« ulilo n... I‘z-‘Il ' 3...! It... .7311? 20."! .. . .. 43h“. .r.flufi(fl.svflt.). tun/'48P“. . H 15.. ct cl... . 1:... .L «r27... . Elna}... {gift 13.13: .1 .1535»?! {cf-Juli; : V 5.... ll...o.(l.es~ls_.¢ 1’; of! .5. J; S... 1. 1.31.9? {no “whiluv'... vb In. I... 7‘rs.§p¢v V..- in“?! . 2. ill!“ . c I. at! a... . 5).... .. i0:- . 001.01 “14!. r 9",...iv3tl {v Kin-.53.”. [313.9914 5 l gaff!) 3 H I. Culle' 1‘1 260.. IV (a . .lf .2... “fr! 15‘ .1355 L. it’ll fl :4 . lQVniu all}: . I32 :‘iE‘SlS STATE U NERSITY LIBRARIES “Milli Illlllllllllllllllllll.lIn 3 1293 00794 98 This is to certify that the dissertation entitled Learning Mathematics to Teach: What Students Learn about Mathematical Content and Reasoning in a Concentually Oriented Mathematics Course presented by Pamela Wallin Schram has been accepted towards fulfillment of the requirements for Ph. D. mgmemTeacher Education Major professor 7" a"? Date QY 3'/’-’ MS U is an Affirmative Action/Eq ual Opportunity Institution 0-12771 LIIRARY "lotus-n State University PLACE IN RETURN BOX to remove thls checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE 5.79276“; “M“. , A 40.99 ,l.;t'uh""‘ ll W9 LU ‘ LW ELllofielfil—iz ‘— iz : ‘I nw ‘ .e-A W~ I!l~. \‘l and . \ w \ MSU Is An Affirmative Action/Equal Opportunity lnetltution . eman-p LEARNING MATHEMATICS TO TEACH: WHAT STUDENTS LEARN ABOUT MATHEMATICAL CONTENT AND REASONING IN A CONCEPTU ALLY ORIENTED MATHEMATICS COURSE By Pamela Wallin Schram A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Teacher Education 1992 ABSTRACT LEARNING MATHEMATICS TO TEACH: WHAT STUDENTS LEARN ABOUT MATHEMATTCAL CONTENT AND REASONING IN A CONCEPTUALLY ORIENTED MATHEMATTCS COURSE By Pamela Wallin Schram Leaders in mathematics education are calling for major reforms in what is valued in mathematics curriculum and instruction--in the roles of teachers and students and in the culture of the classroom. The knowledge, skills and dispositions necessary to teach in this manner require a significantly different experience with mathematics than the traditional K—12 mathematics experiences that most prospective teachers bring to their university coursework. Teacher education must confront students' deeply rooted ideas about mathematics and about teaching and learning mathematics. This study investigated what a group of six prospective elementary teachers came to understand about mathematical content and reasoning in a conceptually oriented mathematics course. Sources of data included four student interviews, two questionnaires, twenty-one classroom observations, and three interviews with the mathematics instructor. Interviews explored what students learned about particular number theory content, relationships, mathematical ways of thinking and problem solving. Findings from the study included: What prospective teachers learn about mathematical content is impacted by many interrelated factors-~prior experience; views about mathematics; patterns of mathematical thinking and problem solving; flexibility in using mathematical knowledge; and habits of reflection about the mathematics one knows and about oneself as a learner. Students often possessed the knowledge needed to solve problems but did not recognize and / or appreciate the power of the ideas they possessed. Many of the students seemed unable to analyze a problem and think about pertinent information that might be helpful in solving the problem or to recognize the relationship between that problem and other mathematical knowledge and understanding in their possession. The process of changing mathematical ways of thinking and dispositions is complex and occurs across time. A single mathematics course is not enough to undo years of mathematics learning; however, a conceptually oriented mathematics course can challenge students' ways of thinking and patterns of reasoning. Students can become aware that mathematics has meaning. Strategies used by students can change from a more technical to a more reasoning orientation. Copyright by IUALJEIJA‘VWAJJLHH’SCITFUARJ 1992 This work is dedicated with love and appreciation to Ken and my dad and in memOry of my mother. ACKNOWLEDGMENTS In one sense this has been the loneliest experience in my life yet, there are numerous colleagues, friends and family who contributed to the completion of this work and deserve a special thank you. To my husband, Ken: Thank you for your patience, support, love, and nurturing throughout the many years of this work. Your constant support, understanding, and faith enabled me to complete this study and maintain some degree of sanity. The work really was "temporary." To my step-sons, Jason and Tim: Thank you for adding joy to my life. For your patience and understanding, for respecting my need for solitude, and for taking wonderful messages, a special and loving thank you. To Ken and Jean Schram: Thank you for supporting my work and making me feel so comfortable and welcome in the Schram family. To my mom and dad, Kathleen and I. D. Wallin: Early in my life you encouraged me to believe that I could do anything that I wanted to do. Thanks dad for your enduring love, confidence, and support. I regret that my mom is not alive but her loving spirit resides within me and continues to guide my growth and development. To my special brother, Ronnie Wallin, wife, Judy, and three extraordinary nephews, Charlie, Michael, and Eric: Thanks for supporting my crazy ideas and always being there when I need you. Also thanks to my wonderful grandmother who encouraged and tried to understand my work. vi v11 To my committee members, each of whom has made a significant impact on my growth and learning as a professional, thank you: Doug Campbell, who taught me ways to observe and question. Susan Melnick, who helped me develop a historical perspective about teachers, teaching, and the teacher's workplace. Perry Lanier, who has provided guidance throughout my time here and encouraged me to stretch intellectually and professionally-- leading to greater challenges and responsibility. Glenda Lappan, my friend and mentor, words cannot adequately express all of the ways you have contributed to my professional and personal growth. In sickness and in health, in East Lansing and in Washington, in your office or in your home, in happy times and sad, you have continued to nurture and support my work. To my colleagues and friends—Kathy Roth, Cheryl Rosaen, Deborah Ball, Sandy Wilcox, Julie Ricks, Steve Kirsner, Trudy Sykes, Margo Stoddard, Patty Noell, Bev Anderson, and Karen Sands -- each of whom provided valuable support in unique ways, thank you. A special thanks to Michelle Parker, my dissertation accomplice and friend. Thank you to the teachers at