substitute x into i t . I: What is the f'(x) at x = -1. Nick: At (a) f'(x) = 0 (at x = 0), and at (b) is 1 (at x = -1), [He referred to the parts (a) and (b) of question #7]. I: Then what is the f'(x) at x = -1. Nick: Should we change i t a l i t t l e bit?! [He meant changing the function.] I: We don't want to change i t . We want to keep i t this way. Nick: I don't know. I: Can you see i t from the curve? Nick: No response. 85 He was stuck. Silence was his answer to the question. Later on he asked for tutoring. In a tutoring session, I explained to him that the derivative of function at x = -1 had two different values. I related this to the definition of derivative and how the limit f(x+h)-f(x) w a s h+0 h varied i f the approach to the point x = -1 were from l e f t or from right. I: Then does f(x) have derivative at this point? Nick: I don't think there is a derivative. I: Why? Nick: Adams*** said that i t is a sharp point, because there is no tangent l i n e . I: Actually you can draw so many tangent lines, because by tangent line you mean secant passing through one point on the curve. Nick: I guess I know i t now. I: What is the condition for a function to have derivative? Nick: Not be sharp, just one tangent l i n e , Adams said, continuous, smooth interval [Professor Adams is teaching Math 100 and math 101 at U.B.C. His calculus book is the text for Math 100 and 101 at U.B.C.]. He referred to Professor Adams many times. He did not have enough self-confidence. 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: "I forgot what the definition of function i s . Is i t for only one x? Is i t one to one correspondence or that doesn't come to it? I would say i t i s " . Professor Adams is teaching calculus at U.B.C. 86 He tried to relate his concept image to definition of function. He was saying whatever he heard about the function without understanding them. Probing questions did not help and he asked for tutoring. The simple examples were used to explain the concept of the function. Also the textbook definition and geometric representation of function were described. His answer to the next question (inverse of the function) proved that he did not only agree with whatever was told in the tutoring, but he rather understood them. Because he could restrict the domain and the range of the circl e in order to have a function. He also said that the semi-circle was not invertible since there would be two y's for one x. 4.2.12 Ted's Profile Ted was a science major. He got 60% in Math 100 but he took i t again. He said that "I really 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 student at high school (B in Algebra 12). He did not know many of the interview questions, yet he was eager to understand them. He chose to begin with the questions on derivative. He did not get stuck t i l l he was faced with the question of slope of the secant l i n e . He then said: Ted: The slope is the... I think, the derivative of the function. I: Without using the derivative. Ted: O.K., Ay Ax I: What do you mean by Ay and Ax? Ted: Changing of rise over the changing of run. 87 Ted: I: What is rise and what is run for PQX? Rise is Qx - P, I say i t ' s 2 units. I: No, Q is an arbitrary point. What Ted said was rather interesting. He measured the distance from 2 (Y(P): The y-component of point P) to Y(Q) (The y-component of point Q). He did not notice that Q, was an arbitrary point. He assumed each space as one unit. He measured the y-differences according to his diagram which was approximately two centimeter and said that "I say it ' s 2 units". The lack of generalizability could be seen in his answer. His response was not the only single incident of this sort. This investigator witnessed many other cases that students' responses were the same with this 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 "rise is Qi - P" which means, he substituted a point i t s e l f for y-component of point. In a tutorial session, the coordinates of point and other related matters of this problem were discussed. After the tu 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 line as: limit (x0+h)2+l)-(x02+l) = limit 2 + h = 2. Ted explained this part h^ O h h+0 rather well. He jus t i f i e d whatever he did and i t was quite convincing that tutorial helped him to f u l f i l l his understanding gaps. The next question was to define the derivative. He said: Ted: Slope of tangent line is 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 is the point...no, the derivative is the...Oh, is the...I guess another definition is the tangent to the slope of the line at point x. He was a bit confused. He knew that the words derivative, slope and tangent were related. His concept image connected them together. His attention was drawn to what he was saying. Ted noticed his mistake and said: "O.K., you got a function f ( x ) , at a point x, the derivative is going to be the slope of the tangent line that touches the graph of the function f(x) at that point". His correct answer was surprising. The goal was to lead his concept image towards an adequate concept definition. The tutorial session helped him to reach the goal which was the correct definition of derivative by him. He had an interesting response to question #7. He said that the f'(x) at x = -1 was 0 and 1. I asked for his reasons. He said that: "Well, I can see i t and i t is 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 line 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 different values for derivative of function at the same point (x = -1). He did not know what he should do. He believed that the both of his answers were correct. 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 different 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, is a corner here [Pointing x = -1 in Figure 3.1]. I asked him the reason that why a function was not differentiable at the corner. His answer was a correct one. That there was no suitable tangent line to the function at x = -1. Interview continued as we moved to the questions on function. He gave correct answers to the questions on functions. He said that: x2 + y2 = 1 is not a function, "because for one value of x is two y values. So i t can't be a function". He had a good concept image of inverse function and reciprocal of function. He showed, geometrically, the necessary condition for a function to be invertible. He asked for more tutoring about the related concepts to the concept of function, (such as one-to-one function). He was eager to know, even though the time was up and there was no more tape to record on. 4.3 Response Category This section is divided into two subsections: 4.3.1 Categories of responses to the question of derivative. 4.3.2 Categories of responses to the concept of function. Each category will be defined and the students' responses will be quoted whenever i t is required. It should be mentioned that probing questions and the interviewer's intervention helped students to improve their concept images. For example, in response to question #7, almost all the students had d i f f i c u l t i e s to understand the concept of di f f e r e n t i a l i t y yet their later answers were mostly correct. 90 4.3.1 Categories of responses to the question of derivative Response categories on concept of derivative and i t s related concepts are discussed in this section. Following sections define these categories: Categories of responses to the definition of derivative. 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 line 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 identified. Category I Definition of derivative - textbook definition Two of the students gave the textbook definition of derivative as: f'(x)=limit f(x-fh)-f(x), Traditionally, using this definition to get h+0 h the derivative of a function is one of the questions on the final exam at U.B.C. (The copies of final exams in Math 100 and Math 101 are available from the Department of Mathematics in U.B.C). Those students who gave this definition, had d i f f i c u l t i e s understanding the concept of derivative. Their understanding was rote. For them this definition was a tool to express the derivative. This concept definition did not lead them to acquire the proper concept image. 91 Category II Derivative as rate of change - velocity Only one of the student referred to "rate of change" to define derivative. This student had a physical interpretation of derivative. In fact his concept image could lead him to the concept definition 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 correctly in terms of slope of the tangent line: "The slope of a tangent line at a given point, is the derivative of the function at that point". While some others' responses were not accurate, for example: "The derivative is the slope of tangent line at point P". Two of the students simply said that, "derivative is slope of the line at a point given". Seven students' responses have fallen into this category. Category IV Derivative as a rule of differentiat ion Students know how to use derivative as a tool to do their computation. One of the students gave the formula as definition of derivative. The following quote is self-explanatory: "No, I don't [defined derivative], I can do i t , f(x) = x2 + 1, f'(x) = 2x. My teacher did the proof, but I didn't understand i t , but I can do i t " [Emphasis is mine.] 92 And later on she said: "in terms of limit?! Oh, Ya, we did the definition of derivative... 