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A. ..|=i|i.svllno. .7‘1..v.- . c! 3. 1o: [I IIJ i lll'l'illllil'ilIllilllllllll,lilllt Illllliill 3 1293 00914 3110 This is to certify that the dissertation entitled Making the Connections: The Mathematical_Understandings of Prospective Secondary Mathematics Teachers presented by Eric Wood has been accepted towards fulfillment of the requirements for Doctoral degree in Philosophy professor M Date 3/15/93 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 _fi‘ _-_‘ _ N ‘ LIBRARY Michigan State UII‘VCI'IIW i K ‘__I' ‘V .PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. ———————-——-—.——————'——_1 DATE U.E DATE DU E DATE DUE 5-13 6:. 711% _ 03 2 1 1 2 1 w_____ L____ii; it'll—”T J MSU Is An Affirmative Action/Equal Opportunity Isothutlon ”39.1 MAKING THE CONNECTIONS: THE MATHEMATICAL UNDERSTANDINGS OF PROSPECTIVE SECONDARY MATHEMATICS TEACHERS BY Eric Frederick Wood A DISSERTATION Submitted to Michigan State University in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Teacher Education 1993 This pa} adtess the * are able to : Ethezatica‘. that, are com The stu interviewed hour) couple the One hOur ABSTRACT MAKING THE CONNECTIONS: THE MATHEMATICAL UNDERSTANDINGS OF PROSPECTIVE SECONDARY MATHEMATICS TEACHERS BY Eric Frederick Wood This paper reports the findings of a study designed to address the way in which prospective mathematics teachers are able to make connections between and among the mathematical concepts that they know, particularly those that are commonly found in secondary school curricula. The study used in-depth interviews (8 students were interviewed 6 times, each interview lasting approximately 1 hour) coupled with a large sample survey (121 students took the one hour survey) to examine how prospective secondary teachers make connections between and among mathematical ideas and other areas of study. The survey responses were coded and analyzed by statistical methods while the interviews were transcribed, coded and analyzed for patterns by more interpretive methods. Many of the ideas that prospective teachers hold are fragmented and disconnected from other potentially powerful mathematical concepts. Even when linkages are drawn they are often made on surface or contextual features of a problem rather than on the mathematical principles underlying the concepts. Furthermore, the degree to which cannections a significant '. teaching exp-e sun'eys and 1 connection me "AF" is O: f"“‘e“a' ch . eerie by peop; exactly the p hence, a stro improve the 11, statements ab When 100 that are made of Ways in uh kinds of Com; rare (Coneept across Studen ac: os . s mauy c iconnections are made does not appear to depend in any significant way on gender, teaching major or practice teaching experience. Inconsistent results between the surveys and the interviews make the relationship between connection making ability and mathematical background problematic. However, incorrect mathematical statements are made by people with weak, moderate and strong backgrounds in exactly the proportions that would be expected by chance; hence, a strong mathematics background does not seem to improve the mathematical accuracy of prospective teachers' statements about secondary school mathematics topics. When looking more closely at the kinds of connections that are made, it is evident that there is a large variety of ways in which students make connections. Although some kinds of connections are common (numerical) and others are rare (conceptual); the connections that were observed vary across students with different backgrounds and widely range across many content areas. The discussion of these results focuses on what these findings mean for teacher preparation and how they might inform dialogue about what is an appropriate subject matter preparation for secondary mathematics teachers. Copyright by ERIC FREDERICK WOOD 1993 This wo intellectual my doctoral 1 decision to 1 because of t] first greete; dissertation and Wiciance rest of my C( Sharon Fem]aI and help in 1 Finally my perSOnal E Years, I eXte ACKNOWLEDGEMENTS This work is the final result of a process of intellectual growth that began in 1986 when I first started my doctoral program at Michigan State University. The decision to pursue my doctoral studies here was made in part because of the friendly and welcoming way in which I was first greeted by Perry Lanier. As my committee chair and dissertation director he garners my thanks for all the help and guidance he has given me over the past years. To the rest of my committee, Glenda Lappan, Robert Floden and Sharon Feiman-Nemser I extend my thanks for their support and help in bringing this work to a successful close. Finally to all the professors who have contributed to my personal and intellectual development over the past seven years, I extend my sincere thanks. 2:5? or TABLE LIST or 25:; CHEER 1 :inooucszori est-em 2 Tm. M’ "R "Hye- S¢0L A‘hb ‘n; Int The Ref Doe Wha Nit-a Why Wha Sun CHAPTER 3 A @2chme The Con C01: COn COn Ped Sum TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . . . . x CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . 1 CHAPTER 2 SITUATING THE RESEARCH QUESTION . . . . . . . . . . . 6 Introduction . . . . . . . . . 6 The Problem with Schooling . . . . . . 7 Reforming Mathematics Education . . 9 Does Content Knowledge Affect Teaching? What Do Teachers Need to Know? . . . . What Do Teachers Know Now? . . . Why Study Connected Knowledge? . What Are the Questions? . . . . Summary . . . . . . . . . . . . O O O O O O O O O N ‘0 CHAPTER 3 A CONCEPTUAL FRAMEWORK FOR CONNECTED KNOWLEDGE . . . 39 39 42 The Range of Knowledge . . . . . . . . . Connecting to the Real World . . . . . . Connecting to Other Subjects . . . . . . 45 Connecting to Other Mathematical Topics . 46 Connections With a Pupil's Way of Thinking 47 Pedagogical Connections . . . . . . . . . 50 Summary . . . . . . . . . . . . . . . . . 52 vi castes 4 5m osszcr CF} TER 5 F.1“INGS (”*47’H rn CEAFTER 6 V "i 1 CHAPTER 4 STUDY DESIGN AND METHODOLOGY . . . . . . . . . . CHAPTER 5 FINDINGS CHAPTER 6 Introduction . . . . . . . . . . . . . Design Issues . . . . . . . . . . . . . The Survey . . . . . . . . . . . . . . Subjects . . . . . . . . . . Instrumentation and Data Collection Data Analysis . . . . . . . . . . The Interview . . . . . . . . . . . . . Subjects . . . . . . . . . . . . Instrumentation and Data Collection Data Analysis . . . . . . . . . . Identifying Patterns and Summary . . . . . . . . . . . . . . . . Introduction . . . . Results from Survey . Interview Results . . . Strong Connections . . . Mathematical Connections Conceptual Connections . Connections to Other Subjc Pedagogical Connections Connections . . . . . . Numerical Connections . Weak Contextual Connections Procedural Connections Pedagogical Connections Summary . . . . . . . . . . . 00000008000000 eeeeeeefleeeeee DISCUSSION OF FINDINGS . . . . . . . . . . . . . Introduction . . . . . . . . . . . Artifact or Fact. Methodological Issues The High School Experience . . . . . . The Curriculum . . . . . . . . . But What Gets Taught? . . . . . . What is a Good Mathematics Student? The Ontario Undergraduate Experience . The Impact of Specialization . . . Disconfirming Evidence: Interviews Versus Surveys . . . The Role of Teacher Education . Painting the Picture . . . . . Summary . . . . . . . . . . . . vii 87 87 89 100 103 103 109 115 117 120 120 125 130 132 134 136 136 139 142 143 146 158 162 163 170 177 181 185 APPLE-IX A Correlation sermon a Survey Iterns APPENDIX c Consent F011 APPENDIX D Card 30m T APPENDIX 2 Card Sort '1 “Farm: X F Card 5 Ort '1 Interview ‘ APPENDIX E {Encoded Tr; ' CHAPTER 7 IMPLICATIONS AND RECOMMENDATIONS . Introduction . . . . . . Connections in Conflict . Developing Connected Knowledge Curriculum Issues . The Role of University Teaching The Role of Teacher Education In Fostering Connected Understanding In Prospective Teachers Looking Back . . . . Looking Ahead . . . . Summary . . . . . . . APPENDIX A Correlation of Standards and Connections in Items APPENDIX B Survey Items and Instructions . . . . . . . . . . APPENDIX C Consent Form for Interviews . . . . . . . . . . . APPENDIX D card sort TaSk # 1 O O O O O 0 O O 0 O O O O 0 O 0 APPENDIX E Card Sort Task #2 . . . . . . . . . . . . . . . . APPENDIX F Card Sort Task #3 . . . . . . . . . . . . . . . . APPENDIX G Interview Coding Scheme . . . . . . . . . . . . . APPENDIX B Uncoded Transcription of Interview #3 - Lynn Allen APPENDIX I Coded Transcription of Interview #3 - Lynn Allen LIST OF REFERENCES . . . . . . . . viii 188 188 189 191 191 199 202 204 206 209 212 213 237 238 240 241 242 245 266 287 Table 4.1 Infomatio Table 4.2 All record Table 5.1 Item Respo Background Table 5.2 Item RESpo; Teaching 0; Table 5.3 Item RESpo} Table 5.4 Item ReSpO: ackground (Stl‘idents 1 Eable 5.5 item ReSpO; Table 6.1 connection {Data FrOm LIST OF TABLES Table 4.1 Information on Interview Sample . . . . . . . . . . . 70 Table 4.2 All records containing CN-DEMO or CN-EXPL or CN-JUST 83 Table 5.1 Item Response Ratings versus Mathematical Background of Respondents . . . . . . . . . . . . . . 91 Table 5.2 Item Response Ratings versus Major Teaching Option . . . . . . . . . . . . . . . . . . . 93 Table 5.3 Item Response Ratings versus Gender . . . . . . . . . 94 Table 5.4 Item Response Ratings versus Mathematical Background of Respondents (Students With More Teaching Experience Removed) . . 97 Table 5.5 Item Response Ratings versus Items . . . . . . . . . 99 Table 6.1 Connections Versus Background . . . . . . . . . . . . 176 (Data From Interviews) ix Fig-ire 3. l " «- cor-“oxen L. Figure 3.2 Geometric . LIST OF FIGURES Figure 3.1 Components of Teacher Knowledge . . . . . . . . . . . 40 Figure 3.2 Geometric Representation of a Binomial Product . . . 51 Figure 3.3 Components of Connected Knowledge . . . . . . . . . . 53 Figure 4.1 Sample Blank Record for Database . . . . . . . . . . 76 Figure 4.2 Sample Record after Data Imported . . . . . . . . . . 77 Figure 5.1 The Pythagorean Theorem . . . . . . . . . . . . . . . 104 Figure 5.2 Graphical Representation of the Problem of an Extraneous Root . . . . . . . . . . . . . . . . . . . 113 Figure 6.1 Sample Textbook Page - #1 . . . . . . . . . . . . . . 147 . Figure 6.2 Sample Textbook Page - #2 . . . . . . . . . . . . . . 148 Figure 6.3 sample TethOOk Page - #3 e e e e o e e e e e o o e e 149 Figure 6.4 Geometric Interpretation of the Tangent Ratio . . . . 155 Figure 6.5 Conceptual Map of Prospective Teachers' Connected Knowledge . . . . . . . . . . . . . . . . . 184 The 1 or she ca The prizna the learn. with contc from the c and adoles reasonable this teach that is to educationa has VaXEd thinkers a OVer t Subject ma underrated J"IT‘EXDI-taIICQ-z Staged a C calls fOr acknowledg missing p 1986b} Emblem mar CHAPTER 1 - INTRODUCTION The idea that a teacher must know something before he or she can teach it to someone else appears self evident. The primary obligation of a teacher is to engage children in the learning of worthwhile content; and, it is this concern with content that makes the teaching relationship different from the other relationships that adults have with children and adolescents (Buchmann, 1984; Confrey, 1982). It is reasonable to assume that it would be impossible to fulfil this teaching function without a knowledge of the content that is to be taught. However, in the history of educational philosophy, the content dimension of teaching has waxed and waned in importance in the minds of many thinkers and researchers (Buchmann, 1982). Over the past 25 years, the importance accorded to subject matter mastery for teachers has been significantly underrated. Recently, however, discussions about the importance of subject matter knowledge for teachers have staged a comeback in the educational press. Fuelled by calls for better academically prepared teachers and acknowledged by leading scholars in the field as the ”missing paradigm” in educational research (Shulman, 1986a, 1986b), conceptual discussions about the importance of subject matter knowledge and empirical studies of the effects that subject matter knowledge has on the way that teachers t9: educational In spit of cont at l mathematics little evidi of the teaci 1989). Alt be explaine “Y in h’hic studies (By been measur r“Khaki-tiCs. 6Wages in PeICentile QCCepted t} bECau5e the 2 teachers teach have become an important branch of educational inquiry. In spite of these recent attempts to examine the role of content knowledge in teaching in general, and in mathematics teaching in particular, there is surprisingly little evidence that better knowledge of content on the part of the teacher results in higher student achievement (Ball, 1989). Although counter-intuitive, this kind of result can be explained by considering the restrictive and simplistic way in which teachers' knowledge has been assessed in most studies (Byrne, 1983). Typically, teacher knowledge has been measured by an examination of the number of courses in mathematics taken at the undergraduate level; grade point averages in college or high school mathematics courses; or, percentile ranks on standardized tests. It is now widely accepted that these are poor proxies for teachers' knowledge because the kind of knowledge that is required for teaching is different in important ways from the kind of knowledge required for performing mathematical calculations or procedures on a test (Shulman, 1987a; Berliner, 1986). A fundamental problem remains, however. If we continue to believe that the knowledge of content that a teacher possesses does have an impact on his or her classroom practice, then how do teacher educators build on and further deveIOP t? do not kn: have? Wha my about effect on subject? This < qaestions. is of prifi particular that the Q Hathezatic advocates very littl understand eSPGCially teachers; . CmCidl fo effective Program 9 (Br 0‘71 and 3 develop the knowledge that students1 bring if instructors do not know what understandings (or lack thereof) students have? What do prospective secondary mathematics teachers know about their subject matter that is likely to have an effect on the way that they might be able to teach the subject? This dissertation addresses the second of these questions. In it I will argue that subject matter knowledge is of primary importance to effective mathematics teaching, particularly if the new vision of mathematics instruction that the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989a) (and other similar documents) advocates is to become a reality. I will also contend that very little is known about the way in which teachers understand the content that they are required to teach, especially in the case of prospective secondary mathematics teachers; and, that an understanding of teacher knowledge is crucial for teacher educators if they are to provide effective interventions as part of teacher education programs either at the pro-service or in-service level (Brown and Cooney, 1982). 1In this dissertation the word ”pupil” refers to a student in an elementary or high school setting, the word "student" refers to a university student or an intending teacher engaged in a teacher preparation program, and the word "instructor" refers to a professor in an education (or other) university faculty. ‘5. one as pec of variou curricula connectio: both survr overcome 1 teacher k: in-depth j analysis C looks like the resear 4 The research study that is reported here focuses on one aspect of the mathematical knowledge of prospective secondary teachers; namely, the way in which their knowledge of various aspects of the secondary school mathematics curriculum is connected together and how they make the connections that are in evidence. The study made use of both surveys and in-depth interviews in an attempt to overcome the weaknesses of more traditional measures of teacher knowledge. The details of the findings from both in-depth interviews and a large sample survey as well as an analysis of what prospective teachers' connected knowledge looks like are reported in Chapter 5. Chapter 2 situates the research within the body of literature on educational reform and teacher knowledge particularly as it applies to mathematics education and provides the argument and rationale for the study. Chapter 3 introduces a conceptual framework of how information about connected knowledge of secondary mathematics might be meaningfully organized and categorized. Chapter 4 includes a detailed description of the study design and instrumentation; and, the method that was employed to collect and analyze the data. A high level of description is included to provide readers with a clear picture of how the personal computer was utilized for both the qualitative and the quantitative aspects of this process and to delineate what was actually done to make a large and amorphous data set more manageable. In Chapter 6 the assertions are discus teacher kn: instructior Chapter 7 1 educational This d instrumenta ComP‘ter in order to g; detracting number of a thrOil's'l‘iout tYpical int procedures; 5 assertions that have been developed from the data analysis are discussed with respect to the various influences on teacher knowledge; for example, undergraduate mathematics instruction and professional preparation. Finally in Chapter 7 the implications of this research for the educational community are considered. This dissertation required the development of its own instrumentation and involved extensive use of a personal computer in the analysis of the data that was produced. In order to give the reader access to this material without detracting from the readability of the report, a large number of appendices are included. These are referred to throughout the text of the report and provide samples of typical interview transcripts (before and after coding procedures), instrumentation, and various other documents that were used in the project. As a content kr qiite rece domain (ma school) th the study consider t a View of ‘. Changes in knowledge. teaching 15 the litéra and second Subject in uh derstand kHOWledge . CHAPTER 2 - SITUATING THE RESEARCH QUESTION Introduction As a line of educational inquiry, studies of teachers' content knowledge, especially as it relates to pedagogy, are quite recent. When restricted to a particular content domain (mathematics) and level of instruction (secondary school) they are also rare. Consequently, in order to place the study in a proper context this chapter will first consider the role of the recent reform movements in defining a view of mathematics teaching that informs demands for changes in teacher preparation and consequently in teacher knowledge. Next, research which links teacher knowledge to teaching practice will be considered. Finally, a review of the literature on teachers' content knowledge in general, and secondary mathematics teachers' knowledge of their subject in particular will highlight the present level of understanding that is available about teachers' content knowledge. Taken as a whole this background will then be used to justify the importance and rationale for the study that is described in later chapters. The p number Of of the Uni R;_s (Nati continuing Associatic American e Pcpulace 5 glohfil ecc Altho different States, tl identical. (1992) ma} bE‘tween 9C Instr e ac 7 The Problem With Schooling The past 10 years have witnessed an unprecedented number of documents criticising the educational institutions of the United States. Starting in 1983 with A Nation at Risk (National Commission on Excellence in Education) and continuing with Time for Results (National Governors' Association, 1986) and others, the message was clear: American education was not doing the job of educating the populace sufficiently well to make them competitive in the global economy. Although the general social structure of Canada is different in many significant ways to that of the United States, the rhetoric of the reform documents has been almost identical. A report by the Economic Council of Canada (1992) makes all of the standard arguments about the link between economic competitiveness and educational achievement. It also expresses concern about Canada's lack- lustre achievement in international comparisons, particularly in mathematics and science. The title of a recent Ontario report, People and Skills in the New Global Economy (Premier's Council, 1990) also makes it clear that the concerns north of the border are much the same as those to the south. The response to these criticisms in both countries has taken several forms, one of which has been to zero in on .1 teachers as education l addressed t any of the have any h: for the 215 Economy, 19 PrePaIEd tea and that on. the Way thA‘ Particular, education m makes the c Prepay-mien This la asgumprion becoming a that seCOnd 8 teachers as the vehicle for improving the quality of education in the public schools. Two significant documents addressed the need to reform the education of teachers if any of the changes that were proposed for schools were to have any hope of success. Both A Nation Prepared: Teachers for the let Centugy (Carnegie Forum on Education and the Economy, 1986) and Tomorrow's Teachers: The Report of the Holmes Group (Holmes, 1986) made it clear that better prepared teachers were the key to educational improvement and that one of the ways of improving teachers was to change the way that they were prepared for teaching.2 In particular, the Holmes Report advocates the abolition of the education major for prospective elementary teachers and makes the case that an academic major is a more appropriate preparation for teaching. This last recommendation brings into focus the assumption that academic study is an important part of becoming a good teacher. Although the report acknowledges that secondary teachers already get such a preparation it calls into question the overall quality of the present undergraduate experience. In spite of this reservation, the idea that some kind of well designed undergraduate program 2In Ontario, a report entitled Teacher Education in Ontario (Fullan and Connelly, 1987) quotes both of these American reports in its rationale. Although some of the arguments raised in the Ontario context are different to those in the United States, many of the issues surrounding the reform of teacher education are the same. of acade: remains a The 1 been disc respect t1 area (see Hollis, L; 1999a; Nat SCienCes E “Stilt of increasing l"Wild requ media atte North Amer 9 of academic study is the best preparation for teaching remains a fundamental position of the report. Reforming Mathematics Education The problems with educational performance that have been discussed in a general way have been re-iterated with respect to mathematics perhaps more than any other subject area (see for example, NCTM, 1980; Romberg, 1984; Dossey, Mullis, Lindquist and Chambers, 1988; Paulos, 1988; NCTM, 1989a; National Research Council, 1989; Mathematical Sciences Education Board, 1990). This emphasis may be a result of the perception that economic success in an increasingly scientifically and technologically oriented world requires a mathematically competent workforce; and, media attention has focused on the mathematical failings of North American students in a number of different international comparisons.3 The National Council of Teachers of Mathematics (NCTM) was the first organized group representing mathematics education to react in a significant way to these perceived inadequacies. Their production of the document the Curriculum and Evaluation Standards for School Mathematics 3Once again the Ontario experience has mirrored that of the United States with many newspaper articles about the mediocre showing in international assessments prompting a call for increased accountability in education. (53TH, lf frameworl diff rent effort t: received subsequen Board, 19 Cent] of what i Unfortuna classroom different. Calculatit algorithm}; memol'izat: teaching I Students 1' the mathen kHOWledge VEry littl 10 (NCTM, 1989a) was an ambitious attempt to lay out a framework for school mathematics that was radically different from what had been the practice in the past. This effort to create a new vision of mathematics teaching has received support from a variety of other documents that have subsequently appeared (Mathematical Sciences Education Board, 1990; National Research Council, 1989). Central to this vision was a very different perspective of what it means to do, learn and teach mathematics. Unfortunately, in the vast majority of mathematics classrooms the three tasks just referred to are clearly differentiated. Doing mathematics usually means performing calculations or carrying out some sort of essentially algorithmic procedure‘; learning mathematics refers to memorization of the rules that govern these procedures; and, teaching mathematics often involves the teacher telling the students how to carry them out. This traditional view of the mathematics classroom, featuring the teacher as the knowledge teller, has been documented many times and varies very little between classrooms (Welch, 1978). 4Carrying out mindless algorithms can occur at all levels of mathematical study. Having pupils completing the square on dozens of quadratic functions without any regard to why they might want to do such a thing is just a more advanced example of the kind of arithmetical calculations that often come under fire in elementary schools. The same could be said of routine differentiation in the calculus classroom at the college or university level. The us away from to a more . participat l Doing maths rather ref methods to conjectures learning ar moves away knowledge 1 which this of meaning matherfiatic. than on re Vhich Can computEIS . 11 The new vision advocated by the NCTM calls for a move away from learning disconnected facts, rules and procedures, to a more integrated approach that allows for active student participation in the construction of mathematical ideas. Doing mathematics is not limited to practising skills but rather refers to the actual process of using mathematical methods to establish the truth or falsity of previously made conjectures. The clear differentiation between doing, learning and teaching is somewhat blurred as the teacher moves away from centre stage as the ultimate arbiter of knowledge to someone who provides the setting and context in which this active student participation in the construction of meaning can occurs. The emphasis in the classroom is on mathematical reasoning, thinking and problem solving rather than on rote learning of outdated procedures and algorithms which can be more efficiently carried out by calculators or computers. Helping pupils to see the connections among and between mathematical and non-mathematical ideas is a major goal of this model of teaching and learning. Although this new vision does not see the teacher as the primary dispenser of mathematical knowledge, it does not imply that teachers will need to know less mathematics than 5For a well documented example of what this kind of teaching looks like in a real elementary classroom see Lampert (1986). Although the content would obviously be different in a secondary school mathematics classroom, the basic principles and approaches that she describes here can still be used for many high school topics. at present the classr role of th of the tea that they of approac. the pupil ‘ where it in. and well i the teache: praCtlce o: the skills Will also I increased ‘ Can-Y out . successful The p: with this Cledrly in fundameDta 12 at present or that they will have a less important role in the classroom. It does, however, imply a rethinking of the role of the teacher and his or her preparation. The ability of the teacher to successfully guide students' thinking so that they spend their time investigating productive avenues of approach presupposes that the teacher understands where the pupil is going with a particular pattern of thought and where it might lead. This ability requires a substantial and well integrated knowledge of mathematics on the part of the teacher. It is reasonable to assume that if the practice of mathematics teaching is to change radically then the skills and knowledge that teachers bring to the task will also need to Change. There will need to be an increased emphasis on strongly prepared teachers who can carry out the demands of the new teaching model successfully. The proposals put forth by the Holmes group, coupled with this new idea of what it means to teach mathematics, clearly imply that a strong mathematics background would be fundamental to a teacher preparation program for secondary mathematics teachers. Indeed, the position of the NCTM and other prestigious groups has long been for well qualified teachers in the mathematics Classroom at all levels -- well qualified teachers being those who have taken a particular listing of mathematics courses either in high school or university. The list of courses that the NCTM (1985) specifies reform doc assumption part of tel the subjec1 teachers vi Both 0 sufficient: assumption teaching e: knle'ledge . and learn: seCtiOD re teacher kn it appliES PrediCateC improves a 13 specifies is bereft of rationale, however, and like the reform documents alluded to above, makes two basic assumptions: (a) that a knowledge of content is an important part of teaching expertise; and (b) that academic study of the subject discipline provides prospective secondary teachers with the content knowledge they need to teach well. Both of these assumptions are plausible but not sufficiently obvious to be taken at face value. The assumption that content knowledge is an important part of teaching expertise is rooted in the belief that the content knowledge of the teacher makes a difference in the teaching and learning that goes on in the classroom. The next section reviews the literature on the relationship between teacher knowledge and teacher effectiveness, particularly as it applies to mathematics. The second assumption is predicated on the belief that advanced study of a discipline improves and refines the understanding of more elementary concepts associated with that discipline. A subsequent section will examine what secondary mathematics teachers need to know and to what extent the academic major is an appropriate way of learning and developing this level of understanding. In the final section I discuss our present knowledge about teachers' ways of knowing and how this study addresses the questions raised in the chapter. As st subject rn significa used as a clearly o of conten teachers 'saturate teachers industriE technica] Fraparat: than a k: PrOCeduri avail-dbl. conteht : Unti dominate. attempts 14 Does Content Knowledge Affect Teaching? As stated at the outset, the perceived importance of subject matter knowledge for teachers has changed significantly over time. Historically, the tests that were used as a way of licensing people to become teachers were clearly oriented towards a traditionally construed knowledge of content (Shulman, 1986a). Dewey (1904/1965) talks about teachers who had little pedagogical knowledge, but were so ”saturated with their subject-matter" that they were gifted teachers anyway.6 However, as the attempt to impose an industrial model on education tried to make teaching a technical science, the importance of subject matter in the preparation of teachers was assumed to be less important than a knowledge of some kind of generic teaching skills and procedures (Buchmann, 1982). As a result, much of the available literature on teacher knowledge is essentially content free. Until the mid—seventies, research on teaching was dominated by the "process—product" paradigm. In their attempts to correlate teacher behaviours with student outcomes, researchers concerned themselves with teachers' actions and ignored factors such as teacher knowledge or 6Even today the awarding of a doctoral degree is a de facto license to teach at the university level. Any university student can attest, however, that mere content mastery does not guarantee teaching expertise. lesson cor advance or the focus 'hat kind from the r This clear in t of the pos mathematic seem to t] curriculm some contl developed lack of t: Curricula Where tEa was Sugge teaCher I S 15 lesson content. Teacher behaviours such as providing advance organizers, asking questions or giving feedback were the focus of study rather than what the questions were or what kind of response the feedback was intended to elicit from the pupil. This belief in content-free pedagogical principles was clear in the implementation plans for the New Math programs of the post-Sputnik reform era. The emphasis on abstract mathematical content in the curricula for pupils did not seem to translate into an understanding on the part of curriculum developers that teachers would also need to have some content knowledge; rather, it was assumed that well developed teacher-proof materials could compensate for a lack of teacher knowledge. Although much of the work in curriculum development was done at the elementary level where teachers were perceived to be less knowledgeable, it was suggested that "even at the secondary level, the teacher's role becomes generally the same across subject matters because the complexities specific to teaching the particular subject are handled by the instructional materials" (Gage, 1977, p. 78). According to the extensive review of this research genre done by Romberg and Carpenter (1986), taken as a whole the results have been inconclusive and unhelpful. The few factors that do seem to be consistently positively correlated with student achievement (giving cognitive clues, V nployin maintain represen been inv« Not investiga (1986) ar produced can help lessons. ducators real teac focusing is 50 rar In SE COncluSio the ‘73er that the 16 employing good classroom management strategies and maintaining student engagement) are almost self-evident and represent a very small return on the time and money that has been invested in the research enterprise (McNamara, 1991). Not all researchers agree, however, that this line of investigation has been more or less a waste of time. Brophy (1986) argues that the teacher effectiveness research has produced substantive findings about classroom action that can help teachers to better structure and deliver their lessons. Furthermore, he contends that mathematics educators need to address themselves to the issues of how real teachers can teach in real classrooms rather than focusing on the ideal image of mathematics instruction that is so rarely observed in classrooms at the present time. In spite of the fact that he does not agree with the conclusions of Carpenter and Romberg (1986) with respect to the value of past research efforts, Brophy (1986) does agree that the content free nature of this research is one of its main weaknesses. It is his contention that the integration of the content dimension should be the next stage in this work. This line of inquiry would seek to answer questions much more closely associated with content knowledge. What makes a good example good or an unusually effective lecture unusually effective? What is it about certain analogies or examples that make them ”just right” for linking new input to students' existing concepts? Are there differences ... between the ways that outstanding teachers present particular mathematical concepts to their students and the ways the ( r vaiousj about tl studies knowledg omission 17 that less effective teachers present the same content? (Brophy, 1986, p. 332) Obviously, although there may be differences of opinion about the methodology and the definitive results of these studies it is fair to say that the absence of a content knowledge dimension is now considered to be a serious omission of the process-product paradigm. Another group of studies which used similar methodologies sought to link teacher characteristics with student achievement; but, because these studies were concerned with characteristics rather than actions they often did examine teacher knowledge as one of the variables. The results were no more helpful than the process-product studies, however (Eisenberg, 1977). Eagle (1978), in an exhaustive summary of research in the area of mathematics education was forced to conclude that although it is commonly held that the more teachers know about the subject the more the effective they are "[T]he empirical literature suggests that this belief needs drastic modification and in fact suggests that once a teacher reaches a certain level of understanding of subject matter then further understanding contributes nothing to student achievement" (Eagle, 1978, p. 51). It might be assumed from this assertion that teachers' content knowledge is not an important area for study; however, the studies on which these conclusions are based suffer from a number of shortcomings. One is the over reliance achieveme measure h students Another d teacher k understan These EXplained in which ' Studies (j been meas- mathernati. aVerageS perCentil aCCepted 18 reliance on standardized tests as a measure of student achievement (Berliner, 1976). These tests do nothing to measure how the answers are arrived at or the way that students think through mathematical problems, for example. Another difficulty even more pertinent to the issue of teacher knowledge is the way in which teacher knowledge and understanding is measured. These kinds of counter-intuitive findings can be explained by considering the restrictive and simplistic way in which teachers' knowledge has been assessed in such studies (Byrne, 1983). Typically, teacher knowledge has been measured by an examination of the number of courses in mathematics taken at the undergraduate level; grade point averages in college or high school mathematics courses; or, percentile ranks on standardized tests. It is now widely accepted that these are poor proxies for teachers' knowledge because the kind of knowledge that is required for teaching is different in important ways from the kind of knowledge required for performing mathematical calculations or procedures on a test (Shulman, 1987(a); Berliner, Stein, Sabers, Clarridge, Cushing and Pinnegar, 1988). In contrast to these empirical studies, a number of more qualitative studies also appeared during the seventies. Using techniques such as stimulated recall, thinking aloud, interviews and participant observation, researchers sought to discover what teachers were thinking about as they conducted thinking, the teache (Clark anc conclude 1 significar The r subs: thoug docurr demar. judg: etfec learr 1988a Taker stUdent P Of COHVi: matter kr weaknesS Bi , dchm‘ifln, ha? e begr k howledgt 19 conducted their lessons. This line of research on teacher thinking, whose goal was to understand what went on inside the teacher's head, has been reviewed by several researchers (Clark and Peterson, 1986; Clark and Yinger, 1978). They conclude that the research has produced substantive and significant results: The research on teachers' thought processes further substantiates that the teacher is a reflective, thoughtful individual. Moreover, the research documents that teaching is a complex and cognitively demanding human process. Teachers' beliefs, knowledge, judgments, thoughts and decisions have a profound effect on the way they teach as well as on students' learning in their classrooms. (Carpenter and Fennema, 1988a, p. 4, emphasis added) Taken together all these results do not provide conclusive evidence that better content knowledge on the part of the teacher produces better teaching and improved student performance or that it does not. Although the lack of convincing empirical evidence that teachers' subject matter knowledge affects student performance remains a weakness in the argument for its importance (Floden and Buchmann, 1989), in the past decade, more and more scholars have begun to argue for the primary place of subject matter knowledge for those intending to be teachers (Anderson, 1989; McDiarmid, Ball and Anderson, 1989). These arguments are based on more recent research which links teachers' Classroom practice with the knowledge of content that they bring to this task. This line of inquiry speaks to the kind of knowledge that teachers have as opposed to the quantity, and seeks as it rel One support t an examir mathemati show real teach an: novices t 55 is the findersta: do all of teaChir‘.g. Special y although mamagamer knowledge BereitEr BerliHEr Smith: is and Bork: filth rOngly \ 7 ‘ A :8th a: matter 3% St 20 and seeks to further refine our notions of teacher knowledge as it relates to pedagogy. One group of research studies that can be used to support the contention that "the subject matters" comes from an examination of expert-novice studies in the area of mathematics teaching. When compared to novices, experts do show real and profound differences in the way that they teach and think about teaching. For example, the ability of novices to construct examples on the spot is very limited, as is their ability to produce alternate explanations and understand student thinking that is non-standard. Experts do all of these things as a matter of course in their teaching. It appears that expert teachers do have some special knowledge that non—experts do not.7 Furthermore, although some of this special knowledge is of procedures and management skills, at least some of the experts' specialized knowledge is subject matter knowledge. (See for example, Bereiter and Scardamalia, 1985; Berliner and Carter, 1986; Berliner et al, 1988; Carter et al, 1987; Leinhardt and Smith, 1985; Leinhardt, 1986a; Leinhardt, 1986b; Livingston and Borko, 1989). Although the results of these lines of inquiry are strongly supportive of the idea that teachers' subject 7A novice is likely to be a non-expert but a non-expert is not necessarily a novice. Not all experienced teachers are considered to be experts, so expertise is not simply a matter of experience. matter kr. (and leav limitatic: subject me ' 4 Studies 01 play and s countermm Play. Kha their mata experts k] nMurally and deveh this rich. Exhibit 9 not? Doe “fly that can teach pmsPECti Uh derSta: The ‘ 21 matter knowledge does affect their teaching, they also raise (and leave unanswered) many important questions. Are the limitations of novices due to their lack of richly connected subject matter knowledge or just a lack of experience? Studies of expert chess players, for example, have shown that experts have a much larger repertoire of patterns of play and so they can quickly relate moves to already known countermoves rather than having to analyze each individual play. What is it about the way in which novices understand their material that makes it different from the way that experts know it? Does this different way of knowing naturally come about through experience or can it be taught and developed prior to teaching? Not all teachers develop this rich, connected, understanding of content that experts exhibit even though they have many years of experience, why not? Does their initial stage of understanding affect the way that their knowledge grows over time? If so, how? How can teacher educators maximize the initial understandings of prospective teachers to ensure that they do develop better understanding of their content with experience? The way in which knowledge interacts with experience is a research question in its own right and perhaps future work will answer some, or all, of these questions. However, my own experience in working with both in-service and pre- service teachers convinces me that experience alone does not automatically produce a deep and richly connected understa: that is t teacher. The of how te the Know; Stanford and Phil} attempted were mpg teachers that beg i the StUde Capable. work and EnGlish a FIESented Wilson an 22 understanding of subject matter. In fact, the only thing that is definitely produced by experience is an older teacher. The last series of studies that speak to the question of how teacher knowledge impacts teaching practice came from the Knowledge Growth in a Profession project undertaken at Stanford by Lee Shulman and his associates (Shulman, Sykes and Phillips, 1983; Shulman, 1987). One part of the project attempted to analyze the various kinds of knowledge that were important for teaching. The study found that expert teachers seemed to have access to special kinds of knowledge that beginning teachers did not have, despite the fact that the student teachers under study were academically quite capable. A.number of interesting papers which describe this work and the findings in the areas of history, science, English and mathematics teaching have been published or presented at scholarly meetings (Hasweh, 1987; Richert, Wilson and Marks, 1986). The results across all subject areas are suggestive in that they make the point that there is a special body of subject matter knowledge (Shulman calls this pedagogical content knowledge) that teachers need to have at their disposal that academics do not. Furthermore, the mathematical knowledge required for doing mathematics does not appear to be the same as the mathematical knowledge required for teaching mathematics (Wilson, Shulman and ‘chert, conclusio . . . 