Elliott - MSU PDS for all of your support. A special note of appreciation to Ramona Berkey who gave me the gifts of "clarity" and "energy." To the teachers and colleagues who helped shape my thinking about mathematics-Janet Hall, Emmett Sams, Ralph DeVane, Charles Boone, Bob Jones, and Barbara Smith and to the many other NC friends who supported and encouraged me—Ivan Randolph, Beverly Wyatt, Barbara Loftin, Jayne Brown, Dennis Hine, Kay Boone, and Melanie Chapman, thank you. To Kara Suzuka and Stephanie Grant who provided invaluable assistance with technical aspects of this work. I greatly appreciate all of the time and energy you devoted to making this work more appealing and acceptable. Thanks to the many transcribers who worked on the interview v111 tapes. Finally, a special thank you to the six participants who gave many hours and much thought to trying to help me understand what they were learning. It was difficult to bring closure to this piece of work. The following quotation from Schatzman and Strass (1973) proved helpful: Having written his final report on his work, the researcher makes a commitment to the validity of the reality he created. He will have to stand by it even though later he will probably change as will his conception of the reality he researched. But if he understands this, then he can also see final closure to his current work, not as an end, but as a single bench mark in an intellectual career course (p. 137). TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES riv YV CHAPTER 1 LEARNING MATHEIfiATICS TO TEACH: DIFFERENT LENS FOCUSED ON PROSPECITVE ELEMENTARY TEACHERS Context for the Research Questions What Prospective Teachers Bring to University Coursework Beliefs about mathematics Prior experience Students' perceptions of formal teacher education coursework Implications for reform in teacher education 10 The Reform Vision 12 Changes in society 12 Growth in knowledge about how students learn 14 Technological advances and the impact on mathematics 16 Implication for mathematics education 19 Subject Matter Knowledge Needed to Teach 76 Research in Mathematics Education - - Problem solving 30 ix Community and discourse Conceptual and procedural knowledge Historical Perspective 1950s and 608: Modern mathematics 1970s: Individualized instruction and "back to basics” 43 1980s and 908: Analysis and reform 47 CHAPTER 2 THE STUDY: PURPOSE, DESIGN, AND ANALYSIS ‘8 Purpose ‘8 Design Cohort profile U1 lb) 33 Datasouroes Analysis 53 Analysis by student Analysis by question Developing a conceptual framework Analysis across participants Analysis of the Math 201 course Subjectivitiy of the Researcher CHAPTER3 MATHENIATICS FOR PROSPECTIVE ELEMENTARY TEACHERS: CREATING AN ENVIRONMENT FOR THE DEVELOPMENT OF MATHENIATICAL POWER Description of Math 201 xi Mathematical and pedagogical decisions R7 Number theory and prospective elementary teachers 90 Math 201 Course Analysis 100 Tasks 101 Discourse 109 Environment -- - 119 Summary 129 CHAPTER 4 LEARNING MATHEMATTCS TO TEACH: WHAT STUDENTS LEARNED ABOUT PARTICULAR NUMBER THEORY CONTENT, RELATIONSHIPS, MATHEMATICAL WAYS 0F THINKING AND PROBLEM SOLVING 131 Students' Views About Mathematics 133 Patterns of Reasoning and Strategies 139 Kim 142 Jason 146 Tim 150 Tamara 151 Linda -- ...... 154 Andrea ...... 156 Knowledge About Number Theory Concepts 163 Factors 163 Multiples 170 Primes and composites - 173 Summary 181 CHAPTER 5 LEARNING MATHEMATICS TO TEACH: RECOGNIZING AND APPRECIATING THE POWER OF VARIOUS MATHEMATICAL IDEAS 183 Flexibility in Using Number Theory Ideas 187 Structure of number ...... 188 Relationships 221 Summary -131 CHAPTER 6 WHAT PROSPECTIVE ELEMENTARY TEACHERS LEARN ABOUT MATHEMATICAL CONTENT AND REASONING IN A CONCEPTUALLY ORIENTED MATHEMATICS COURSE 936 Purpose of the Study 916 Summary and Discussion of Main Findings--- 7'37 Prior experience 239 Views about mathematics 240 Patterns of mathematical thinking and problem solving 942 Flexibility in using mathematical knowledge... 244 Habits of reflection 350 Time and Complexity of Change 253 Unique opportunity for additional interview---- 2. Patterns of mathematical thinking and problem solving 2'15 Views about mathematics 262 Flexibility in using mathematical knowledge .................. 266 Habits of reflection ...267 Time and complexity 269 xiii Number of Zeros Question Revisited. 270 Summary of Research Questions 279 Significance and Implications - 282 APPENDIX A CODING CATEGORIES: REASONING AND TECHNICAL STRATEGIES 288 APPENDIX B A AND B INTERVIEWS 295 APPENDIX C MIDDLE SCHOOL TEXTBOOK ANALYSIS 313 LIST OF REFERENCES 318 Table 2.1 Interview Matrix ....... LIST OF TABLES xiv ................ 61 LIST OF FIGURES Figure 2.1. Number Theory Concept Map 58 Figure 4.1. Kim: Percentages of Technical / Reasoning Strategies 143 Figure 4.2. Jason: Percentages of Technical / Reasoning Strategies 147 Figure 4.3. Tim: Percentages of Technical/ Reasoning Strategies 150 Figure 4.4. Tamara: Percentages of Technical/ Reasoning Strategies 152 Figure 4.5. Linda: Percentages of Technical / Reasoning Strategies 155 Figure 4.6. Andrea: Percentages of Technical/ Reasoning Strategies - 157 Figure 4.7. Factors Interview Questions 164 Figure 4.8. Participants' Strategies for Factors Questions - 165 Figure 4.9. Multiples Interview Questions 170 Figure 4.10. Participants' Strategies for Multiples Questions 171 Figure 4.11. Primes and Composites Interview Questions - -174 Figure 4.12. Participants' Strategies for Primes and Composites Questions ...... 174 Figure 5. 1. Multiplicative and Additive Structure Interview Questions 189 Figure 5.2. Participants' Strategies for Multiplicative and Additive Structure Questions ............. 190 Figure 5.3. Structure of Numbers in Unfamiliar Contexts Interview Questions ............................ .205 Figure 5.4. Participants' Strategies for Structure of Numbers in Unfamiliar Contexts Questions 205 Figure 5.5. Relationships Interview Questions 273 Figure 5.6. Participants' Strategies for Relationships Questions 794 XV xvi Figure 6.1. Framework: What Students learn About Mathematical Content 7- - Figure 6.2. Jason: Percentages of Technical] Reasoning Strategies 1988 and 1989 956 Figure 6.3. Linda: Percentages of Technical/ Reasoning Strategies 1988 and 1989 257 Figure 6.4. Tamara: Percentages of Technical/ Reasoning Strategies 1988 and 1989 .................... .258 Figure 6.5. Tim: Percentages of Technical/ Reasoning Strategies 1988 and 1989 959 Figure 6.6. Andrea: Percentages of Technical/ Reasoning Strategies 1988 and 1989 .................... .260 Figure 6.7. Kim: Percentages of Technical/ Reasoning Strategies 1988 and 1989 261 Figure 3.1. Grid Paper Rectangular Shapes 797 Figure 3.2. Graph of GCF of 6 and 1-18 798 Figure 3.3. Graph of GCF of 5 and 1-18 298 CHAPTER 1 LEARNING MATHEMATICS TO TEACH: DIFFERENT LENS FOCUSED ON PROSPECTTVE ELEMENTARY TEACHERS Too many Americans seem to believe that it does not really matter whether or not one learns mathematics. Only in America do adults openly proclaim their ignorance of mathematics ("I never was very good at math") as if it were some sort of merit badge (National Research Council, 1989, p. 76). Changing this perception requires changes in thinking about teaching and learning mathematics. Changes in what it means to know mathematics and what it means to do mathematics are required. Thinking and practice related to the education of prospective teachers must be redefined. "Learning mathematics to teach" takes on a different persona. This study investigated what a group of six students learned about the mathematical content in a mathematics course for prospective elementary teachers. The course was designed to reflect the changes in teaching and learning of mathematics called for by the recent reform documents (e.g., National Council of Teachers of Mathematics, 1989, 1991; National Research Council, 1989). The main question driving the study was: What do prospective elementary teachers learn about mathematical content and reasoning in a conceptually oriented mathematics course? Other questions include: In what ways and to what extent do students understand the mathematical content presented in Math 201? What strategies and patterns of reasoning do students use when solving problems related to the mathematical content in Math 201? In what ways and to what extent do 2 students apply their knowledge of this mathematical content to solve problems or create new mathematical knowledge? The proposed reform changes in teaching and learning mathematics represent a dramatic shift from traditional mathematics instruction. Mathematics education has a history of less than successful reform efforts in its past. The history of teacher education also is important as efforts are made to change mathematics education. There are many reasons that support such a significant change in mathematics education including changes in society, growth in knowledge about how students learn and a technological explosion that has changed what is valued in the mathematics curriculum. This chapter is divided into six major sections. The first section provides a brief description for the context of the research questions. The second section focuses on three aspects of what prospective teachers bring to university coursework: beliefs about mathematics; prior experiences as learners of mathematics and as observers of teachers of mathematics; and perception of formal teacher education coursework. The third section explores three themes of the current reform efforts: changes in society; growth in knowledge about how students learn; and technological advances and the impact on mathematics. The fourth section focuses on the kind of subject matter knowledge (i.e., mathematics) prospective elementary teachers need to teach. The questions driving some of the major areas of research in mathematics education are problem solving; community and discourse; and conceptual and procedural knowledge. A review of the research related to these three topics is described in section five. The last section contains a brief historical review of reform in mathematics education: 1950 and 60s, modern mathematics; 1970s, back to the basics and individualized instruction; and 1980 and 905, analysis and reform. 3 Context for the Research Questions Leaders in mathematics education are calling for major reforms in mathematics instruction (Mathematical Association of America, 1991; National Council of Teachers of Mathematics (NCTM), 1989, 1991 ; National Research Council (NRC), 1989). Recommendations include substantial changes in the curriculum, moving from a focus on mechanics and computation to an emphasis on sense-making and reasoning (e.g. Burns, 1986; Cobb, 1989; Madsen-Nason & Lanier, 1986; Mathematical Sciences Education Board, 1990; Schoenfeld, 1987). Advocates argue for focusing on a wider range of mathematics content-including statistics and probability, for example-and to do so with a greater emphasis on mathematical reasoning. Moreover, beyond topics and presentation, the reforms imply a dramatic shift in instruction—in the roles of teachers and students, and in the culture of the classroom (NCTM, 1991). A typical mathematics class begins with checking the previous day's assignment. Troublesome problems are worked by the teacher or a student. Then the teacher briefly explains the next piece of material and the remainder of the time is spent with students working independently on the next assignment (Conference Board of Mathematical Sciences, 1975; Davis 8: Hersh, 1981; Peterson, 1988; Stodolsky, 1988; Welch, 1978). The images portrayed by this profile of mathematics instruction is not consistent with the ”vision" described by reform advocates. One reform advocate argues, "The traditional situation described is organized, routine, controlled and predictable—an unlikely environment for the creation of knowledge." (Romberg, 1988b, p. 10). Traditional mathematics instruction has promoted a narrow view of mathematics as a static discipline consisting primarily of arithmetic skills and 4 procedures in which success is measure by technical competence. This type of mathematics instruction is not conducive to the development of an inquiry orientation to learning and doing mathematics advocated by current reform efforts. A significant change is required in the way in which mathematics is organized and taught (NRC, 1989; NCTM, 1989, 1991; Romberg, 1988a). Teaching mathematics in a way that is consistent with reform recommendations depends on teachers' own understandings and conceptions of mathematics (e.g., Carpenter, Fennema, Peterson, & Carey, 1989; Cohen 8: Ball, 1990; Thompson, 1984). Most prospective teachers come to university mathematics courses with deeply rooted ideas about teaching and learning mathematics largely shaped by their own experiences in mathematics classes that offered traditional instruction. Because the knowledge, skills and dispositions necessary to teach in the manner advocated by reformers require a significantly different experience with mathematics than the traditional K- 12 mathematics experiences that most prospective teachers bring to their university coursework, teacher education needs to change as well (see, for example, Ball 8t McDiarmid, 1990; Schram, Wilcox, Lanier, & Lappan, 1988; Tirosh & Graeber, 1990). Most elementary teachers can solve the computational