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 th i s , you do i t so mechanical [Eemphasis is mine.] Categories of responses to the concept of slope of the tangent line 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 line as derivative of function Almost half of the students used the formula to get the derivative which they then call that the slope of the tangent l i n e . "OK, f(x) = y = x2 + 1, f'(x) = dy/dx = 2x... There is only one point, I can't use the same thing here, but I can give you the definition of derivative. This is giving you a right answer". It is interesting to know that the above quote belongs to one of the students that gave the textbook definition of derivative. If he understood that definition, he would not say that "I can't use the same thing here". He defined derivative as: "f'(x) = limit 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 limit in this formula. Category II Slope of the tangent line as the limit of slopes of other secants Many students had d i f f i c u l t y in understanding the concept of tangent l i n e . Only two of them responded correctly. "I guess h approaches to 0, then i f we take the limit f(x+h)-f(x) while h-*0 h 93 we can get the slope of tangent l i n e " . Three others responded correctly after the tutorial session. Some of the students had an idea that the slope of the tangent line should be the limit of the slopes of other secants. Their stumbling block was the lack of understanding the concept of limit and tangency. Categories of responses to the question of di 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) is not differentiable. Category I The right limit and l e f t limit are not equal Only one of the students' responses f a l l s into this category. Oh, Ya, you have to do limit 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 exist, no?! Because approaches to two different numbers. The above quote shows that he understood the concept of derivative because a function is differentiable i f derivative of function- exists at that point. By definition of derivative, this existence is equivalent to the existence of limit f(x+h)-f(x)t He knew that for existence of l i m i t , h+0 h the right limit and l e f t limit should be equal. Category II There is more than one tangent line "No, there isn't, because there are too many tangent lines". "No, i t doesn't have derivative at that point then?!...because i t has two different tangent lines at one point". 94 The responses of this 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 derivative. Yet the right answer obtains from both categories. Category III Function is not differentiable at sharp points "It is a sharp point, because there is no tangent l i n e " . "Probably, because i t ' s not a smooth curve. Two curves chain together". "It is a sharp point, because there is no tangent l i n e " . These responses stem from a practical and concrete understanding of the concept of di f f e r e n t i a b i l i t y . It is more convenient for students to look at the graph of function to see that whether i t is smooth or i t is sharp at some points. 4.3.2 Categories of responses to the questions of function Students' responses to the question of function is categorized into four major ones. Category I Some elements of the formal definition of a function "It's not a function, because for one value of x is two y values, so i t can't be a function". "It's not a function, because for every x there is two y, perpendicular 1ine". "It's not a function, because i f we draw a perpendicular l i n e , there is two value of y for one x". "Yes, and i t intersects the y axes at two points, so i t ' s not a function". 95 "71-x2 or yi-x2, two different functions put together". "Something defines the y values, certain values for 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 of function by means of an example. The responses show that these students have some proper concept images of function which could lead to their developing of an appropriate concept definition; yet the above responses in themselves contain only some of the elements of the concept definition of function. Category II Function as a relation between two variables \ This category represents a function as a relation between two variables. In fact function is a relation between x and y such that for each x there is only one y. The following quotes show that the concept images of these students see a function as a relation without having the restriction 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?!! I don't know, the equation is a function". Category III Function as an algebraic term, an equation The concept images of those students whose responses f a l l into this category, view the function only as an algebraic function. Although they have studied the transcendental functions ( i . e . , non-algebraic function such as exponential function, trigonometric function and logarithmic function). 96 "You mean i f I solve i t for x? or i f I solve i f for y? It is 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". "Function is just an equation". Category IV Idiosyncratic responses This category consists of a variety of responses. The following quote: "I know that i t ' s a business of drawing l i n e " , shows that her concept image of function was limited to a 1ine without even remembering that whether the line was vertical or horizontal to the graph. Although drawing a vertical line is a good test to check that whether or not a relation is a function ( i f vertical line hits the graph of relation at two points, i t shows that a relation is not a function). The following is an example of another idiosyncratic responses to the question of function: Owen: "No, i t doesn't, y = /1-x2, y1 = Ml-x2)~% (-2x) = ) (1-x2)3* I: Why did you take derivative? Owen: "I don't know, just instinct". "Yes, i t is a function, because i t has a certain number of points?!!" His response (as he called i t instinct) is reminiscent of a quote from Kalmykova who was interviewing a pupil. He said that (the pupil) "When I cannot arrive at the answer to the problem, I begin to add, subtract, multiply, or divide the numbers until I obtain the right answer" (Kalmykova, 1975, p.2). 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 . Five of the interviewees thought that f1 and 1/f. are the same while four of them said "I don't know". Two of the students did not have extra time to finish this part. Only one of the students responded correctly. Except for two students who did not finish this 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. 4.3.3 Summary of Results Calculus courses are mostly designed in a way to cover many topics in a limited time. In calculus classes there is not enough time for mutual discussion between the instructor and the students. Many concepts are considered to be known by the students prior to their 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" for 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 subtraction: (x + h) - x = h. Probing questions and tutorial sessions helped them to understand how to find the y-component of Q. Later on a l l of them got the slope of PQ correctly. Another d i f f i c u l t y for many students was seeing the tangent line as the limit of other secants and seeing its slope as the limit of the slope of other secants. When Q moved along the graph and got closer and closer to p, one of the students said, "The secant has disappeared, i t ' s really 98 disappeared". In his discussion of students' misconceptions of the tangent l i n e , Orton (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 differentiated f(x) to get the slope of the tangent l i n e . They were advised to follow the same procedure that they did for getting the slope of the secant lines. Only a few of them, then, got the slope of the tangent line as the limit of other secants. In order to get the slope of the line tangent to f(x) = x2 + 1 at P(l,2), a l l the students offered to either give the definition of derivative or differentiate f(x) = x2 + 1, yet they mostly did not know the relation between derivative of function and slope of line tangent to i t . Many of the students viewed the derivative as a useful tool with many applications. Three out of twelve students did not believe that the derivative of a function at different points has different values. Some