1 US pt merel maths exten To he think solve 1988, Anoth involved 6 that the d of the C03 their teac 1985) . In SHbStantia. Very Open ‘ interpretat Despite he: little fem 23 Richert, 1987). Some scholars have reached the further conclusion that ... teaching mathematics, for example, in contemporary US public schools, is so complex in comparison to being merely a practising mathematician who solves mathematical problems, that at least two complex and extensive knowledge domains must be used at all times. To be an expert teacher of a subject matter is, we think, far more complex than to be an expert problem solver in that subject matter domain. (Berliner et al, 1988, p. 92.) Another important finding from this work, which involved extensive case studies of beginning teachers, was that the depth of knowledge that these student teachers had of the content they were teaching had a direct effect on their teaching practice (Steinberg, Haymore and Marks, 1985). In one case an English teacher, Colleen, who had a substantial knowledge of English literature, taught in a very open ended, interpretive and inquiry oriented fashion. She encouraged participation and was open to alternative interpretations of text if justified by the students. Despite her rich background in literature, Colleen had little formal instruction in prescriptive grammar. A typical high school English curriculum, however, does contain some lessons in formal grammar and when faced with the prospect of teaching grammar, something that she felt unprepared for, she became a totally different teacher. Discussion was replaced by lecture, inquiry with prescription -- ”Socrates replaced by DISTAR" as the researcher observing the lesson puts it. She deliberately encourai Sir: science physics. other. backgrou that the less lik They werl Planned ‘ activiti‘ am“ so: teemlers dEteCt ar less knOu miSCoanp smarizE this way: 24 avoided eye contact with pupils because she was afraid that they might ask her questions and she did not want to encourage them (Shulman, 1987a). Similar effects were also reported in the case of science teachers who were required to teach both biology and physics. Some had strength in one discipline some in the other. Hasweh (1987) found that the teachers with a strong background knowledge were better able to connect the ideas that they were teaching to other concepts and that they were less likely to accept a textbook approach to the topic. They were also more likely to make modifications as they planned their lessons and were able to design their own activities. Although both groups had incorrect knowledge about some aspects of the topics they were teaching, teachers who were more knowledgeable were better able to detect and correct pupils' misconceptions. By contrast, less knowledgeable teachers actually reinforced pupils' misconceptions in some cases. Fennema and Franke (1992) summarize their conclusion about this line of research in this way: The studies of the mathematics, science and history teachers possessing a rich, integrated knowledge of subject matter lend credence to the belief that teacher knowledge can influence instruction. In these studies, the content of instruction appeared to be at least partially dependent on teacher knowledge, as did the discourse of the class. Knowledge did not dictate precisely what was done ... [H]owever, the richness of the material being taught appeared to be directly related to the subject-matter knowledge of the teacher. (p. 151) Othe: Anderson about stu in the su hackgroun lot of th ability t somewhat 1 assertion of examin; the issue this know; Counts. ] ... a rich under thEm, teach PeOpl relat relat aCCeS This and how t} the SeCOnc rhetoric: is,“1 app: unclear th hematic, for teach- ‘ i1 25 Other researchers have had somewhat different results. Anderson (1989) provides a number of interesting vignettes about student teachers, all of whom have an academic major in the subject that they will teach. In spite of this background, these student teachers did not seem to have a lot of the skills that are important for teaching -- the ability to see multiple representations, for example. This somewhat contradictory evidence gives support to the assertion that researchers must find better and finer ways of examining teacher's knowledge. Having knowledge is not the issue in teaching, rather it is the ability to transform this knowledge and make it accessible to students that counts. Furthermore, Anderson (1989) contends that the ... academic disciplines are systems of knowledge so rich and complex that no one could ever fully understand all the structural relationships within them. The transformation of disciplinary knowledge for teaching, however, can be accomplished well only by people who are aware of a rich array of structural relationships within a discipline and who can use those relationships to reorganize their knowledge and make it accessible to students. (p. 95) This quotation raises the issue of what teachers know and how they know it; and, this question brings us back to the second assumption which underlies much of the reform rhetoric: namely, that academic preparation in a discipline is an appropriate background for teaching. In fact it is unclear that students learn anything at all in university mathematics courses that is perceived by them to be useful for teaching mathematics at the high school level. One study co: involved mathemat: practice Others p1 believe i improving Thes: value of for teach. HOWeveI-I preparati of knowle (1989) be may be uh will 1.1an 1Whether we Prospecti‘ whether c] The f illustratJ diSCusSinC . thematic 6 SemeSter quefined 1 26 study concluded that the prospective teachers who were involved in the project believed that their university mathematics courses were essentially irrelevant to the practice of teaching high school mathematics (Owens, 1987). Others provide evidence that prospective secondary teachers believe in the importance of mathematics courses as a way of improving their teaching (Wood and Floden, 1991). These studies relate to peoples' perceptions about the value of studying university mathematics as a preparation for teaching; and, perceptions are not always accurate. However, it is not at all clear that the present academic preparation of secondary teachers does provide for the kind of knowledge of "structural relationships” that Anderson (1989) believes to be important for teaching. Indeed, it may be unrealistic to expect that many (or any) teachers will have this level of knowledge; still, we must decide whether we are doing the best job possible of preparing prospective teachers with respect to content knowledge or whether changes for the better can be made (Kennedy, 1990). The following personal anecdote provides a good illustration of the reason for some of my reservations. In discussing division by zero with a group of secondary mathematics education majors (all of whom had taken at least 6 semester courses in mathematics with some having as many as 30) I asked how they might justify the fact that 1+0 was undefined to a student in ninth grade. After much discussion ve talke‘ equivalen asked: Wh precisely for x the that 1+0 Befo. responses conventio, 27 we talked about letting 1+0 = x and then getting the equivalent expression 0.x = 1.8 The question was then asked: What number when multiplied by 0 produces the value 1? No one was able to answer this question; and, it is precisely because no one is able to give a numerical value for x that there is no defined value. It is for this reason that 1+0 is considered to be undefined. Before this line of argument had been developed typical responses from the class were based on mathematical convention or definition. A few students tried to use the idea of limits in spite of my statement that we were teaching in a grade 9 class. Interestingly, no one in the class was able to come up with the justification given above by themselves and when asked if 0+0 was undefined they quickly agreed that it was -- without going through the same, or any other, logical argument’. ‘When the same line of reasoning produced the statement that 0.x = 0 and hence the conclusion that any value of x is acceptable, they were dumbfounded to learn that 0+0 is considered to be indeterminate. Some even made statements such as: ”My high 8This transformation is easily argued by considering the question 8+4 = 2. This statement is true because 4 x 2 = 8. Similarly 1+0 = x is true if 0.x = 1. 9This willingness to accept outside authorities (i.e. my argument) for the validation of knowledge rather than using personal logic is a troubling, but common, characteristic of prospective teachers. school te undefinec This calculus use it on x indetermi] I one exampy‘ the class how an arg 01‘ falsit The p. deep Under Come about branches C as in this I want to in faVOur teachers; Preparatic possib1e t may not he PthaE prospectiw 28 school teacher told me that division by zero was always undefined!" This revelation is made even more surprising by the fact that, when questioned, every student in the class admitted that they remembered studying L'Hopital's rule in calculus and was able to tell me that it was appropriate to use it on indeterminate forms. When pushed to explain what indeterminate forms looked like they typically gave 0+0 as one example! Apparently not one student out of the 23 in the class had ever wondered what an indeterminate form really was, why it was called that in the first place and how an argument could be structured to establish the truth or falsity of the suggestion that 0+0 was undefined. The point to be emphasized by this example is that a deep understanding of some reasonably simple ideas does not come about automatically from the study of more complex branches of mathematics -- even when the idea is revisited, as in this case. I do not want to belabour this point nor do I want to leave the reader with the impression that I am not in favour of a strong content component for mathematics teachers; however, unless the kind of mathematical preparation as well as the quantity is addressed it is possible that requirements for more university preparation may not have the intended effect. Perhaps the first step in addressing the issue of where prospective teachers' acquire (or ought to acquire) their knowledge need to k mathemati It 9: minimum, ' teaching . understan: they must 29 knowledge of mathematics is to examine what teachers might need to know to teach according to the new vision of mathematics instruction promoted by the Standards. What Do Teachers Need to Know? It goes without saying that teachers must, at a minimum, be able to do the mathematics that they are teaching. They must know rules and procedures and understand the underlying theory behind them. Simply stated they must know the content that they are teaching both instrumentally and relationally (Skemp, 1978). It is reasonable to assume that a student would not be able to complete a mathematics major if he or she could not do algebra, analytical geometry, trigonometry or calculus. However, given the significant change in approach to mathematics instruction that is presently being advocated, courses in traditional mathematics, taught in the traditional way, may still not be enough (Committee on the Mathematical Education of Teachers, 1991). Recent work by the NCTM to produce a set of professional standards that will inform the preparation of teachers has moved far beyond the earlier lists of required courses. Their position paper Professional Standards for Teaching Mathematics (NCTM, 1989b) spells out in much more detail the kind of richly connected knowledge and flexible teach eff Teachers understar. other ext able to 5 generally help stud. interconn: The E a number c it is not conneCted Would need ideas to h various tc knowledge it becomes with the c on a knowl hderStand heed. 30 understanding that someone needs in order to be able to teach effectively according to their vision of teaching. Teachers need to be able to construct mathematical understandings of their own rather than relying on texts or other external arbiters of knowledge. They will need to be able to structure tasks, assignments, discussions and generally orchestrate classroom discourse in a way that will help students to see how various mathematical ideas are interconnected. The Professional Standards (1989b) and the Curriculum Standards (1989a) are important documents but they do leave a number of significant questions unanswered. For example, it is not completely obvious what is meant by the term connected knowledge, although it seems clear that a teacher would need to have a connected understanding of mathematical ideas to be able to empower pupils to find linkages between various topics and ideas. Once again the role of teacher knowledge becomes highly significant at the same time that it becomes problematic. Any attempts to provide teachers with the content knowledge that they require must be based on a knowledge of what they know now and a concomitant understanding of how they can acquire the knowledge they need. Def. understa possible A nur-a'oer seems cl) kind of ; useful f: Specific functions understap investiga teaChers Conclusic that teac PTOCedure the <2011Ce (Miami consisteD that many candidate 31 What Do Teachers Know Now? Definitive statements about the kinds of mathematical understandings that typical secondary teachers hold are not possible based on the research that is presently available. A number of researchers are pursuing this question and it seems clear that even mathematics majors do not have the kind of firm grasp on many content areas that would be useful for a high school teacher. Studies that address specific topic areas such as Evan's (1989, 1990) work on functions; Fisher's (1988) study of prospective teachers' understanding of ratio and proportion; and, Lee's (1992) investigation of the way in which prospective mathematics teachers view the limit concept have all come to the conclusion that there are important gaps in the knowledge that teachers have. Typically, their knowledge of procedures and algorithms is good but their understanding of the conceptual basis of these processes is often weak (McDiarmid and Wilson, 1990). These findings are also consistent with work by Ball (1988b) which makes it clear that many elementary and even some secondary teacher candidates do not have a deep understanding of very simple mathematical concepts such as division, fractions, or division by zero (Blake and Verhille, 1985). Although my intuition and experience with mathematics teachers, both at the pre-service and the in—service level, tells me prospect the rich cannot b studies of teach universi‘ undergrac 32 tells me that these authors may well be correct and that prospective teachers' "understanding [I assume this means of the richly connected flexible kind] of school mathematics cannot be assumed” (NCTM, 1989b, p. 65), the scarcity of studies in this area does not allow for detailed discussions of teachers’ mathematical knowledge. As McDiarmid (1989) rightly points out, we simply do not know what it is that university graduates learn about specific subjects in their undergraduate experience. Why Study Connected Knowledge? If the level of knowledge about teachers’ content mastery in mathematics is as weak as earlier sections have suggested then studies of teachers' content knowledge could focus on many different aspects of that knowledge. Why then should this study focus on the way that prospective teachers make connections between and among mathematical topics? The answer lies in a closer examination of the various reform documents that call for changes in the way that mathematics is taught as well as changes in what is taught. It is not surprising that the majority of beginning mathematics teachers will have gaps in their knowledge. It is unrealistic to expect that students understand everything that they have been taught throughout university. Even experienced teachers may be called upon to teach new topics hat the geometry not diff demonstr. traditiOJ topics tl What which the important how pros; This lacu ChOSen to rather th FUrtI PerVasiVE documents programsh § (1989b) a ConneCtin 33 that they have not previously had experience with, fractal geometry, for example. Strict deficiencies in knowledge are not difficult to fix -- a mathematics major has at least demonstrated his or her ability to learn mathematics (in the traditional way) and the teacher can learn or relearn new topics that have been added to the curriculum. What is more troublesome is the fact that the way in which the teacher knows his or her content is just as important as what they know; and, we know very little about how prospective mathematics teachers' know their subject. This lacuna in our knowledge is one reason that I have chosen to examine an aspect of teachers' ways of knowing rather than a particular mathematical topic. Furthermore, the connections theme is one of the most pervasive in the various mathematics education reform documents. It is also commonly featured in conference °, professional journals11 and books”. The programs1 Standards Document (1989a), the Professional Standards (1989b) and A Call for Change (COMET, 1991) all perceive the connecting of various high school topics to other areas both 1”For example, the NCTM regional conference held in Montreal in August, 1992 had as its theme Connecting Mathematics to the Real world. l'1The most recent issue of the Illinois Mathematics Teacher, for example, features connections as its major theme. 12Willoughby's book Mathematics Education fer a Changing WOrld (1990) dedicates an entire chapter to the topic of connections. inside a teaching Math of t defi and them math (NC'I’) Clea. fundament able to s Connectic teachers reSpect t Node} maths-man major Str .. . 1 reSui than< lntel 501v; teac] that app); (Fem hm inStr 'le Examp adVocated Any 6 weYs 34 inside and outside of mathematics to be fundamental to good teaching. Mathematics content knowledge is an essential component of teachers' knowledge. ... The ability to identify, define, and discuss concepts, structures and procedures and to develop an understanding of connections among them and eventually, appreciate the relationship of mathematics to other fields is critically important. (NCTM, 1989b, p. 71) Clearly a teacher who does not see mathematics as a fundamentally connected body of knowledge is unlikely to be able to structure experiences for pupils to bring these connections into focus; hence, knowing how prospective teachers do understand high school content specifically with respect to its connectedness is vitally important. Modern writing about the nature of the discipline of mathematics also emphasizes the interrelationships of its major structural elements. However, although ... the consideration of the nature of mathematics has resulted in demands that the mathematics curriculum be changed to emphasize the process of mathematics, the interrelations of mathematical ideas and problem solving in the real world, the nature of mathematics has not been adequately considered in many studies of teacher knowledge of content. Needed are procedures that measure the interrelations of ideas, the applications of ideas and the processes of mathematics. (Pennema and Franke, 1992, p. 152, emphasis added) The instruments that have been developed for this study are one example of the kinds of new procedures that are advocated. Any attempt to develop new ways of knowing in the teaching force must begin with a knowledge of what those ways of knowing are now (Fennema, 1981). Given the wide sc0pe of focus on fill an knowing educatio. complemei knowledg or at ot 1988b; Ba McDiarmid 35 scope of this study with respect to content and a specific focus on intending secondary teachers, the research will fill an important void in our understanding of the ways of knowing mathematics that prospective teachers bring to their educational studies. Being subject specific it will complement the work of other scholars studying teacher knowledge in more general terms, in other subject domains, or at other levels. (See for example, Ball, 1988a; Ball, 1988b; Ball, 1988c; Ball, 1988d; Ball, 1988e; Ball and McDiarmid, 1989; Ball and McDiarmid, 1990). What Are the Questions? The fundamental purpose of this study is to find out about the way in which prospective secondary mathematics teachers understand some of the significant strands of the mathematics that they will be teaching at the secondary school level. It is not to find out if they can do the mathematics that they will teach; but, to establish if they know it in the way in which they might need to know it for the purpose of teaching secondary school pupils to appreciate the interconnections embedded in the subject. The study addressed two primary research questions: 1. Do prospective secondary mathematics teachers have a connected understanding of the content that they will be teaching in typical high school mathematics classes? A number research (a) Can ' together the curr: (b) Can t that pupj (c) Can t (d) Can t knowledge Pupils? (e) Can t. high SChOc deriVEd . 2- Is the about PrOs knowledge: Taken provide a teacheps a Mthematic own knO’W‘lg ClaSSI-Oom 36 A number of related questions are nested in this overarching research question: (a) Can they connect various mathematical principles together so that pupils see the inter-relatedness of much of the curriculum? (b) Can they connect mathematics with other school subjects that pupils are familiar with? (c) Can they connect mathematics with real life phenomena? (d) Can they connect their own mathematical knowledge to a knowledge of pedagogical principles and a knowledge of pupils? (e) Can they connect their knowledge of theoretically derived university mathematics with the mathematics of the high school curriculum which is in many ways empirically derived. 2. Is the concept of connectedness a useful way of talking about prospective secondary teachers' mathematical knowledge? Taken as a whole, the answers to these questions will provide a detailed picture of how prospective secondary teachers are able to connect their knowledge of various mathematical ideas together; and, how they can connect their own knowledge with the knowledge that pupils bring to the classroom. that the; the cont instruct content have a r' content Procedure kind of k important to includ The a improving prediCatec teachers It tEach and than a fra are only a Pr05pectiv‘ little abs 37 Summa; 2 The knowledge of subject matter that teachers bring to their classroom teaching has a significant impact on the way that they teach. Recent reform initiatives to change both the content and the methodology of secondary mathematics instruction put an even greater emphasis on the mathematical content knowledge of the teacher. Teachers will need to have a richly connected and flexible understanding of the content they are teaching rather than a knowledge of procedures and algorithms only. The question of where this kind of knowledge base is likely to be developed is an important one as teacher preparation programs are modified to include more academic preparation. The ability of these programs to be successful in improving the knowledge base of prospective teachers is predicated on a better understanding of how prospective teachers know the mathematics that they are preparing to teach and the extent to which it forms a connected rather than a fragmented body of knowledge. Unfortunately there are only a few studies of the mathematical knowledge of prospective secondary teachers and we know comparatively little about what people learn in their undergraduate experience. This study addresses the issue of teacher knowledge especially as it relates to the way that “V ‘ :1. OS‘I nathl 38 prospective teachers are able to connect the various mathematical ideas that they hold together. CHAPTE The organizi question to Miles I 'explains factors, among the frame the Mthwolc I CHAPTER 3 - A CONCEPTUAL FRAMEWORK FOR CONNECTED KNOWLEDGE The design of any research study requires some kind of organizing framework to allow the researcher to focus the questions, design the instrumentation and so on. According to Miles and Huberman (1984) a conceptual framework ”explains ... the main dimensions to be studied -- the key factors, or variables -- and the presumed relationships among them" (p. 28). This chapter presents the conceptual frame that was used to plan the study and inform the methodology and instrumentation that were employed. The Range of Knowledge Using their recent review of some of the relevant literature on teacher knowledge, Fennema and Franke (1992) have tried to develop a cognitive model of teacher knowledge. The integration of Shulman's (1987a) work on pedagogical content knowledge; Peterson's (1988) ideas about teacher cognition; Leinhardt's study of expert teachers (1985, 1986a, 1986b); and, the perspective of Elbaz (1981) with respect to situated knowledge and practical personal understanding, leads them to conclude that teachers' knowledge is a dynamic and multifaceted construct. The diagram in Figure 3.1 (taken from Fennema and Franke, 1992, p. 162) illustrates the various components of teacher 39 knowled; diagram differen model of it does . can be f1 40 knowledge and how they are interrelated. Although this diagram is useful as a way of illustrating that many different factors need to be considered when developing a model of teacher knowledge, it is overly simplistic because it does not show how each of the various kinds of knowledge can be further subdivided. Beliefs Mathematics Content '0le Figure 3.1 - Components of Teacher Knowledge The category designated as knowledge of mathematics, for example, has a number of different dimensions. Typical categories of knowledge classified under this one title could be, knowledge of the structure of mathematics; knowledg knowledg how vari each of organizi Conn a number x Particule example; topic the “Y of U and HUbez developed Study; ar are inter framewc)“< “Gouda“ S ubsequer 41 knowledge of mathematical procedures and principles; knowledge of the history of mathematics; or, knowledge of how various ideas in mathematics are interconnected. Within each of these sub-categories are nested still other organizing structures. Connections, the subject of this study, can be made in a number of different ways. A teacher could take a particular mathematical idea and connect it to a real world example; to another subject area; to another mathematical topic that the pupil has previously studied; to a pupil's way of thinking; or, to a pedagogical principle. As Miles and Huberman (1984) suggest, such categories are often developed from experience and the general objectives of the study; and, getting clear about what they mean and how they are interrelated is a fundamental purpose of a conceptual framework. In this case, my own experience as a successful secondary mathematics teacher and the reflection that I have subsequently engaged in as a teacher educator grappling with the issue of how people learn to teach, has led me to develop these connection categories as a way of classifying teaching techniques that are useful when helping pupils to develop mathematical understanding. In order to have a clear idea about what these various connections might look like in practice, the next sections will cor. that pro study; a ‘fficul' vignettes that all: hence se'. illustra: of these connectic intended exhaustiv 42 will consider several descriptions of teaching situations13 that prospective teachers were given to examine in this study; and, how a knowledgeable teacher might make various kinds of connections as a way of responding to the difficulties that pupils appear to be having in the vignettes. It is very difficult to choose one situation that allows for every category of connection to be made, hence several different vignettes will be used in the illustration of the various categories of connections. Many of these vignettes do have several different categories of connections associated with them; however, the examples are intended to be illustrative of the category rather than exhaustive. Connecting to the Real World It has been argued that pupils often perceive mathematics as irrelevant because it is rooted in school learning rather than situated in a context that relates to their own lives (Resnick, 1987). It seems reasonable to assume therefore that connecting mathematics to the real world is a useful kind of connection to be able to make. Consider this description of a typical teaching situation: 1":"The role of these vignettes, which formed a major part of the instrumentation, is discussed in more detail in Chapter 4. '3! 43 The following question is in a homework exercise that you gave your grade 9 pupils: A woman drives from here to Toronto (a distance of 200 k) at a speed of 100 k/hr. She immediately turns around and drives back but because of the traffic she can only drive at 80 k/hr. What is her average speed for the whole trip? You notice that two pupils are arguing as they do their work and you go to see what the controversy is about. It turns out that they used different approaches and got two different results: Pupil #1: .Average speed is (100 + 80)/2 = 90 k/hr. Pupil #2: Time to drive to Toronto is 200/100 = 2 hrs. Time to drive home is 200/80 = 2.5 hrs. Average speed = distance/time = 400/4.5 = 88.89 k/hr They want you to judge which solution is better. How would you respond? Mathematically, the second is correct. To understand the problem, imagine that the same trip was taken but this time the driver travelled at 100 k/hr in both directions except for the last 5 minutes when construction forced a slowdown to 10 k/hr. If the assumption is made that whenever it is necessary to find the average of two values they are simply added up and the result divided by 2, then the first method of solution is appropriate. This method would predict an average speed of (100 + 10)/2 = 55 k/hr; however, this result clearly does not represent the reality of the journey. The difficulty with using a simplistic understanding of the notion of average in this question has to do with the fact that the speeds that are being averaged do not take place over equal segments of time and so the ‘- I'r'“II 44 effects of the two speeds must be weighted. Using a weighted average with time as the weighting factor does produce the correct answer.14 In some sense this problem already represents a ”real world” situation; however, the "real world" connection that pupils often understand best has to do with school itself. For pupils of this age, school does occupy a large segment of their lives and they do perceive it (although probably not mathematics class) to be their “real world" in many ways. Pupils who found this question puzzling might be asked what their final mark would be if they got 5/10 (50%) on a short quiz and 90/90 (100%) on a major examination. The "simple average” (50% + 100%)/2 = 75% is clearly not an accurate representation of the individual's mark and most pupils intuitively realize this fact. They can be pushed further to explain that the 100% mark should have more weight because it is worth more and that their mark ought to be (50% x 0.1) + (100% x 0.9) 5% + 90% = 95%”. Once this connection is made the teacher can return to the previous problem and consider the situation where the speeds y l".Alternately it could be argued that speeds are rates rather than scalar quantities and so the "normal" rules do not apply . 1'E‘Pupils can also get the same results by simply calculating a total score of (5 + 90) out of a possible (10 j- 90) to give 95/100 or 95%. It is interesting to note that le this case, adding the numerators and the denominators of the fractions that represent their marks (another thing that People are told they can never do)is a correct way to Calculate the final result. 45 vary as suggested. Now, because the 80 k/hr speed takes place over a longer period of time it has more weight and the solution to the problem needs to take this weighting into account. Connecting to other Subjects Connecting mathematical ideas to other subject areas helps pupils to see where mathematics can be applied and at the same time makes use of intuitive ideas and knowledge that they may have developed in other classes. Using the same example as above, the teacher might ask a pupil who is having difficulty to think about the problem in terms of molecular masses of chemical compounds. For example, if carbon consists of two isotopes, one with an atomic mass of 12 and the other of 14, it does not follow that the mass of a carbon molecule will be 13 because the two isotopes do not occur with the same relative abundance in nature. If, for example, only 1% of the carbon atoms in the molecule are Carbon-14 and the rest are Carbon—12 then the actual mass of the molecule would be very close to 12 and only slightly influenced by the higher mass of Carbon-14. Once again a Vneighted average is an appropriate way to calculate the “kiss; and, this is the method that would normally be used in 1flae chemistry classroom. Obviously it would be of little Value to use this kind of example with pupils who did not 46 take chemistry; however, for pupils who do, the discussion makes a nice link with another subject area. Connections to other Mathematical Topics Making connections with other mathematical topics can help pupils to see how one piece of the mathematical picture relates to the whole. To illustrate how this kind of connection can be made consider this common teaching situation: The following conversation takes place between a grade 11 pupil and her teacher: T: What is a2 +b2 ? S: I know, that's easy, a + b. T: No, that's not right. S: Yes it is! an Could you explain please. S: Sure. You taught that when there are several operations exponentiation comes first and you also taught us that square root is the same as an exponent of 8 so I did the square root first. Th Hm... How would you respond to this pupil? The difficulty that the pupil is experiencing in this problem relates to his or her misinterpretation of the rules about order of operations and what it means to ”do" the square root. From the conversation it appears that the pupil is thinking of this problem as (a2 + b2)“Q. One way to relate this situation to another area of mathematics would be to ask the pupil to evaluate a similar expression 47 with an exponent but to choose an exponent with which they are more familiar, for example (a + b)2. This question can be done by rewriting the original expression in the form (a + b) (a + b) and then evaluating the answer as a2 + ab + ab + b2 = a2 + 2ab + b2. The answer if done by the pupil's method would be a2 + b2 and so there is a clear problem. The connection with another situation involving exponents and brackets which has its own special procedure for evaluation illustrates to the pupil why exponents cannot be distributed and makes use of a previously developed idea. However, the difficulty can also be handled by the teacher's attempt to analyze what conceptual error in the pupil's thinking is leading to the problem. Putting oneself in someone else's mode of thinking is a difficult but useful 8kill for a teacher; and, it allows the teacher to connect with the pupil's way of thinking. Connection with a Pupil's Way of Thinking In the example discussed above the pupil is making a fundamental conceptual error. It is true that multiplication is distributive over addition in the real number system, so a(x + y) = ax + ay. However, the belief that an exponent can be distributed in the same way that a nuluber multiplied by a bracket can be distributed, is rooted .L‘Ll eve the log und the the) brac 48 in the belief that all functions are linear.16 In fact, this conceptual misunderstanding surfaces throughout mathematics courses in many different guises. When pupils, even in first year university mathematics classes, write that sin (A + B) = sin A + sin B or log (x + y) = log x + log y, they are exhibiting the same lack of understanding. Alternatively it may be that in this example the pupil has not made the connection between brackets as they are normally used, for example in (x + y)2, and brackets as implied in the expression W .17 The teacher must first connect with the pupil's way of thinking to be able to help him or her to solve this l;roblem. If the difficulty is that the pupil does not tinderstand that there are brackets implied in a square root tsign then one way of making this connection would be to Jrewrite the question as (a? + b2)”2 so that the use of karaokets becomes explicit. Now the analogy with other Similar questions becomes more obvious and the reasons for the error in logic can be realized. \ 1"‘One of the properties of a linear transformation is 1311£rt k(f(x) + g(x)) = f(kx) + g(kx). This property is not t1Tue for non-linear relationships. let is interesting to note that in old mathematics teaxtbooks, I assume because of the difficulties with :5lflpesetting, square roots were actually written like this —- ( a3 + b3). In some ways this method is an aid to understanding what the symbols mean. real the C033 what in t. afte. appl: methr Conn: many case) was 1 Itasc implj where SGVQI implj aPPrc and t frdct Lieu Hith i 49 If the response of the pupil is that he or she did realize that there were implied brackets and that they did the brackets to get their answer, then the teacher must connect with this way of thinking somewhat differently. It is at this level that arguments need to be made about under what conditions the distributive property holds and whether in this case those conditions have been satisfied. Only after a realization of why the property that they wish to apply is incorrect can pupils then realize the correct method of approach . This example illustrates that these categories of «connections are not mutually exclusive; rather, there are Inany interrelationships between them, For example, in this <3ase, a teacher who connected to the fact that the student seas not interpreting an implied bracket correctly might reason that the pupil needed to see other examples of dmelied brackets. Questions involving complex fractions Vibere both the numerator and the denominator consist of several fractions that are added or subtracted also have iquplied brackets. It is for this reason that the typical £1E>proach is to evaluate the numerator, then the denominator and then divide these two fractions. In this case, the fI‘action bar (like the top of the square root sign) is an ijuplied bracket. This connection could be considered to be a connection wj~‘t:.h.the pupil's way of thinking but it then leads to a 50 connection to another mathematical topic. In some sense these kinds of interrelationships are a product of the desire on the part of the teacher to have the pupil understand the mathematics that they are teaching and they are all rooted in pedagogy. Pedagogical connections, therefore, need to be considered in more detail. Pedagogical Connections The philosophy that a teacher holds can influence his (or her teaching in profound ways (Ernest, 1988). If, for example, the teacher believes that it is important for students to understand even routine procedures and why they evork, then he or she will teach differently to one who Inelieves that conceptual understanding is unimportant in learning procedural aspects of the subject. Consider the following case in point: You have taught the pupils to multiply two binomials by using the distributive property. Although most of the pupils seem to understand what you have taught, a couple just don’t seem to be able to understand how to go about it. How would you explain the problem (x + 7)(x + 3) to such a pupil? A teacher who wanted the pupils to understand this EPITCJcess would likely try to find a way of representing the I?1?