problems presented in K-6 textbooks, but this level of knowledge is not sufficient to teach K-6 mathematics to children in ways that are consistent with reform efforts. Teachers should understand why problems can be solved in various ways. Teachers also need to be able to comprehend how various mathematical concepts relate to the larger field of mathematics (For further elaboration of the significance of the "bigger picture" in mathematics see e.g. Steinberg, Haymore, 8: Marks, 1985). If a teacher understands how individual concepts relate to one another then he/ she is better able to provide 5 experiences that enable students to construct appropriate links between prior knowledge and new concepts. The proposed reforms in mathematics education are concerned with what mathematics citizens will need to be successful in the Information Age. Preservice teachers have logged many hours of experiences in traditional mathematics instruction which developed in an industrial society. The focus of this instruction was computational and procedural. Preservice teachers expect future mathematical experiences to fit their beliefs about what it means to know and do mathematics. Reform efforts that argue for significant changes in K-12 mathematics teaching and learning extend to teacher education. Authors of one reform document argue: Teachers themselves need experience in doing mathematics - in exploring, guessing, testing, estimating, arguing, and proving - in order to develop confidence that they can respond constructively to unexpected conjectures that emerge as students follow their own paths in approaching mathematical problems. Too often, mathematics teachers are afraid that someone will ask a question that they cannot answer. Insecurity breeds rigidity, the antithesis of mathematical power. (NRC, 1989). Teacher educators must confront the deeply rooted ideas about teaching and learning mathematics that prospective teachers bring to their professional studies. What Prospective Teachers Bring to University Coursework Beliefs about mathematics. It is difficult to distinguish between beliefs about mathematics and knowledge about mathematics. In recent years research on teachers’ mathematical beliefs and in particular the interaction 6 between beliefs and teaching practice has received increased attention (see e.g., Brown, 1985; Cooney, 1985; Dougherty, 1990; Peterson, Fennema, Carpenter, 6: Loef, 1989; Shaw, 1989; Thompson, 1984). If one thinks about a continuum for beliefs about mathematics as a discipline, on one end are beliefs about mathematics as a static, technical, black and white discipline and on the other end are beliefs that mathematics is a dynamic, creative, and uncertain discipline. The former view emphasizes mastery of symbol manipulation while the latter focuses on the process of doing mathematics. Researchers have categorized these extremes in various ways. Skemp (1978) refers to these views as "instrumental mathematics" and "relational mathematics." He argues that the differences in teachers' underlying views of mathematics are fundamental to what guides mathematics teaching in a teacher’s classroom. Ernest (1988) identifies three categories—"the Platonist view" (similar to the static, technical view), "the problem-solving view" (similar to the dynamic, creative view), and "the instrumentalist view or mathematics as a bag of tools" (p. 10) to be used to solve mathematics problems. Regardless of the "label" to describe the particular belief system held by a teacher, there is wide-spread agreement that teachers' belief systems influence their mathematics instruction (Cooney, 1985; Hersh, 1986; Hoffman, 1989; Peterson, et a1. 1989; Thompson, 1984). Beliefs about mathematics also influence learning mathematics (see e.g., Resnick, 1988). If one believes that mathematics is a dynamic discipline, learning is a process of sense-making. Resnick (1988) argues that Students who understand mathematics as a domain that invites interpretation and meaning construction are those most likely to become flexible and inventive mathematical problem solvers (p. 33). 7 Prospective teachers' beliefs about mathematics teaching and learning emerge as a result of their schooling experiences as students (see e.g., Ball, 1988b; Bush, 1983; Lampert, 1990; Thompson, 1985). Prior experience. Imagine six hours a day, 180 days a year, for thirteen years (14,040 hours). That is the approximate amount of classroom hours logged by a student by the time he/ she graduates from high school. It is not surprising that the process of schooling is one of the few topics about which the general public feels confident and knowledgeable to discuss, debate, and evaluate. Upon closer examination it is apparent that the intricate facets which combine to form the organization called "school" are much more complex than the experiences as students would reveal. As a subset of the general population, preservice elementary teachers have experienced an equal number of hours in schools. Lortie (1975) describes these prior experiences as an "apprenticeship of observation." He suggests that these prior experiences influence prospective teachers in ways they do not perceive. These many hours of previous school experiences were viewed from a naive student's perspective. Lortie argues that students are unable to "place teachers' actions within a pedagogical framework. What students learn about teaching is intuitive and imitative rather than explicit and analytical" (p. 62). Sarason and Klaber (1986) describes the difficulty involved when it becomes necessary for preservice teachers to "unlearn" attitudes acquired in their years of experiences as students. Ball (1988b) extends this notion to "unlearning" mathematics to teach. Feiman-Nemser (1983) argues that prospective teachers should gain "cognitive control" over these prior school experiences. Dewey (1904) suggests that a student's own direct and personal experience is the "greatest asset in the student's possession" (p. 153). He calls for helping prospective 8 teachers to place these experiences in light of theory and principles that are included in teacher preparation courses. Formal preparation of teachers should not ignore these early informal influences. Indeed teacher education must seek ways to challenge students to examine the beliefs developed during previous school experiences. Feiman-Nemser (1983) summarizes the importance of considering prior experiences when planning preservice courses: The likelihood that professional study will affect what powerful early experiences have inscribed on the mind and emotions will depend on its power to cultivate images of the possible and desirable and to forge