of the students were given the graph of f(x) = x2 + 1 which had many tangent lines drawn on i t . They had trouble finding the derivative of f(x) = x2 + 1 at different points of tangency. One of them said: "shouldn't [there] be one derivative for one function"?! It was hard for them to conceptualize that the function of derivative remained the same while its numerical values were changed by changing the point of tangency. Students' d i f f i c u l t i e s in understanding function, hindered them in their understanding of derivative. They had d i f f i c u l t y finding the y-component of Q on the parabola (f(x) = x2 + 1) with arbitrary coordinates, since a majority of them did not know the behavior of this function. In the question on di 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 rule of correspondence (see question #7, Chapter three). Also, i t was hard for them to understand the concept of function. The graph of function (#7) helped some of the students to answer the question correctly. For those subjects who had d i f f i c u l t y in understanding the concepts of tangency and l i m i t , grasping on the concept of di f f e r e n t i a b i l i t y was harder. None of the students had d i f f i c u l t y in doing straightforward computation of the derivative (#6). The equation of a c i r c l e (x2 + y2 = 1) was given and the subjects were asked to determine that whether or not the c i r c l e represented a function. The immediate responses of seven students was "yes" to the question. Three of them drew vertical lines and said that c i r c l e does not represent a function since the line 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 is therefore not a function. After the tutorial sessions, most of the subjects restricted the domain and the range of x2 + y2 = 1 (circle with radius 1) in order to have a function. They mostly chose the upper semi-circle as their desired function. Two interesting answers of the same nature were given for this question. One of the students said: "make i t [circle] a l i n e , that's the only way" and the other one said: "you could expand i t [circle] just like that" _. They both thought that as long as the length of circumference of c i r c l e was conserved, they were allowed to make any changes to the graph. They did not understand the behavior of x2 + y2 = 1 which means that this relation between x and y should be conserved. 1 0 0 Later research questions drew students' attention to the definition of inverse function. They were asked to check that whether or not the inverse of the semi-circle was a function. A number of subjects said that the function and its inverse are mirror image of each other. Three of them said that the domain and the range of the function must be switched with each other. Those who used the term "mirror images", said that function and its inverse were "mirror image" of each other. Although what they said was true, using this term does not imply that students understood the concept of invertable function. They were asked to show and explain their responses by using the graph of semi-circle. They gave incomplete explanations as to what they meant when they used the term "mirror image". After the tutorial and discussion, many of them restricted the domain and the range of semi-circle and determined the quarter-circle to be a function whose inverse is also a function. After they did i t , this writer told them that they had now defined the invertable function as one which has to be a one-to-one function. They were very surprised by this result. For most of the interviewees, the difference between the inverse of a function f"1) and reciprocal of a function (1/f) was not clear. Most of them could not distinguish between 1/f and f"1. Their concept images dealt with 1/f and f"1 as i f they were the same. One of the students said that f- 1 is like x"1 which can be written as 1/x. 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 derivative, function and their other related concepts. B. To develop a category system for 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 year* university students were interviewed in this study. The collected data were analyzed and the individual profiles were produced for 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 definition. 5.2 Method The purpose of present study has been to investigate the nature of students' understanding of the concepts of function, derivative, and other related concepts. The researcher's aim has been to choose a method of collecting data that enables her to see the students' processes in constructing their mathematical knowledge. What students did in solving the problems discussed during the interview was not the investigator's only concern, but she also was interested in how they did 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 c l i n i c a l interview, provided the opportunity to address this concern. The tutorial sessions and probing questions helped students to improve their concept images. 5.3.2 Tutorial Session This writer believes that teaching has an essential role in students' formation of concept images particularly in an area like calculus. The tutorial sessions, in which the researcher acted as a teacher and provided some instructions for the students, seemed to be effective for revealing the students' concept images of function and derivative. Some of the students preferred to quietly write the answers to the questions, or more often, their immediate responses to specific questions were simply "I don't know". While some aspects of students' concept images were revealed by probing questions, in other instances some students appeared to be quite confused about the issues being discussed. In such a situation, the tutorial sessions were provided to enable the students to better understand the mathematical concepts being discussed. These sessions were judged to be useful i f they helped the students to acquire the adequate concept images. 5.3 Conclusions of the Study A number of tentative conclusions are offered in this section. They are presented in two different sections in the same order of the research questions. 103 5.3.1 The nature of the students' conceptual understanding The following conclusions are summary statements 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 meaningful understanding of the concept of derivative as was evident in terms of the types of responses given in the interview setting. 2) The students, except one, did not have a physical interpretation of derivative. 3) For some students, the algorithm of differentiating the function became the definition of derivative. 4) Some of the students did not believe that the derivative of a function at different points has different values. In general terms, then, their concept images were based upon the notion of derivative as a rule which assigns a number to each function even though derivative is a rule which assigns a new function f"1 to each function f . The nature of the students' conceptual understanding of the slope of the tangent line 1) Almost half of the students believed that there was a relation between the slope of the tangent line and derivative. Although most did not know that the slope of the tangent line is the derivative of a function at point of tangency. 2) Some of the students viewed the tangent as the disappearance of the secant l i n e . 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 dif f e r e n t i a b i l i t y was caused by their lack of understanding of the concepts of tangency and l i m i t , also by their lack of knowledge of the properties of a function and, in particular, the constant function. 2) Those who said that a function was not differentiable at sharp points and that the smooth function was differentiable, appreciated the geometric representation since their geometric intuition helped them to do so. The nature of the students' conceptual understanding of function 1) A number of students held proper concept images of function which should lead to the development of an appropriate concept definition. 2) Few of the students, understood function as a relation between two variables (without having the restriction that for each x there is 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 conflict between the students' concept images and the concept definitions of inverse function (f- 1) and reciprocal of function (1/f). 