<:blem.so that the pupils could visualize what was being <1<>Ile in a more concrete way. One simple way to do this is 1:<> represent the problem as that of finding the area of a ‘3‘3<:tang1e with sides of length (x + 7) and (x + 3). In this case repre 3.2). Conn Conn Cons hath teat Stud Stil 51 case the rectangle can be divided into four separate areas represented symbolically by x2, 7x, 3x and 21 (see Figure 3.2). x 7 2 x x 7x 3 3x 21 Figure 3.2 - Geometric Representation of a Binomial Product Once again the clear categorization of this kind of It isa connection as a pedagogical one is not possible. connection that the teacher has made based on pedagogical considerations and yet it does relate the problem to another mathematical idea, that of area. It is the ability of a teacher to represent and transform knowledge that makes 8tudent learning possible; so, pedagogical connections are 811:111 highly significant. 52 Summa; 2 This brief overview of the kinds of connections that can be made between and among mathematical ideas has illustrated that there are many different ways of making connections. It has also shown that the kinds of connections that are made are not necessarily discrete and that some attempts at connecting topics together can encompass several different categories of connections. Given that teaching is fundamentally a process of connection :mmking between the teacher and the pupil (or perhaps between 'the pupil and the content), pedagogical considerations are at the heart of most connections that teachers are likely to Inake in the course of their daily classroom lives. This situation is illustrated diagrammatically in Figure 3.3. Connections with other areas of mathematics Pedagogical connections Connections wlth students' ways of thinking Flea] world connections Figure 3.3 - Components of Connected Knowledge 53 CHAPTER 4 - STUDY DESIGN AND METHODOLOGY Introduction One of the primary arguments that has been made for the importance of studying the subject matter knowledge of teachers is that the knowledge they have of subject matter has an impact on their teaching. Still, links between subject matter knowledge and student achievement have been consistently difficult to find. If one reason for this apparent contradiction is the simplistic way that teachers' lknowledge has been measured in earlier studies then it is {important that techniques other than traditional measures be «developed for judging what teachers know about their subject and how they know it. The considerations that went into this decision are presented in the first section of this chapter. Subsequent sections deal with instrumentation, research subjects, data gathering methods and analysis Strategies . Design Issues This study sought to understand something about tleachers' subject matter knowledge particularly as it aPplies to teaching. I decided, therefore, that the kinds of questions that needed to be asked must be contextualized 54 55 for the respondents so that they could think about the responses that they might make in a teaching situation. In order to accomplish this goal I designed a number of teaching vignettes, each of which had a subject matter component nested in it; and, which I knew could be connected with other areas of mathematics, with other subjects or with other representations of the problem. A deliberate attempt was made to sample widely the major content strands that are given in the Standards. (See Appendix A, for the relationship between the items that were constructed, various content strands and kinds of potential connections). These vignettes asked students how they would react as teachers in a classroom setting to a variety of teaching situations; and, because their predisposition might be to simply tell the pupils the answer, it was quite possible that they would not focus on making connections even when the interviewer tried to move them in that direction. Consequently, to provide for the greatest possible opportunity for them to be able to talk about how they did make connections, three card sort tasks that further focused the respondents' thinking on connection making were Provided. Stake (1978) suggests that "... useful understanding 13 a full and thorough knowledge of the particular. . .". This kind of understanding is the kind of knowledge that can best be established by an in-depth interview. However, althc indix the I allo‘ COESc was sax; Simi chos some kind infc C0n1 Con: var: kin: 56 although interviews were the logical way to find out what individual people thought about these teaching situations, the labour intensiveness of the interview process does not allow for a large number of subjects to be interviewed, and consequently does not allow for more general conclusions about prospective secondary mathematics teachers to be drawn. In order to be able to generalize to some degree, it was decided to combine the in-depth interviews of the small sample with survey data from the larger but demographically similar sample from which the interview subjects were chosen. The two methods of data collection, although they used some of the same items, provided different (but related) kinds of information. The surveys were designed to give information about students' connected knowledge and the connections that they do make. The interviews were employed to provide a more detailed picture of the kinds and range of connections that individuals typically make on a wider vanriety of items and tasks; and, also to illustrate the kinds of connections that people can make when prompted to do so. In one sense the surveys were designed to examine a"indents' connected knowledge (static at any point in time) While the interviews were developed to also illuminate their ability to make connections (dynamic and changing over time) . Given that the purposes of the two types of data Collection and the methods used to analyze the data are 57 different, they are treated separately in the discussions that follow. The__S_tir_v_ex Subjects The survey subjects were 127 (71 males and 56 females) prospective secondary mathematics teachers enrolled in three different teacher preparation programs at a large Canadian university. All of the programs are consecutive programs with some variation in the timing of student teaching; and, in the number of weeks of practicum that are required. One group in a co-operative arrangement with school boards had taught for at least two separate four month work terms before their entrance into the formal teacher education phase of their degree program (38%). The structure of the second program provides for a two week orientation session followed by sixteen weeks of practicum experience which precedes the major portion of their on-campus studies (31%). The final group has a ten week practicum experience distributed throughout a full year program in three shorter segments (31%). The subjects ranged in age from 23 to 48 years with a median age of 25 years. All had completed a baccalaureate degree in a subject discipline (for 85% this was their only degree) although not necessarily in mat] had ens. rel sex: as : we: Wee sci. suk tEi 58 mathematics. Seventeen also held Masters degrees and two had completed a doctorate. Their subject matter preparation in mathematics varied enormously. Approximately 28% of the sample had a relatively weak background in mathematics (less than 10 semester courses); 26% had a background that was classified as moderate (between 10 and 18 semester courses) and 46% were classified as strongly prepared candidates who had taken 20 or more semester courses in mathematics. However, because of the competitive nature of entry into faculties of education in Ontario, their overall level of academic achievement, as measured by traditional means, varied from very good to outstandingh18 The reason that some students could have relatively weak backgrounds and still be intending to teach secondary school mathematics is that the Ontario regulations specify that secondary teachers must be certified to teach in two subject areas. Consequently, for many prospective secondary teachers, mathematics is a second teaching option. The minimum requirement for a second teaching option is six semester courses in mathematics. By contrast a student with an honours mathematics degree could have as many as 25—30 semester courses in mathematics after a fifth year of high school which offers three mathematics courses, each of which 18It is typical to have 5000 applications for the 650 positions at this College of Education. is rough; American Maths teaching in second options we Other (199} accountinc enClineerir that lasi they had i teacher 13: In 0] two Sets 59 is roughly comparable to freshman level mathematics in an American university. Mathematics was the most commonly occurring first teaching option (65%) with physics (17%) and chemistry (12%) in second and third place. The most common second teaching options were computer science (40%) and mathematics (35%). Other degree majors represented in the sample were french, accounting, data processing, general science, biology and engineering. Approximately 20% of the respondents received their last degree in 1988 or earlier which suggests that they had worked in business or industry prior to entering a teacher preparation program. Instrumentation and Data Collection In order to provide a focus for respondents' thinking, two sets of related instruments were developed for this study. One was a survey which asked prospective teachers to respond to a number of classroom scenarios. The original items were first piloted with two classes of students in the 1989-90 academic year who were not otherwise involved in the project. In addition to answering the questions these students were also asked to comment on any items that they thought were poorly worded, unclear or vague. As expected, some items were ambiguous, some did not elicit the kind of information that they were intended to elicit and some were simply nc thinking.' the items extensive tasks des ones were The were sort« that each I “Minis- beginning about the them that the Proje instructi think 596 60 simply not interesting in what they revealed about students' thinking. Based on this preliminary work, the majority of the items were re-written (some slightly, others more extensively); some were moved into the various card sort tasks described later; a few were discarded; and, some new ones were added. The 21 items that were the result of this pilot project were sorted into surveys using a matrix sampling method so that each student was asked to respond to five items. When I administered these surveys, I spent a few minutes at the beginning of the lecture hour telling the group a little bit about the research and its purpose. I clearly indicated to them that they were under no obligation to participate in the project if they did not wish to do so. They were given instructions orally before starting the survey and asked to think specifically about ways of making connections. These instructions were provided in written form as part of the survey package which also contained a consent form and a questionnaire asking for a variety of demographic information (age, gender, university major, teaching options and so on). A typical survey consisted of the cover pages given in Appendix B and five of the items that follow these introductory pages, chosen at random. The consent form served the dual purposes of obtaining informed consent for the survey and at the same time establishing a pool of students who would be prepared to take part project. during a were free participat one lectui completedl intellectx think aboi Vere unaca the summe the items In I'eSI-XNISeg, interesti kinds of least 30 61 take part in the interviews in the second phase of the project. The surveys were administered to complete classes during a regularly scheduled lecture period. Although they were free to decline, only a few students decided not to participate; and, they were able to complete the surveys in one lecture period. Many commented as they handed in their completed surveys that it had been a difficult but intellectually stimulating task that had forced them to think about their teaching knowledge in a way to which they were unaccustomed. In fact, one student wrote to me during the summer and asked if I would send him a copy of all of the items so that he could further reflect on them. In order to be able to say anything at all about responses to specific items (if analysis revealed an interesting pattern between males and females on certain kinds of questions, for example) it was necessary to have at least 30 responses to each item. This restriction meant that the number of items times 30 would give an acceptable total number of responses; and, if each person had five items to answer, the sample size should be approximately (21 x 30) + 5 = 126. Consequently, the data was collected in several stages because various groups were on campus at different times of the year; and, in order to get a sample of this size it was necessary to spread the project over two academic years. Initially, two classes were given the surveys at the end of the 1990-91 academic year and then two other grc 1991-92 a in early campus . Data Anal 62 other groups responded to the survey at the beginning of the 1991-92 academic year. The last group completed the surveys in early January when they began their final semester on campus. Data Analysis The quantitative analysis of this data was done with a microcomputer running the program SPSSPC+. This program is the microcomputer version of the Statistical Package for Social Sciences (SPSSX) program. It was chosen because it is just as powerful as the mainframe product for a dataset of this size; but, it is significantly more user-friendly and convenient. It features pull down menus and help screens to aid the user in writing the programs needed for the analysis. In addition, the almost instantaneous results that it produces makes the analysis process very efficient. Furthermore, in this kind of research, it allows for the _kind of exploratory analysis that would be more difficult on a remote mainframe system. The first step in preparing the data for analysis was to assign a code number to each student. This code number was then written on each page of each survey so that the surveys could be separated and sorted by item but still linked (anonymously) to the student who had responded. All of the demographic data was entered into the computer and a file was characte: was pose: female In“ either as The independe establish strong Cc Connectic Category 1 ‘ Stro The 31'.ch item “it “Ethernet this 18‘ Humbe); . Common '. (See AP r‘epreSE metaphc Dibble] 2 s We The St 63 file was created that allowed for the examination of various characteristics of the sample population. For example, it was possible to select all of the respondents who were female math majors and then analyze their response patterns either as a whole or item by item. The survey responses were coded by two raters independently. The first stage of the coding process was to establish a five category system. The five categories were strong connection, weak connection, correct but no connection, no idea and incorrect. The meanings of these categories are detailed below: 1 - Strong Connection The student has clearly linked the content embedded in the item with some other mathematical or real-life example. The mathematics used is relevant and reasonable for students at this level -- for example, they do not try to use advanced number theory but rather talk about remainders and lowest common multiples as a way to solve the problem in Item 18 (see Appendix B). Alternatively, they may have chosen to represent the problem situation differently by using a metaphor or analogy to connect their understanding of the problem with that of the pupil. 2 — weak/Unclear Connection The student tries to connect the mathematics in the item to something else but the explanation is not coherent; however, the gene his or h I 3 - No Cc The stude linking ;' subject al refers to example , rise/run 1 0f lOgica mathemati the guide setting a ‘ ‘ Unabl The StUde 64 the general direction that the person is trying to take in his or her response is evident to the reader. 3 - No Connection The student re-explains the concept without any attempt at linking it to a different mathematical topic or other subject area; or, simply uses a rule or explanation which refers to some kind of mathematical convention. For example, the definition of slope is just considered to be rise/run by agreement rather than as the result of some kind of logical argument. The explanation given, however, is mathematically correct and is detailed enough to show that the student does clearly understand what the question is getting at. 4 - Unable to answer The student does not appear to know what the question is about and either leaves it blank or states explicitly that they are unsure about how to proceed. 5 - Answers Incorrectly The student does provide an explanation of what he or she would say but it is mathematically incorrect or imprecise. The answer given may also indicate that the student did not have a good idea of what mathematical ideas were embedded in the original item. The original decision to use a five category scale was made on the basis of the fact that SPSSPC+ would allow for easy collapsing of the categories without manual recoding if they tur subdivid computer categorie was carrJl| The decided 2 help imp: a SL1bfiamp Although agreement more subt rated. I COding Ca Were not conseqUet was meant clarifies 65 they turned out to be too fine; alternatively, trying to subdivide categories after the data had been input into the computer would have been impossible. I then discussed these categories with the other rater and the initial trial rating was carried out. The surveys had been sorted by item and so it was decided to rate all responses to the same item as a block to help improve consistency. After using this coding scheme on a subsample of the items, the two raters compared results. Although some items had an extremely high degree of agreement (greater than 90%), typically the items that had more subtle ideas embedded in them were not as consistently rated. It was clear during this consultation that the coding categories and the mathematics nested in the items were not understood in the same way by both of the raters; consequently, a substantial amount of discussion about what was meant by each of the categories took place. This clarification of ideas allowed the raters to refine their understandings of the categories and the mathematical principles that were under consideration in the various vignettes. All items were then independently re-rated by both raters. It became clear as the rating was being done that the open ended nature of the items and the free format of the responses made judging between a strong or weak connection, for example, very difficult. Students did not always thought judge wh some rest technical one. I u rater tent decided t< (COded as but no cox resulted .‘ The p as the in: with them reaSOned ‘ respolmes rater Pro SYStem . an“is. is 66 articulate their thoughts well in their written work and the raters were often left to make assumptions about what they thought the student meant. Similarly it was difficult to judge whether a respondent had just forgotten to mention some restriction in their explanation (which made it technically incorrect) or whether they even knew there was one. I usually rated these as incorrect while the other rater tended to be more lenient. For these reasons it was decided to collapse the five categories to three: connection (coded as the number 1 for later analysis purposes), correct but no connection (2) and error (3). This coding system resulted in an overall inter-rater agreement of 80%. The purpose of this double rating was to ensure that, as the individual who developed the items, my familiarity with them would not prevent me from being able to make a reasoned decision based on some criteria about student responses. The reasonably good agreement with the second rater provided some measure of confidence in the rating system. The final ratings used throughout the rest of the analysis procedures were the ones that I had assigned. The ratings for each item were then typed into the computer and this process produced a file which linked each item and its connection rating with a particular student. This file and the demographic file were printed out and checked manually for any errors. Then, in order to be able 'to make comparisons between various subgroups of the populati the file each stu< swsspc+ 1 This 636 ratiI which was Character for the s reSPOUden had diffe Compare. analYsis to be ext linked to infoI'Itlati Connec3tio examPle . 67 population with any particular characteristic, this file and the file which contained the demographic information for each student were matched and joined into one file by the SPSSPC+ program. This last manipulation produced a file which contained 636 ratings of responses to individual items”, each of which was linked to a respondent and his or her demographic characteristics. This process defines the unit of analysis for the survey as the response to an item rather than as the respondent. This distinction is important because students had different survey packages and so are not easy to compare. Using each individual item response as the unit of analysis does provide the potential for useful information to be extracted from the data because each response is linked to a student and to that students' demographic information. Consequently it is possible to compare the connection ratings of items completed by students with different backgrounds, genders or teaching options, for example. The final step in this stage of the analysis was to select subgroups of students with various characteristics and look for patterns in the data. Cross-tabulations were used extensively for this purpose and those that suggested 19Each survey had five items except for one which had six. The total comes from the product of the number of students (127) and the number of items per survey (5) plus 1 extra. interest square t the effe the deve‘ connected detailed interview 68 interesting avenues of pursuit were accompanied by chi- square tests to establish the statistical significance of the effects that were observed. This process allowed for the development of a number of themes about students' connected knowledge in general that complement the more detailed descriptions of connections that came from the interviews. The Interview Subjects The sample for the interview was not chosen at random; rather, it was selected purposefully to ensure that males, females and students with a variety of mathematical backgrounds were represented. Furthermore, because the purpose of the interviews was to get a detailed picture of the kinds of connections that prospective teachers typically make, it seemed prudent to try to pick people who were likely to be able to make connections that they could then talk about. Prior to selecting the students for the interview sample, the overall connection score (using the rating values discussed above) for every student who responded to the survey was calculated. These values, along with identification numbers, were then printed out in rank order using th students someone w Connectio resPondeci in this a certain b For eXam; that they Iflake any “the-man they made errors. a Willing easy to R” likely tc 69 using three categories (weak, moderate and strong) for students' mathematical background. Students who had not agreed to be interviewed were crossed off this list and the remaining students then formed a pool of potential candidates from which the interview sample was chosen. Obviously the process is not foolproof. Because a strong connection was rated as a one (1) it is clear that someone who got a total score of only five is good at making connections (at least on the particular items that he or she responded to) and a score of 15 demonstrates a lack of skill in this area. What a score of 10 means is not at all certain because a person could get 10 in a variety of ways. For example, if someone received five ratings of two, the 10 that they would get as a total would mean that they did not make any connections but they were consistently mathematically correct. A score of 10 could also mean that they made two connections, were correct once and made two errors. Fortunately, 60% of the survey sample had indicated a willingness to be interviewed and so it was relatively easy to work down the list selecting the students most likely to be able to make connections in each category (males and females; strong, weak and moderate backgrounds). In fact, even the weakest of the choices had been able to make at least one connection in the survey. The eight people who comprised this interview sample were contacted and asked once again if they still wished to take par in Appen agreed t: openly e: summary c students 70 take part in the project. in Appendix C. All of the candidates who were contacted agreed to participate and in almost all cases they were openly enthusiastic about being given the opportunity. A summary of the information collected about these eight students is given in Table 4.1. The letter they received is given Student Gender Age Degree First Second Math Option ggption Background Dana" Female 25 PhysEd PhysEd Math Weak Bill Male 36 Physics Physics Math Moderate Greg Male 27 Chem Chem Math Weak Dawn Female 24 Chem Chem Math Moderate Kirk. Male 29 Engin Physics Math Moderate Lynn Female 24 Math Math Comp Sc Strong Josh Male 30 Chem Chem Math Weak Jeff Male 24 Math Math Comp Sc Strong Instrumentation and Data Collection Six interviews with each student, each lasting Table 4.1 - Information on Interview Sample approximately one hour, were conducted during February and March of 1992. However, because one student withdrew from the program due to ill health I was only able to do 46 ‘NSame gender pseudonyms are used for all respondents throughout this dissertation. intervie' been ana prelimine approprie others . P001 of 2 three int first int Second ir. the focus the items Possibil j all of ti Candidate To p connecti‘ on Conne. The3e an Task One were Wri procedu- so that in the s Perceive conneCti all Of 1 71 interviews (rather than 48) in total. The survey data had been analyzed before the interviews were conducted and this preliminary work revealed some items that were not as appropriate for eliciting information about connections as others. Consequently, 15 items from the original survey pool of 21 were selected to form the basis for the first three interviews. Items 1, 2, 3a, 3b and 4 were used in the first interview; items 5, 7, 8, 9, and 11 comprised the second interview; and, items 12, 14, 16, 19 and 20 provided the focus for discussion in interview number three. Because the items were the same for all candidates and there was a possibility that some items might provide clues for others, all of the items were discussed in the same order with all candidates. To provide the respondents with an opportunity to make connections with some items that were slightly more focused on connections, three card sort tasks were also produced. These are included in Appendices D, E and F respectively. Task one provided students with a series of cards on which were written various mathematical topics, formulae, and procedures. They were asked to take some time to sort them so that items which they believed to be connected were put in the same pile. They were then asked to discuss why they perceived these items to be connected and to explicate the connections that they had identified. This task required all of interview four. we: war a d car the pil The thi int int 4' int Ba; the 'A U] 72 In the second card sort task, students were presented with cards which contained various mathematical definitions and they were asked to sort them into two piles once again. In this case they were trying to decide if the definitions were arbitrary and conventional or whether they were warranted in some way with a reason behind them. Once again a discussion followed the sorting. Task three provided cards which had a variety of common mathematical rules on them. Students were asked to sort these cards into two piles: rules they would teach and rules they would not. They were then asked to explain why they had sorted them in this way. These last two card sort tasks were combined in interview five. A final interview was conducted to allow the students a chance to talk about themselves and their ideas about mathematics and/or their participation in the project. All interviews were audiotaped and all of the students' notes, rough work and so on were filed with the interview tapes. These written records were used to help clarify any ambiguities that arose when transcribing the interviews. Data Analysis The first step in analysing the data was to transcribe the tapes of the interviews. The word processing program WORDPERFECTo was used for this purpose. This task, although 73 tedious and time-consuming, is a valuable first step in the analysis process because it familiarizes the researcher with the data much more intimately than reading someone else's transcriptions. After the initial transcription of each interview was finished, the complete tape was replayed while the transcript was corrected for any errors or omissions. During the transcription process, tentative themes began to emerge and these were noted briefly in short analytic memos (Bogdan and Biklin, 1987). These memos later informed the development of the coding system for the interview data and the framework which was used to help organize the presentation of the findings. The process that I used for developing a coding scheme was essentially that outlined by Tesch (1990). Three transcripts out of the 46 were chosen and each was blocked into coherent thought units. These divisions were often difficult to decide upon because the respondent might have been talking about one topic for three or four pages and I did not want to have blocks of text that long; however, usually when the interviewer asked for clarification these interjections allowed for a convenient break in the dialogue. The interviews, now blocked, were then re-read and I wrote down on a separate sheet of paper how I would describe what each block of dialogue was about. For example, if the student was talking about how to illustrate a point about 74 triangle congruency I might have noted down the word "proof”. This process was continued for all three interviews, at which time I had a list of categories on several sheets of paper. I then put the interviews to one side and went over the list of categories looking for duplications (same idea but a different word), deciding at the same time which word to use for that particular idea. This process produced a list of categories to which I then assigned codes. The process was not totally static, however. The later interviews involving the card sort tasks required a few new categories to be created that did not occur in earlier ones. (The final list of codes along with the meanings associated with them is provided in Appendix G). Once the categories and their codes had been established I went back to the beginning and blocked all of the interviews (including the original three) and then assigned codes to each of these blocks. Any extraneous discussion about non-relevant issues was also removed at this stage. The blocking and coding process was an evolutionary One. Students often started their responses in a rambling, 1T-entative fashion but became more focused as I asked for cIlarification or amplification. Consequently, one long Segment of an interview transcription about a single item might have been broken into several text blocks with different codes throughout as the student rambled around 75 (CN-NR - close but no connection) and finally focused on a mathematical link (CN-MTH - mathematical connection). To provide an accurate sense of this process without causing data overload, I have included a typical transcription as it appeared before the coding process in Appendix [-1. Appendix 1: contains the same transcription after it has been blocked and coded. The purpose of this blocking was to allow the transfer of the raw transcriptions into a database program that would then permit rapid manipulation of large chunks of textual data. The large size of most of the thought units required a database that was not overly restrictive with respect to field size. The database program DATAPERFECTO allows for very large text fields and it is a WORDPERFECTO compatible Product. These two features meant that I could design a Custom database using DATAPERFECTO and then import all of the coded transcriptions into this database with only a Small amount of extra typing. To understand how this process was accomplished, note that, in Appendix I the first thing that appears in the coded transcription is the character string SREF"R12"R. Also note that at the end of the phrase "[laughs]" is the character string “E. These codes were added to the original transcription to mark the beginning and the end of the first block of text. In addition, the first group of codes was Selected to let the database know what this text block was 77 records were erased and the panel was redesigned. The final blank form that the database would display on the screen for each record21 is shown in Figure 4.1. I typed the student's name, major, background and so on into the appropriate fields in the database and then had the computer import the entire transcription of that particular interview. Assume, for example, that I was importing the interview given in Appendix I into the database. r1 MEWS-9 5 1 lane: Allan Date: 02/20/92 Intorviewh 3 Background: Strong Stutypo: W Gender: F Age: 24 Major: Math Minor: Comp Sc ID: 0642 Codes: SRBF Item 12 fit: Lynn: I had this one earlier ... Eric: Oh, did you, on the survey? Lynn: Well on the survey yeah and I realized what I said wrong then now. Eric: I have no idea because I don't remember what everybody said. A Figure 4.; - Sample Record After Data Imported When the importation takes place the program interprets the SREF as a code and places it in the code field; it interprets the 12 as the item number and places it in the item field; and, it interprets everything up to the "E as a thought unit and places it in the text field. All of the Other information about that particular student and \ . 21In database terminology, a record'is a unit of JJlformation which contains several fields. 78 interview that is common for all text blocks was automatically put into this record by the importation program and then the process was repeated until all text blocks had been imported, with their codes and other relevant information, into separate records (see Figure 4.2). Each interview had the number of text blocks that it contained (typically 20 to 30) counted manually and this value was checked with the computer's count of how many records it had imported. If these numbers did not agree it meant that some data had been lost or misplaced. The reason for the problem was found and the data was re-imported correctly. This system of cross checking was necessary because the actual importing of files is essentially transparent to the user and even small errors in coding can result in big problems. One missing “E or “R would get the computer out of step With the records and fields that were being imported and it Would not be able to interpret the incoming data properly noz- associate it with the correct fields in the database. consequently, unless some checking strategy is in place there is no way (other than brute force checking of the err'T-ire dataset) to tell if all of the data that is supposed to be in the database is actually there; or, if the data that is there is properly situated for later searches. I a180 programmed the database to assign a unique number to 76 about (SREF - reflection about own knowledge) and which item (Item #12) was being discussed. The character string "R was used to separate the fields so that the database program would know when the information in one field ended and another began. In order to now import this coded transcript into the database program the following procedure was used. The database was custom designed to contain all of the demographic information that I thought I might conceivably need in the analysis process. Each of the various student characteristics had its own field, name, age, and gender, for example; and, other fields were established for the COdes, text blocks, items, interview number and soon. This panel, as it is called, was then used as a template for importing a couple of interview transcriptions. 1W0). 