commitments to make those images a reality (p. 154). The literature on the impact of professional study on teachers' beliefs points to the difficulty in overcoming ingrained notions developed during previous school experiences (Ball, 1988b; Feiman-Nemser, 1983; Tabachnick, Popkewitz, 8: Zeichner, 1979-80; Zeichner, Tabachnick, 8: Densmore, 1987). Students' perceptions of formal teacher education coursework. Consideration for the tremendous number of years of "apprenticeship of observation" provides support for studies that reveal preservice teachers' confidence in their ability to teach prior to formal preparation (e.g., Book, Byers, 8: Freeman, 1983; Freeman 8: Kalaian, 1989; Fotiu, Freeman, 8: West, 1985). Book, et a1. (1983) found that almost 90 percent of a group of entering preservice teachers exhibited moderate to high degrees of confidence in their ability to teach. Lortie (1975) suggests that the ease of entry into the field of teaching as well as the ease with which students can complete preservice education signals to students that little new knowledge is required to be a good teacher. 9 Many students appear to view teaching as a continuation of parenting rather than an intellectual pursuit. Lanier and Little (1986) suggest several factors that contribute to this belief including students' personal experiences as camp counselors, church school teachers, and volunteer student aides. Prospective teachers have limited expectations for formal coursework to contribute to their knowledge of teaching. Lortie's study of practicing teachers revealed similar feelings among practicing teachers. The only component of formal teacher education that teachers valued was the student teaching experience. Students view a knowledge base as an unnecessary component to preparation to teach. There is evidence (Finkelstein, 1982; Gage, 1978, 1985; Lanier 8: Little, 1986; Turner, 1975; Warren, 1982) that students strongly desire techniques—"What to do and how to do it" and that some teacher education programs try to comply. As a result of these factors, some teacher preparation programs have become more technically oriented. The issue is, of course, much more complex (e.g., the historical development of schooling, societal demands on schooling, characteristics and background of the typical teacher education faculty). Traditional mathematics instruction has embedded the image of teaching mathematics as a technical (how-to) model. Prospective elementary teachers have experienced mathematics as a series of rules and algorithms which must somehow be "memorized" and filed for future reference. These experiences provide a view of mathematics as an abstract, mechanical, and meaningless series of symbols and rules. Mathematics courses for prospective elementary and middle school teachers should attempt to offer alternative ways of thinking about learning and teaching mathematics. 10 The changes recommended for mathematics instruction must include changes in teacher education. Feiman-Nemser, McDiarmid, Melnick, and Parker (1989) describe the strong notions that prospective teachers bring to their university coursework: Spelling tests and reading groups, workbooks and recess, raised hands and reprimands are typical details in the picture of teaching that most students come with. These strong and enduring notions about teaching constitute a lens through which teacher education students perceive and interpret the preservice curriculum. Consequently their learning during teacher preparation is an interaction between the conceptions they bring and the knowledge and experiences they encounter (p. 1). Implications for reform in teacher education. In 1984 Goodlad argued that "teacher education programs are disturbingly alike and almost uniformly inadequate. The conventions to be broken and the traditions to be overcome in developing the needed programs are monumental" (p. 315). The education of prospective mathematics teachers has received limited attention from researchers (Sowder, 1989; Ball 8: McDiarmid, 1990). The attempt to change from the traditional view of mathematics instruction to the vision imagined by the authors of the various reforms cannot be viewed as an "add on" to existing mathematics instruction. The proposed reforms offer a completely different way of thinking about learning and teaching mathematics. The proposed reforms in mathematics education are concerned with the mathematics citizens will need to be successful in the Information Age. Preservice teachers have logged many hours of experiences in traditional mathematics instruction which developed in an industrial society. The focus 11 of this instruction was computational and procedural. The preservice teachers' perception of what mathematics is and what it means to do mathematics constitutes a knowledge constraint. Preservice teachers expect future mathematical experiences to fit their beliefs about what it means to know and do mathematics. The National Research Council (1989) argues for teachers to experience mathematics in ways that are consistent with reform recommendations: Teachers themselves need experience in doing mathematics - in exploring, guessing, testing, estimating, arguing, and proving - in order to develop confidence that they can respond constructively to unexpected conjectures that emerge as students follow their own paths in approaching mathematical problems. Too often, mathematics teachers are afraid that someone will ask a question that they cannot answer. Insecurity breeds rigidity, the antithesis of mathematical power" (p. 65). Research studies (e.g., Shulman, 1986a; Zeichner, 1985) provide skepticism considering the impact of entire teacher preparation programs. It would be naive to expect "miracles" from a single course. Analysis of any single element which affects the relationship between educational reform and the structure of schooling reveals a complex subset of intricate and interwoven elements (e.g., working conditions of teachers, tradition, policy makers, standardized tests, accountability, availability of appropriate textbooks). The teaching and learning of mathematics is but one aspect of the complex structure of schooling. It is difficult to isolate and "reshape" a single aspect of such a complex structure but the process has to begin somewhere. An analysis of the proposed reforms in mathematics education provides a framework for thinking about the kind of subject matter knowledge (i.e., 1 2 mathematics) prospective elementary teachers need to teach in ways that promote the ideals described in the reform vision. The Reform Vision Unlike the reform efforts in the 19505 and 605 that focused on the elite (i.e., the best and