5.3.2 A system for categorizing students' understanding 1) The 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 was their degree of closeness of the responses to the concept definition. These categories appear to be useful from an instructional point of view for two reasons: (a) they provide the researcher with some insight into understanding the conceptual d i f f i c u l t i e s experienced by the students in the interview setting and hence enhanced the fruitfulness of the tutorial sessions; (b) they appear to make some intuitive 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 - Significant 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 is an intuitive understanding, to enable them to understand derivative. This writer believes that limit should not be presented by i t s concept definition. The abstract definition in the absence of students' proper concept images will not be of great help to students. Math 100 is concerned primarily with the derivative and i t s applications. It enables students to do a l l sorts of differentiation and to mechanically substitute different values of x in the derivative function to get different derivatives. None of the students failed to answer question #6 where they had to use the chain-rule to compute the derivative. The question aimed to check on the students' s k i l l in applying differentiation rules. The subjects seemed to be quite confident working with the derivative. To them, the derivative was a useful tool to do a l l sorts of things. 106 In the Math 100 class, the concept of function and limit are usually taught in a short period of time. About two weeks of class is devoted to the teaching of li m i t , function and a l l their related concepts (see Appendix C). Function and limit are p i l l a r s of calculus. Without them, students will not be able to understand such concepts as derivative. In the interview section on derivative, the students experienced great d i f f i c u l t i e s because of their lack of understanding of function. One of the educational implications of this study is for instructors to spend more time on the teaching of function and limit in calculus classes. Furthermore, they should use meaningful examples for teaching the concepts. This study shows that students are frequently unable to generalize their understanding since only few routine examples are used by instructors. Instructors usually only lecture and mark the final exams. Beyond doing the repetitive calculus problems, and getting good marks, students need to be confronted with more challenging and interesting problems. This cannot be done without direct involvement of students and instructors' in the problem-solving process. In this process, the instructors should be focusing on students' d i f f i c u l t i e s and looking for their causes. Being aware of the nature of students' d i f f i c u l t i e s , may direct their attention to the possibility of altering their teaching method. It is important for a teacher to realize the students' prior beliefs and how they might affect their further understanding. Ausubel (1968) says: 107 If I had to reduce a l l of educational psychology to just one principle, I would say this: the most important single factor influencing learning is what the learner already knows. Ascertain this and teach him accordingly (p. v i ) . The information of students' background and their prior knowledge of concepts should guide an instructor in choosing the way in which to present those concepts to the students. For instance, function is taught in high school in a different way than i t is presented in university. It is hard for students to see a connection between the high school definition of function and the one that is presented to them in university. The formal definition of function is not easy for students to understand since for the most part their concept images of function are not adequate. Concepts are usually presented by their formal definitions. Vinner (1983) said that i t is wishful thinking for "the concept image [to be] formed by means of the concept definition and under i t ' s control" (P. 295). This expectation is an idealistic view since many studies show that students usually forget the abstract definitions and, since they do not have a well established concept image, a l l that remains for them is the technique and some procedural knowledge so that they can do some computational work. For a majority of students, definitions in calculus are unrelated to their intuition. An example of this is the student who said that, for her, the inverse of a function ( f1) and the reciprocal of a function (1/f) are the same even though she believed that since there are two different names for them, i t means that they are not the same. This contradiction will clearly prevent her from obtaining any real understanding of these concepts. 108 There are conflicts between students' concept images and concept definitions. An example of this is a student who was asked to inspect that whether or not x2 + y2 = 1 was a function. She said that the circl e (x2 + y2 = 1) is a function without giving any reason for i t . The interviewer suggested to her to do the "vertical line test" which is used in calculus classes to check whether a graph represents a function. She failed to do the test. The researcher did the test to show her that the ci r c l e (x2 + y2 = 1) is not a function since the line intersect the ci r c l e at two points, which means there are two y values for one x. Her response was quite interesting. She drew the vertical line further (with x >1). Her response revealed two aspects of her concept image of function: 1. She ignored the fact that the projection of the vertical line on the x-axis should be in the domain of function. In other words, f(x) can be defined for x in the domain of function. 2. She did not know that this "line test" must be true for a l l lines not some lines. In other words for every x in the domain there should be one y such that y = f (x). It is 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 for some x in domain (or outside of domain) there is one and only one y in range. One of the stumbling blocks of students of calculus is their lack of self-confidence in dealing with mathematics. A student computed the derivative of f(x) = x2 + 1 (#5) at x = 0 by using the definition of 109 derivative. He did i t correctly but he crossed out his work even though, in question #2, he correctly used the definition of derivative to differentiate f(x) = x2 + 1 at x = 1. He said he was wrong because the answer should be 2. 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 different points. Many students complained about the class size. In large classes, there is l i t t l e mutual discussion between instructor and students. Students passively take their notes and their questions and concerns remain unaddressed. Small classes could provide the opportunity to both students and teachers to have better communications. An alternative model would be a mixture of large classes and tutorial sessions. In this researcher's opinion, the tutorial sessions will be more helpful i f they are offered by the same instructor of the large class. 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 calculus. The following are recommendations for further study which aim to identify students' understanding of calculus. 1. None of the students of this subject had taken calculus in high school. The existence of calculus in high school, has been controversial in British Columbia (B.C.) within the last few years. Many studies should be conducted in this area to give some insight into the question as to whether or not calculus should be taught in high school in B.C. 2. The possibility of having a pre-calculus course in high school should be examined. A pre-calculus course could prepare 110 students for calculus. Since i t presents fewer concepts, therefore instructor has more time to elaborate on questions and 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 is a necessity. Research is needed to investigate the teaching of function in high school. I l l REFERENCES Adams, Robert A. (1983). Single Variable Calculus. New York: Addison-Wesley Publishers Limited. Ash, Carol, Ash, Robert, & VanValkenburg, M.E. (1985). A Sensible Approach to Calculus. Selected Papers on the Teaching of Calculus. Aquirre, J . (1981). Students' Preconceptions of Three Vector Quantities. Unpublished Doctoral Dissertation. University of British Columbia, Vancouver. Ausabel, D.P. (1968). Educational Psychology: A Cognitive View. New York: Holt, Rinehart and Winston. Bishop, E. (1967). Foundations of Constructive Analysis. New York: McGraw-Hill. Cobb, P. (1987). Information-Processing Psychology and Mathematics Education - A Constructivist Perspective. Journal of Mathematical Behaviour. 6, (pp. 3-40). West Lafayette. Cobb, Paul and Steffe, Leslie. (1983). The Constructivist Researcher as Teacher and Model Builder. Journal For Research in Mathematic Education. 