3...: Data: Interview! : Background: Stutypo: Condor: Ago: » Maj or: Minor: ID: Cod... : Ital: 1'21; 8 \ a Figure 4.1 - Sample Blank Record for Database 3 I then experimented with the system to see what I could do with the data. Weaknesses were noted, the experimental 79 each of these records so that they could be easily identified later on if necessary. The overall result of this process was to produce a file consisting of 951 records each of which was independent of the other and each of which contained information in a variety of fields in addition to the blocks of text. The information that would appear on the computer screen when the first record from the interview in Appendix I is displayed is shown in Figure 4.2 This graphic was produced by capturing the screen display in the database program and importing it into this dissertation. Consequently it shows the screen display exactly as it would appear. Although the entire text block is not visible in the bottom window it is still accessible and does appear in its entirety when a record is selected and printed out. The database program could now be used to select records based on certain criteria. For example, all records containing the code CN-MTH (mathematical connection) could be selected. Or, all records which contained the string CN- MTH in the CODES field and the word "Strong" in the BACKGROUND field could be selected and examined. Combinations of codes in the same field are also possible; so, for example, all blocks containing both of the codes ERR (error) and FLCON (flawed conception) could be selected. All normal Boolean operators were permitted in combining the search conditions so the overall result was a system that 80 was extremely powerful and flexible. While selecting the groups of records to study further, the database was able to sort the entire list of records according to one or more specified fields. The ability to sort records is very useful because it makes concentrations of common values or words in a particular field easier to see. The use of this feature in identifying themes is detailed in the next section. Identifying Patterns and Themes This study was designed to find out something about what students could and did do when searching for connections among and between mathematical concepts and not to be a litany of their weaknesses. This goal suggested that I should try to look for patterns of commonality between students and connections. This commonality can take two forms: many students might respond in a similar way to an item; or, very few might be able to respond in a particular way. Consequently, the search for patterns focused on items and codes that were frequently or rarely associated together. Developing what themes and patterns were embedded in the data was an evolutionary process which made extensive use of the ability to select and simultaneously sort the records in the database. It took the computer approximately 81 'two minutes to complete one search of 951 records, select the appropriate ones and write them to a disk file for later examination. Overall I did dozens of these searches looking for themes and patterns. The basic strategy was to work from a relatively atomistic view and then make the searches more and more inclusive until something became apparent. Initial reports that were produced were of records containing individual codes; however, these only rarely illustrated any patterns with respect to the items that were being listed. The next level of search would combine codes that seemed closely related. This second pass would produce a larger number of records in total with larger numbers of items associated with them. As the number of items associated with various codes increased in overall number, differences in the frequency of occurrence of items would become apparent and then some notion of consensus would sometimes emerge. To illustrate a typical search, assume that a selection for the CODE field equal to CN-DEMO or CN-EXPL or CN-JUST is sorted on the ITEM field. This procedure produces a list of certain important fields of all the records that contain any one of the codes above, sorted according to item, name and background. This sorting is done in a hierarchical way so that all of the occurrences of each item number are grouped together and then within this grouping all names are grouped together and so on. Examining this list then allowed me to 82 see which items seemed most amenable to eliciting this particular category of connection. A listing of this report is given in Table 4.2. It is immediately obvious that some items have many more occurrences of a particular code associated with them than others. This fact does not necessarily mean anything, however, because some students often changed thoughts while talking and this habit made the size of the text blocks for their interviews smaller and more numerous. Others were more articulate and coherent and this type of interview tended to contain fewer, longer text blocks. Consequently, there may be many text blocks associated with a particular code and item but they could easily have come from the same section of one interview. However, if a particular code is associated with an item frequently and different students' names are grouped with this item then it appears that this item.has some kind of common response pattern. For example, in Table 4.2, Items 1, 8 and 12 look interesting not only because they have quite a few text blocks associated with them but because these blocks do not all represent portions of a conversation with the same student. It was for this reason that I always printed out the students' names with the codes. In addition, the identification numbers can be used to indicate whether the blocks of texts are contiguous because these numbers were assigned sequentially when the records were imported. For 83 exanlple, Item 12 has three blocks of text attributed to Brown with identification numbers 514, 515 and 516. The Code- Item ID# Student Background CN—EXPL CERT 1 0147 Allan Strong CN —E XPL 1 0 2 2 8 C larkson Strong CN-EXPL RULES 1 0229 Clarkson Strong cauc CN-EXPL 1 0125 Smith Weak RULES AHA PEDR CBNC CN-EXPL 1 0120 Smith Weak CDT—DEMO 1 0084 Roberts Weak CN-EXPL 1 0080 Roberts Weak MPUSH CN-EXPL 1 0081 Roberts Weak MPUSH CN-EXPL CN-MTH 1 0082 Roberts Weak CN-MTH CN-EXPL ERR 1 0048 Brown Moder Chi-JUST CN-NUM CN-CNTR 12 0539 Grant Moder CN—JUST CN-CNTR CN-ST 12 0588 Jackson Moder CN-ST CN-DEMO CN-EXPL 12 0644 Allan Strong CN-M'rn CN—NUM CN-JUST 12 0567 Roberts Weak REP cuqmno 12 0558 Roberts Weak (HM-JBJKPL 12 0516 Brown. Moder CN-NUM CN-JUST CN-CNTR 12 0514 Brown Moder RULES CN-EXPL 12 0515 Brown Moder (IN-Nun CN-DEMO CN-JUST 2 0031 Grant Moder CN-N‘UM CN-JUST 2 0007 Jackson Moder “PUSH MCLAR CN—NUM CN-DEMO CN-JUST 2 0239 Clarkson Strong CN-NUM CN-JUST CN-CNTR LINTR 4 0043 Grant Moder CN-NUM CN-CNTR CN-CXT LINTR 4 0171 Allan Strong LIN'I'R CN-MTH CN-DEMO CN-JUST 4 0256 Clarkson Strong Chi—Nun CN-CNTR CN-DEMO 4 0113 Roberts Weak CN-CNTR CN-JUST CN-SUB 5 0434 Grant Moder CN-s'r CN-CNTR CN-JUST 5 0467 Jackson Moder Ct""~7l'TJS'.'l‘ 5 0285 Allan Strong elm-ST CN-EXPL 5 0406 Brown Moder €3~Mu~n CN-DEMO CN-JUST 7 0445 Grant Moder “1‘2er REP RULES 7 0295 Allan Strong CN“EXPL 7 0267 Clarkson Strong CN‘E XPL 7 0 3 6 1 Smith Weak CN‘Eer 7 0362 Smith Weak cur-Mira CN-NUM CN-CNTR PEDR CN-JUST 7 0411 Brown Moder mist. CN-NUM CN-DEMO 8 0227 Jones Weak Dr '1‘ REP RULES CN-DEMO 8 0449 Grant Moder CNAG CN-MTH RULES CN-EXPL CN-DEMO 8 0482 Jackson Moder CN‘JUST RULES 8 0296 Allan Strong Ragntm CN-JUST a 0299 Allan Strong can (ZN-RUM CN-JUST 8 0297 Allan Strong CN‘JTJs-r CN-ST 8 0365 Smith Weak can RULES CN-DEMO CN-EXPL 8 0394 Roberts Weak ‘JTJST CN 8 0391 Roberts Weak Table 4.2 - All records containing CN-DEMO or CN-EXPL or CN-JUST . 84 fact that these numbers are in order tells me that all three of those blocks came from the same larger chunk of text in one interview. Consequently, the fact that there are three blocks of text for Brown probably is a function of the way that he responded and the way the text was blocked. However, the fact that there are five different students who responded to Item 12 with the same group of codes still suggests that there might be something in Item 12 that makes this kind of response common. Typically, I focused on code-item combinations which had five or more (out of a maximum of 8) different student's responses linked with them. The entire records (including text blocks) linked to such patterns were then printed out and examined to try to find anything that the responses had in common. Sometimes this reading suggested other groups of records to select or other items to examine and the process Was then repeated. One of the most useful features of this system is that it allows the researcher to look for patterns without having to wade through the actual content of the text blocks that are in each record. Given the fact that the dataset for just the interview portion of this project consisted of a‘E’Zproximately 700 single spaced typed pages of transcripts, any patterns that are embedded in the data would have been impossible to see without some way of eliminating a portion of the data from the initial analysis. Looking at the codes 85 associated with text blocks allowed for this focusing of attention on specific portions of the data. Once patterns had been discerned, further analysis of the content of specific text blocks then provided for the descriptions of students' knowledge that follow in the next chapter. Summagj 2 The methodology employed in this study was chosen to make it possible to find out something about one aspect of teachers' mathematical knowledge. In order to help overcome the traditional problems associated with investigating teachers' knowledge, both surveys and in-depth interviews Were employed. In one sense the surveys are designed to examine students' connected knowledge (static at any point in 'time) while the interviews can also illuminate their ability to make connections (dynamic and changing over The instrumentation used did not ask for answers to time) . nu3-"2‘-111ematical questions but asked students to reflect about how to connect mathematical ideas together for the purpose of teaching. The survey results (636 item responses rePresenting surveys completed by 127 students) were a‘t‘4la1yzed using crosstabulations to reveal themes inherent in the data. The interviews (46 hours of interviews with eight reSpondents) were transcribed, coded and imported into an electronic database so that large blocks of textual data 86 could be easily sorted and intermixed. This method allowed for 700 pages of transcriptions to be managed in a way that made possible the development of patterns in the data. These patterns then provided the focus for the discussion presented in subsequent chapters. CHAPTER 5 - FINDINGS Introduction The methodology employed in this study provided for two different kinds of information to be collected with respect to prospective secondary mathematics teachers' connected knowledge. The survey results allow for general comments to be made about the large sample while the interview analysis provides a much more detailed examination of what various kinds of connections looked like in practice, based on data collected from this smaller subsample. These two data gathering strategies yield different kinds of information and so they should not be expected to produce identical results; however, given that they are both focused on the Same general area of study, it is reasonable to expect that the results be roughly comparable. The focus here is on a description of what was observed, with the analysis and jJ““JE>Zlications of these findings in subsequent chapters. The surveys asked students to respond to teaching B(r-‘eharios without any possibility of further probing or follow-up. Consequently, these responses provide it:‘fermation about the connected knowledge that prospective teachers readily have at their disposal. The next section of this chapter discusses this ability (or perhaps pI‘edisposition) of prospective teachers to make mathematical 87 88 (and other) connections; and, the relationship between various demographic characteristics of the students in the sample and this ability. The interviews can also provide information about specific teachers' connected knowledge but because of the ability of the interviewer to question and ask for clarification, this data gathering strategy also provides some insight about how students make connections that might not be immediately obvious. The second portion of the chapter will be devoted to an examination of the kinds of connections that students commonly made and how these connections were manifested in various different items and card sort tasks. These discussions will be illustrated with quotations from the interview data and where appropriate triangulated with information from the surveys. In order to give the reader a true picture of the <-3°1':atext of students’ responses it is sometimes necessary to include long passages from the interviews. Short phrases r adt‘ely give an accurate portrayal of the extent to which the 8t7-11dent is answering the question or whether the interviewer has pushed the student into responding in a particular way. These quotations should not be interpreted to be isolated examples of the connections under discussion but rather representative of a class of responses that had a common pattern. The decision about which quotation to include from this class of responses was made on a variety of factors, 89 including the length of the text block, whether that particular item had been used as an illustration before, the clarity of the student's response and whether that student had been quoted previously. In addition, in order to ensure that the data speaks for itself, only minor editing has been done to improve the readability of these text segments. Results from the Survey The collection of information about gender, mathematics background, type of program, age, major teaching option, undergraduate degree major and so on, allowed for an examination of various factors that might have had some influence on students' connection making ability.22 Crosstabulations of all demographic characteristics (gender, background, teaching options and so on) were done with the three possible ratings (connection, no connection but cc->:I':rect and incorrect). Although most crosstabulations revealed little of interest, some surprising differences in the way that prospective teachers were able to make col'lnections were evident between certain groups. s‘tatistically significant differences were apparent with \ 22I will no longer differentiate between a student's a*titility to make connections and his or her predisposition to Given the fact that they were asked to think of ways (19 so. Qt making connections orally before the survey was handed ‘11“. and as part of the written instructions that accompanied <3 it , I can only assume that if they were able to make a Qanection they did so. 90 respect to gender, mathematical background and specific teaching options. Not all of these characteristics are independent of one another, however, and further analysis will show that the differences that were observed are actually the result of a fundamental explanatory factor. Before discussing these differences, consider the survey results as a whole. Out of a total of 636 answers, given by 127 respondents, 216 (34%) were classified as demonstrating some sort of connection (either strong or weak) while 146 (23%) were considered to be mathematically correct but not connected to any other mathematical topic or to another subject. In 274 (43%) of the responses, students were not able to answer; or, they gave a mathematically incorrect explanation . In total, 57% of these responses indicated that the stilldents making them had sufficient command of the mathematical content of these questions to be able to e3‘Kplain them to pupils either with or without making some l{filled of connection. It is puzzling, however, to note that 4 3 % of the responses revealed a lack of understanding of 1"igh school content, even at the most fundamental level of Qerectness. What is most surprising is that the distribution of responses in the incorrect or unable to atlswer category is not related to any other factors, most I"‘Qtably, mathematical background. This last assertion is 91 illustrated by the crosstabulation of item response ratings versus mathematical background shown in Table 5.1. fl I Number of Responses BACKGROUND OF STUDENT MAKING RESPONSE Number Weak Moderate Strong Expected by Chance (e) R Connection 72 56 88 A 59 (e) 56 (e) 100 (e) T I Correct 26 35 85 N 40 (e) 38 (e) 68 (e) G I ncorrect 7 7 7 5 1 2 2 75 (e) 72 (e) 127 (e) Pearson Chi Square p = 0.006 Table 5.1 - Item Response Ratings versus Mathematical Background of Respondents This table represents the distribution of responses (N = 636) according to students' mathematical background and the kind of response that they were able to make. For exaIllple, 72 responses that were rated as connections came from students who had a weak background in mathematics. Beflatlse there are unequal numbers of students with various backgrounds, and different numbers of ratings in each of the three categories, it is to be expected that the values in each of the cells of the table will different. 92 In this case there is a total of 175 responses from students with weak backgrounds, and a total of 216 responses that were judged to be connections. The chance probability that a response comes from a student with a weak background is 0.275 (175 + 636) and the chance probability that a response is rated as a connection is 0.340 (216 -.- 636). The chance probability that a response would come from a student with a weak background and be rated as a connection is the product of these two probabilities. Therefore, by chance alone we would expect that 9.3% (0.275 x 0.340) of the responses would fall in the weak background-connection Consequently we would expect there to be 0.093 x c ategory . In the table this is 636 = 59 responses in this category. the quantity that is referred to as the expected value. It is interesting that the differences in the expected v'aZLue and the count for the “incorrect" category are extremely small even though logic would suggest that Students who have an extensive mathematics background should be Wrong less of the time (especially when discussing high 3<31‘lczol mathematics) than those who are weakly prepared. <"’cz’l‘lxrersely, there are statistically significant differences with respect to mathematical background and ”connections" versus "no connections”. The notable fact is that these differences are exactly the opposite of what might have been 93 predicted.23 The strongly prepared students are less likely to make connections than would be expected by chance while the weakly prepared students are more likely to make connections. The exact opposite is true of the ”correct but no connection" category . Number of Responses STUDENT'S MAJOR TEACHING OPTION Number Mathematics Physics Expected by Chance (e) R Connection 125 38 A 135 (e) 28 (e) 1' I Correct 112 11 N 102 (e) 21 (e) G Incorrect 173 37 174 (e) 36 (e) Pearson Chi Square p = 0.006 Table 5;; - Item Response Ratings versus Major Teaching Option The pattern observed in Table 5.2 is consistent with the overall results reported earlier with respect to Ina~‘tl‘1ematics background -- obviously mathematics majors would Inca‘l: often be in the strongly prepared category and the Ina~fl<>rity of physics majors (including engineers) are in the weak or moderate category. The observed effect then is \ w 23The first thing that I did when this result appeared as to go back to my data to check if I had got the coding Cheme switched around. The data was coded correctly. 94 ZLikely due to differences based on mathematics background, aalthough there is no way to tell if their ability to make <:onnections is a function of the fact that they haven't Estudied a lot of mathematics or the fact that they have studied a lot of physics. The gender differences are more subtle; but, upon (zloser examination they are still consistent with the crverall results in Table 5.1. The sample had more male students in the weak and moderate categories than we would expect by chance and more females in the strong category than would be expected by chance. Number of Responses GENDER OF STUDENT MAKING RESPONSE Number Male Female Expected by Chance (e) R Connection 132 84 A 121 (e) 95 (e) T 1 Correct 70 76 N 82 (e) 65 (e) G Incorrect 153 121 153 (e) 121 (e) Pearson Chi Square p = 0.047 Table 5.3 - Item Response Ratings versus Gender These differences are statistically significant (p = 0.037). Therefore, because there are more females in the 95 strong category than would be expected by chance and because strong backgrounds tend to be associated with poor (zonnecting skills it is reasonable to assume that low connection scores would also be associated with the females in the sample. Also, crosstabulations of gender versus connection rating, holding the mathematics background constant, Showed no differences between males with weak backgrounds and females with weak backgrounds or males with strong backgrounds and females with strong backgrounds. Although weak and strong backgrounds Show no differences with respect to gender, in the moderate category there were statistically significant (p = 0.019) differences between males and females with males more likely to make connections and errors while females were less likely to make connections or to be incorrect. This apparent anomaly is consistent with the findings about physics majors being good at making connections but less likely to give a correct explanation. The physics majors are mainly in the moderate category and they are mostly males (14 males versus 3 females). Hence, the difference in connection making ability in the moderately prepared group is not due to gender but to the teaching major. The gender differences evident in Table 5.3 are once again attributable to the fact that female and male differences are linked to strong and Weak backgrounds respectively rather than gender; and, the 96 patterns are what could be expected according to the previous results. One of the groups of students in this sample has had a significant amount of teaching experience (at least eight months) prior to their teacher preparation program. The majority of this group also have a strong mathematical background and they represent a major portion of the students in the sample with a strong background. Consequently any effects that are conjectured to be attributable to the strong backgrounds of the students making the responses could, in fact, be a function of the teaching experience this group has had rather than the background in mathematics that they possess. In order to establish if this difference in experience had any impact on their ability to make connections (especially in simulated teaching situations) when compared to other strongly prepared candidates, these students were removed from the sample and crosstabulations again computed. If this group is different from other strongly prepared candidates by virtue of their teaching experience, the Previous pattern whereby strongly prepared candidates were less likely to make connections than weakly prepared candidates should no longer be apparent. However, the Pattern in Table 5.4 is exactly the same with the respect to 97 «connection making ability and mathematical background as was evident in Table 5.1 where they were included in the sample. Number of Responses BACKGROUND OF STUDENT MAKING RESPONSE Expected by Number Weak Moderate Strong Chance (e) Connection 72 52 61 (e) 55 (e) Correct 26 32 34 (e) 30 (e) Incorrect 77 72 _80 (e) 72 (e) Pearson Chi Square p = 0.037 Table 5.4 - Item Response Ratings versus Mathematical Background of Respondents (Students With More Teaching Experience Removed) (Zonsequently, it seems reasonable to conclude that (iifferences in teaching experience (four weeks versus 8 Inonths in some cases) do not have any significant impact on the ability of these students to make connections. Another factor that was of interest was the way in IVhich the content of the items interacted with connection Inaking ability. The crosstabulation of ratings with items lis given in Table 5.5. It is clear from this table that (although the overall level of connection making is low, there are some items for which the connection rating is 98 quite high (2, 4, 10 and 16) and others where the rating is substantially lower than the average (3b, 5, and 9 are notable examples). A second observation can be made based on the information in Table 5.5. Every item on the survey did get at least one response that was classified as a connection. This observation supports the assumption that the items as constructed did contain potential connections that were reasonable to expect students to bring out. Although the items were not equally easy to connect, the results overall point to a somewhat fragmented knowledge of the subject with the same individual sometimes making very clever connections on a difficult item and then being unable to make any connection on another item that was routinely connected by most people. Why a strong mathematics background seems to work against the ability of people to make connections and has little effect on their ability to be correct is a difficult question to answer. In addition, it is puzzling that a Substantial amount of practice teaching experience does not appear to be helpful either. It is also interesting to speculate about why the content of some items allows for Connections to be made more commonly than in others. The inability to ask people about these issues in the survey format makes a definitive answer impossible at this stage. However, as the interviews were analyzed, the items that 99 <:ommon1y elicited connections, or rarely did so, were given special attention to see whether the same pattern was ’ 1 ' Item! Percentage Percentage Percentage : of Responses of Responses of Responses i rated as a rated rated ; Connection Correct Incorrect I 1 48 16 36 2 70 30 0 3a 50 13 37 5 3b 10 3 87 1 4 93 7 0 1 I 5 31 35 35* i 6 23 20 57 l 7 21 35 45 g a 27 57 17 V 9 3 13 84 I 10 71 10 19 t 11 13 3 84 12 52 42 7 13 13 65 23 14 26 32 42 ' 15 19 42 39 i 16 65 13 23 1 17 29 10 61 5 18 14 10 76 19 23 0 77 ‘ 20_11 _ 13“ f _ -27 60 i 71 1 _11_ 1i— 1 7 711111. * Some rows.may not add to 100% because of rounding. Table 5.5 - Item Response Ratings versus Items 100 evident; and, if it was, to try to understand what makes some items more or less amenable to connection making than others. Some tentative thoughts on these issues will be provided in a later chapter, after the interview data is presented. The Interview Results This section presents the findings from the interviews. I begin with an examination of some of the more general features of the interview data after which the details of the findings will be discussed. In particular, the extent to which students were able to make connections can be roughly gauged by the number of text blocks that contain 3Cutie kind of connection code as compared to the total number of text blocks. One caveat is in order, however, before any interpretation of these numbers is attempted. This value prOvides a rough gauge only and it should not be considered to be any more than that. The number of text blocks that an interview produced had a lot to do with the way that the il"(lividual being interviewed was able to articulate his or her thoughts; and, consequently, how easy it was to break the dialogue into coherent units. For some students there Vere many short segments while for others there were fewer longer blocks of text. Consequently the number of blocks of text is a function both of response frequency and how the interviews were blocked. 101 There were 951 blocks of text, 425 (47%) of which were coded with one or more connection codes. This fraction is somewhat higher than the 34% representing the proportion of responses classified as connections in the survey, but is consistent with the fact that the interviewees were picked in part because of their better than average ability to make connections. In addition, because the interviews allowed for analysis of subtle nuances of students' statements and further probing by the interviewer, the notion of what was a connection was somewhat more fine grained than was possible for the survey data. The result of this closer analysis was that many things Wh—‘Lch would not have been judged to be connections on the 8‘~1|-II':\reys were judged that way in the interviews. For eJ‘a-I'nple, Item 3b was rarely connected in the survey (10% of the responses exhibited a connection) but frequently c3<>rllr1ected in the interviews (7 out of 8 students made a connection). This item provides a good illustration of the differences between the data gathered in the surveys and the interviews. Although many students in the interview did make a connection in this item, most came to this connection as a function of puzzling over questions of clarification that were asked during the interview rather than as a well known idea that they volunteered without much prompting. Thus the results of the surveys and the interviews taken as a Whole are roughly comparable, as would be expected. 102 The more detailed findings that are reported here were established by using the searching and sorting techniques clee£3<1ribed in Chapter 4 (see page 80). In trying to find tv11£31:'was common about certain text blocks that seemed to be frequently linked to similar types of connections or rarely associated with connections of any kind, broad categories of connections began to emerge. It became clear that some kulzncis of connections turned out to be much more common than others (for example, numerical justifications) particularly when considered in interactions with content. Conversely, some kinds of connections were rare (for example, conceptual cOnnections) and were only reported for a few items. The Patterns reported here focus on commonalities in the data in telTmns of connections that were often made and those that were rarely evident. The common connections identified were grouped into two broad (and fuzzy)24 categories -- strong connections and "€3iiJ< connections. These categories grew out of an e3‘&.l'nination of the way that connections were commonly made in text blocks that had common codes associated with them. s"thong connections were typically similar to those discussed when laying out the conceptual framework. They exhibited a clear link among and between mathematical concepts, other Bllbject areas or pedagogically powerful ways of thinking. 2"This word is used in the ethnographic sense to refer to categories that do not have clear boundaries. 103 Weak connections were often ones that had not appeared obvious to consider at the outset; but, ones that became more obvious as they continued to appear in the data. Within the strong connection category there were four subcategories identified: mathematical connections; conceptual connections; connections to other subjects; and, pedagogical connections (these included connections to pupils’ ways of thinking). Weaker connections were subdivided into four subcategories also: numerical connections; contextual connections; procedural Connections; and, pedagogical connections. Each of these Various categories will be briefly explained and then illustrated with examples from the interview transcripts. Strong Connections Ma‘tl‘aematical Connections Mathematical connections were connections where a clear link was established between two mathematical principles wt‘1i.ch at first glance seem to be separate. For example, in one of the card sort tasks, students were asked to group together the topics that they thought were connected. Two or the cards referred to the pythagorean theorem and the COSine law. Students made a mathematical connection between tllese two concepts when they saw the pythagorean theorem as 104 Figure 5.1 - The Pythagorean Theorem a special case of the cosine law where the angle in the formula is 90° . One form of the cosine law is given by 02 e a2 + b2 - 2ab.cos C, where C is the angle between the sit'les a and b and directly opposite side 0 (see Figure 5.1) . S~‘5-I1ce cos 90° = 0, in the case of a triangle with a right angle at C, the formula reduces to c2 = a2 + b2 which is the p1"tllagorean theorem. This particular connection was made by most of the students with varying amounts of probing by the it‘l‘llerviewer. One student put it this way: Josh: That's not how you write the cosine law”, you write it as a2 = b2 + c2 - 2bc cos A. And that's how I remember it. And the reason I remember it that way is because then you can, by taking an appropriate angle, produce the pythagorean theorem from the cosine law. By choosing ... in my particular diagram here I chose this triangle 25The cosine law was presented in a non-standard format 30 that the link would not be so obvious. 105 here and C is my right angle. So now = az-rlf - 2ab cos C but cos C is 90 ... so that's 0 ... well C = 90° so cos C = 0 so that part drops out and I'm back to my pythagorean theorem... .Another common mathematical connection was made in reference to Item 16 (this result is confirmed by the survey). This item asked students to respond to an advertisement for pizza which claimed that two 18 inch diameter pizzas would be equivalent to one 36 inch diameter pizza. Again, all students were able to make some sort of mathematical connection. An atypical response came from Bill, who used the idea of direct proportionality to resolve the difficulty: Bill: The value of pizza would be how much area you get, so we'll just figure out the area for each one of them and just check if this is more or less pizza. And if it is more you're actually better off, if it is less then it isn't and I don't think I should say much more about it and leave it to the student to work it out. If he doesn't remember the formula then you can refresh his memory ... Eric: Do you have an intuitive guess before you actually do any work with it? Bill: Well let's see ... so this is 36 and the area goes like the distance squared so the ratio between ... so if I look at one pizza ... okay so I'm doubling this so that's going to be a factor of four here so this is four times as much as this one and then you take two of those and it's going to be still only half of that so in fact it isn't the same it's ... it's less value. Most students made a connection with a visual representation or actually calculated the areas. Bill’s 106 explanation was less common than most because he established 1:11¢e :relationship between area and radius without doing a direct calculation. It was also interesting because this EI‘VEBIIUG of approach was not his first idea; rather, he too suggested a straightforward calculation using the formula :f<>J: 'the area of a circle. However, when pushed to try to connect this problem with other things that pupils might be :fatmndLliar with he was the only one who spontaneously generalized the problem to figures other than a circle. Eric: Bill: What if you had this problem and the pupils had not done anything with proportions, they hadn't ever figured out the formula for the area of a circle either. Can you think of any way of illustrating the problem without making use of either of those techniques? [pause] Okay ... [pause] ... would it make sense that the question might also be answered if you used square pizzas instead of round pizzas with the same sort of situation? So instead of ... taking one square 36 by 36 and you take two 18 by 18 and then this certainly could be worked out easily ... so just compare the areas of that and see ... see in fact ... I guess the issue is when you add these two, two times ... two times 18 is 36 and that's the same as down here so is that same relationship between the dimensions here also manifested in the area? That's really what you're getting at so ... so if that's not the case here [points to the squares] then it might be reasonable that it would also not be the case in the case of the circle one. In spite of this initial attempt to generalize, neither Bill nor any other student was able to make the general Connection between this situation and the relationship between the areas of similar figures, even when prompted by the interviewer. 107 Pupils in grade 11 are routinely taught that the areas of similar triangles are proportional to the squares of corresponding sides; and, I am certain that all of the students in this sample would have been exposed to this theorem in high school. Still, they were not able to see that the mathematics nested in this vignette is just another manifestation of this situation. Indeed, most seemed surprised that the result did hold for squares, rectangles and triangles of the same shape but different size. Other interview items that elicited discussions of mathematical connections from a large fraction of the interview sample were Items 3b, 4, and 7. In Item 3b, which asIced whether two angles and a non-contained side were Bufficient conditions for congruency, most students made a connection with another mathematical theorem and how it could be used to prove that triangles satisfying these c°hditions were congruent. Although the majority of students worked their way through this problem logically and did. make some links, many were surprised to find out that the side does not need to be contained. They had learned cgngruency theorems in high school by rote, it seems, and szt linked the side-angle-side situation where the angle “ulst be contained to the angle-side-angle case where the Side need not be. Being pushed to look for a connection with another theorem in this case helped them improve their Own understanding . 108 Item 4, which involved a pupil’s misinterpretation of the method for simplifying the expression W produced a number of connections based on order of operations, implied brackets and the non-distributivity of exponents (see page 46 for a more detailed discussion of these connections). Item 7 asked students to explain to a pupil how extraneous roots could occur in the solution of radical equations. Graphical illustrations and simpler counterexamples that pupils were more familiar with were most often used to explain this situation. Other items, for example, Item 2, produced only a minimal number of mathematical connections. This simple item involved the multiplication of two binomials. Only one st-‘—I-1c1ent was able to make a link between the multiplication of the two binomials and the area of a rectangle (see Figure 3 a 2 , page 51). Furthermore, when the card sort task asking for connected topics to be grouped was done, no one made the Ina‘tlhematical linkage between the multiplication of two binomials and the formal multiplication algorithm for nulnbers which pupils are taught in elementary school. Even Vwhen pushed, students typically thought that these were eBsentially two different processes rather than two different ways of looking at the same thing. One student in 109 particular came back to this question several times but still did not see the connection.26 Conceptual Connections Given the rule oriented way that mathematics is often taught, it is not surprising that some students did not look at the difficulties in the teaching scenarios as conceptual problems. Indeed, conceptual connections were rare, with only a few good examples in the data. One item that did generate a strong conceptual connection was Item 11, which asked students to respond to a pupil's query about the definition of the derivative of a function. In this item a pupil is wrestling with the fact that the derivative is a limit where the denominator of a fraction is approaching zero and he or she knows that division by zero produces an undefined result. There were few connections of any kind with this item (9 out of 425 text blocks, again consistent with the survey) but Lynn was able to link the problem that the pupil was having with the derivative to a limit and the notion of indeterminacy.’7 2:‘Students did group these two topics together but for other than purely mathematical reasons. These will be discussed in a later section. The reasons were essentially the same as those in the surveys, so the overall level of connection was still high. 2“’For a discussion of division by zero and the difference between an undefined quantity and an indeterminate one, see pages 26-28. Lynn: Eric: Lynn: Eric: Lynn: Eric: Lynn: Eric: Lynn: Eric: Lynn: Eric: Lynn: Eric: Lynn: 110 ... Now in this case a 0/0 is different than a case of 3/0. In what way? If you start with a/b = c then that just says that a = bc. In this case if we want this to equal some letter and cross multiply we'd get k.0 = 3, right, if you could just cross multiply this. Well there's no k that would make that statement true. Alright. Alright, so this is obviously an undefined ... ... because you can't find the value ... ... because you can't find a value that would work. If on the other hand you start with 0/0 = k [very sure of herself here] that just tells you that 0.k = O ... that means that any value of k would give you a true equation. Like, I mean, you can't divide by zero type of thing but if you look at this equation you could find a solution. Okay, you could take k = 4, k = 24 whatever ... Well how many solutions are there? There's just an infinite number there ... Alright ... The problem with the 0/0 case is that it does have this infinite number of solutions if you look at it in this form so you have to do something else to find what the actual answer is. And hopefully they've done the indeterminate limits before they get to the derivatives ... What was the word you just used? Indeterminate? Tell me about this ... [laughs] Okay ... okay, indeterminate was always explained to me that there's an answer there but what you're given so far you can't tell what it is. 111 Eric: You can't determine it? Lynn: You can't determine it yet. You've got to do more things before you can determine it. Undefined is different. Undefined is just that there is no answer, there is no definite solution to this problem, you can't consider this as mathematically correct ... but the indeterminate -- just in the form it is -- you can't do anything about it. The thing that comes to mind most I guess is the limits where you have to factor the numerator and denominator and worry about, you know, taking out the common factor and then you can find out what the actual limit is. The idea of the derivative is the fundamental concept underlying the calculus. Calculus is one of three mathematics courses taught in Ontario high schools and the one most frequently taken by pupils. Furthermore, it almost always forms the basis of first year mathematics courses in university. It is, without doubt, a subject that everyone in the sample had studied regardless of the relative Strength of their mathematical background. The rarity of this kind of response to the pupil's problem prompted me to 53k Lynn what led to this way of thinking about derivatives. 391‘ response was that she had come to this understanding in the previous fall, during her practicum. She was teaching calculus and had been reading the textbook when she saw a discussion about division by zero. The link with the derivative as an indeterminate limit which may or may not be defined for a particular function was then made. Another rare conceptual link was made by Josh, a Student with a weak mathematics background. When discussing 112 Item 7 (extraneous roots) in the second interview he had not been able to figure out why one of the solutions did not work when substituted back into the original radical equation. Sensing his frustration we left the item. Two days later he left me a note with a well written explanation for the difficulty which essentially took solving a radical equation and changed it into a conceptually different problem. Most students looked at solving the radical equation as an algorithmic procedure of squaring both sides. They reasoned (correctly but vaguely) that it was this squaring process that caused the problem of extraneous roots. Josh interpreted the problem quite differently as the intersection of a straight line and a parabola. He conceptualized solving the single radical equation x+2= x+5 +3 as finding the points of intersection of a curve y=(/x_+5_+3 and a line y = x + 2. He drew a graph of the situation (see Figure 5.2) and the problem of the extraneous root became clear. The solution to this system of equations represents the intersection of the top half of a parabola and a straight line, In Figure 5.2 it can be seen that the top half cuts the line once. However, once the squaring has been done, the new system is equivalent to an entire parabola and a line. It is reasonable that we will get two roots for this 113 y - ./x + 5 + 3 ('593) (4.1) """ / x y - J; + 5 + 3 x - -1 Is the extraneous I’OOt Figure 5.2 - Graphical Representation of the Problem of an Extraneous Root new system.28 However, because the original system CODSisted only of the top half and the line, only one of these solutions will work in the original equation. Moving from an algorithmic squaring procedure to this leVel of conceptual understanding is a big leap and one that no One else made. When we discussed his findings in a \ 28As it turns out, a full mathematical analysis of this Problem is much more complicated that either Josh or I J‘magined and is well beyond the scope of this dissertation. 