brightest), the 1980's reform advocates argue that the proposed changes in mathematics education must include all students. At least three common themes emerged from the current reform reports (NCTM, 1989, 1991; NRC, 1989): (1) The United States has shifted from an Industrial Age to an Informational Age; (2) the field of research and understanding (e.g., cognitive science) about how students learn continues to evolve and grow; and (3) the field of technology has exploded producing powerful tools for doing mathematics. Mes in society. Formal schooling in the United States developed during the early 1800's. Given that 70 percent of Americans were involved in farm related work in 1820 (Trow, 1977, p. 106), it is not surprising that only the affluent had the opportunity to attend schools. Other sectors of society were more concerned with working and meeting basic needs and viewed school as a luxury. Schools existed to produce the leaders of our country and offered little for those who were not candidates for those positions (Kliebard, 1985). As technology and industry developed, America changed from an agrarian to an industrial society. This shift in the economic and occupational structure created a demand for more highly skilled and educated workers. As a result, a greater portion of society had more time and a greater need to become educated. Larger numbers joined the school population and education of the masses resulted. The needs in the work force were for adults with minimum literacy skills and basic computational skills. Only a small portion of the school-age population received instruction that was geared 13 toward more scholarly thinking. Thus, the Industrial Age was embedded in a "tracked" school system (Oakes, 1985) . Technology and other societal changes thrust the United States into the Information Age (see e.g., Fey, 1989; Naisbitt, 1982; Toffler, 1985). Adults who are proficient in "shopkeeper arithmetic" are no longer a sufficient outcome for mathematics instruction (NCTM, 1989, 1991; NRC, 1989; Sowder, 1989). It is impossible to predict the content of problems that students will face in the 21st century. Krulik and Rudnick (1986) argue: Students will face problems and will still be required to think logically, creatively, and critically as they are called upon to make decisions. Much of the content currently stressed will probably diminish in importance, but the problem solving skills will surely remain vital (p. 47). Technological advances and the resulting changes in societal needs require different outcomes for citizens of the 21 st century. Steen (1990) describes the ways in which mathematics interacts with many facets of our daily lives: We must understand mathematical concepts to appraise the flood of news and information we receive each day, from the weather report to the latest tax regulation. Mathematics affects everything from the food we eat, to the investments we make, to the size and shape of the cities we build. It provides the tools for coping with the technology that increasingly penetrates our lives (p. i). It is not only the discipline of mathematics that recognizes a change in the type of mathematics that students need to function in our society today. Computer technology enables other disciplines and business fields to mathematize problem situations. Romberg (1992) offers support for this assertion: 14 The study of advanced mathematics has tended to be necessary exclusively for the physical sciences and engineering. Today, the computer has made it possible to quantify and use mathematical procedures in an ever-widening variety of disciplines and applications. Furthermore, the mathematical ideas that are important in these new areas of applications (e.g., the handling of large sets of data with statistical procedures) are not those that have traditionally been emphasized (p. 27-28). Proficiency in shopkeeper arithmetic is no longer adequate. Citizens who are good problem solvers, who are able to mathematize situations appropriately, and who generally are mathematically literate are needed to successfully function in the let century. If this transition from proficiency in computation to developing mathematical power is to occur, considerable changes in mathematics teaching and learning are required. §r9v_vth in knowledge about how students learn. Proposed reforms emphasize active student construction of knowledge. Cognitive psychologists and many mathematics educators (see e.g., Bereiter, 1985; Cobb, 1990 ; Kilpatrick 1987a; Putnam, Lampert, 8: Peterson, 1990; Resnick 8: Ford, 1984; Resnick, 1983; Romberg 8: Carpenter, 1986; Shuell, 1986; von Glasersfeld, 1987) support the theory that students do not passively receive knowledge or make mental duplications of knowledge possessed by teachers or contained in textbooks but actively "construct" knowledge for themselves. Knowledge is not a static product that can be moved from one person‘s mind and "placed" in another. Some type of processing must occur. There are many definitions and degree of implementation related to constructivism. Rather than dwell on those distinctions, two descriptions will provide a sense of the spirit of constructivism. Cobb, Wood, Yackel, Nichols, Wheatley, Trigatti, 8: Perlwitz, (1991b): 15 From the constructivist perspective, mathematical learning is not a process of internalizing carefully packaged knowledge but is instead a matter of reorganizing activity, where activity is interpreted broadly to include conceptual activity or thought (p. 5). Prior knowledge also impacts the interpretation made by a learner and the constructions that develop. Putnam, et a1. (1990) illustrate: . . . the learner plays an active role in interpreting and structuring environmental stimuli. Rather than passively receiving and recording incoming information, the learner actively interprets and imposes meaning through the lenses of his or her existing knowledge structure, working to make sense of the world. . . learning or development takes place . . . through the modification and building up of the individual knowledge structure (p. 87). Romberg (1983) borrows terms from Dewey to argue for a distinction between "knowledge" and "the record of knowledge." The former requires active student construction of meaning while the latter implies absorption of someone's work. Traditional mathematics instruction places an emphasis on the absorption of the record of knowledge. Proposed reforms emphasize active student construction of knowledge. George Polya argued the value of engaging in mathematical problems solving in a spirit consistent with the way mathematicians do mathematics thus providing first-hand experience of mathematical discovery (Stanic 8: Kilpatrick, 1988). Stanic and Kilpatrick describe Polya's distinction between the "finished face of mathematics" and "mathematics in the making": Polya's experience as a mathematician led him to conclude that the finished face of mathematics presented deductively in mathematical journals and in textbooks does not do justice to 16 the subject. Finished mathematics requires demonstrative reasoning, whereas mathematics in the making requires plausible reasoning (p. 16). There is a tension between allowing students to construct understanding for themselves and wanting students to acquire particular knowledge (Wilson 8: Ball, 1991). Cobb, Yackel, and Wood (1988) describe that tension: On the one hand mathematics education involves acculturating students to a shared mathematical reality. On the other hand, it involves guiding individual students' subject construction of mathematical knowledge (p. 100). As mathematics educators and psychologists have increased our knowledge and understanding about how children learn, the focus of instruction has shifted from the teacher as the dispenser of knowledge to teacher as facilitator and provider of rich mathematical environments in which students have the opportunities to construct individual understandings in a social context. Technological advances and the impact on mathematics. Scientific calculators, graphing calculators, and computers are powerful tools that can be used in mathematics. Tedious calculations and much of the symbol manipulations of the past can now be quickly and easily performed using technology. How many adults use paper and pencil to do three-digit by three- digit multiplication? long division? decimal computation? Anderson (1982) compares the kinds of mathematics that is emphasized in classroom instruction versus the kinds of mathematics that is used in daily life: Quantitative understanding and thinking will be more important than ever before. In an increasingly complex technological society, numbers and their uses become more fundamental for more people. . . . 93% of students' learning of 17 arithmetic is with paper and pencil and 90% of peoples' uses of arithmetic already does not involve paper and pencil. (p. 2, 3). Even though calculators are readily available and relatively inexpensive, wide-spread use in elementary classrooms has occurred slowly. Many teachers and parents fear that the use of calculators may discourage students from learning the "basics." Hembree and Dessart's (1992) meta-analysis indicated that the use of calculators did not negatively affect students' computational skills and in fact improved problem solving skills. Consistent with advice for utilizing many tools, Hembree and Dessart suggest that teachers consider the context of calculator use as a factor in decision-making about calculators. Calculators can do much more than perform the four basic operations of whole numbers and decimals. Formulas can be pre-programmed and accessed by simple keystrokes. There are now hand-held calculators that can perform "virtually every mathematical technique taught through the sophomore year in college both in purely symbolic form. . .and in the numerical forms that scientists need" (Steen, 1986, p. 3). In the last five years graphing calculators have begun to appear in some mathematics classrooms. Graphing calculators enhance work with solving equations, analyzing functions, and data analysis (Barrett 8: Goebel, 1990). Demana and Waits (1990) describe the value of graphing calculators: Computer graphing utilities make graphing a fast and effective problem-solving strategy. With this power we can now graph numerous functions quickly and establish common properties of classes of functions, have students explore and discover mathematical concepts, and use graphs to solve problems. . . . We can use realistic applications involving complicated algebraic representations (p. 212). 1 8 The applications and uses of graphing calculators should continue to evolve as teachers continue to explore their uses in upper elementary, middle, and secondary classrooms. The use of graphing calculators could greatly alter the curriculum but it is difficult to predict the actual length of time required for such changes to occur. Computers are another technological tool that have changed what is valued in mathematics curriculum. Steen (1986) contends that "computers change both what is feasible and what is important in the mathematics curriculum" (p. 3). Romberg (1992) supports Steen's argument about the ways in which computers are changing what is valued in mathematics curriculum: Given that computers can perform many calculations quickly, it is no longer adequate to consider one computational procedure. Instead, such questions as is this the best procedure? How can we prove it is best? What do we mean by best? lead to important, unsolved mathematical problems. To reiterate, the computer is changing mathematics and, in turn, it is making certain topics within the discipline more important for school mathematics and others less important (p. 29). Instructional time once used for students to perform tedious calculations can now be used to allow students time to explore complex mathematical problem situations and to conjecture about mathematical patterns and relationships. Technological advances have changed the discipline of mathematics. Steen (1990) compares the changes to the transformation of Newtonian analysis: Not since the time of Newton has mathematics changed as much as it has in recent years. Motivated in large part by the introduction of computers, the nature and practice of mathematics have been fundamentally transformed by new concepts, tools, applications, and methods. Like the telescope of 19 Galileo's era that enabled the Newtonian revolution, today's computer challenges traditional views and forces re- examination of deeply held values. As it did three centuries ago in the transition from Euclidean proofs to Newtonian analysis, mathematics once again is undergoing a fundamental reorientation of procedural paradigms (p. 7). Willoughby (1990) argues that "simple symbol manipulation can be done effectively by machines, but higher-order thinking skills and the ability to communicate intelligently about mathematical situations are still uniquely human skills" (p. 4). While technology has opened new worlds of mathematical exploration, humans will continue to be an integral component in changing mathematics instruction. Implications for mathemgtics edupation. Changes in society, changes in the discipline of mathematics as a result of technological advances, and growth in knowledge about how students learn underlie the proposed changes