14(2), 83-94. Cobb, P., Wood, Terry, & Yackel, Erna. (1987). In E. Van Glaserfeld (Ed.), Constructivism in Mathematics Education, (pp. 1-30), Holand: Reidel, in press. Davis, P.J., & Hersh, P. (1980). The Mathematical Experience. Boston: Birkhouser. Davis, R.B. (1985, D e c ) . Calculus at University High School. Selected Paperson the Teaching of Calculus. Epp, Susannas. (1985, Dec). The Logic of Teaching Calculus. Selected Papers on the Teaching of Calculus. Erickson, G. (1987). Constructivist Epistemology and the Professional Development of Teachers. Paper Presented at the Annual Meeting of the American Educational Research Association. (pp. 1-44). Washington, D.C. Ginsburg, Herbert. (1981, March). The Clinical Interview in Psychological Research on Mathematical Thinking: Aims, Rationales, Techniques. For the Learning of Mathematics, 1(3): (1985). Montreal, Quebec Goodman, N. (1983). Reflections on Bishop's Philosophy of Mathematics. Mathematical Intelligencer. 5(3), pp. 61-68. 112 Gorodetsky, M., Hoz, R., & Vinner, S. (1986). Hierarchical Solution Models of Speed Problems. Science Education. 70(5). (pp. 565-582). Jones, Sue. (1985). Applied Qualitative Research. Ed. by: Robert Walker, Great Britain. Kalmykova, Z.I. (1975). Analysis and Synthesis as Problem-Solving Methods. (Volume XI of Soviet studies in the Psychology of Learning and Teaching Mathematics). Chicago: University of Chicago. Kenelly, John W. (1985, Nov.). Calculus as a General Education Requirement. 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. Kline, Morris. (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. Kline, Morris. (1985). Mathematics and the Search for Knowledge. New York: Oxford University Press. Lax, Peter D. (1985, Dec). On the Teaching of Calculus. Selected Papers on the Teaching of Calculus. 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: Springer-Verlag. Magoon, A.J. (1985, F a l l ) . Constructivist Approaches in Educational Research. Review of Educational Research. 47(4), pp. 651-693. Maurer, Stephen B. (1985, Dec). Reflections of an Ex Foundation Officer. Selected Papers on the Teaching of Calculus. Monk, S. (1987). Students7 Understanding of Functions in Calculus Courses. Seattle. Nichollas, J . , Cobb, P., Wood, Terry, Yackel, Erna, & Patashnick, M. (1988). Goals and Beliefs in Mathematics: Individual Differences and Conseouences of a Constructivist Program. (pp. 1-34). West Lafayette. Piaget, J . The Child's Conception of the World. London: Routledge and Kegan Paul, 1929. (Reprinted-Tonowa, N.J.: L i t t l e f i e l d , Adams and Co., 1969). 113 Pines, A., & West, L. (1986). Conceptual Understanding and Science Learning: An Interpretation of Research Within a Sources of Knowledge Framework. Science Education, 70(5), (pp. 583-604). Polia, G. (1945). How to Solve It, A New Aspect of Mathematical Method. 2nd edition. Renz, Peter. (1985, D e c ) . Steps toward a Rethinking of the Foundations and Purposes of Introductory Calculus. Selected Papers on the Teaching of Calculus. Rodi, Stephen B.J. (1986, June). Some Systematic Weaknesses and the Place of Intuition and Application in Calculus Instruction. A Contributed Paper to the Sloan Foundation Conference on Calculus Teaching: New Orleans. Sinclair, H. (1987). Constructivism and the Psychology of Mathematics. In J . Bergeron, N. Herscovies, & C. Kieran (Eds.), Proceeding of the Eleventh International Conference, Psychology of Mathematics, 1, (pp. 28-41). Montreal. Steen, Lynn. (1985, D e c ) . 20 Questions for Calculus Reformers. Selected Papers on the Teaching of Calculus. Stein, S.K. (1985, D e c ) . What's All the Fuss About. Selected Papers on the Teaching of Calculus. Stevenson, James R. (1985, D e c ) . Physical Science and Introductors Calculus. Selected Papers on the Teaching of Calculus. Thorn, R. (1971). Modern Mathematics: An Educational and Philosophical error? American Scientist. 59, pp. 695-699. Vergnaud, G. (1982). Cognitive and Developmental Psychology and Research in Mathematics Education: Some Theoretical and Methodological Issues. For The Learning of the Mathematics, 3(2), (pp. 31-41). Montreal. Vergnaud, C. (1987). About constructivism - A Reaction to Hermine Sinclair's and Jermy Kilpatrick's Papers. I.J. Bergeron, N. Herscovies, & C. Kieran (Eds.), Proceeding of the Eleventh International Conference, Psychology of Mathematics. I, (pp. 42-54). Montreal. Vinner, Shlomo. (1975). The naive Platonic approach as a teaching strategy in arithmetics. Educational Studies in Mathematics: Dordrecht-Holland, 339-350. Vinner, Shlomo. (1982). Conflicts Between Definitions and Intuitions -The Case of the Tangenet. Proceedings of the Sixth International Conference for the Psychology of Mathematical Education: Antwerp. 1 1 4 Vinner, Shlomo. (1983). Concept Definition, 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 British 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 calculus. Selected Papers on the Teaching of Calculus. 1 1 6 APPENDIX B Oral Consent Form (To be read or given to students prior to interview) This project is 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 for 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 Topic Sections Hrs 1 D I F F E R E N T I A T I O N a functions, domains, ranges, graphs, compositions, odd and even symmetry, 1.2 2 inverses b limits (intuitive - t-6 not necessary), properties of limits, one sided limits, 1.3-1.4 2 infinite limits, limits at infinity c continuity, intermediate value theorem (statement) 1.5 1 d tangents, normals, derivatives and differentials 2.1-2.2 2 e differentiation rules 2.3-2.4 3 f interpretation of the derivative, rate of change, velocity and acceleration, 2.5 1 marginals g higher order derivatives, differential equations, initial-value problems 2.6 1.5 h implicit differentiation 2.7 1.5 i the mean-value theorem (omit proof), increasing and decreasing functions 2.8 2 j antiderivatives and indefinite integrals 2.9 1 II E L E M E N T A R Y F U N C T I O N S a trig functions and their derivatives, projectiles, simple harmonic motion 3.1-3.2 4 b inverse trig functions and their derivatives 3.3 2 c natural logarithm and exponential functions 3.4-3.5 3 d general exponentials and logarithms, logarithmic differentiation 3.6 1.5 e exponential growth and decay, logistic growth 3.7 1.5 III A P P L I C A T I O N S O F D I F F E R E N T I A T I O N '""! a local and absolute extreme values, critical and singular points, first deriva- 4.1 1 tive test b concavity and inflections, second derivative test 4.2 1 c asymptotes, formal curve sketching 4.3 2 d optimization problems 4.4 3 e related rates problems 4.5 2 f tangent line approximation, error estimate, Newton's method 4.6 3 8 indeterminate forms, l'Hopital's rules 4.7 2 Total 43 Tests and leeway 6 Approximate number of class hours 49 O P T I O N A L T O P I C S (if time permits — not examinable) hyperbolic functions and their inverses 3.8 parametric curves 5.1-5.2 vector velocity and acceleration in the plane 5.3 • Text sections refer to SINGLE-VARIABLE CALCULUS, Revised Edition, R. A. Adams, Addison-Wesley Canada, 1986. • Calculators, while useful for some parts of the course, w i l l not be required for the final examination, • Term marks for the course should be based all or mostly on in-class tests, and should be scaled within sections (when the course is finished) so that the final marks for a section have the same median or mean as final exam marks for that section. M A T X 1215 Local 3782 1 1 8 APPENDIX D Written Work From One of the Students' Profi1es A1- -• 4 7 - -- • ^ .L ^ r.±l - * * 1 rti/ -. ^ -. . "n y \ -\ 1 ' ; — ~ V - X" • • • - g •-J3 1 1 9 APPENDIX E Exemplary Transcript From A Student Interview Analyst Comments I: Interviewer Barbara: Subject I: Before you start, I like you to read this consent form. I: How did you do in Math 100? Barbara: Like marks? You know is good, but... I: What did you get? Barbara: I got 75% for the term. But that really doesn't reflect l i k e . . . you know, so... I: What about this term? Barbara: This term? I think [Math] 101 is a bit harder, because you know, in [Math] 100, you just...like chain rule does a chain rule, power rule is power rule, you just gonna do i t . That's not so hard, because there is something specific, 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 well, you will make i t . I: What is your major? Barbara: I don't know. I think I ' l l gonna go to Physics though. I: So you will 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 like Biology and I don't really like Commerce because I'm not really into the money, so...Math is like...I don't really like i t that much, but 1 2 0 i t doesn't bother me, but i t ' s more like a tool, you know, when you need i t , for...1 ike Physics. I: You didn't have calculus in high school, did you? Barbara: Ya, actually I did about a month of i t . My teacher wanted to introduce us to these stuff. I: Was i t in Algebra 12. Barbara: Ya, just the rich Algebra or some-thing, but he only gave us a month and gave us the test. Because we did differentiating and integrating and i t was too much for us, too much. I have forgotten a lot of i t . I: O.K., Now, I like 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 is the other choices? I: Leave i t and go [Laughing]. Barbara: Is i t [Laughing]?! I: I was just kidding. You may start with derivative. 