114 subsequent interview, Josh's excitement at his discovery was palpable: Josh: Okay, what happened was I ... this was just as I was waking up one morning, it must have been on the ... I gave this to you on the Thursday so it must have been Thursday morning. I woke up thinking what if I made one side of that equation equal to y. I went well then you’ve got, you know, a linear equation or some sort of ... and the one side I said ... I made the square root side equal to.y and then I said x+2 I can graph that, I know I can graph that, so why not do it the other way around and then graph the other side as well. So I said okay what happens if I do that and then I started graphing. So I graphed y'= x+2, I recognized that line immediately and just threw it down and then I started into the other side and I said okay, well obviously won't go below x + ... I said it won’t go below -5 because you can't take the square root of a negative number. So I started graphing and got this weird line, it started to curve away and I thought hm ... this doesn't look reasonable and as I was doing it I thought oh what if I took the negative square root now. And at this point then I just went look I've got a parabola. That's when I saw where the solution came from because as soon as I did that I said oh now you've got the line intersecting this parabola at two different points and you're going to have two different roots. Considering the time and intellectual energy that Josh had expended on this problem it is clear that these kinds of conceptual connections are difficult for most students to make . 115 Connections to Other Subjects Connections to other subjects were perhaps the easiest to categorize because the students made a clear link with another discipline other than mathematics or a ”real world" example. This category, however, has few examples with only 17 text blocks out of 425 exhibiting this category of connection. The item that was most commonly discussed in this context was Item 529, with links being drawn with batting averages, marks and atomic mass. Greg's discussion was the most lucid: Eric: Could you think of another situation where this kind of problem might arise? Greg: [Answers instantly] Weighted averages. Eric: Can you give me a specific for instance? Greg: Chemistry. Say for instance you wanted to calculate the average atomic mass of an isotope of chlorine. Chlorine for instance has two isotopes, one has a mass of basically of basically 37 the other has a mass of 35. Two isotopes of chlorine and this one the lower mass one is three times more ... abundant as the other one so you end up with a weighted average of about 35.5 as the average mass. Eric: How would you get that ... how would you come up with that, the weighted average? Greg: Well if I were to ... this is the way I did it in class. I was using percentages and say you had 100 of these atoms you have to add up ... if you had a 100 you have to add up 75 of them at 29This item involved finding the average of two speeds. See page 43 for a more detailed discussion of potential connections. 116 this weight so it'd be 35 + 35 + 35 + blah, blah, blah and I started writing across the board and they were kind of looking at me funny like he's actually going to write 75 of them. I said is there a better way I can write this and it's just (75)(35). So it'd be (73)(35) or (3)(35) in this case but that's not the one. And then (25)(37) because I have a 100 of them here so I have to divide by 100. Although Greg was quickly able to relate the problem of driving at varying rates of speed to the question of average atomic mass he was unable to get the problem in Item 5 to work out to the correct answer by using a weighted average. He seemed to know that the same principle was behind both situations but the knowledge required to make the original one fit his chemical analogy (as a chemistry major he was more familiar with this situation) seemed to elude him. Some students were able to relate the problem to a weighted average and get the correct answer; however, non of these students could see how to connect the problem to another subject area. No other items from the survey commonly elicited this category of connection from students. Only the card sort task asking them to discuss warranted versus conventional knowledge in mathematics elicited a number of linkages to real world settings. The question, which asked if slope as defined to be rise/run was arbitrary or logical, did produce a number of good connections with our intuitive real world notion of steepness and how it relates to slope. Kirk explained it this way: 117 Eric: Well what if ... just for the sake of argument ... what if slope was defined as run over rise would that not be common sense ... Kirk: No that wouldn't be the same thing. Eric: What would be the problem with that? Kirk: Um ... because that would no longer really be slope ... because slope has to do with steepness and that would be the inverse of steepness so then as our line was getting steeper, say the side of the mountain was getting higher and higher our slope wouldn't be increasing it would be decreasing ... again it has to do with common sense. A slope ... something is getting steeper we’d like our number to get bigger rather than smaller. [Emphasis added] Kirk's discussion makes the point that although slope is a definition it is not a definition without reason. If we wish to quantify something in the real world then our numerical intuition tells us that if more of the thing that we are measuring is present then the number that represents this quantity should be larger. This definition of slope allows for the confirmation of what our senses tell us and so it is more useful than run/rise, for example, which would give small values for extremely steep slopes. Pedagogical Connections The teaching context that underlies most of the instrumentation naturally focused the students' thinking on how to teach things; consequently, it is not surprising that connections with pedagogical ideas and pupils' ways of thinking were more common than the previous categories. 118 Fifty-four text blocks were classified as having to do with a connection with a pupil’s way of thinking. For example, in Item 1, pupils were trying to solve a quadratic inequality by using a method that is appropriate for linear inequalities. The pupil's confusion arises from the fact that they think they have followed the correct procedure as given previously by the teacher but it seems to produce an answer that is incorrect. Dawn was able to figure out how the "rule" that the pupils were invoking could be applied in the quadratic situation if it was interpreted properly and hence make a connection with both the pupil’s way of thinking and with the mathematics that the pupils already know. Dawn: Well, the first thing that I thought of... I don't know if it works... well x2 is x.x ... greater than 25 ... well if x were -5 and then you would ... if you're solving for x so you divide both sides by x and if x were -5 then you are dividing by a negative so x ... has to be less than -5. Eric: Well if you divided this side by x what would you be dividing this side by? Dawn: x which is -5... if x = -5 and you're dividing by a negative so you would use the same rule ... so then you would have that -4 is not less than -50 Eric: So you're saying that if you really interpret what you're doing properly the rule still holds. Dawn [laughs] ... yeah... In this discussion Dawn is tacitly using a definition of the square root of a number that is perhaps rarely stated; namely, that the square root of a number is one of 119 two equal factors of that number. Consequently, dividing a number by its square root produces the square root. She has then shown how this method allows the rules for inequalities that pupils already know to be applied; and, in this way the pupil is not left with any contradiction. In spite of the fact that many times the students could relate to what the pupil in this vignette was thinking, there seemed to be a general tendency not to start with the pupil's way of thinking in their explanations. It was more common to find students providing an alternative approach to the problem; re-explaining the situation using essentially the same ideas; or, making use of another rule or procedure to try to clear up the difficulty. Compare Jeff's response to this same item with Dawn's: Jeff: Well, if you have x2 > 25 ... you square root both sides, you can get : X'on one side, and i 5. And, you'll end up getting all these situations ... [writes four inequalities] and only two of them are correct. [selects two] this is correct ... and this is incorrect, and this will end up being correct after you go to the point where it's x [changes the sign]. That will be correct ... and this will be incorrect. Eric: That's interesting ... why would you ... what do you think this young person's going to think about this ... that some of these solutions are correct, and other ones aren't? Jeff: I think you have to explain to them that when you do certain operations, like square rooting and squaring, in particular ... actually all kinds of operations, that sometimes you’re going to get solutions that aren’t logical. .And you have to go back and check. It’s also a good thing to point out to all of the pupils that it ... that their answers won’t always be correct, 120 that there’ll be some slight error, and they should always go back and check it anyway. (Emphasis added). Clearly, Jeff knew precisely how to solve the problem and did understand how the student was thinking about it; however, he was not able to connect this knowledge to the student's understanding of linear inequalities and how to solve them. In addition, he ends up his explanation with some general procedural rules that he feels to be characteristic of mathematical methodology -- ideas that would likely be hotly debated by mathematics educators everywhere! Weak Connections Numerical Connections It is true that pupils in secondary school, particularly in the lower grades, have a much better intuition about what is mathematically appropriate when working with numbers than they do when they are working with variables. Many pupils who would not dream of adding two fractions by adding the numerators and denominators will cheerfully use this technique if the denominators are replaced by letters. It is not surprising, therefore, that students wishing to make a connection with a pupil’s 121 knowledge should make use of numerical arguments as a way of explaining algebraic problems and difficulties. Although a substantial number of text blocks (85 out of 425) were classified as containing a numerical connection, this connection was rarely a direct mapping of the numerical situation onto the algebraic one or vice versa. Rather than building on the knowledge of number that pupils possess by trying to generalize this knowledge to algebraic situations, it was much more common for respondents to use numbers as a way of producing a counter-example to show why a particular method of doing something algebraic did not work. Alternatively they used numbers to construct an example to justify that a procedure did produce the correct numerical result. This specific use of a numerical analogy was very prevalent in Item 12 where pupils were trying to simplify a rational fraction by cancelling the individual terms in the numerator and denominator. The item was constructed in such a way that this incorrect method did produce the correct answer. Consequently, just putting a number in place of the variable (which most of the respondents tried to do at first) and using the pupils' incorrect method still produces the right answer because the variables just represent numbers. This discovery came as quite a revelation to the respondents although most later realized that they needed to create a different algebraic example and then use a 122 numerical substitution to justify or refute the pupil's method. Kirk's response was typical: Eric: Okay let's go back to your other example though where you picked x2-6/x+2, the student’s method would give x-3 and now what you want to do is to convince them that that's not the correct answer. Can you think of a way of convincing them that it's not the correct answer? Kirk: Yeah, putting in numbers would be the best way. So if you put in ... if you put in one ... -6 ... okay put in one as an example you would get -5/3 and here if you put in one your answer would be -2 so you get two entirely different answers so they can't be equal. So that's probably ... Eric: So that convinces me that either one method is wrong or the other method is wrong. Now what's going to convince me that it's my method that's the one that's wrong as opposed to your method of working out the top and working out the bottom and then dividing. Kirk: Sure ... the fact or the proof would be if you start with this question and you plug a number in it and follow the operations you end up with -5/3. Now his solution ... he's done some work with it he's changed it somewhat to find a simplified form so the original one that you start with is the original correct one I mean there's been no manipulation whereas his something has been manipulated. So really there's no question ... I wouldn't say it's valid to say whether this side's right or that side's right I mean the first side with no manipulation is where you're starting from you're just plugging in numbers and come down to the solution and that's the answer. Now the question is, is his simplification correct? And you can just establish it by putting in numbers if it's the same. Other items where this method of numerical justification was commonly used were Items 2 (multiplication of two binomials); 4 ( ¢a2hb2 ; and, 7 (extraneous roots). 123 One item that did elicit a common numerical connection based on a parallel understanding of numbers and variables was Item 8 (x9 = 1). A number of students used numerical patterning as a way of explaining the difficulty to the pupil. Jeff's explanation is somewhat atypical because he uses base 10 rather than the base two that most students chose to use for some reason; but, the fundamental method of demonstration that he talks about was commonly employed: Jeff: Ahh.. and this is a good way to introduce the ah.. negative exponents. And you say well what is 10 to the fourth? That's 10 times 10 times 10 times 10 ... and that's 10,000. Eric: Right.. Jeff: Yeah.. Okay.. equals 10,000 . And you say, well okay that's this column. And 10 to the third is a 1000 ... that's this column. Ten squared is 100, under the 100 column ... Ten to the one equals 10, in the tens column.. Well what's 10 to the zero going to be? If we keep this pattern going? They'll say.. Zero?.. and you’ll say.. well what is the next column? And they’ll say ones. So what will this be? How are we going - What do we do each time? Well we're dropping a zero off each time.. so it's going to be one. In some ways this might be considered to be a strong connection because the student has used the pupil's prior knowledge to help he or she understand the value of x”. However, it is precisely because the connection is aimed at producing an answer to a question that it represents a weaker level of mathematical connection than one that seeks to produce connected thinking in the pupil. 124 A reading of the item shows that the pupil in the vignette is searching not for an answer but for a meaning that he or she can associate with the symbolism of a zero exponent. The pupil already has a meaning that can be associated with positive exponents and he or she wants a comparable meaning for the case of a zero exponent. A strong connection would show how the pupil's prior understanding of meaning in terms of exponents could be carefully used to generate a meaning for x” that is consistent with their prior understanding of other exponents. This kind of connection is not aimed at producing an answer but at connecting together various pieces of the pupil's understanding. To clarify the distinction between the two kinds of connection consider how else this problem might be resolved. A teacher wishing to make this kind of strong conceptual link could remind the pupil that 1:2 means 1.x2 or 1.x.x and that x means 1.x. The exponent, therefore, tells us how many factors of x there are being multiplied by 1. Consequently it is correct to say that xpzmeans there are no factors of x in the expression. This fact does not mean that the answer is 0, however, because even though there are no factors of x there is still the 1 that was the coefficient. A further advantage to this kind of argument is that it can be extended to generate a meaning for negative exponents also. For example, if an exponent of 3 125 on a base of x tells us that we have 3 factors of x being multiplied by 1, then an exponent of -3 (the opposite to 3) ought to tell us we have 3 factors of x being divided (using the opposite operation to multiplication) into 1; and, in fact this way of thinking does produce the correct value for an expression with a negative exponent. Now the overall understanding of exponents does not require a special status for zero or negative exponents because the meaning for these exponents has been directly connected to the meaning for whole number exponents. Contextual Connections Contextual connections are connections (often correct) that are made on the surface features of a problem rather than on the mathematical principles underlying it. As a subcategory it was quite common with 67 text blocks having this classification as one of their codes. These kinds of connections typically arose in the card sort task which asked for connected mathematics topics to be grouped together. For example, some of the people who correctly grouped the pythagorean theorem with the cosine law did so, not because they perceived them to be mathematically related, but because they both contained the sum of squared terms. 126 These contextual connections illustrate the contention, made earlier in the chapter, that in the interviews the students were sometimes making connections in their own minds as they went along rather than just reporting the connected knowledge that they held. A weak contextual connection was sometimes the precursor to the development of a stronger link. Indeed, some of the students who began by linking the pythagorean theorem with the cosine law because they looked similar, were later able to prove one to be a special case of the other. Another pair of items on the card sort task referred to exponential growth of bacteria30 and the compound amount of moneyn'invested at a particular interest rate. These two formulae are closely connected mathematically because the exponential growth formula is the limiting case of the compound amount formula when the compounding of interest is instantaneous. This derivation does not require anything more sophisticated than an understanding of limits as studied in introductory calculus. It can be easily demonstrated with a calculator by using both formulae to 39A population of bacteria will grow exponentially according to the formula y =P0 e“t where k is the rate of growth, P0.is the initial population and t is the elapsed time after the initial population measurement was taken. 31The compound amount of P dollars invested at an interest rate of 1% per interest period for n interest periods is given by'A = P(1 + i)”. calcule interes 1(1 + . $1.0512 the cal interes number compoun = 1(1 + QrOWth 1.05127 before; jEllI’ther Case of even af‘ inter-Vi) were CO] 127 calculate the value of 1s after 1 year assuming say 5% interest compounded yearly. The two calculations give 1(1 + .05)1 = $1.05 (compound interest formula ) and Le05 = $1.0512711 (exponential growth formula). If we now repeat the calculation assuming daily compounding of interest the interest rate per interest period is .05/365; and, the number of times it is applied in one year is 365. The compound interest formula now gives A = 1(1 + (.05/365)):365 = 1(1 + .000137)365 = 1.0512674. Using the exponential growth formula for this same situation still gives 1.0512711. These answers are substantially closer than before; and, more frequent compounding narrows the gap even further. Hence, it is easy to see how one is the limiting case of the otherfi32 No one was able to draw this link even after considerable questioning and guidance from the interviewer, although most students still believed that they were connected.33 :”Some banks actually advertise that they compound their interest instantaneously and this is how they do it. :”One student seemed upset by her inability to make this link and so I explained it to her after the interview. She responded that it was "neat”. I found that with some students I needed to be sensitive to the fact that I was often putting them into a state of cognitive dissonance where they were becoming concerned about their knowledge of mathematics. I felt an ethical obligation to encourage and reassure them so that they would not feel that their participation in the project had undermined their own self confidence. However, this tension sometimes meant that I had to pull back a bit in some interviews as the anxiety level rose. A var together; Greg: Other stuc were both problems : Kirk: 128 A variety of other reasons were given for grouping them together; for example, they both contain powers: Greg: Well I grouped a pair -- one computing the compound interest and another one exponential growth of bacteria both together. Basically because they both deal with powers. [pause] That's about all I have to say on that one. Other students argued that they were connected because they were both formulae which could be used in application problems: Kirk: I grouped them together ... because they're applications I suppose. They’re both specific applications with formulas and quite often ... quite often they're given or the way they're used they're also practical, you know, formulas that are used in everyday ... well not our everyday life but somebody's everyday life [laughs] ... Even the students who recognized that they both represented some kind of exponential growth were not able to show how the two were related, other than to suggest that they both had variables for exponents. This kind of contextual fixation occurred in other pairings as well and often prevented students from seeing much deeper mathematical connections. For example, consider the following three items from the same card sort task: (a) The area of a triangle is given by the formula A = % bh where b is the base and h is the height; (b) The area of a circle is given by the formula.A=n12; and, (c) A function is defined by f(x) = x2. What kind of curve will it produce when graphed? Al because calcula despite essenti parabol more us relatio reSpons Dar Eri Dan Eri Dana Eric Dan. C 129 All of the students grouped (a) and (b) together because they were both formulae used for the purpose of calculating areas. No one grouped (b) and (c) together despite the fact that the ”formula” in part (b) is essentially a quadratic function and will result in a parabola when graphed just like f(x) = x2. It is, in fact, more useful to look at this formula as a functional relationship between two variables, area and radius. Dana's response was typical of the students: Dana: Well these all have to do with ... familiar shapes, one's a triangle, one's a rectangle, and one's a circle ... and ... we're having to use information about the outside measurements to find the area. In this case, it's the maximum area. In this case it's just.. the formula for any particular circle, and in this case it's just the formula for any particular triangle. So ... they're all related because of their area. Eric: So you feel they're all related because of area. Are there any other connections that may exist between them? Dana: The fact that they're all.. like the three most basic shapes that you'll ever deal with. Like ... circle, rectangle ... Like you don't have abstract shapes like an octagon, or something bizarre. These are very common, even in small children they can pick out what a circle is, what a ... Eric: They can pick those out ... Okay ... So the connections, let me get this right, are that they're dealing with area, and that these are basic shapes. Dana: Yes.. Eric: Okay.. any other connections you can think of? or relations? Dana: [no response] Thi several given or ideas if ensued. well the eventual somethir is fair the stu< finding: What the disCuSs: ProceC111: 130 This quotation, which is actually a concatenation of several text blocks, also illustrates how students were given opportunities to reassess and change their original ideas if something else occurred to them as the discussion ensued. As a teacher I can question people sufficiently well that they will tell me almost anything I want them to eventually; so, putting words in people's mouths was something that I deliberately guarded against. I think it is fair to say that although the data does represent what the students' thinking was (as opposed to mine), these findings do not represent their first quick impressions but what they had to say after considerable thought and discussion. Procedural Connections Another kind of connection which was made on the basis of surface features was sufficiently common to be classified separately and this is the category of procedural connections. In this kind of linkage students related one item to another on the basis of the procedure that was being carried out rather than the idea embedded in the procedures. Many students linked together the cards that contained a binomial multiplication and the multiplication of two numbers using the formal multiplication algorithm; but, only because they were both multiplication problems. Students seemed t mathemat Kiri 131 seemed to look at these as strictly procedural without any mathematical reasoning behind them: Kirk: These ... well I suppose I would group these together because of the operation. They're both multiplication. This one is of two specific numbers and this one is also of two numbers but they're written in ... the two numbers are written in a different form ... a variable form. x+3 is in effect in essence one number and x+4 is another number so it's just going about the multiplication in a different manner. Both of these are illustrating procedures. This is how you do this procedure and this is how you do the other procedure. And here as well you end up with two numbers that you have to add together in the end to get the final product and here as well when you're doing the distributive property you have a couple of terms to add together to give you the final number and instead of there being one number because there's a variable in there it's, you know, it's not a final number it's got a variable still left. A few students were able to eventually see the parallels between the two questions but only after a substantial amount of questioning by the interviewer. Students also made procedural connections when they used rules that were appropriate for one topic to justify their use for a different topic. Falling back on rules and algorithms was a common occurrence whenever the solution to a pupil’s difficulty was not immediately obvious. In Item 8 (x0 = 1) Bill connected the previously learned rules about exponents to this new situation: Bill: I think it pretty much works consistently with everything else that we do with it with exponents ... so if you have something like this a5 over az‘we can apply our straightforward definition of saying this means four factors of a, this means two factors of a, and we know what to do with this. We can cancel two of those and It is C] that the inSiSter make Sen C°nnecti the PUpi Pedagogi In E th°u9ht . students their rer understar fail to u 132 we' re left over with a2 . So what you then have is a over a2 is a2 or in other words if I subtract these two exponents I get the answer because 4 - 2 = 2. And you can do this with more examples and you find that it works with all of those. So ... I guess the thing that we insist on in mathematics is that when we have a principle like this that we try to carry it as far as possible so when we come up with a situation like this we still like to be able to do the same thing so we have az/az here, I ’m going to insist I still can apply that rule by subtracting exponents so that should be a22 so this gives you a0 using the laws of exponents fbr combining these two here. On the other hand this is also equal to a times a over this [points to denominator in factored form] so it's also equal to one. So if you insist that the law of exponents still applies here then you come up with 1 = a0 . [Emphasis added] It is clear that Bill is trying to connect with knowledge that the pupil already possesses; however, this blind insistence that rules must work, regardless of whether they make sense, just for mathematical consistency makes this connection conceptually weaker than other ways of addressing the pupil's search for meaning. Pedagogical Connections In some sense weak pedagogical connections can be thought of as missed strong connections. That is, the students’ comments are rooted in pedagogical concerns and their remarks may reveal a good insight about pupils' flawed understanding; but, they are relatively weak because they fail to make a clear conceptual connection based on mathemat of teach In 1‘ error in function assumpti operator root Ope multipli this con Eric Bil: Eric Bill 133 mathematical principles between the pupil's ideas and a way of teaching that might be pedagogically powerful. In item 4 ( [/a2+b‘2 ) for example, the fundamental error in thinking that is often made by pupils is that all functions or mathematical operators are linear. This assumption leads them to believe that the square root operator behaves like multiplication and that the square root operation can be distributed over addition just as multiplication distributes over addition. Bill almost makes this connection but stops too soon: Eric: Bill: Eric: Bill: You said that in this question that you didn't feel it was a question really about square roots the student was having but it was really a problem of understanding exponents more than the square roots, is that correct? Yes, that's what I said. OK. Can you think of any other situations, different grade levels whatever, I mean it doesn't matter, where a similar kind of misunderstanding occurs? [pause] Ok, so what do we have here, you have to do this operation and then do this operation. [pause] Well I think in a sense it is a similar situation that you get with differentiation of more complicated expressions, the sort of situation where you use the chain rule. So, for example, the structure is a little different but I think the sort of.misunderstanding is the same. If you have something like [pause] well just to take something simple f times 9, f(x) times g(x), if you want to differentiate that, it is not going to be simply f’ times 9’ which is probably what intuitively the student would probably want to write down. There is a specific rule for dealing with those sorts of things that you can get again by going back to the basics so go to the definition of what it means to differentiate. Then you get the Bill cox error at where tt another than ta} Pro anthemat and inte might be Which Cc teaching reguts mathemat 134 correct result but it isn't what you might expect it to be. I think that's similar to this because here you think that this is going to cancel in the same way as if this other term wasn't there. [Emphasis added] Bill correctly analyses the pupil's fundamental conceptual error and even correctly relates it to another situation where the same kind of conceptual error would lead to another common pupil mistake in calculus. However, rather than taking this idea a little further and trying to figure out what is the root of this error in thinking, he chooses to try to solve the problem by reverting back to a rule that governs the procedure. Once again a potentially powerful link has been missed. In general there were many situations where students came close but failed to make a connection. The two codes CBNC and CN-NR were both used to categorize this situation and they occurred in 90 blocks of text. Summag Prospective secondary teachers can, and do, make mathematical and other connections; however, both surveys and interviews confirm that they do not do it as often as might be hoped or expected. Furthermore, the degree to which connections are made does not depend on gender, teaching major or practice teaching experience. The survey results suggest that it is related in an inverse way to mathematical background with strongly prepared candidates making f] prepared not show mathemat. moderate that won mathemat mathemat about Se Wher that are of “3178 kinds Of rare (Co acrosS s acrgss It IlSed to knowledg almoSt a the Card some 0f and anal 135 making fewer connections than expected by chance and weakly prepared candidates making more. The interview results do not show this same pattern. In addition, incorrect mathematical statements are made by people with weak, moderate and strong backgrounds in exactly the proportions that would be expected by chance; hence, a strong mathematics background does not appear to improve the mathematical accuracy of prospective teachers' statements about secondary school mathematics topics. When looking more closely at the kinds of connections that are made, it is evident that there is a large variety of ways in which students make connections. Although some kinds of connections are common (numerical) and others are rare (conceptual); the connections that were observed vary across students with different backgrounds and widely range across many content areas. The quotations that have been used to illustrate various aspects of students' connected knowledge come from all students in the subsample and tap almost all of the items from the survey as well as many from the card sort tasks. Still, there are puzzling aspects to some of these findings. These will be more fully discussed and analyzed in the next chapter. Thi of the v are able reveals connecti ability strongly and Inle 01' Conne Fur with (111 were Cle Students with a V Consists strongl) with a V help ex; does not tYPical the two kinds Of CHAPTER 6 — DISCUSSION 0? FINDINGS Introduction This study has attempted to provide a detailed picture of the way that prospective secondary mathematics teachers are able to make connections. The picture that has emerged reveals that although students can and do make individual connections with a diverse range of topics, their overall ability or predisposition to make connections is not strongly developed. Also, connections based on procedures and rules are more commonly made than conceptual connections or connections to other subjects or mathematical topics. Further, there are observed differences among students with different mathematics backgrounds. These differences were clear in the survey data where strongly prepared students appeared less likely to make connections than those with a weaker background. However, this finding was not consistent with the findings in the interview data where strongly prepared students made more connections than those with a weaker mathematics preparation. One factor which may help explain this inconsistency is that the survey format does not allow for the kind of deeper probing that was typical in the interviews; and, as was suggested earlier, the two data gathering strategies give somewhat different kinds of information. Furthermore, a simple reading of the 136 data doe the habj ability to deter consider results project connecti educatic reality making 1; Ever reformer that mgr will 18a teachers Provides rich, Co: teach We. 2' if th‘ mathemat: interemu In t explanatj 137 data does not allow for a distinction to be drawn between the habit of making connections (disposition) and the ability to do so. Consequently, further analysis is needed to determine if this distinction is an important one to consider in understanding the findings; and, if the survey results are credible. Still, the data gathered in this project does suggest that the current emphasis on connections as an important theme in the mathematics education reform documents is not characteristic of the reality of prospective teachers' knowledge and skills in making them. Even tentative results such as these make it clear that reformers need to examine the validity of the assumption that more university study, of the kind presently available, will lead to better subject matter knowledge in prospective teachers. It does not appear that such study automatically provides prospective mathematics teachers with the kind of rich, connected and flexible understanding that is needed to teach well (McDiarmid, 1992). And, as was argued in Chapter 2, if the various attempts to reform the way that mathematics is taught and learned are to be successful, this interconnected knowledge of content is essential. In the discussion that follows, a number of possible explanations for the observed ability of students to make connections, and the nature of many of those that were made, will be advanced. I first consider how the study design could he be confj on what typical seconda: backgrm teachers of what and/or F then rec diSCUSsj between Strongly Prior ec‘ Variety diSCusse the repc although are hard rammed Will be prOSpect 138 could have influenced the results and to what extent we can be confident of these findings. The discussion next focuses on what is known about the educational experience that typical prospective mathematics teachers have had, both in secondary school and university. Further analysis of this background shows how it may have affected prospective teachers' beliefs about the subject and their understanding of what it means to know and do mathematics. The ability and/or predisposition to make connections in mathematics is then reconsidered in light of these factors. This discussion helps illuminate the apparent inconsistency between the interview and the survey data with respect to strongly and weakly prepared candidates. The interaction of prior educational experiences with teacher education (and a variety of participants' attitudes towards it) will also be discussed in the attempt to develop a cogent explanation of the reported results. These arguments will show that although the findings of this study may be disquieting they are hardly surprising. Finally, a synthesis of the findings reported earlier and the arguments advanced in this chapter will be used to sketch a more complete picture of these prospective teachers' connected knowledge. 