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. It's hard to do i t mentally. Question 1: a) Find the derivative of She started f(x) = x2 + 1 at x = 1 to read the b) Sketch the graph of question out f(x) = x2 + 1 loud. Barbara: I love sketching the graphs! I'm good on that. 121 I: Do you like to go through a l l of the questions and then come back and discuss each one of them or do you like to finish 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 like to see how you are doing i t . Barbara: To see how I do i t ! 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 this [question 1], O.K., so...I guess, f(x) = x2 + 1, f'(x) = 2x at x = 1, f ' ( l ) = 2. I like that [laughing]. O.K., sketch the graph of i t . I: Yes. Barbara: O.K., I call i t y, because I like y [She wrote y = f(x) = x2 + 1] and this is a parabola [She sketched the graph correctly]. I: Right on! 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 directions. Such lines across a Parabola are called secants, and some examples are shown in diagram. a) How many different 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 PQ1 Find the slope of PQ2 d) Find the slope of L at point P = (1,2). She f i r s t read the question. Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: Barbara: I: Lines PQ's? Yes, PQj, PQ2, PQ3... What do you mean [line extended from] both directions? I mean i t ' s not bounded, i t ' s not limited, you know, you can extend as [I was interrupted]. Oh, Just a line with arrows. It doesn't have two arrows... Oh, O.K., not a segment, a l i n e . Yes. Such lines across a parabola are called secant [reading a part of the question again], O.K., that's true, ya...How many different secants could be drawn in addition to the ones already in the diagram? Through point P you mean? Yes. How many?! I guess...a lot [Emphasis is hers], infinite numbers I guess, because you can just...you know, angle i t or something. Do you want me to write that down? 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 Barbara: Barbara: Barbara: Barbara: I: Barbara: I: Barbara: I: Barbara: becomes more like a tangent [laughter]. That's what I think any way. O.K., that's correct. Find the slope of...here to here?! [She showed PQ: on the graph]. Yes. O.K., what is this point [Pointing P]. This is point P = (1,2). That's (1,2)? O.K., O.K., so this minus this. What is the height of this [Pointing QJ. The length of this [QJ is x0 + h. This distance? [she showed xQ + h on the graph, but she was not sure]. Ya, the whole thing. O.K., I just redraw i t . And you want the slope of this [PQJ? Yes. Oh, i t ' s that the function? [She meant f(x) = x2 + 1 Ya, the function is f(x) = x2 + 1. The diagram shows the graph of the above function which is f(x) = x2 + 1. [The figure was in front of her], O.K., so I guess y at Qj would be equal...Just wild guessing now...I guess y of Q;. I have no idea...well, I guess i t will be difference between these two points I guess. Many guessings are going on here [laughter]. The copy of her work is included, (see Appendix F) That's [slope of PQ,] Ay = + 1 - 2 Ax 1 + h - 1 What do you mean by Qx2 + 1? 1 2 4 I: Barbara: I: Barbara: What is this [Q:2 + 1] for...?! I: Qj is the point right? Then what do you mean by + 1? Barbara: Oh, Ya, you're right, you have to put x in eh...O.K., so that's (1 + h)2 + 1, ya, that's more like h something that we were doing. That's i t . This is [(1 + h)2 + 1] height of Qr Ya, ya. Then what would be this distance [the amount of r i s e ] . See what did you get for i t [she got (1 + h)2 + 1). Barbara: Oh, I forgot -2, O.K., so: Ay = (1 + h)2 + 1 - 2 Ax h I also make lots of middle errors... [laughter]. Do you want me to finish it? I: Yes, please. Barbara: Slope?...Last term memories are coming back. We take the limit I guess. I: What is this [I showed her what she got as rise = (1 + h)2 + 1 - 2] run h Barbara: Oh, that's the slope. I: O.K., then... Barbara: That's i t ? ! I: Are you looking for something else. Barbara: I don't know! 125 I: Barbara: Because you told me that this is [(1 + h2 + 1 - 2 ] slope. You just h finish i t up. Oh, Oh, I know what are trying me to do, l2 + 2h + h2 - 1 = 2 + h?! I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: O.K., that's fine. What would be the slope of PQ2. [Laughter] No! No! PQ2 is another secant. Are they a l l about the same?! Usually?! What do you mean that they are all the same? What would be the x part of this part? Just another h. xQ plus another h. We can name i t hp h2, whatever, just another h. Oh, different h?! We named this distance xQ + h, so when we move down, you know, then this distance will be changed. So this is not the same h, i t would be another one. Ya, so I guess slope of line [PQ2], will be 2 + hr I call i t hr O.K., then what will be the difference between 2 + h and 2 + h The h part. What happened to h part. It gets smaller. O.K. Now find the slope of L at P(l,2). She under-stood this part rather well. 126 Barbara: I: Barbara: I: Barbara: I: Barbara: Barbara: Find the slope of L?! O.K... f'(x) = 2x and f ( l ) = 2. No, you already got that. Oh, Oh, you want me to actually do i t ! O.K., I guess h is approaching zero, so i t would be one of these things again, so i t would be 2 + h. We take limit as h approaching zero, because i t ' s too close like point [Q] coming down down down and then get 2. [Emphasis is hers.] O.K. So I pointed up there. O.K., Now, can you give the definition of derivative. Oh, O.K., what I do... *f'(x) = f(x0 + h) - f(x) (x + h) - h It's not xQ, because you said f'(x) Ya, O.K., O.K., inconsistency! That was 1 so there is only x, so f'(x) = f(x + h) - f(x) (x + h) - x f(x + h) - 1 *I intentionally did not say that her definition of derivative was wrong. I did not want to confuse her by the notion of limit at this stage. I: Remember what did you do here! Barbara: Oh, Ya, where am I?! I: You just simplify i t . It was (1 + h)2 + 1 - 2, you just add them up h together and got that. I: I like you to give me the definition of derivative in general. Barbara: Oh, in general, i t is f(x + h) - f(x) (x + h) - x I guess which is really h at the bottom. 127 I: What is 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 is the relation between this curve and it ' s derivative? Barbara: You mean the parabola?! I: Yes. Barbara: And i t ' s derivative!...Well, the derivative will give you so like 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 like to the curve. I: You mean slope of a point? Barbara: Yes. I: Does a point have a slope. Barbara: Not really, but...the slope of the curve at that point. I: Does a curve have a slope? Barbara: I guess not. [Laughter.] 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 is equal to the slope of what? By definition, slope of the curve at point P is the slope of the tangent line at the point. Her answer was correct. I just asked this 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?! They are two points on the curve?! I: It wasn't the derivative. 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 limit of h approaches to zero. I: O.K., then what is L, what's the position of this line to the curve. Barbara: I really don't know [laughter]! I: You know! Barbara: I do?! You think I know?! I: Ya, what's the position of this line to this curve [I showed her the tangent l i n e ] . Barbara: I see the l i n e . I: How this line and this curve are related? Barbara: How are they related?! It touches i t at one point only. I: O.K., what does that mean? Barbara: I don't know what I mean [laughter]! I: You said i t touches i t at one point, then what we call this line? Barbara: What we call the line?! Oh...I 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? behind i t . By this line line of reasoning she seemed to get confused. I should be more ex p l i c i t . The term position might be misleading here. Tutorial Session Barbara: What happens to the secant?! 1 2 9 I: You said i t by yourself. As Q gets closer and closer to P, then what happens to the secant? Barbara: The secant is disappeared, does disappear, i t really 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 lines are extended from both directions. So i t doesn't mean that i t disappears. The secant ya, PQ will be disappeared, but the line is there. P is fixed and we are just moving Q towards P, so i t means that i t is the same line which has rotated around P. It just hits the curve at one point. Barbara: Ya. I: Then what we call this line? Barbara: I call i t the tangent, but... I: O.K., back to here again. You found the slope of the tangent line at this point [P = (1,2)]. You got the slope of L [tangent line] by taking the limit of a l l these slopes. You said that i t approaches to its l i m i t . Then you gave the definition of derivative. Barbara: That! [Pointing f'(x) = lim f(x + h) - f ( x ) ] . h-0 h I: Ya, how did you get this definition? Barbara: How did I get the definition?! I: Yes. Barbara: I just did this, sort of, only more general term. [She meant f(x + h) - f [ x ) h h 1 3 0 I: O.K., then what is the relation between this curve and i t ' s derivative? Look at here that what did you get for the slope of the tangent l i n e . Do you see any kind of relation. Barbara: I should! [Laughter]... I: How did you come to the idea that this limit [lim f(x + h) - f(x)] i s f (x) . h+0 h Barbara: I: I: Barbara: I: Barbara: I: Barbara: I: It looks to me...I always call slope sort of that line that P like about almost very close. O.K., then call i t again... i t the and Q O.K., you are saying that the derivative...you call that the derivative i t slope of what? The slope of the secant which I guess is almost like a tangent at a point. 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 line] as the limit 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 is it? Derivative [laughter]. What is this derivative? What i s i t [Emphasis is hers]?! Do you want the definition?! No, you already gave the definition as f'(x) = lim f(x + h) - f(x)] h-0 h which is quite right. Then what is this derivative? You said that this derivative is sort of slope. Slope of what? 1 3 1 Barbara: Slope of the tangent?! To a point on the curve! I: O.K. Yes. Barbara: [Laughter] O.K., that eased lots of pain. I: O.K., this is the f'(x) at what point? Barbara: Any point along the curve. You just pick i t and plug i t i n . I: You say any point along the curve. But at what point did 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 for any point. I: You said that f'(x) is the slope of the tangent line at any point. Let's change the point P = (1,2) [Point of tangency] to P, = (0,1). Then what would be the derivative of function at this point? [I sketched the graph]. Barbara: Oh! I: You said that at any point. But at this point [Px = (0,1)] you got 0 for derivative and at P = (1,2) you got 2 for derivative. 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 the curve?!...[Laughter]...ask me the question again. Tutorial Session 1 3 2 I: O.K., you said that this [f(x)=lim f(x+h)-f(x)] is the h-0 h derivative of function at any point. But we said that this derivative at length of any point is equal to the slope of the tangent line at that point. Barbara: Ya, at that point. I: O.K., what is that any point and what is this specific point? How can you relate these two points together? At any point and at that point [Emphasis is mine]. You said any point and that point. How are these two related? Barbara: That is the specific of any which is the general term. That part of that group [Emphasis is 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 is equal to the slope of the tangent line 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 is equal to the slope of the tangent line at point p = (1,2). If we change This is a very common mistake to say that "derivative of function at any point is the slope of the tangent line at that point". "The derivative at the x-component of 1 3 3 the point, we can see that the value of derivative will be changed when the point of tangency is 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 different values for derivative. Barbara: Oh, what a struggle any way... I: Do you know that where have a l l differentiation formulas come from? Like chain rule etc. Barbara: Ya, that's the whole bunch of these stuffs. I: We derive a l l these formulas from this definition of derivative [pointing f'(x) = lim f(x + h) - f(x)], h-0 h I: Now, compute the derivative of above function [f(x) = x2 + 1] at x = 0 by using the definition of derivative. Barbara: lim f(x + h) - f(x) = h-0 h the any point". I did not pick on this since they realized the difference. Just the wording was not accurate. lim ((x + h)2 + 1) h-0 h (x2 + 1) That's really a definition?! I: Why you doubt it? Barbara: I don't know, I have mental elapses right now [laughter]. I: Just look at the curve again and check this definition. 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 is f ( x ) . Then you want to find this [f(x + h) - f ( x ) ] . 1 3 4 lim (x2 + 2xh + h2 + 1) - x2 -1) = h-0 2x + h Then f'(x) = 2x at x = (0) = 0. We should have done this [interview] at the end of December. There was some kind of memory. Right, you check i t by the graph too. This derivative is the slope of the tangent line at x = 0 which obviously i s . . . A f l a t angle. And i t s slope is 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 definition of derivative]?! Oh, No, No. Using the formula. =50 (1 + 1/77 - 5x)49 d/dx (1/77 - 5x) What is this formula called? The one that I'm using it? Ya. Chain rule, O.K., Woh, what a struggle to remember that stuff. But you are using i t again, in Math 101. Ya, Ya, we are in series now, power series, woh! O.K., I have to differentiate this as well, so d/dx (1 + 1/77 - 5x) = (7 - 5x)-1/2"2/2 then d/dx (1 + 1/77 - 5x)50 = 1 3 5 4 9 Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Thanks. That's fine. Now compute the at: a) x=-2 b) x=-l c) x=-l/2 d) x=l/3 e) x=l f) x=10 [See Figure 3.2 for the diagram]. 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 is the same thing, x<-l again and f(x) = 0. c) [I did not let her to continue] what about here [I showed her that f(x) = x + 1 at x = -1 as well]. Oh, Jee, two points! The derivative is f'(x) = 1. O.K., so what is the derivative of function at x = -1? ...[silent] You said 0 and 1 right? Actually there is nothing to plug i n , that's the real thing. Why there is nothing to plug in? The derivative of this is 1. Is that for any x? 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? derivative of f(x) = fO - x+1 -x+1 0 x<-l -l<x<l 0<x<l x>l Barbara: How we can interpret this?! 136 I: Yes. Look at the function again. f(x) = 0 at x = -1, but f(x) = x + 1 at x = -1 as well. Barbara: Oh...what a question! Something strange is coming. Could i t be two different answers?! I: You t e l l me. Barbara: Because there is two function for... I: There is not two functions... Barbara: One function. I: This function is defined this way. Barbara: Ya, i t ' s defined this way. Two definitions 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 this?! I: Ya, that the function has two different derivatives at one point. Remember we said that the derivative of a function at a given point is the slope of the tangent line at that point, right? Barbara: Ya. I: 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 like i t sort of [see Figure 3.2]. I: Well, you said there are two tangent lines. I'm arguing that there are 1 3 7 Barbara: I: Barbara: I: Barbara: I: more than two you said two but I like to have this one and this one [I drew many tangent lines to the curve such that they a l l passed through x = -1]. Then what is the derivative of function at that point? I guess from 0 to 1. Such a thing. So I guess derivative is from 0 to 1 like the range goes from 0 to 1. What does that mean, derivative? What is the I don't know, - I I guess. No, I mean you said that derivative at a length of a given point, is the derivative of function at that point. Oh, Oh, What a fun! When you say that derivative is from 0 to 1, you mean that the slope of the tangent line is between 0 and 1. Barbara: Passing through that point. I: I showed you that we can have many more tangent lines which they pass through that point, they're not only the tangents with the slopes of 0 and 1. Barbara: I think these may be the highest tangent and lowest tangent, like highest slope and lowest slope. I: Have you remembered something like 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 differentiable at x = -1. Barbara: Yes, something exists, so much exists [laughter]. Her response was quite interesting. She is defining derivative as a range of slopes. In advanced mathematics when derivative does not exit, a subderivative is defined as a set of points like an interval. For example f'(x0) is value of subderivative and in this case is [0,1]. If f is differenti-able at point Pxn, then f i x . ) = {c}. She tried to complete her definition later on. I was not aware of this definition, when I conducted the interview. 1 3 8 I: What is the condition for a function to be differentiable? In what condition the function is differentiable? Barbara: Oh...I guess at point x, there should be only one answer for the derivative [Emphasis is hers]. I: What is that answer for f'(x) at x = -1. Barbara: What is that answer?! We call the slope of the tangent at that point. I: So you are saying that there must be one tangent line at that point. But there is not a suitable tangent line at that point. There are so many, so is this function differentiable at x = -1 or not. Barbara: That's weird! I like 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 differentiable [no trust on her words]?!...Yes i t i s , here is the answer. She liked challenging questions. Why i t is [differentiable] and why i t is not. Barbara: Oh, I guess i t i s , because... What does the differentiability 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) is differentiable at x = -1 or not]. 