139 Artifact or Fact: Methodological Issues One of the things that the survey data presented in the previous chapter suggests is that more intensive mathematical study may have a negative influence on prospective teachers' ability to make connections between mathematical topics and other areas of mathematics or other subject areas. Fennema argues that one of the values of mathematics education research is that it can provide ”systematic description of specific situations to see what an objective examination of reality reveals." Furthermore, “[R]esults from such status studies are often surprising and in conflict with widely held beliefs." (1981, p. viii). However, when the results are in conflict with widely held beliefs, they need to be examined carefully to establish whether they are real or merely an artifact of the experimental method used in the study. One possible, indeed obvious, explanation for this finding in the survey is that because the respondents all had different question packets and some survey items were substantially more difficult to see connections in than others, it is just an unlucky coincidence that the strongly prepared candidates tended on average to get questions for which connections were more difficult to find. A chi-square test done on a crosstabulation of items versus mathematical background showed that there was no relationship between the 140 two variables and that the observed distribution of questions that was used in the surveys would occur by chance 93% of the time. It is fair to argue, therefore, that the efforts at randomization were effective and that some other factor or combination of factors is responsible for the findings. Although the data from the survey is clear in this regard, the interview data does not show that stronger students are less able to make connections. There is a number of possible explanations for this inconsistency and these will be discussed in a later section of this chapter; however, it is important to note that the two data gathering methods are quite different in what they can and cannot tell us about what a student does or doesn't know. The survey is useful because it allows data to be gathered from a large sample; but, its weakness lies in the extent to which a researcher can then be sure about what the responses tell him or her about that student's connected knowledge. The interviews only give information about a small number of prospective teachers but because of the intensive nature of the interview, the information that is gathered is much more likely to be a valid reflection of the student's ability to make connections. The interviews also provided data about the quality of the connections that were made; and, in general, weak connections were more common than strong connections. A 141 possible explanation for the relatively weak nature of reported connections could be that the items and tasks that were used did not accurately tap the construct of connected knowledge as developed in the conceptual framework. However, every item in the survey produced some responses that were judged to be connections of this kind; and, all of the anticipated connections (along with many new ones) appeared in the student responses. When picking blocks of text to illustrate various points I tried not to use the same student all the time but I did not deliberately try to ensure that all students were represented. Still, every student in the interview sample ended up with some portion of an interview quoted to illustrate a particular category of connection. Furthermore, the triangulation of survey data with interview data where appropriate, and the agreement obtained, adds credence to these results. It could be argued that the coding of the interviews was open to interpretation and that different people would have coded the various text blocks differently. This is certainly possible; however, the reasonably good level of inter-rater agreement that was achieved during the coding of the survey items helped provide a consistent baseline for coding the interviews. Furthermore, the large number of codes that were used in the interviews allowed for some leeway in coding. For example, it is possible that one person might have coded a block of text with the code CN- 142 JUST (connection - provides a justification) while a second individual would use the CN-EXPL code (connection - provides an explanation). However, it is unlikely that one coder would assign the code CN-JUST to a text block and another would choose the FLCON code (flawed conception). When the analysis was done, similar codes were collapsed; so, subtle differences in coding patterns would not have changed the overall conclusions that were drawn. It is reasonable to conclude that the findings reported here do make a valid statement about the connected knowledge, or the ability to make connections, of this sample of prospective secondary teachers generally. However, a clear differentiation between the way that strongly prepared and weakly prepared candidates make connections is not possible without further analysis. The next, and subsequent, sections will discuss those factors that allow for an understanding of the results that were observed. The High School Experience The view of mathematics that many of the students in this study demonstrated points to a picture of a subject dominated by rules, procedures and conventions. Getting the answer is more important than getting the idea and connections are rarely pursued as a matter of course. Many 143 of these attitudes and ideas are shaped by earlier educational experiences. Although it is possible that people's ideas about mathematics are formed even in elementary school, because students perceive the elementary curriculum to be arithmetic (rightly or wrongly), they associate mathematics more closely with those topics that they have studied in high school. This section explores the role that the high school program might play in explaining how these attitudes are established and consequently how they affect students' connected knowledge. The Curriculum In Ontario, the high school program consists of five grades, nine through 13. The curriculum is set at the provincial level with individual school boards having the right to make small modifications to accommodate the perceived needs of their particular student population. This system results in some measure of consistency with respect to the written curriculum; however, the taught curriculum often varies considerably from school to school and district to district. Although only two years of mathematics are mandatory, anyone who has university aspirations typically would study mathematics every year because university entrance requirements demand that all candidates have studied either 144 a foreign language or mathematics at the grade 13 level. Furthermore, many university programs, even non—mathematical ones such as psychology and sociology, have a prerequisite of grade 13 mathematics. Consequently, few university bound students drop mathematics before grade 13 graduation. In grades nine through 12, the mathematics courses are integrated ones that are designed around a spiral curriculum. Each year, the pupils study many of the same topics but in increasing depth. Algebra, deductive geometry, analytic geometry, functions, sequences and series, transformations, probability and elementary trigonometry are typical of the topics studied in these grades. In grade 13 there are three different mathematics courses: Calculus, Algebra, and, Relations and Functions. The level of the grade 13 courses would roughly approximate freshman mathematics at an American university. Any pupil who is planning to specialize in either mathematics, engineering, physical science or any closely related field would normally take all three.34 Pupils planning to enter other fields would typically take Calculus and/or Relations and Functions. The calculus course consists of a fairly full treatment of single variable calculus (including :“Strictly speaking the system is not presently organized exactly as I am describing it; however, this description is accurate for the time period when the students in this study were pupils in high school. 145 trigonometric and exponential functions) up to and including an introduction to integration. Relations and Functions includes work on conic sections, transformations, parametric equations, trigonometry and analytic geometry. Algebra is an advanced course which contains material on probability, permutations and combinations, linear programming, the binomial theorem, vectors, matrices, complex numbers and polar co-ordinates. The fact that the vast majority of the students in this study have come through this kind of integrated high school mathematics program makes their weak ability to make connections even more puzzling than similar findings would be for students from other jurisdictions in both Canada and the United States where algebra, geometry and so on are treated as separate courses. Certainly if they had studied algebra and geometry as separate courses we could expect that there would be little attempt to connect mathematical topics from these different courses together; however, it appears that integrating various topics into the same course does not have much effect either. As every teacher knows, it is not so much the formal curriculum that influences what students know and can do in mathematics but how they are taught; and, how they are tested. It has been suggested that textbooks can play a significant role in defining what gets taught. In this case, the textbooks are written specifically to meet the content requirements of the 146 provincially set curriculum standards and are approved by the Ministry of Education; consequently, one way to gain some insight into this issue is to examine a number of typical textbooks. But What Gets Taught? The samples in Figure 6.1 (Kelly, Alexander and Atkinson, 1987); Figure 6.2 (Dukowski et al, 1987); and, Figure 6.3 (Ebos, Tuck, Schofield and Hamaguchi, 1987) show the first page of a section of work from three popular textbooks. The intent of this section is to introduce pupils in tenth grade to the concept of slope of a line segment. Although a few "real world" examples of where slope occurs are briefly mentioned, there is no conceptual development of the idea of what slope might mean in any of these samples. Furthermore, the definition is just stated without any discussion of why this ratio might be a reasonable definition; or, whether other definitions are also possible and defensible. Pupils could profit greatly from a consideration of these issues. For example, even if we accept the notion of rise and run as socially constructed knowledge, there are 147 6-3 SLOPE OFA LINE SEGMENT The pitch of a roof, the steepness of a ski run, or the gradient of a moun- tain road are all examples of slope. n'se - - 810 e— — — In each case, the slope IS the ratio p run of the rise to the run. Example 1. The diagrams below show how the rise and run of a water wave are defined. Oceanographers have found that a wave tends to fall over, or break when its slope becomes greater than %. For the waves shown, determine if either one should break. a) _ b) 0‘5 m \A/ 2.8 m L9 m 0.6 m Solution. Find the slope of each wave. a) 510P6=m ri—S: b) Slope=n.—S:l =92 . __.Q-_6n 2.819 £0178 =0.3l6 Since % i 0.286, the wave in (b) should break, and the one in (a) should not. Fi ure 6.1 - Sam 1e Textbook Pa e - 1 148 4—4 Calculating Slope In many places, the law has specific requirements regarding the steepness of a wheelchair ramp, as a ramp that is too steep is very difficult to maneuver. Suppose a wheelchair ramp rises 42 cm for every 560 cm of horizontal Rise distance. The slope of the ramp is a measure of its steepness and is equal to the ratio —R . _ un Rise 42 3 . =—=——=— Rise=42cm Run = 560 cm To calculate the slope of a non-vertical line on a coordinate grid: a. choose two points on the line; b. determine the rise and the run between the two points; c. find the rise-to-run ratio, expressed as a fraction. EXAMPLE 1: a. Find the slopemof XE. Form,m=B'—m- 4-(-2) Run = 4 - (-2) x 3 - (-1) ' =2 4 = .3. ’ 2 The slope is positive. It slants up and to the right. b.. Find the slope m of 235. o—+ Rise For PQ, m = m _ 3 - (—1) ’(—o-4 =3-F” _:L = ’—8 __l ’ 2 The slope is negative. It slants down and to the right. Figure 6.2 - Sample Textbook Page - £2 4.9 149 Working With Slope In the previous section, you worked with an important property of a line segment, namely, its length. Another property of a line segment, or a line, is its slope. The term slope is often used when describing steepness. When a cottage is built the roof must have a certain amount of steepness to it so that the snow will slide off, rather than building up on the roof. A jet, as shown in the photograph, must maintain a certain slope for a successful take- ofi". .-x--- .c -. .. __.-.._.L.... f ’ ":.,.. . ‘. -"'. .W' i. .' V. - u" \-.'~ . . . :4 E; r" _ _ .r.; .g {rt-"- ‘rwyaf‘rii‘t‘f 13¢ " :7) 4;; ' 1". $33”. ~ A 'i.‘ _‘l _ 1m": " ’32—.» >. -.‘- a ...-, _- The Lockheed L-1011 is a wide- bodied jet that carries 256 passengers It must maintain a certain slope for a successful lake- of. This jet is so large that there is a galley in which the meals are prepared to be taken up to the passengers by 2 elevators. it: In calculating slope, you need to know the terms rise and run. D Rise 6m Rise 3m A C Run 6 in Run 3 m Rise of AB Rise of CD Slope OfAB — m Slope Of CD — m -3 _ 6 _ 6 ' 3 _1 = 2 ” 2 From the diagrams, it seems CD is steeper than AB. How do their slopes compare? Fi ure 6.3 Sam 1e Textbook Pa e — 3 150 still many ways of combining these quantities to come up with a numerical indicator for the concept of slope or steepness.35 Why not use (rise) + (run)? What about (rise) - (run) or (rise) x (run)? What makes any of these possibilities any less palatable than the one that we accept as being correct? Unfortunately, the definition of slope as (rise) + (run) has some problems. For example, it produces a zero value whenever a line has the rise and run numerically equal but opposite in sign. Such a line would be inclined at a 45° angle to the horizontal and our intuition clearly tells us that lines like this one have more slope than zero. In addition, adding the rise and run would give numerically equal values for pairs of lines where the rise and the run were interchanged; and, such lines would have vastly different slopes. Multiplication of the rise and run would also give numerically equal slopes in the same situation. In fact any way of combining these quantities that makes use of a commutative operation will exhibit this same dilemma; consequently, it is easy to argue that division or subtraction are the only reasonable alternatives.Subtraction 3SIt is arguable that other references could be used in place of horizontal and vertical lines with no loss of mathematical generality; and, hence the notions of rise and run are socially constructed. Still, gravitation plays a role in making things stand vertically and the horizon is a constant reference in our world. Consequently, I would argue that the use of these references is not totally arbitrary either. 151 is not appropriate because it gives a zero slope for lines with equal rise and run, for example, again a contradiction to intuition. Even finding the ratio as run/rise gives numerical values in contradiction with what we know about steepness because an infinitely steep line would have a slope of zero while a line that was horizontal would have an infinitely large slope according to this definition! Therefore, the decision to define slope (in terms of rise and run at least) is not arbitrary; rather, it is the product of experimentation and logical analysis. It is also the product of a particular orientation to mathematics on the part of the teacher; and, a knowledge of content that allows for this kind of exploration. The ability to teach in this way presupposes that [A] teacher's grasp of subject matter must extend beyond the conventional image of mathematics ... What is at stake is not the ... end-product that is usually called mathematics, but .. the whole domain in which mathematical ideas and procedures germinate, sprout, and take root ... (Hawkins, 1972, p. 114) It is this sequence of false starts and refinement that is characteristic of how real mathematics gets done. The neat and precise organizing structures that we learn in school mathematics are often the product of much conjecture, testing and experiment; however, in many cases the only portion of this process that we see as learners of mathematics is the final product. The obvious opportunity to give students a sense of the actual process of coming to a logical decision based on 152 guesses, false starts, conjectures and experimental trials has been completely missed by all of the textbook authors cited here. It would not have been difficult to start this section with a short exercise where pupils were asked to examine the rise and run of a number of given lines after :making some conjectures about what their slopes ought to be relative to one another. They could try combining rise and run in several ways and come to the conclusion that finding the ratio of the two, in the order that we now conventionally use, gives a numerical value that is most descriptive of the physical reality of the situation. Unfortunately, it seems that the primary goal in each of these sections is to get on to the numerical calculation of slope as quickly as possible. Indeed, in the sample in Figure 6.1 there is a total of only 37 words before the first example which involves a calculation using the given formula. So much for the contention that pupils ought to learn mathematics by reading. My intention is not to single out these authors as bad writers or poor teachers. They are hardly unique in the style of presentation that they use; and, if they are guilty of anything it is conformity. Indeed, this method of giving the formula and then following almost immediately with examples is a common occurrence throughout these textbooks, and others, regardless of the secondary grade level at which 153 they are targeted. I do not believe that this situation is an accident . Textbooks are written for the market that exists and a considerable amount of research goes into finding out what the buyers want. If the books do not sell they are revised in subsequent editions to make them more palatable to the consumers. In fact, one publisher's series that preceded the editions from which these examples were taken did try to incorporate some questions that required student discovery as part of the more traditional exercises. I was told informally by one of the authors that their inclusion had been criticized by teachers. The lack of this kind of material in the new release gives further evidence to the argunlent that teachers influence what goes into the texts; what they appear to want is more practice questions The similarities and, r ather than more conceptual development. in all three of the series presented here makes it reaSonable to assume that this kind of approach is what the tea<-‘-hers, and perhaps even the pupils, want. The result is that what gets taught is the formula for slope and the ptic><=edure for calculating it. Unfortunately, however, when pupils are exposed to this kind of model in high school mathematics classes year after Year, it is unlikely that if they become teachers they will Bllqdenly change what they think it means to "do" mathematics and start searching for ways to connect ideas and principles v? A! o r 1 154 together. It is more likely that they will think of mathematics as a series of arbitrary definitions to be learned and then perhaps applied. Evidence to support the claim that these students hold this view of mathematics can be found in the interview responses to the second card sort task (see Appendix E) vfl1jL<:h asked students to classify knowledge as warranted or cxaraxrentional. In this task students were asked to state whether they thought each item was arbitrary (conventional) or' 14f there was some reason behind it (warranted). One item presented students with the typical definition of the tangent ratio in trigonometry as opposite/adjacent. Although all eight of the students in the interview sample Perceived this to be an arbitrary definition, in fact it is not as arbitrary as they seemed to think. laistorically the Greeks used geometric methods for most of: their mathematical applications, even when doing trigonometry. To understand the link between the term ta“agent as studied in trigonometry and the word tangent as Stu‘d-ied in geometry, consider the diagram in Figure 6.4. In this diagram our modern definition of the tangent ratio, using A OAB, would give that tan 6 = opposite/adjacent = Am3"51- or just AB. The line AB, however, is the tangent line tn) the circle with centre 0. Hence the word tangent as 155 Figure 6.4 - Geometric Interpretation of the Tangent Ratio used in trigonometry class is closely related to the word tangent as used in geometry class (Eves, 1976).“5 None of the students in the interview sample knew this relationship; and, all believed that the definition of the tangent ratio \ t ”I myself came upon this information as the result of wiying to figure out the answer to a pupil's question about Stet-her these two words were the same or not. Like the th‘EGents in this sample, I too had never been exposed to 13 connection. 156 was essentially arbitrary with no reason behind it what soever . 37 The approach illustrated in these textbooks; the commonly held belief on the part of the students that much mathematical knowledge is conventional rather than warranted; and, the reported research on the character of classroom instruction (Welch, 1978) make it reasonable to assume that the high school experience of these students did not emphasize a strongly connected view of mathematics. Although it may not be true that all students will teach in the same way that they were taught, we cannot ignore Lortie's (1975) apprenticeship of observation and the impact it has on pupils in developing ideas about what it means to learn and to teach mathematics. Given many hours of Observing the kind of mathematics teaching that focuses on anSwers rather than questions and on procedures and rules r ather than conceptual development, it is hardly surprising that the kinds of connections that were observed in this study were often somewhat shallow. Few conceptual connections were evidenced in the data because students do not associate the content that they have learned in high School with conceptual thinking. And, it is not surprising t hat even when pushed to make connections they commonly made \ ex 3“’In fact all of the trigonometric ratios have similar tafilanations for why their ratios are what they are. I did th e the time to explain this linkage to the students after exe interviews were over. Most were surprised and visibly cited by how "neat” the connection was. 157 linkages based on procedures and context, features of high school content with which they are more familiar. Although prior socialization into the norms of high school learning and teaching are no doubt part of the problem, it is also true that despite the fragmented knowledge the students in this sample often demonstrated, they must have had some success at the subject. It is unlikely that these students would have chosen to study mathematics at the university level and that they would aspire to teach it if they did not believe themselves to be reasonably competent in high school mathematics. We often equate being a good mathematics student with being a Successful mathematics student; however, the terms are not necessarily synonymous. It is worth examining in more detail what a successful pupil of mathematics looks like in a tyPical high school classroom; and, if this profile autcmatically means that the pupil is good at mathematics. 158 What is a Good Mathematics Student?38 A successful mathematics student is typically one who achieves high marks on tests and examinations -- evaluation instruments that rarely include process components. Does this mean that a good mathematics student, as measured by traditional means, is also one who takes good notes, copies down and learns examples, and memorizes definitions without question? Certainly these features would help students to be successful in a mathematics program that stresses the acquisition of knowledge and skills. And, to be fair, these are all important components of learning mathematics. Students can't apply knowledge that they don't have and it 13 not possible to divorce the mathematical process from the Content that results from the process. The terms good and successful are not synonymous, h(“areven I would argue that a good mathematics student is a successful mathematics student who, in addition, has deVeloped a habit of inquiry and a way of thinking about problems that makes use of mathematical methods as well as mat-1'.lematical knowledge. Unfortunately, in most schools, it \ th 3"I have deviated, out of linguistic necessity, from Ina convention of using the word pupil for a school student. in this next section the word student is broadly used to The lude anyone at any level who is studying mathematics. ate context makes it clear that I am talking about school c “dents rather than intending teachers so there is no Onfusion. The original differentiation between pupil and :tudent that has been used throughout this dissertation e8limes in the next section. I a” Dull- lei ,: be ill: I’ r it 159 seems that the balance between the two components is somewhat skewed in favour of the skills, procedures and defijlitions of mathematics rather than an exploration of the rictl set of interconnections that it exhibits. However, for the purpose of teaching it is just as important to be good at Immathematics as it is to be successful at mathematics. ‘This discrepancy between what it often means to be successful at school mathematics versus what it means to be 900d at mathematics for the purpose of teaching it was brought out very clearly by one of the people in the interview sample in the following spontaneous comment: Josh: 0k, because, I mean, I'm stuck with this thing of I thought I knew math, I really did. I'm very good at it, I manipulate things, you know, well and show me an algorithm and I catch it and I'm gone. I can reproduce very well. Show me where you're going even, like that one question with the averages, give me a hint where you want me to be and I'll catch it, I’ll jump on the train and I'll get there. And I don't ... I've always thought that I did that all my life with mathematics ... but now I'm starting to go well what underlies all of this. What are the, what am I REALLY dealing with and the questions ... I've really got to look into what does a square root mean? You know because it's always been a functional thing for me, square root is not an abstract mathematical thing, it's tangible. You take 2 numbers and multiply them together and so you've got to be able to unpack that and say what are the 2 things that ... so when you end up with those ... what are they called ... extraneous roots and things like that it doesn't make sense it's ... it's illogical because I know you've got to be able to take things apart as well as put them together. Um ... but it's very much a manipulative thing, give me an expression, tell me where to go and I'll go there, I’ll get there. 160 The language used by this student as he explains his views about mathematics and how they have changed, is striking. Words and phrases like "reproduce”; "get on the train"; "tell me where to go and I’ll go there”; all give the sense of a procedurally driven view of the subject. Clearly, this student was successful at mathematics; however, I would suggest that he is now beginning to question whether he was ever good at it. Although this particular student had a weak background in mathematics, the strongly prepared students made similar, although not so emotive, comments from time to time. These Comments were always offered spontaneously because there were no questions in the first block of interviews which aSked them to talk about their views of mathematics. The fact; that most felt a need to bring the topic up suggests that they were beginning to question these deeply held beliefs. The intent of this discussion is not to cast blame on 8econdary teachers. After all, they are doing the right thing according to their view of mathematics; and, the view of mathematics that is often implied by textbooks, administrators, parents, provincial tests and curriculum gui<:1es. It follows that the disconnected and fragmented l“lo‘niledge of high school mathematics that these students Often exhibited is not at all surprising; and, in fact, is consistent with what they were taught in high school. It 161 seems clear that the way that mathematics is taught, learned and tested at the secondary school level does not promote a connected view of the subject. Because the high school background of these students is so similar, it is not as surprising as might be expected that the differences among them with respect to their understanding and knowledge of high school mathematics are small, despite university study. It does not seem reasonable, however, to assume that the mathematical knowledge base with respect to high school mathematics of prospective teachers is fully formed when they leave high school. Nor is it easy to understand why extensive mathematical study appears (from the survey data at least) to impede their ability39 to make connections. Furthermore, connections with other subjects were rarely mede by any subgroup of students. We know that all the Participants in this study have studied mathematics at the university level as a preparation for teaching in secondary 8911001. What we don't know is what they learned about mathematics as a result of this further study. What is it about more intensive study that has apparently affected the Way that they make connections? The next section examines the Undergraduate mathematical experience and how it also has a role to play in explaining this group of results. \ Co 3”The distinction between a student's ability to make lannections and predisposition to do so will be discussed th er. For now, I will continue with my earlier assumption at if they could make connections they did. 162 The Ontario Undergraduate Experience In Ontario, the university system is somewhat different to that of the United States and the rest of Canada. Ontario is the only province that still has thirteen grades (a remnant of the old British sixth form) and consequently it takes only three years to earn a general baccalaureate degree.‘0 However, many students do not elect to take this kind of liberal arts program; rather, they opt to enter the more intensive, specialized and rigorous honours programs. Honours programs are four years in length and consist of Courses which are substantially more difficult than those Studied in general degree programs. In addition, students take more courses in the chosen area of specialization throughout the four years of study. Prospective teachers ar- e given priority for admission to teacher preparation pr c>9rams if they have completed these more demanding degrees (presumably because they are assumed to provide a better g”(handing for teachers than general degrees); consequently, it is rare that intending secondary teachers do not have an h()‘lours degree in either their first or second teaching opt-ion. In this sample, all of the strongly prepared \ 11°“ “Technically, grade 13 was abolished a few years ago; ever, it was replaced by the requirement that pupils had in complete six advanced courses after grade 12 graduation ch order to qualify for university. Although the name has a s nged most students still spend five full years in high Q1'ltool. 163 candidates had honours degrees in mathematics. Those in the other categories also had honours degrees but in a discipline other than mathematics.“ The Impact of Specialization The argument presented in Chapter 2 with respect to the need for a strong knowledge of subject matter for teachers is still valid; however, just what kind of preparatory educational experiences might produce this kind of richly connected knowledge is becoming more problematic as the discussion ensues. Even scholars who strongly believe in the importance of content knowledge for teachers remain unconvinced that the understanding and knowledge acquired in tYpical university courses is adequate for the task of teeChing (McDiarmid, 1992). The results of this study do not convincingly confirm the widely accepted idea that a highly specialized and focused mathematical preparation pr<>duces teachers with a more connected understanding of high school content than a less intensive, more general, pItogram of mathematical study. \ b “This assumption about the superior preparation provided My an intensive honours program is also manifested in several cihistry regulations. Mathematics department heads and a thultants are required to have a honours degree as well as advanced educational credentials; and, teachers with honours dzgrees earn substantially more than those with general eQrees. 164 In Ontario the liberal education that was long ago envisaged as the most appropriate preparation for teachers (Borrowman, 1965) has been replaced (especially in the sciences) by a highly technical and very narrow program of studies. This kind of preparation, although possibly appropriate for those students who intend to pursue academic careers in research may not be the best possible background for those intending to be secondary mathematics teachers (Peters, 1977)."2 A number of Holmes Group institutions have developed interdisciplinary majors for teachers because of this very PrOblem. They believe that a solid background in a single Bubject complemented by a wider range of studies in other areas will give teachers a broader base which will allow them to better integrate their major subject into other fields of study. Virginia Commonwealth University has deVeloped an interdisciplinary science program for middle 8"3110(31 teachers that features a broad but fundamental preparation in science. They have also developed two new thathematics courses of an interdisciplinary nature, but iJ:‘<>liically they are designed for students who are not \ b 42Not everyone is convinced that a narrow, specialized Qaekground is a good thing, even for potential researchers tfid scholars. Many would argue that some of the problems a fit we face today with respect to galloping technology are 9 result of scientists not having a sufficiently broad erspective about the world in general. 165 mathematics majors (A. McLeod, personal communication, January 25, 1993) .43 The focus on specialization at an early point in a student’s academic career has a direct bearing on the results of this study. The strongly prepared students in this sample have, for the most part, had a program so heavily weighted towards mathematics that they have studied little else. Of the 40 semester courses required for a degree, as many as 30 of these could be courses in mathematics or mathematical computing. There is very little that could be said to be liberal about this education. At the most simplistic level this fact might help explain why they were not good at making connections with other s'Jlleects. The ability to make connections outside one's own discipline presupposes that the individual has some understanding of, and knowledge about, other disciplines; and, this is not always the case with students who have 3Pecialized very early. Perhaps the reason that the physics Illaficrs were better at making connections than any other S"1t>group was precisely because they have a substantial an“bunt of mathematical knowledge and more knowledge about a mathematically related field than mathematics majors \ <1 ”A science program along these lines is being leveloped at the University of Western Ontario. (interestingly, the mathematics courses that are being esigned for this program are the kind that I would like to use prospective mathematics teachers take; but, they will 0 t be eligible to register in them (University of Western htario, 1990). 166 themselves. It may even be the case that the study of other mathematically related subjects increases the ability of the student to connect ideas within mathematics because they see the material in different contexts and because they develop a habit of mind towards connection making. The role of habit will be further discussed in a later section when the contradictory results with respect to strongly and weakly prepared candidates evident in the survey and interview results are reconciled. Over-specialization, however, is not the only problem. It is still reasonable to expect that students who have done lots of mathematics but little else should be able to Connect things better within the discipline of mathematics than more weakly prepared candidates. This common sense expectation was not unequivocally confirmed by the data. The compartmentalized way in which mathematics is often taught at the university level and the somewhat idiosyncratic nature of some of the programs can result in 8t“dents acquiring lots of facts and information but little Betlse of how the various pieces fit together (Kennedy, 1991) . Students simply do not perceive the various courses E‘t1lanations; and, to what extent they would help the pupils in the scenarios to understand what was the root cause of the difficulties, it appeared clear that the strongly prepared students had no such initial ambivalence. \ ”Listening to the tapes, however, was prompted by a comparison of interview transcripts and noting that some had ny more pauses and breaks indicated in them than others. 172 This certainty about knowledge is the result of a number of factors, notably, the student’s confidence in his or her mathematical knowledge and the perception about mathematics as a discipline that he or she holds.46 It is hardly surprising that students who have been successful in a university mathematics program renowned for its difficulty would be confident that they know high school mathematics; after all, they likely needed good marks in high school mathematics courses in order to get into their university program in the first place. Although beliefs in their ability and knowledge appeared to play a role in how students responded, their beliefs about mathematics as a discipline also seemed to have a role to play. One dominant feature of students' knowledge that was a£>L>arent from both the pattern of quick responses and the Content of these responses was the perception that mathematics is a subject which has definite answers to Problems; and, one that has rules and procedures that must be obeyed. Because they know these rules and procedures it fol lows that they would be able to respond quickly to the items presented in the interviews. The content is high 8cl'lool level material, which they believe they know; and, \ t “Personality also comes into play here; however, over he course of the project the level of rapport that I was aJule to develop was high and I do not think that anxiety was an important factor in how individuals responded. 173 they have the belief that most mathematical errors are the result of rule infractions to which they can quickly attend. This initial certainty quickly disappeared, however, as more probing questions were asked. The strongly prepared students found that there were conceptual holes in their knowledge just as there were for the students with weaker mathematics backgrounds. The strongly prepared candidates also searched, when pushed, for ways of connecting ideas together as the interviewer continued to ask for examples and illustrations. Attempts to compare the amount of time it took various students to complete their surveys would have had little value because different surveys had different items; and, the items were not equally easy to analyze. However, there is no reason to believe that stamdents would have responded in a different way to the 8lil.:l':vey items than they did to those same items in an interview setting. I am not suggesting that the content of their responses would necessarily have been constant but that the manner of response would likely have been similar. fact, students sometimes made comments in the interviews The In a*bout items that they had seen before on the surveys. fj-Jrst text block in the sample interview transcription (see Appendix H) is precisely this kind of remark; and, Lynn is a"’axe that the response she gave at that time was not what 3tie would now say after some thought. This data suggests that she may indeed have given a quick but perhaps facile The: 5250. :repe DIOCE 174 response when she did the survey and this statement helps to confirm the contention that the strongly prepared students may have responded to the survey in the same way that they initially responded to the items in the interviews. Therefore in the survey situation it is doubtful that this second level of response was reached by the strongly prepared candidates; and, they only gave their initial quick procedural response without the benefit of any follow up. This response pattern in the interviews is quite different from the way that less strongly prepared candidates reacted to the same items. Generally they were much more tentative in their initial answers and often followed a kind of exploratory self questioning process to arrive at a response. This perception, arrived at from 8:LIuply listening to the interviews, is also confirmed by the faot that the number of text blocks containing the code for uIl<=ertainty occurred 30% more frequently than expected by Even chance for the students who had a weak background. taking into account the previous proviso about over- interpreting the meaning of the number of blocks of text, this result does buttress the previous, more qualitative, ‘3 eertion. It was common for weaker students to think longer about the scenarios that they were asked to respond to; and, rather than stating quickly what they would say they often started by asking themselves things such as ”Now let me see 175 if I understand what's going on here . . ."; or, "Let me write that down and see if I can find what the problem is . This investigative process often revealed the real difficulty that was embedded in the question and illuminated how a connection could be made. In the surveys, if weakly prepared students acted the same way, they would naturally make more connections than the strongly prepared candidates who reacted quickly with an overly simplified view of the problem. This result would not be because students with a weaker background know more or are intrinsically better at making connections; but, because they used a different process to establish their final written response to the Survey items . Certainly the small number of students in the interview sample does not make generalizations possible; however, it is instructive to look at the differences in the number of text blocks coded as connections versus student background. The number of text blocks, classified 'with one or more cohnection codes, was greater than would be expected by chance for both the strongly and moderately prepared s‘t-mdents; and, less than would be expected by chance for the weakly prepared students (see Table 6.1). If there was no difference in the way that strongly Dl'i‘epared candidates and weaker candidates responded with rQSpect to making connections then the 425 text blocks that represented connections should be distributed among the 176 three groups in proportion to the number of text blocks that each category of student contributed to the total. a: student Number of Blocks Number Percent Background Text Blocks with CH Expected Deviation Strong 260 127 116 + 9 Weak 323 122 145 - 16 Moderate 368 176 164 + 7 Totals 951 425 425 0 ll Table 6.1 - Connections Versus Background (Data From;Interviews) Consequently, the number expected by chance (expected value) was calculated by multiplying the fraction of text blocks for the entire dataset contributed by each student category by the number of text blocks that represent connections. For example, if 260/951 = 27.3% of the text blocks came from 81:1'ongly prepared students we would expect that by chance 116 of the text blocks classified as Although the alone 0.273 x 425 = cOnnections would come from this same group. d1 fferences between the expected value and the observed value are not large, they do suggest that the strongly prepared candidates in the sample are somewhat more able to Ina-ke connections than the weakly prepared candidates, when pushed to do so. The previous discussion suggests that they may not perceive the need to follow the kind of investigative process that helps in making connections, witzhout being encouraged. This argument helps explain the 177 contrary findings of the survey where further questioning was not possible. The Role of Teacher Education“7 In one sense, the discussion about making connections and students' connected knowledge is akin to discussions about pedagogical content knowledge. Connecting various mathematical topics together or connecting to a pupil’s way of thinking in a pedagogically powerful way requires more than just knowledge of content. The interaction between knowledge of pedagogy and content that Shulman (1987a) talks about is at the heart of this way of thinking; still, Students do not appear to think about content in this Pedagogical fashion to the extent that might be hoped. Even when pedagogical connections were made they were often weak, with links to previously studied rules and procedures rather than to concepts and their underlying principles. Before they were asked to think deeply about some of these issues, prospective secondary teachers did believe that they knew their material, although there is little tendency to link what they have learned in university mathematics courses with what they are teaching in high school. It is only as \ "Strictly speaking everything that someone learns lDefore he or she begins teaching could be considered to be teacher education. In this context, however, teacher eClucation refers specifically to professional teacher education rather than disciplinary studies. 178 they grapple with non trivial pedagogical issues that they begin to realize that their own knowledge and understandings may be superficial. These findings are particularly relevant to the role that teacher education plays in helping to explain the results of this study. In some ways formal teacher education is an unwitting participant in the development of what can be called the Two Solitudes of teacher preparation.“8 The secondary teacher preparation institutions of Ontario have always admitted students who have completed their baccalaureate degrees; and, there has always been an implied assumption in this System that teacher preparation programs (at the secondary level at least) need not be concerned with teaching content.‘9 Content competency has always been perceived to be the responsibility of the university departments that granted these individuals their university degrees; while methodological competency has been the responsibility of the teacher educators; a bifurcation that is also prevalent in “The late Hugh MacLennan, one of Canada's most famous writers, used this phrase as the title of a novel which eJilzamines the endemic tensions between English and French cenada. The expression has now become part of the language; and, I believe it to be most appropriate here. ”The normal school tradition of teacher education for elementary teachers was ended in Ontario in the early seventies when the education of teachers was handed over to lie universities. Prior to this step prospective secondary Eaachers were prepared in different institutions to those ho were intending to teach elementary school. Elementary eachers were not required to have university degrees and QOnsequently it was assumed that although they needed nstruction in content, secondary teachers did not. 179 the minds of policy makers in the United States where recent reform initiatives seek to separate professional and disciplinary studies (Kennedy, 1990). The proceedings of a recent conference sponsored by the Council of Ontario Universities, entitled To Make a Difference: Teacher Education for the 1990s (Van Fossen, 1990) made it clear that the arts and science departments on many university campuses know little about Faculties of Education or what they do; and, that historically they have not seen their role as one of providing the people that they graduate with the content knowledge that they might need to teach well. In many cases their graduates do not go on to teacher preparation programs in the universities where they receive their degrees because not all Ontario universities have such programs. Arts and science faculty perceive the job of teacher preparation, in all its forms, to be the job of the education school, although historically universities did have a significant role to play in the preparation of teachers (Sewall, 1889/1965). Their role is to induct new Inembers into the community of scholars to which they t hemselves belong . The students also hold views about role responsibility Partially shaped by their years at university prior to Galitering their professional education studies. They believe that they know their material when they begin their teacher Preparation programs and so they do not perceive that they 180 need to think about content issues as part of their preparation to teach. Although this perception is less pronounced with elementary candidates, many of whom do fear certain subjects, by and large prospective secondary teachers think they have the content knowledge they need. The result is that discussions of content, in many cases, simply do not take place -- a contention that is supported by the fact that all of the students who were interviewed commented at some point during the process that they were not used to thinking about content issues in this way. This problem is similar to the other world pitfall of experience that Feiman-Nemser and Buchmann (1985) talk about when discussing the role of practical classroom experience in teacher education. If the students perceive classroom work as belonging to another world different to the one in which their teacher education courses reside, they will not link them together. The same thing happens with university mathematics courses and teacher education programs. One is rooted in the world of scholarship and the other in the messier more problematic world of the school classroom. How could they have anything to do with one another? How can we expect students to make connections between what they have learned in one world with what they are learning in another When they perceive the worlds to be essentially different? The rarity with which university studies are mentioned in 181 any of the interviews, despite the fact that they focus on mathematics, further corroborates this assertion. aintinq,the Picture The analysis of the results undertaken in the preceding sections has provided a firmer foundation for conclusions to be drawn from the data. Affective factors such as disposition and habit appear to have had an influence on what students had to say about the items in the interviews and sometimes even how they said it. It follows, therefore that any meaningful picture of the findings of this study will need to include more than just the categories of connections that were observed. Furthermore, because many of the items involved teaching situations, in many cases the students put themselves in the role of the teacher and talked to the interviewer as if he were the pupil. In fact, many of my probes started with phrases such as "If I were this pupil and I said ..." because I wanted to try to prompt a more detailed discussion of the item without the student getting the impression that his or her mathematical competence was somehow being evaluated by someone who is a mathematics methods instructor.50 In this sense, the 50Although I did not use students from my own methods Class in the interview sample, the students who were interviewed knew who I was and still perceived me to be "the expert." 182 observed connections can be thought of as links between the pupil and the teacher that are built of mathematical or pedagogical ideas. In the discussion that follows, the findings, and the analysis of these results, are used to generate a picture of connected knowledge as it exists in these prospective secondary teachers. The ability of a student to make connections in a pedagogical situation appears to be influenced by many factors in addition to his or her knowledge of mathematics. Students must understand how pupils of this age typically think about mathematical concepts; and, what kinds of concepts are especially difficult for them. They need to know what the content of the curriculum is so that they can make judgements about whether pupils can be reasonably assumed to know certain ideas or not. They need to have an understanding of where mathematics can be found in the real world of the pupil and how to make the links between these situations and the ideas under discussion. Furthermore, they need to understand pedagogical principles and how these principles are related to the content that they are trying to teach. The interaction of all of these various components of the prospective teachers' knowledge then influences the decisions that they make about the best way to resolve a pedagogical dilemma. Although a mathematical Connection may be one way of resolving this dilemma, other kinds of connections are also possible. 183 The diagram in Figure 6.5 is a conceptual map of the connected knowledge of this group of prospective teachers. It includes the components of the organizing framework that was used to structure the presentation of the findings in Chapter 5; however, it has been modified to include other factors that may influence the students' thinking. The diagram illustrates that the knowledge that the teacher holds interacts with the knowledge that the pupils bring. These two different sources of knowledge both contribute to the ability of the teacher to make a connection with the pupil in order that learning can take place. The discussion of the results in the previous sections advances the argument that the tendency of students to make connections, and the kinds of connections that they make when they make them, is influenced by their beliefs about mathematics as a discipline, their attitude towards it and their predisposition to teach mathematics as a coherent and connected subject as opposed to a series of disconnected facts and procedures. These factors are included to more accurately portray the complex and multifaceted nature of students' knowledge and how other factors interact with this knowledge to influence how they make connections. \ 86:8 33.3.. l & Bdhfll L 184 (Msm*xm (Numanmmn Ohemummi swoon 7‘... [Dispositions I (Inmflmmmhs (Ism*xu (NemeMm cummuma OhedumMi E Figure 6.5 - Conceptual Map of Prospective Teachers' Connected Knowledge This stu findings whic ELY-ever, a dq experiences ‘ teacher prep. The higr procedurally what mathema Textbooks, a testing PrOg SubjECt. pu mathematics Biking and t persPECtive 185 Summer y This study has reported a number of provocative findings which at first glance seem to be counterintuitive. However, a deeper examination of the various forces and experiences that shape students' thinking before they begin teacher preparation makes the findings less surprising. The high school experience with its rule and procedurally oriented methods gives pupils a false idea of what mathematics is like as a discipline of inquiry. Textbooks, administrators, curriculum guides and provincial testing programs all contribute to this image of the subject. Pupils go on to university with a view of mathematics that does not appear to emphasize connection making and there is no evidence to suggest that this perspective is changed by university study. University studies are often highly specialized and sometimes poorly taught. Ironically the students who have the strongest preparation in mathematics also know the least about anything else. This specialized study makes them certain about their mathematical knowledge and so they tend to give quick answers in response to pedagogical problems -- answers that are often based on procedures or rules. They are not predisposed to think about connections and cOnsequently they do not make them unless pushed to do so. Students their mathem reams; and, for this tea as being be1 students. ‘ interview d other ways the PEdagog The in faculties and each 9 the PIOCes teachers j Way Prior 186 Students with weaker backgrounds tend to establish their mathematical ideas through more tentative, exploratory means; and, in so doing make or discover connections. It is for this reason that the survey results show weaker students as being better at making connections than stronger students. This result from the surveys is not clear in the interview data where all participants were pushed to explore other ways of thinking in trying to establish how to solve the pedagogical dilemmas embedded in the scenarios. The interaction between university arts and science faculties and the education schools is often very limited and each group assumes things about the other's role. In the process, the content knowledge of prospective secondary teachers is assumed and is not examined in any meaningful way prior to teaching. Students do not perceive a connection between the world of the scholar that they have just left and the world of the practitioner that they are about to enter. Consequently they do not try to make links between the mathematics that they have studied in university and the content that they are teaching in secondary school. The result of this separation in the students' minds between school mathematics and university study, coupled with the fact that content issues are rarely addressed in any meaningful way with prospective secondary teachers, is that the ability to make connections between and among mathematical this group I 187 mathematical topics does not appear to be well developed in this group of prospective teachers. CHM This st kind of aca mthematics education 6 mergradue Chapter the changes in Finally, a been raise. FirstI Erich of th “thematic C-Onceptual the“ Pros SeCohd’ t} Etudetits E CHAPTER 7 — IMPLICATIONS AND RECOMMENDATIONS Introduction This study has raised important questions about the kind of academic preparation that prospective secondary mathematics teachers receive prior to their formal teacher education experience, both at the high school level and as undergraduate students studying mathematics. In this chapter the implications that follow from this work and the changes in practice that they suggest will be considered. Finally, a number of further research questions that have been raised by the findings will be presented. First, I examine the view of connections that permeates much of the reform literature and how this view of mathematical connections, as delineated in the earlier conceptual framework, is not congruent with the way that these prospective teachers went about making connections. Second, the small differences between various subgroups of students with respect to connection making ability, regardless of mathematical background, will be discussed. Next I consider the lack of consistent results for some aspects of the study between the survey and the interviews and what this finding might imply for further studies of teacher knowledge. A number of recommendations for practice are.made based on these implications. The chapter concludes 188 with a retrospe was designed tc answered by the further researc raised by this The mathem reference to t] alld mong math: Connection Insight de develop the griEty of of arithmet ic . ind 80 On. Al 189 with a retrospective view of the questions that the research was designed to address; and, to what extent they have been answered by the study. Finally, a number of suggestions for further research will be made, based on questions that are raised by this work. Connections in Conflict The mathematics education literature makes frequent reference to the importance of making connections between and among mathematical topics. Connections give mathematics power and help determine what is fundamental. Pedagogically, connections permit insight developed in one strand to infuse into others. Multiple strands linked by strong interconnections can develop mathematical power in students with a wide variety of enthusiasms and abilities. (Steen, 1990, p. 7). Teachers ought to connect algebra with geometry, the ideas 0f arithmetic with the ideas of algebra, probability Concepts with area models, mathematics with other subjects and so on. Although these are all valid examples of good mathematical connections they only reflect one facet of the kind.of connected thinking that prospective teachers eXhibited; and, these strong connections were not commonly made by many of the teachers in this sample. The results t'POrted here illustrate that the match between the kinds of d°Birable connections that mathematics educators believe to be useful and t lake is not goo The lack 0 between well pr ability and/or the discipline implies a recon of the kind pre lcomected vie between the su that studies of ”5191‘ to conce instruments the Suited to the t that might be e example, the ro students to flirt Erhaps tacit) u on the results. dispositions to lathematiCal knc surveYs and the 190 be useful and the kind that prospective teachers typically make is not good. The lack of consistent and appreciable differences between well prepared and weakly prepared students in their ability and/or disposition to make connections, both within the discipline of mathematics and with other subjects, implies a reconsideration of the role that intensive study of the kind presently available plays in the development of a connected view of the subject. The lack of consistency between the survey results and the interviews also implies that studies of prospective teachers' knowledge may be easier to conceptualize than operationalize. Even instruments that have face validity and appear to be well suited to the task may not provide the kind of clear results that might be eXpected or desired. In this study, for example, the role that communication played in allowing students to further develop and refine their own prior (or Perhaps tacit) understandings did have a substantial impact on the results. In addition, the interaction of students' dispositions to make connections and their beliefs about mathematical knowledge made interpreting the results of the BurVeys and the interviews problematic. Still, if the ability to make connections, and the habit of doing so, is as important as reformers suggest, this study prompts us to Consider how a more connected view of mathematics could be developed. Some suggestions that might help to move the vision of Literatur: connected Curriculm 191 vision of connected knowledge that is featured in the reform literature closer to the reality of prospective teachers' connected knowledge are considered in the next section. Developinngonnecteg,Knowledge Curriculum Issues The ability to generate mathematical connections and thereby develop connected knowledge requires that teachers have the desire to develop this level of understanding and that they have a sufficiently strong background that they know concepts, principles and ideas that can be connected. For example, the interesting connection discussed in Chapter 5 with respect to compound interest and exponential growth would have been impossible to make if students did not know one or both of these formulae and where they can be applied. Furthermore, if a teacher is to be able to give pupils a feel for the dynamic nature of mathematical knowledge (another important strand in the mathematics reform literature) she or he must be knowledgeable about recent developments in mathematics. Chaos theory, fractal geometry, the use of bar code check digits; or, the RSA cipher system, are all good examples of modern mathematical developments that are accessible to students in high school. If teachers are going to be successful in giving pupils a sense of th they need t pupils in a the prospec of mathemat Further students' p to make use Of the know Prepared ca but they ma- t-0 act effe learning Co to be able so; henCe, rlec‘issary b of a rich a teachers Unfort‘ 192 sense of the changing and evolutionary nature of mathematics they need to be able to relate topics such as these to pupils in a meaningful way. Such an ability requires that the prospective teacher have a solid and thorough knowledge of mathematical content. Furthermore, the ability of any intervention to modify students' perspectives about mathematics and to allow them to make use of the knowledge that they possess is a function of the knowledge that they bring to the task. Weakly prepared candidates may change their ideas about the subject but they may not have the mathematical background to be able to act effectively on this new vision of what teaching and learning could be. Strongly prepared candidates did appear to be able to make connections more often when pushed to do so; hence, a strong preparation can be considered to be a necessary but not sufficient condition for the development of a rich and connected knowledge base in prospective teachers. Unfortunately, many university mathematics departments have constraints on what they could do to help this situation, even if there was a will to change. For example, most departments have a large proportion of their available resources committed to providing service courses for other disciplines such as engineering, physics, or economics (Madison and Hart, 1990). It is not realistic to expect them to mount expensive new programs just for students who think that th the three role the developme1 support for 0‘ teachers, is 1 number of inii the concerns 2 One Chang tremendously \ mathematician: It is still c, dePal'l'.mel'lts w; t0 use calcula beliEVe it re] away from rea. reports the n gathematician: their ideas a] ‘h 193 think that they might want to teach. This tension between the three roles of the undergraduate mathematics program in the development of new mathematics scholars; providing support for other disciplines; and, preparing prospective teachers, is not easily resolved. However, there is a number of initiatives that could be undertaken to address the concerns raised by the findings of this study. One change that would cost nothing and help tremendously would be a change in the attitude of many mathematicians towards non-traditional mathematics courses. It is still common to hear that some university mathematics departments will not allow students in undergraduate courses to use calculators on examinations and tests because they believe it represents a lowering of standards and a move away from real mathematics. A recent paper by Mura (1992) reports the results of a research project in which she asked mathematicians at a number of Canadian universities to give their ideas about mathematics as a discipline. Although there was a diversity of opinion, logic, rigour and formalism were overwhelmingly the most common characteristics associated with mathematics by the 173 faculty members who responded to her survey. If this view of mathematics is held by the majority of university teachers, it is not surprising that many university mathematicians refuse to accept the view that advanced study of elementary mathematical topics is just as appropriate in a universi topics . Mathe curricului being wor“ especial 1' Professio 194 a university setting as elementary study of more advanced topics. Mathematical topics that are part of the high school curriculum are not accepted by most university faculty as being worthy of study in any undergraduate program, especially an honours programmu However, as the Professional Standards (NCTM, 1989b) make clear, ... because their understanding of school mathematics cannot be assumed, the preservice and continuing education of teachers of mathematics must include both learning higher mathematics at an advanced level and revisiting the content of school mathematics. (p. 65) The results of this study have confirmed that connected understanding cannot be assumed, regardless of academic background; and, the findings provide further evidence for this assertion. A close examination of the mathematical structure of the real number system that we all use (without much thought) could lead to a better understanding of many of the properties of number that are taken for granted; and, this improved understanding would then allow teachers to connect some of the more abstract mathematical principles that they teach to various arithmetic ideas with which pupils are more familiar. For example, Item 12 (see Appendix B) in the 51There is no such thing as remedial mathematics in Ontario universities. The lowest level courses would be equivalent to grade 13 mathematics and are open only to students from other provinces or Ontario students who have not completed grade 13 mathematics before entering university. interviews Pr‘ kind that pup. spite of bein« interviews wa: context is in: fact that the numerically p. even suggestec resulted in a then had stud, given me the . °f a group or unable t0 see aPhlies to th« The role Rathematics (3. teachers. kno‘ changes Collld naming the 0p] tangent and 1; 210581? Conne< aline that i] auseful and 1 '1 title to expla' J 195 interviews presented a situation where cancelling of the kind that pupils often attempt provides a correct result, in spite of being mathematically incorrect. No one in the interviews was able to explain why cancelling in this context is incorrect because their usual recourse to the fact that the procedure produced the wrong answer in a numerically parallel example could not be utilized. A few even suggested that the method must be acceptable because it resulted in a correct answer. Despite the fact that many of them had studied abstract algebra and could probably have given me the technical definition for the identity element of a group or what the properties of a field were, they were unable to see how the concept of the identity element applies to the situation under discussion. The role that an understanding of the history of mathematics could play in the development of prospective teachers' knowledge is another good example of where useful changes could be made. Earlier I explained the reason for naming the opposite/hypotenuse ratio in trigonometry as the tangent and illustrated that the name is not arbitrary but closely connected to the geometrical meaning of a tangent as a line that intersects a circle in only one point. This is a useful and relevant bit of background information to be able to explain to pupils when introducing trigonometry; and, it comes from a knowledge of the history of the development of mathematics. Knowledge of this kind is useful to tea sense of how historical de that pupils c Unfortune and definitic do not get th they think th Ironically, 1; content is of ~- courses th f0! Credit. to be watere d certainly not notion of wha university le rigour and f0 alluded to ab 196 useful to teachers because it allows them to give pupils a sense of how mathematics developed and in some ways this historical development parallels the development of ideas that pupils often experience as they learn the subject. Unfortunately, if the teacher simply knows the rules and definitions and gives them without explanation, pupils do not get the idea that mathematics ever developed; rather, they think that it was just discovered somewhere, intact. Ironically, this kind of interesting and yet elementary content is often studied in liberal arts mathematics courses -— courses that mathematics majors are not allowed to take for credit. Historically, such courses have been perceived to be watered down versions of real mathematics courses and certainly not worthy of the mathematics specialist. The notion of what makes a real mathematics course at the university level, however, is governed by the beliefs about rigour and formalism held by university mathematicians alluded to above. Unfortunately, the prospective teacher may never learn many of the interesting and useful things for the purpose of teaching that can be found in many liberal arts mathematics courses. The notion of a liberal arts mathematics course is somewhat ill-defined; however, to me it suggests a course where the technical detail is less important that the level of mathematical discourse in which students are engaged. Obviously discourse should not displace content, and talking about one's mathematics. acquisition discussion 0 better artic connections to doing mat communicate We wish to h interrelatio The idea tha written work Concept at p I would SPeCialist . S tepicS revig prospectiVe 197 about one's ideas is appropriate for anyone studying mathematics. However, it is the balance between the acquisition of mathematical skills and concepts and discussion of those ideas in some depth that needs to be better articulated. The evolutionary nature of the strong connections that students did make suggests that in addition to doing mathematics it is also useful to have students communicate with others (either verbally or in writing) if we wish to help them develop a rich set of interrelationships between and among mathematical topics. The idea that a mathematics course could require thoughtful written work about mathematics is something of an alien concept at present -— perhaps it shouldn't be. I would argue that at least part of the mathematics specialist's program could contain elementary mathematical topics revisited from a more advanced viewpoint so that prospective secondary mathematics teachers can learn more about the underlying principles of the content that they take for granted. The focus here would not be on the acquisition of new mathematical topics but a deeper and richer understanding of topics that students already know in a more traditional sense. The role of discussion, argument and communication of mathematical ideas both verbally and in writing would be key features of such a course. If a course such as this one were coupled with a history of mathematics course which focused on the development of mathematics it would help to knowledge, a report Moving 1991). Designing take is not t The number of take it diff i subjects such geography and Mathematj from ar9E haviora Pursue Substantj 1991, p. dependent 0n topiCs to (30:) 198 would help to broaden the base of prospective teachers' knowledge, a position endorsed by the recommendations of the report Moving Beyond the Myths (National Research Council, 1991). Designing new courses that prospective teachers could take is not the only change that could be made, however. The number of compulsory courses in typical honours programs make it difficult for the mathematics major to study other subjects such as physics, chemistry, economics, geology, geography and so on. Mathematics teachers at this level should explore other disciplines in which mathematics is used. Course work from areas in the physical, biological, social and behavioral sciences and in business and finance should be pursued so that prospective teachers will encounter substantial applications of mathematics ... (COMET, 1991, p. 34) Just as the ability to connect well in mathematics is dependent on the student having knowledge of different topics to connect, the same could be said for the ability to make connections with other subjects. I am not suggesting that all of a mathematics specialist's courses should be high school content revisited or general interest introductions to another discipline of study; but, rather that we need to find a balance between breadth and depth that will encourage the development of a more connected view of the subject. This tension between the mathematical preparation that potential mathematicians need and the different background that prospe one (Howson teachers mi fundamental present mat respected m the followi Univers see an with a mathema from ur aPparer Structr underg: the fee While c apply 1 reflect teach j intEQI-E that be These 5 lt- iS r apPrecj and whe uECESSari 1y important f The RC’le of The tea the Value tl 199 that prospective mathematics educators require is not a new one (Howson, 1984). Neither is the idea that prospective teachers might need to spend more time in reflecting on fundamental ideas. Fifteen years ago (long before the present mathematics reform movement began) David Wheeler, a respected mathematics educator and mathematician, provided the following cogent summary of the problem: University mathematicians look at education courses and see an apparent lack of structure and rigour together with a plenitude of non-refutable theories; university mathematics educators look at the students emerging from undergraduate mathematics programmes and see the apparently deadening effects of a training dominated by structure and rigour... How does one give undergraduates an education in mathematics which suits the few who need a solid foundation for graduate work while catering to those who need some mathematics to apply in another field and those who need time to reflect on some of its fundamentals so that they can teach it better? How does an education programme integrate rather than juxtapose the many considerations that bear on the theory and practice of teaching? These are not easy questions for anyone to answer and it is not helpful merely to criticize without an appreciation of the difficulties. (Coleman, Higginson and Wheeler, 1978, p. 56, emphasis added). It appears that the identification of a problem does not necessarily make its solution any clearer; still, as Wheeler suggests, an appreciation for the difficulties is an important first step. The Role of University Teaching The teaching of mathematics at the university level and the value that is accorded to it also needs to change (National Res committed inc students witl reward struct attractive. be rewarded 1 and teach inr structure the published reg faculty eSpec of work. The that it is it“. they Can of F illPc‘lct of 90c like” Pay di mmeers of me 200 (National Research Council, 1991). Although there are many committed individuals who work hard to provide undergraduate students with the best possible mathematical experience, the reward structures of universities do not make such actions attractive. Instructors with expertise in teaching need to be rewarded for the extra effort that is needed to develop and teach innovative courses. However, with the reward structure that is presently in place and the emphasis on published research, there is not enough incentive for junior faculty especially to concentrate their efforts on this kind of work. The university community at large must realize that it is in their own best interests to do the best job they can of preparing prospective teachers, because the impact of good teachers in the high school classroom will likely pay dividends for them in the future with increased numbers of motivated students who are interested in pursuing studies in the mathematical sciences. If more value was placed on teaching, and interested faculty members were able to consider the needs of prospective teachers seriously, it would be useful for prospective teachers to be able to take one or two courses that provided a different orientation to the teaching of the subject. The normal lecture method needs to be replaced with more investigation and an opportunity for the students to participate in the active construction of mathematical ideas. In 0rd env151 opport mathem collec proble Teache origin Realis way; and, t thing that about. How the methods asked to te higher prob in their ow t° 8llggest about learn taught in t 1988), Still, Radargradua teaching me addreSSing mtential r cannected u 5 L ‘5 «e next Se. 201 In order for teachers to implement the curriculum envisioned by the NCTM Standards, they must have opportunities in their collegiate courses to do mathematics: explore, analyze, construct models, collect and represent data, present arguments, solve problems. (Committee on the Mathematical Education of Teachers (COMET), 1991, p. 1, emphasis in the original). Realistically, not all courses could be offered this way; and, to be fair, preparing teachers is not the only thing that university mathematics departments have to worry about. However, if intending teachers were able to practice the methods of mathematical inquiry that they are being asked to teach in secondary school there would be a much higher probability that they might at least try to use them in their own teaching. Furthermore, there is some evidence to suggest that prospective teachers' ideas and beliefs about learning mathematics can be influenced by courses taught in this manner (Schram, Wilcox, Lanier and Lappan; 1988). Still, even changes as dramatic as these in undergraduate programs, degree requirements, curriculum and teaching methodology do not absolve teacher educators from addressing content issues as part of their mandate. The potential role for teacher educators is helping to foster a connected understanding in their students is discussed in the next section. AlthOI mostly in ignore the have to wo prospectiv background secondary hawe conce Procedural DL‘es'fiege th in the Uni“ 1990; Kenn that altho COrlnection think abou of “theme need to re 202 The Role of Teacher Education in Fostering Connected Understandingyin Prospective Teachers Although teacher preparation programs see their mandate mostly in terms of preparing people to teach, they cannot ignore the realities of the knowledge base with which they have to work. Although it has long been assumed that prospective elementary teachers have weak mathematics backgrounds, this study has shown that even prospective secondary teachers with a discipline based honours degree have conceptual gaps in their knowledge of mathematics and a procedural and rule oriented perspective on teaching, a message that is consistent with the findings of researchers in the United States (Ball, 1988b; Ball and McDiarmid, 1990; Kennedy 1990; McDiarmid and Wilson, 1990). It appears that although students do have the knowledge to make connections in many cases, they have not been shown how to think about making them or ways of linking their knowledge of mathematical topics together. ‘Teacher educators first need to realize this problem and then devise strategies to address the difficulty. One way to help begin the process of helping students develop an orientation to inquiry in their teaching might be to introduce the case study method in teacher preparation programs. Educators have recently begun to promote the case method of teaching as being particularly appropriate for teacher education (Shulman, 1986c; Kleinfeld, 1991; Kleinfeld, 19S| 1991); and, it are just mini- cases with a n with a framewc mathematics at shown that the paradoxes can the understanc teachers eXhik Hovshovitz-Hac this research alth°ugh somet of pedagogical did not expel-j as the int ervj hiQh level of explicate PUp 203 Kleinfeld, 1992; Goldman, 1991; Shulman, 1989; Merseth, 1991); and, in some sense the vignettes used in this study are just mini-cases. The development of good pedagogical cases with a mathematical basis could help provide students with a framework for discussion of both pedagogy and mathematics at the same time. Furthermore, research has shown that the cognitive dissonance induced by mathematical paradoxes can be a powerful way to change ideas and refine the understandings of mathematical concepts that preservice teachers exhibit (Movshovitz-Hadar and Hadass, 1989; Movshovitz-Hadar, 1991). Certainly the students involved in this research study found the interviews to be a useful, although sometimes challenging experience; and, the mixture of pedagogical thinking with mathematics was something they did not experience anywhere else. It was very clear to me as the interviewer that some of them were experiencing a high level of cognitive dissonance when attempting to explicate pupils’ difficulties. Although it was not always comfortable for them at the time, they did resolve the contradictions in their minds and went away with a much improved level of understanding. This st of primary : view of what exemplified also argued Prospective been studied connected kr recollllllendatj amount of ac improve teac in derlopir Althoug results of 1 Provide evic Programs an Connected v. Rudy's res Wficatio Situation: Pro I- . 