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 like t h i s . It is good stuff. Yes or No?! No I guess. 1 3 9 I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Then why no? If i t ' s no, why not. If i t ' s yes, why yes. I think, i f i t ' s yes, because...it existed like you calculate i t . O.K., but we calculated and we came up with two different answers. Oh my Lord... like a range tangent line. and...it seems to of slopes of that be What do you mean by "the range of slopes"? O.K., O.K., I say no, there isn't, because there are too many tangent lines [laughter]. O.K., the f(x) is not differentiable at x = -1, because when we move from l e f t to right which means from -» to -1, the slope [of the tangent line] would be 0, right? Ya, Ya, right. But when we move from right to l e f t , the slope [of the tangent line] would be 1, which is the slope of this line [pointing on the graph] including x = -1. Then this function is not differentiable at this point. Because the limit from l e f t is not equal to the limit from right. Oh, Oh, that... Because the derivative is the limit of this slope. When that distance [ i t was showed on the graph] approaches to 0. But from here, [right limit] we get two different l i m i t s . Which mean two different values for limit at this point. So the function is differentiable all along this line [pointing line f(x) = 0] and all along that line [pointing line f(x) = x + 1], but function is not differentiable at this point. She is s t i l l thinking about her definition of sub-derivative. Tutorial Session 1 4 0 Barbara: Ya, because the l e f t and right limit 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 this function was not differen-tiable at x = -1. Barbara: I: Because, l e f t and Barbara: I: Barbara: .didn't you t e l l right limits are me?! The not equal. An in a very simple way, those functions are differentiable that are smooth. Smooth functions are differentiable. Remember the parabola [f(x) = x2 + 1] we could draw tangent line at any point on the curve. Consider this one. This is differentiable at a l l the points of the tangencies [I sketched two smooth curves]. But consider this [a graph with a pick] Is this function differentiable at that point [pointing the pick]. Oh, at the pick?! Yes. I guess not, because this side is like this and this side is like t h i s . These two are straight lines. [So function is not differentiable at the pick]. That's right. When a graph of a function is sharp at some points, so function is not differentiable at picks. And the same thing here again. For example y = |x| is not differ-entiable at x - 0. When the function is smooth, then i t is differentiable like lines. Barbara: This is not continuous. 141 I: Barbara: Barbara: I: No, i t is 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]. Oh, Oh, you don't stop. We didn't stop here [at the pick] but...this is a sharp point. To be differentiable, function must be smooth, without a sharpness. And we didn't stop here [at the pick] but...this is a sharp point. To be differentiable, function must be smooth, without a sharpness. O.K. So far so good! that. Any questions on Barbara: That's neat! I never knew something like 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 logic. Because I'm interested on your responses on function and we don't have more than 10 minutes. Barbara: I: Barbara: Barbara: Oh, Oh, I have physics [class] right now. But who cares, I ' l l stay. I won't take more than 5-6 minutes of your time. "Consider the equation x2 + y2 = 1. Sketch the graph of this equation! O.K., this is a circl e with radius 1 [She sketched the graph]. Is that i t ? ! Yes, that's fine. "Determine whether the above equation represents a function y = f(x) or not". Oh...strange! y2 = 1 - x2 so y = ± 71 remember. 7. Is i t +?! I can't 1 4 2 Ya. Barbara: I: Barbara: I: Barbara: Barbara: Barbara: I: Ya, better be! Is is a function or not...[silent]. Can you say i t from the graph! No! It's not a function. Why? Because...for certain point of x there are two y's, y value. In function, for a point x, we have to have only one y value. That's right. "Determine..." didn't I answer that [question 2 on functioning]. Yes, you did. O.K., "If not, determine a domain and a range such that the above equation is a function". O.K., y...Just the positive values, y > 0, so restrict to only positive y's. Then what is the domain and the range of this function [I meant upper semi-circle]. She started to read the next question, but she stopped. Barbara: Domain i s . . , from 0 to 1. •1 to 1 and range is Right Barbara: O.K., "What is the relation between the domain and the range of a function and it s inverse and it s reciprocal [l/f(x)]"? I guess you can say either you f l i p the variable around in an equation, for example y = x + 1, i t ' s inverse will be x = y + 1. I: O.K., now what's the difference between f(x) and l/f(x)? 1 4 3 The difference between this Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: [y = 71 - x2 which is upper semi-circle] and l/f(x)? The inverse is when you switch x's for y's and y's for x's. But l/f(x) is the reciprocal, and that's not necessarily equal, so...what am I going to conclude [laughter]?! [I repeated question 4 again]. Oh, they are not the same, are they?! Look at the graph. This is 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? Can I redraw it? [She correctly did i t . ] So was not a function any more i f we invert i t . Good. O.K., O.K., then no domain! no range! O.K., how we can restrict the domain or the range of the function [y = yi - x2] in order to have an invertable function. Say i t again. You said that i f we f l i p i t over, i t is not function any more. Now what can we do to the domain and the range of the semi-circle [y = yi - x2] to make i t an invertable function. Barbara: O.K., Let's cut off 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 restrict the domain to just positive x's or just negative x's. 1 4 4 I: Now you just defined the invertable function. It is a function that for every x [I was interrupted by her]. Barbara: There is only one y and... I: Remember you restricted again. Barbara: Then...you mean a function is 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 different y's and you restricted to quarter of the c i r c l e . Then i t means that before you cut off semi-circle you had two x's for one y. But after you cut off one quarter of i t , then we have what? Barbara: One x for one y. I: O.K., and one y for one x [She got this from the beginning]. We call this [pointing quarter of circle] one-to-one function such that there is one y for every x and there is one x for every y. I: Now what is the domain and range of inverse function. Barbara: Oh, [0,1] and [0,1]. I: Let's see what is 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) is D and range of f(x) is R, then what would be the domain and range of inverse function? Barbara: [laughter]...backwards! I: What do you mean by backwards? Barbara: Oh, switch them around. I: O.K., which means... 1 4 5 Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: I: Barbara: Ohm...What does that mean?! I'm not very good at words [laughter]. You said backwards. What do you mean by backwards? I guess domain and range... D[domain] becomes R[range] and Range becomes Domain. 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)? 1/D, 1/R?!...D and R?! Do you have any restriction for it? The bottom [f(x)] better not be 0. Better not be or should not be 0. The f(x) should not be 0. O.K., that's fine. Now compute one of the following please [composing of functions]. "If f(x) = /x and g(x) = x + 1, compute one of the following: a) fog(x), b) gof(x), c) gog(x), d) fof(x)". Oh I remember doing that. That was a long time ago, few months or so. I ' l l try one...woh . . . i t ' s ancient history now. O.K., a) f(g(x)) = !x~ + 1. Thank you very much for your time. You're welcome. 1 4 6 APPENDIX F Written Work from the Exemplary Student Interview 3) f ( ^ V-y* H '~a*'*-. T5T -2X AT / / / ) r r -1 J V \ V \ ' -i 1 1. SC. \ v r v 1 ; - z ^ <\* — — 1 W \*\\ -_ t /A . < _ 1. a / • o / \_SN-. — —) A- W , W *«,».! it*. 1 4 7 1 4 8 149 1 5 0 NAME DATE 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. TELEPHONE CALL NO. AUTHOR/TITLE PAGES REQUESTED TO DE PICKED UP TO BE MAILED MAILING ADDRESS (Surcharge will be added) (For Office Use Only) OVERSIZE SPECIAL INSTRUCTIONS EXPOSURES PRICE $J_ MAILING CHARGE TOTAL $ PAID •• STAFF •_ NOT PAID APPROVED EU-12 (5/86)
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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 |
AggregatedSourceRepository | DSpace |
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