9 ms 11: .ecommendat he . S\t 204 LookingfiBack This study began by arguing that content knowledge is of primary importance to teaching, particularly if the new view of what it means to learn and teach mathematics as exemplified in many reforms documents is to be realized. I also argued that although the subject matter knowledge of prospective secondary teachers is often assumed, it has not been studied in any concerted way; and, that with respect to connected knowledge very little was known. Furthermore, the recommendations by various groups that seek to increase the amount of academic study for intending teachers as a way to improve teaching practice assumes that such study is useful in developing the kind of knowledge that teaching requires. Although it would be dangerous to over-generalize the results of this study it seems fair to say that it does provide evidence that traditional university mathematics programs are not necessarily as effective in promoting a connected view of the subject as would be hoped. The study's results do not suggest that appropriate modifications to these programs would not improve the situation; and, it is the call to re-examine existing programs in light of these findings that is the most salient recommendation that follows from this work. The Standards, although an influential voice in the attempts to reform mathematics education have not been without thei many of the without citi research to they make wa He argues th carefully at teaching. Therefo: agenda ; generat; as they Precoll. emphasi: intellectual StUdentS ' an in the we needed for c the Cage Of SECOndary m a as- ~ allable’ i the neXt We: 5 address :tudy are Dr 205 without their critics. Silver (1988) has suggested that many of the recommendations of the Standards were made without citing research evidence and that in many cases research to confirm or contradict many of the claims that they make was simply not available when they were produced. He argues that the research community needs to look more carefully at mathematical thinking and how it influences teaching. Therefore, one item on the transformative research agenda is the examination of the nature of these more generative aspects of mathematical thinking, especially as they are represented in the intellectual activity of precollege students and their teachers (p. 341, emphasis added). This study addressed one aspect of mathematical thinking and intellectual activity in prospective teachers of precollege students, and it helps to provide credence to the assertion in the Standards that subject matter expertise, of the kind needed for connected teaching, cannot be assumed, even in the case of well prepared secondary candidates. Although the research reported here has provided a more detailed picture of the connected knowledge of prospective secondary mathematics teachers than was previously available, it still leaves many questions unanswered. In the next section, a retrospective look at the questions that were addressed as well as new ones that were raised by the study are presented. This 5 questions: have a com be teaching (b) Is the about pros; knowledge? questions j kind seem 1 answer. T1» Emassures 01 connections understandj connections generating the various Clan-1y de] 206 Looking Ahead This study was designed to address two primary research questions: (a) Do prospective secondary mathematics teachers have a connected understanding of the content that they will be teaching in typical high school mathematics classes; and, (b) Is the concept of connectedness a useful way of talking about prospective secondary teachers' mathematical knowledge? Looking back it seems that the answer to these questions is -- yes, sometimes. As always, studies of this kind seem to raise more questions than they definitively answer. The survey results do give some rough quantitative measures of how frequently students are able to make connections; and, the interviews have added to my understanding of the way in which students are able to make connections and the kinds of thinking that they often use in generating examples of connected knowledge. Furthermore, the various dimensions of connected knowledge are much more clearly delineated in my mind as a result of this research. Taken as a whole the results do answer the general questions that the study was designed to address. However, a number of new questions have developed over the course of this study and these are ones that would provide interesting avenues of inquiry for further research. The role that experience plays in the development of connected knowledge was raised as an open question by one of the students kind of rich have at his teaching exl kind of kno between var I started t facts than knowledge ! done 80? Can this is Curriculun their 11181 likElY to experienc backgrouh 207 the students in the interview sample. He believed that the kind of richly connected knowledge that he did not presently have at his fingertips would naturally come about through teaching experience. I myself know that I did not have the kind of knowledge about mathematics and the interconnections between various aspects of the subject that I now have when I started teaching, although I did know more mathematical facts then than I do now. Why did I develop this kind of knowledge when many of my colleagues do not seem to have done so? What factors contribute to this change over time? Can this way of thinking about mathematics be developed by curriculum materials that teachers use in the course of their instruction? Is the strongly prepared teacher more likely to develop this rich understanding through teaching experience than one who has a weaker mathematical background? One way to answer these questions would be to give this same survey to a group of experienced teachers and to follow this first wave of data collection with a similar set of interviews. The results could then be compared to the findings of the present study. The findings of this study in concert with the companion study of experienced teachers would provide valuable information for those designing teacher preparation programs. The role of experience versus university study in learning to teach has always been a contentious development Another knowledge of In this expe prospective pupils in th quite possib would. Anot teaching pra connected kn fragmented u in knowledge diffeI‘ences examine. If discourse, 1') knowledge so The role 208 contentious issue and the impact of each component on the development of the teacher is still problematic. Another question that could be pursued is how the knowledge of the teacher is actually used in the classroom. In this experiment all that I can report is what these prospective teachers told me they would do or say to the pupils in the scenarios. In a classroom situation, it is quite possible they would not react as they claimed they would. Another study could seek to examine differences in teaching practices between teachers with a strongly connected knowledge of mathematics and those with a more fragmented understanding. Whether or not these differences in knowledge would be translated into pedagogical differences is an open question that is important to examine. If differences are apparent in classroom discourse, how do these teachers transform this connected knowledge so that it does become accessible to pupils? The role that communication plays in the understanding and development of mathematical connections is another interesting question. Does articulating an idea improve one's understanding of it or not? Does this articulation produce a change in knowledge or just make explicit those tacit understandings that were previously present? The keeping of journals, although a common procedure with some groups, is rare with prospective secondary mathematics teachers. Research that would show the value of such an enterprise wc technique Wit These ar evolved over this work pr teacher know and how it, :' 209 enterprise would be the first step in the adoption of this technique with such students. These are but three of the many questions that have evolved over the course of doing this research. I hope that this work prompts researchers interested in the field of teacher knowledge to examine other aspects of the problem and how it impacts teaching practice. 3111111118.; 2 The results of this study imply that there is a gap between the kind of connected knowledge that appears desirable and the level of connected knowledge that prospective mathematics teachers exhibit. Although there does not appear to be a clear difference among students with varying amounts of mathematical preparation; beliefs, dispositions and attitudes do have a role to play. Furthermore, being required to communicate mathematical ideas seems to have an impact on connected knowledge and understanding. However, despite the fact that this study used two different data gathering strategies and non- traditional techniques for ascertaining prospective teachers' knowledge, finding out what people really know about a subject remains problematic. Given these implications it seems reasonable to suggest that if teachers are to develop a connected understanding of the subject. secondary 126 example, hor allow for s; narrowly foc elementary c of study cou high school are learning flexibility subjects tha Althoug] hard to prov POSSible mat universities 210 the subject, some changes in the way that prospective secondary teachers are prepared need to be made. For example, honours courses could be made more flexible to allow for specialists to study some courses which are not narrowly focused on technical detail; but, which consider elementary content from an advanced perspective. This kind of study could be focused on helping students to link their high school mathematics knowledge to the material that they are learning in university. Students also need sufficient flexibility in their programs to allow them to study other subjects that are mathematically related. Although there are many committed individuals who work hard to provide undergraduate students with the best possible mathematical experience, the reward structure of universities does not make such actions attractive. Professors who are interested in developing innovative courses and teaching methods need to be rewarded for these efforts otherwise they will remain the exception rather than the rule. Teacher education professors need to realize that there are strengths and weaknesses inherent in the knowledge of all the students they teach, regardless of mathematical background. Discussions of pedagogy should be integrated with content issues so that weaknesses can be at least partially addressed. Using cases that generate cognitive dissonance may be one way of having students develop more sophistica‘ often hold Furthe how experir how these i in daily c the develo interestin 211 sophisticated understandings of the naive concepts that they often hold, even after university study. Further research efforts might focus on understanding how experienced teachers are able to make connections and how these differences in connected knowledge are manifested in daily classroom practice. The role of communication in the development of connected knowledge is also an interesting question to pursue. APPENDICES APPENDIX A ‘ [_____ tandard Problem Solving Comnicat L‘ Reasoning " Algebra r.\ Functions \ SYIIthetic Tri gonome t] \ StatistiCS Pmbfibilit) P‘AthEmatic S calculus \ s tica % 212 Correlation of Standards and Connections in Items Other Real life Students University School Standard areas of applications and mathematics subjects math pedagogy Problem 18 6, 18 18 Solving Communication 2 13 Reasoning 1, 8, 9, 5 1, 4, 6, 20 20 rAlgebra 2, 7, 8 2, 7, 12 Functions 16 16 16 Synthetic 3, 17 17 3 Geometry Analytic 10 10 10 Geometry Trigonometry 14 Statistics 15 15 15 15 Probability Discrete 19 19 Mathematics Calculus 11 11 Mathematical 9, 12 13 Structure E== APPENDIX B 213 Mathematical Connections Project Pilot Survey This survey is part of a research project that I am carrying out to try to better understand the way in which teaching knowledge is different from other kinds of knowledge and understanding. As prospective teachers, you are being asked to help in this research by filling in this survey and also by considering participating in the second wave of data collection which will consist of a series of about 10 interviews over a period of several months. If you agree to participate in either of these aspects of the research your responses will remain completely confidential and you will not be able to be identified in any way in any published work that may come from the analysis of the data. Furthermore, you are free to decline the invitation to participate or to change your mind about participating at any time without any kind of penalty. I hope that you are willing to participate in the survey. If so, please sign in the appropriate place below and begin. You will find instructions on the page preceding the items. Please read them carefully before trying to answer any of the questions. If you would also be willing to be considered as an interviewee please sign in the appropriate place below. In signing below you do not commit yourself to a series of interviews but only agree that you are willing to be asked at a later date. If you do sign you will be contacted later and formally asked if you are still agreeable or you will be informed that you are not needed for interview purposes. Those people who are interviewed will receive a small gift as a token of appreciation. I hope that you are able to help with this important research. I agree to complete the survey I am willing to be interviewed 214 Demographic Information Remember that this information is for the eyes of the research director only. All responses are completely confidential. Name: Gender: M F Date of Birth: Day Month Year First Teaching Option: Second Teaching Option: Degree(s): University(ies): Year(s) received: Major: Program (check) Regular PEM Waterloo Number of full math courses "'taken as an undergraduate/graduate student: ** If you took half courses give the full course equivalent above. 215 Instructions This survey consists of five classroom scenarios, each of which requires some sort of response from the teacher. You are asked to imagine that you are the teacher and to figure out how you might respond to each situation. There are a total of 20 survey items being used in the research but you have only been given five so that you have ample time to think about what is being asked and how you might react. You have about 45 minutes so you can afford to spend about 9-10 minutes on each item. Please try to be as accurate and complete in articulating your responses as possible; and, if other thoughts occur to you as you go along please include them. This is not a typical 'right' or 'wrong' kind of situation. In some sense there are no correct or incorrect responses, rather, what matters is what you would think about and say in each of these circumstances. Remember also that in teaching you are always trying to connect your knowledge with other mathematics that the student already knows or with some other way of looking at the problem that they may be able to relate to. In fact it is the way that you make connections with other mathematical topics or other ways of thinking, representing or explaining these difficulties that is my primary interest in this research. Consider this point as you do the survey. Once again, thank you for helping with this work. 216 Item #1 The following conversation takes place between a bright student and a teacher in a grade 10 classroom. S: T: Yesterday you said that when solving inequations we could follow the same procedure as we had already learned for solving equations except that if we multiplied or divided by a negative number we had to change the direction of the inequality sign. Yes, that’s right. Well it doesn't work! What do you mean? Just out of interest I tried one different to the ones in the text. I tried to solve x? > 25 by finding the square root of both sides just like you do with equations. So what happened? Well I got that x > 5 or x > —5 just as in the case of an equation I would have got x = 5 or x = -5. Then I tried a number greater than 5 like 6 and it checks because (6)2 >25. But when I picked a number greater than -5 like -4 it doesn't work because (-4)2 < 25. What's going on? am... How would you respond to this student? 217 Item #2 You have taught the students to multiply two binomials by using the distributive property. Although most of the students seem to understand what you have taught, a couple just don’t seem to be able to understand how to go about it. How would you explain the problem (x + 7)(x + 3) to such a student? 218 Item #3(a) In a grade 10 geometry class you have introduced the students to deductive reasoning by using congruent triangles to prove a series of related properties about parallelograms. You have encouraged them to work in pairs so that they can discuss their ideas and you notice that there seems to be a dispute going on between two girls at the back. You go over to investigate. 81: That's not a valid proof! 82: Yes it is -- I've got 2 sides and an angle in both triangles so they must be congruent. 51: That's not good enough, the angle must be scrunched between the two sides to get congruent triangles. 82: What's the difference? They both notice you and look up. 82: Who's right? Th She is, it does have to be the contained side. 32: But why, what difference could it make? How would you respond? 219 Item 3(b) Assume that these same two girls are having the same argument later in class about whether the side must be contained if you have two angles and one side equal in two pairs of triangles. 82: Does the side have to be contained? it: Yes, just like the other one. 82 Why, I still don't see why it should make a difference? How would you respond? 220 Item #4 The following conversation takes place between a grade 11 student and her teacher: T: What is W 2 S: I know, that's easy, a + b. 1r: No, that's not right. S: Yes it is! an Could you explain please. S: Sure. You taught that when there are several operations exponentiation comes first and you also taught us that square root is the same as an exponent of 8 so I did the square root first. T: Hm... How would you respond to this student? 221 Item #5 The following question is in a homework exercise that you gave your grade 9 students: A woman drives from here to Toronto (a distance of 200 km) at a speed of 100 k/hr. She immediately turns around and drives back but because of the traffic she can only drive at 80 k/hr. What is her average speed for the whole trip? You notice that two students are arguing as they do their work and you go to see what the controversy is about. It turns out that they used different approaches and got two different results: Student #1: Average speed is (100 + 80)/2 = 90 k/hr. Student #2: Time to drive to Toronto is 200/100 = 2 hrs. Time to drive home is 200/80 = 2.5 hrs. Average speed = distance/time = 400/4.5 = 88.89 k/hr They want you to judge which solution is better. How would you respond? 222 Item #6 In an attempt to provide a "real world" problem for your students to illustrate the value of solving systems of equations you assign the following question: A manufacturer makes 4 wheel wagons and 2 wheel scooters. Both use the same wheels. Each also has a fancy crest with the manufacturer's name on it. He has lots of other parts but he only has 31 crests and 102 wheels left, and he wants to use them all. How many scooters can he make? Most of your students do a solution similar to this one based on two equations in two unknowns: s + w = 31 2s + 4w = 102 where g and w represent the number of scooters and wagons respectively. These equations produce the answer that there are 11 scooters. One student has a different approach: 102/2 = 51 51 - 31 = 20 Therefore he can make 11 scooters. How would you respond to such a solution? 223 Item #7 The teacher has taught the students to solve radical equations by squaring both sides and so on. One question in the exercise, x+2=.x+5+3 produces an extraneous root. The following conversation occurs between the teacher and a student: S: T: How come we get an answer that works but is wrong? What do you mean? Well it works if I substitute in after I have squared but not if I check in the original equation. How is this possible? Does this happen any other time? am .0. How would you respond? 224 Item #8 The following conversation takes place between a teacher and student in grade 9. S: T: It doesn't make sense! What's the matter? Well you just taught us that a0 = 1. Is there something that you didn't understand in my explanation? Well I thought I understood until I started to think about it. Explain what's confusing you. Well yesterday we learned that x5 meant that you had.4 x’s, x cubed meant that you had three x's and x squared meant that you had 2 x's didn't we? Yes. So if you have x? doesn't that mean that you have zero x's and so the answer should be zero? “.0. How would you respond? 225 Item #9 A student correctly solves the linear equation 5x — 2 = 12 + 7x and gets x -7. However when he comes to verify he does the following 5(-7) - 2 = 12 + 7(-7) -35 - 2 = 12 - 49 -37 = -37 Therefore x = -7 is the correct solution. The following conversation then transpires: 1r: You have done a good job of solving the equation but you haven't verified it correctly. S: What's wrong with it? 1r: Well you need to consider the left side and the right side separately. S: Why? 1r: Because you are assuming what you are trying to prove. S: Don't we often assume things in mathematics and then see what happens? In this case if I assume they're equal and if it's right I'll get them equal and I'll know my assumption was correct. On the other hand if I assume them to be equal and they aren't I'll get a statement at the end that is not true and I'll know that my original assumption was false. T: Hm... How would you respond? 226 Item #10 You have just introduced graphing in the x—y plane to a grade 9 general level math class. The following conversation ensues as you are teaching the lesson: S: This stuff sucks! T: Why do you say that? S: Well nobody would ever use this stuff for anything if they weren't a math teacher! How would you respond? 227 Item #11 You are teaching an introductory calculus class and you have taught the fact that the limit as mx-+ 0 of ay/ax is dy/dx, the derivative of y with respect to x. The following question is raised: S: T: In this case isn't y a function of x? Yes. And doesn't a little change in x produce a little change in y? True. Well if ax goes to zero doesn't ay go to zero as well? Well, yes but ... But we can't divide by zero so doesn't this mean that the derivative will be undefined? am... How would you respond to this series of questions? 228 Item #12 ‘A student simplifies the rational expression (x2 - 4)/(x + 2) by cancelling the x into x2 to get x and the 2 into the - 4 to get -2 for a final answer of x - 2. The following conversation ensues: S: I have an easier method than the one you showed us. T : What is that? S: well the one that I used on this question here ... T: I'm.afraid that isn't correct. S: I checked my answer in the back and it's right! T: Hm ... How would you respond? 229 Item #13 A student is helping to simplify an expression on the board that you are using as an example and she says that "the +3 and the -3 cancel out.” Later on another student says that "to simplify 3x/x you just cancel the x’s." A third student then raises his hand and says ”Our teacher last year told us that we should never use the word cancel and you haven't said anything to these other two about it. Is last year's teacher right or is it OK to use the word 'cancel'?” How would you respond? 230 Item #14 You have given the following question on a trigonometry assignment: In a aABC, BC = 10 cm, AC = 7 cm and AB = 30°. Find AA.accurate to the nearest degree. Two students do this question by using the sine law but one gets sin A = 0.7142857 and concludes AA = 46° while the other concludes that AA = 134°. They compare solutions and start to discuss who is right. After some arguments back and forth they call on you to settle the question. How would you respond? 231 Item #15 You have been using simulations of various games of chance to teach some basic probability concepts in a grade 11 class. You deliberately pick games that can be analyzed theoretically to provide the opportunity to link the experimental results with theory. As a further exercise you ask the students to figure out the probability that a thumbtack that has been tossed up lands point upwards. The students have a lot of fun with this and come to some conclusions. Then at the end of the class one of the students raises her hand and says "What's the correct probability for a thumbtack landing with the point up?” How would you respond? 232 Item #16 One of your students comes into class with a newspaper ad which advertises 36 inch pizzas for $18.95. He says that on the weekend he and some friends called for one and they were sent 2 18” pizzas for the same price. He isn’t sure if this is alright or not, how would you respond? 233 Item #17 One of your students comes into class one day a bit puzzled. You ask her what's the problem and the following conversation takes place: S: T: Well I went to the copy centre to do some photocopies for my History project and I wanted to enlarge some pictures. Yes, what was the difficulty? The sign said that I could get a 100% enlargement but when the technician set the machine to an enlargement factor of 1.41 I think I only got a 41% enlargement. Maybe I got ripped off. m ... How would you respond? 234 Item #18 One of your students works in a restaurant after school. He comes in one day puzzled. The following conversation ensues 3 S: T: S: What a night last night! Why what went wrong? Well this big company party was to be held but they didn't know how many people were coming and they didn't want anybody to have to sit alone either. SO? We tried to seat people in groups of 2, 3, 4, 5, and 6 but we always had 1 left over. Finally we found that 7 to a table worked. It sounds a bit confusing but I still don't see the problem. In all the confusion we forget to get a head count and I get a quarter per head bonus! Is there any way to figure out how many were there? How would you respond? 235 Item #19 One of your senior students has been playing with the square root key on her calculator and she noticed that no matter what number you start with, if you keep pressing the square root key you always get 1 eventually. Furthermore she claims that after the number gets below 1.1 all you have to do to get the square root is divide the decimal part in half. She wonders why this works this way. How would you respond? 236 Item #20 You have a very bright student in grade 11. One day he asks you the following series of questions: S: T: Is 1/3 = 0.33333... ? Yes. Exactly? Yes. That's a bit weird because that means 3(1/3) = 3(0.3333...) and that means that l = 0.999... . Is this true? am 0.. How would you respond? APPENDIX C 237 January 27, 1992 Dear : Earlier this year you filled in a survey which asked you to react to various teaching situations. The data (some 127 cases in all) has now been examined in some detail and we are ready to continue with the second stage of data collection. The second stage involves interviewing a smaller sample of students based on various different factors such as mathematical background, gender and so on. The sample, consequently, has not been chosen at random but purposefully. You indicated that you were willing to be interviewed and we would very much like to include you in our sample. At this time we estimate that you will be interviewed approximately 6 times, with each lasting approximately 1 hour. You do not need to do anything to prepare for these interviews so hopefully you will not find the time commitment to be excessive. In order to help me in scheduling could you please place an ”x” in each time slot where you presently have a lecture or tutorial and return this sheet to George at the end of class. The interviews will all take place at a mutually convenient time (not Friday at 5:00!) I will be in touch later on this week. Thanks in advance for helping with this project. Yours sincerely, Eric Wood —__<><1___u'the respondent: One of the key aspects of teaching mathematics at the secondary level is the ability to provide students with some sense of the connections between various mathematical ideas and principles. You are provided with a series of cards each of which contains mathematical topics, theorems, definitions etc. Please sort them into piles that contain information that you believe to be connected. Do not try to look for extremely tenuous connections, it is possible to link together much mathematics but it takes a huge amount of work —- this is not the intention here. Rather I want you to think about connections that you could and would make in a high school classroom. Do not assume that any of these topics are connected or that they are all connected -- just sort them as you see fit. After you are fiJnished we can talk a bit about what you have done. Give students time to sort through these cards and then go through the piles and discuss their logic or thinking about the various statements. Card 1 The pythagorean tzheorem: In a right angled triangle a2 + b2 = c2 . Card 2 The compound amount of P dollars invested at an interest rate of i% per interest period for n interest periods is given by A = P(l + 1)“. Card 3 To multiply two numbers the following procedure is followed: 1478 37 10346 4434 54686 Card 4 A population of bacteria will grow exponentially according to the formula y =P0 e“ where k is the rate of growth, P0 is the initial population and t is the elapsed time after the initial population measurement was taken. Card 5 To find an angle in a scalene trianzgle on can use the cosine law, cos A = (a2 c)/(- 2bc). ("I Card Card Card Card Card Card Card Card 6 10 ll 12 13 239 Multiplication of algebraic expressions uses the distributive property. To figure out the product (x + 3)(x + 4) you would compute x.x and x.4 followed by 3.x and 3.4. Then the answer would be x2‘+ 7x + 12. The area of a triangle is given by the formula A = 8 bh where b is the base and h is the height. The area of a circle is given by the formula A=nr . A rectangle has a perimeter of 400 metres. Find the maximum possible area of the rectangle. A function is defined by f(x) = x2. What kind of curve will it produce when graphed? When we write a a b we say that a is directly proportional to b. What does this mean? The limiting value of (x2 - 4)/(x - 2) as x approaches the value 2 is ...? The equation of a straight line through the origin is given by y = mx. APPENDIX E 240 Card Sort Task #2 There are many kinds of mathematical knowledge but one method of categorization could be between knowledge that is conventional and knowledge that is warranted. Conventional knowledge is agreed to by consensus just to make it convenient for everyone to use the same method for doing things, for example, the numbers to the right on the number line are positive and the numbers to the left are negative. Warranted knowledge is knowledge that has some logical, historical or commonsensical reason behind it and it would not make sense to change it. For each of the following rules, procedures or mathematical facts please put the cards in three piles, conventional, warranted or unsure. Afterwards we can talk a bit about what you thought. Give students time to sort through these cards and then go through the piles and discuss their logic or thinking about the various statements. Card 1. When multiplying two negative numbers the result is positive. Card 2. To divide fractions invert and multiply. Card 3. Sec 0 = hypotenuse/adjacent. Card 4. Anything raised to the power zero is 1. Card 5. Slope = Rise/Run. Card 6. The x-axis and the y-axis are perpendicular. Card 7. Tan 0 = opposite/adjacent. Card 8. Division by zero is always undefined. APPENDIX F 241 Card Sort Task #3 Mathematics is often taught as a series of rules or procedures to be followed. Each card contains a rule that you may be familiar with. Please sort the cards into two piles, those you perceive to be useful "rules of thumb" to teach students and those you don't. Give students time to think and sort the cards, then for each ask them why they think it is a good rule or a bad rule. Also try to ascertain whether they believe these rules to be correct or not as stated. Card 1. Two negatives make a positive. Card 2. Anything raised to the power 0 is 1. Card 3. When solving equations if you move something to the other side you have to change the sign. Card 4. When you multiply, the answer is larger than what you started with and when you divide the answer is smaller than what you started with. Card 5. To factor trinomials such as x2 + 3x + 2 just find two numbers that add to give the number in the middle and multiply to give the last number. Card 6. Division by zero is always undefined. Card 7. Multiplying entire radicals together is just like multiplying numbers except at the end you have to remember to stick in a radical sign. Please note: The order here is not the necessarily the Order in which the cards were presented, they were randomized each time they were given to the students to Sort. APPENDIX G 242 CODING SCHEME Connections C)! General connection ell—8T A connection with the student's way of thinking Chi—SUB A connection made with another school subject other than mathematics or alternatively with a real life example ell-MT]! A connection with another mathematical topic, correctly developed CN-NR Student is making a connection or is very close to making a connection but they do not appear to realize what the connection is or that they are on the verge of making it CN-WRG A connection is made but it does not seem to me to be accurate or appropriate to the topic under discussion cal-con A link is made with concrete materials or perhaps a diagrammatic approach cn-mm Example is explained by using a numerical case CN-DBHO A demonstration by the teacher CN-EXPL An explanation by the teacher Chi-JUST A justification or verification by the teacher C8-CXT Student uses contextual or situational clues as a way to establish linkages -- often very shallow perhaps just looking at surface features cN-CWTR Connection but uses a counterexample CBNC Close but no connection -- student appears to be on the track of making a connection but does not quite get to it and then has nothing further to add even when I give them one last chance to respond. This also refers to a missed connection when I hint rather broadly and they still don't see it 1‘131P Student essentially repeats what has been said in the description of the scenario -- no new mathematics or pedagogy is brought out 131313 Dead end in thinking without successfully resolving the conflict SESHntent Issues Ana PR: l"Loon Algorithm Notion of proof, what it means to prove something Flawed understanding of content in question HHIIJH! liflfli IhIJITR 1&1)! (ZIHRT fiflISINT azunur ‘CNJNTRA IIIJNE 243 Student resorts to rules in resolving the problem Student makes a mathematical error in his/her explanation Content is fundamentally about linear transformations Content is about limits broadly defined Student gives signs of having an "aha" experience where something suddenly occurs to them as a function of their own discovery or my questioning Student makes reference to their undergraduate mathematics experience as part of a discussion of an item Student makes reference to own high school mathematics experience as part of answer to a question Student appears certain about their knowledge of the concepts in question -- they need not be correct as long as the student gives evidence in tone or approach that they are confident about the question Student misinterprets the item and really doesn't figure out what the mathematics contained in it is Thoughts about mathematics as a discipline Thoughts about teaching mathematics or general philosophy about teaching Contradiction with beliefs as stated or ideas expressed earlier -- not a mathematical contradiction Comments about importance of history in study of mathematics ‘PedagogicalZCognitive Issues 8R2? “coo Talus Penn Reflection on own knowledge about the subject or way of thinking about the subject or teaching Metacognition, thinking about their own thinking processes Student makes reference to his or her teaching experience as part of their discussion of a particular item Pedagogical reflection -- thinking about ways of teaching this topic or problems that students might have with it Appears that they know something but their ideas are poorly articulated -- I think I might understand what they mean but a student at this level certainly would not -- this a tough judgment call! 244 AffectiveLcher Categories UNCENME BBDG» BKOFT' NERV' Student appears to be uneasy with the interview item Student is exhibiting uncertainty about their answer either in words or in body language, tone of voice, stumbling speech and so on ... Hedging because they are not completely sure about the answer that they have given and they don't want to be wrong or simply going around in exploratory circles without really getting anywhere Me reassuring them about the questions so that they don't lose confidence and/or interest in the project I back off on a question because the student seems anxious about the direction things are going Student appears nervous ProcessZMethodological Issues “PUSH! HELP’ "CLAIR HBXPE. BIG» DIRK; Me pushing an idea because they have not completely explained it or because I think they may know more than they are able to articulate I give them some help to get them out of a dead end or similar situation I clarify something to the student I explain something to the student Background information about the project Need to look up diagrams in the interview envelope to see what is being said in the interview -- this is a notation to me Lew Categories Created for Card Sorts CKHWVEQI "ARR Tues "TEACH Goon Ban Chem: OTHER ERIE: Conventional knowledge Warranted knowledge Would teach this rule of thumb Would not teach this rule of thumb Good rule Bad rule Changed mind about previous answer Something else other than a defined category but not frequent enough to define a separate code Something special to note APPENDIX H 245 Lynn Allan Interview #3 February 20, 1992 Eric: Lynn: Eric: Lynn: Eric: Lynn: Eric: Lynn: Eric : Lynn: We're on. Okay there's the first one ... pen, paper I had this one earlier ... Oh, did you, on the survey? Well on the survey yeah and I realized what I said wrong then now. I have no idea because I don’t remember what everybody said. Actually I have no idea what the survey’s individual people said because I coded them with numbers and then took them all apart so it was completely anonymous so ... For some reason I just ... well I wrote this great big long page and then went oh stupid! [laughs] Um I guess the first thing that I'd stress about this is that you have to think about ... like you have to think about the numerator and denominator having brackets like they're a whole term. So you just can't start cancelling this way ... What is it that tells you the brackets? That tells you? I guess like the fraction symbol with the division separates the numerator and the denominator so you have to consider whatever's on top as one full term and whatever's on the bottom. They're not really written but they should be implied. Okay. Um ... the other thing the way they cancelled they said that ... like x goes into itself and just leaves an x there, well they forgot there's a 1. Like if you're going to do x +x you've got a 1 in the denominator so even following their reasoning they're not ... not leaving these 1’s. Like if I agreed that you could ... that you could cancel this term this term with this term which you can't but to try to go in the middle with them saying if that worked then 4 or 2 would go into 4 twice 2 would go into 2 once, Eric: Lynn: Eric : Lynn: Eric: Lynn: Eric: Eric: Eric : Lynn: Eric: Lynn: Eric: Lynn: 246 you’ve still got this 1+1 on the bottom that they seem to just forget about. So you've got that problem. What if they said okay the answer should be x-2/2? Okay then I could go to numbers. I think would be easiest. So if we started with 32-4/3+2 ... now if you do it just doing numerator and then denominator so this would be 9-4 is a 5/5 which is just a 1 and that's just doing the numerator doing the denominator. If you want to go their way this would be 3-2/1+1 so it would be 1/2 so this number using their method and the number you get doing whole numerator whole denominator are very different. What if they ... they went back to their original method you know just saying well just 3 into 32 and 2 into 4 Then they'd get this 1 ... Which is ... Because they would have 3-2 and then they completely forgot about the denominator and so they would have the 3-2 which is 1. Um hm. So would that convince me do you think that that method was wrong? Ah ... not exactly because you could do the exact strict method -- that's the problem I had when I first saw this plugging in any numbers if you use this strict method you'll always get the same thing. And when you say using this strict method what do you mean? Well cancelling ... like cancelling the 3 ... or like common factor 3 out and then forget that you have a 1 there. So that would always work? Any of the ones I tried did ... Or what ... can you remember what kind you tried? ... but that was eliminating the2 denominator. Um even just of this formsomething2 - 4 over that something + 2. Eric : Igynnn: lazrjxn Lynn : IEzrixz: Icyranu Er:i<:: IEYIIII: Eric: LWHQII: Eric; Lynn : Eric 3 Lynn : 247 What about things of other forms? I’m sure you could find a fraction that it wouldn't work. If you did ... oh 6+25 over 3+5 or something like that that should ... Now their method would say 3 goes twice so it's 2+5 ... Um hm ... ... so they'd get an answer of 7. Now if you just make this 31/8, 31/8 doesn't equal 7. Okay. So if you go to ... I guess this set up with the squared I guess because of the way it factors it's working ... in this case. So is this the exception or is this the rule. It's ... well it's the exception for sure. Alright can you think of any other way of convincing me? You see I guess the problem I would have as a student is let' s say you wrote down an example like x2+9/x+3. So I' d say well that' 3 easy I can do that that’ s x+3 and you'd say no no that's not right because I can't ... I can't factor the numerator. But you see I would say but I have no way of knowing that your method is better than my method and my method gives me an answer and your method you can't even do the question. Right. So have would I now know that my method's wrong. If this divided by this does equal this then you should be able to ... to expand like figure that as a denominator one you should be able to cross multiply and get an equation still. If this is true then you should be able to go from here and get a left side equals right side. If you do (x+3)(x+3) you' re going to get x +6x+9 and then in this other side you 'd have ){2+9 so this really doesn' t work ... Okay ... ... and that was doing again like last time doing everything mathematically correct and getting ... and getting an impossibility. Eric : ijflln: ZEJrjnc: lxywirl: Erti