in mathematics teaching and learning. Response to the shift from an Industrial Age to an Informational Age is more obvious in a discipline such as science than in a discipline like mathematics. Comparisons of the content in current science textbooks with science textbooks in the 1940's would reflect an information explosion. The media and society in general would acknowledge this change in science content as well. A similar comparison of content in elementary and middle school mathematics textbooks would reveal quite different results-the basic content remains the same. Many people view mathematics as a static discipline. In reality it is a dynamic and growing discipline (see e.g., Barbeau, 1989; Davis 8: Hersh, 1986; de Lange, 1987; NRC, 1989; Smith, 1987; Steen, 1990). Over half of all mathematics has been invented since World War H (Davis 8: Hersh, 1981). Steen (1990) describes the recent growth of mathematical disciples: 20 In the last century alone, the number of mathematical disciplines has grown at an exponential rate; examples include the ideas of George Cantor on transfinite sets, Sonja Kovalevsky on differential equations, Alan Turing on computability, Emmy Noether on abstract algebra, and, most recently, Benoit Mandelbrot on fractals (p. 1). Yet, few lay persons would predict that the following quote came from a document written 33 years ago: Mathematics is a living, growing subject. The vitality and vigor of present-day mathematical research quickly dispels any notion that mathematics is a subject long since embalmed in textbooks. mathematics today is in many respects an entirely different discipline from what it was at the turn of the century. New developments have been extensive; new concepts have been revolutionary. The sheer bulk of current mathematical development is staggering (College Entrance Examination Board, 1959, p. 1). Romberg, (1987) argues for changes in teaching and learning mathematics to meet the needs of an information society: Current educational practice is based on a coherent set of ideas about goals, knowledge, work, and technology based on a set of "scientific management" principles that grew out of the industrial revolution of the past century. These ideas about schooling practice need to be challenged and replaced with an equally coherent set of practices in light of the economic and social revolution in which we are now engaged. The shift from an industrial to an information society has some immediate consequences for our views of schooling and, in particular, for our views about the teaching and learning of mathematics" (p. 9). 21 It is not sufficient for students to possess a large number of facts and computational skills. Students must develop ways of thinking about problem situations which enable them to do something with the facts and skills. The authors of Everybody Counts (NRC, 1989) describe this change as a transition - "The teaching of mathematics is shifting from preoccupation with inculcating routine skills to developing broad-based mathematical power" (p. 82). Traditional mathematics instruction and textbooks focus on proficiency in computation rather than genuine problem solving. Concern about adults' abilities to solve problems also surfaces in the world of business (e.g., Bernstein, 1988). Henry Pollak, a retired employee of Bell Laboratories has discussed some of the struggles young adults experience as they begin work in Bell Laboratories. One of the major difficulties involves learning to participate in group problem solving. He contrasts applied mathematics with our usual mathematics teaching: Thus, the heart of applied mathematics is the injunction: "Here is a situation; think about it." The heart of our usual mathematics teaching, on the other hand is: "Here is a problem, solve it" or "here is a theorem; prove it" (Pollak, 1970, p. 328). Encouraging students to think about when, how, and why the mathematics works should improve their ability to think about applications for mathematics. Pollak adds to this statement: "When one applies mathematics in practice, the problems which arise are not exactly the problems in the textbook" (p. 329). Examination of typical elementary and middle school textbook series reveal a similar format for all of the major textbook series. As early as the kindergarten level, children are expected to operate at an abstract level. As children progress to higher grades they are greeted with page after page of 22 problems involving mathematical symbols which provide drill and practice for the various algorithms. The experiences students typically have in mathematics classes K-12 are not conducive to the "vision" described in current reform efforts. Research studies (Conference Board of Mathematical Sciences, 1975; Davis 8: Hersh, 1981; Peterson, Swing, Stark, Waas, 1984; Stodolsky, 1988; Welch, 1978) confirm the similarity of mathematics classes that students experience K-12. The following description drawn from NSF case studies (W elch, 1978) illustrates well the consensus of these studies: In all the math classes I visited, the sequence of activities was the same. First, answers were given for the previous day's assignment. The more difficult problems were worked by the teacher or a student at the chalkboard. A brief explanation, sometimes none at all, was given of the new material, and problems were assigned for the next day. The remainder of the class was devoted to working on the homework while the teacher moved about the room answering questions. The most noticeable thing about math classes was the repetition of this routine (p. 6). The mathematical environment suggested by these studies promotes an image of teaching mathematics as a technical (how-to) course. Steen (1990) describe the consequences of traditional mathematics instruction and argues for change: Traditional school mathematics picks very few strands (e.g., arithmetic, geometry, algebra) and arranges them horizontally to form the curriculum: first arithmetic, then simple algebra, then geometry, then more algebra, and . . .calculus. This layer-cake approach to mathematics education effectively prevents informal development of intuition along the multiple roots of mathematics. Moreover, it reinforces the tendency to design 23 each course primarily to meet the prerequisites of the next course. . . . To help students see clearly into their own mathematical futures, we need to construct curricula with greater vertical continuity, to connect the roots of mathematics to the branches of mathematics in the educational experience of children (p. 4). Romberg and Carpenter (1986) concur with Steen's description of traditional mathematics curriculum as fragmenting mathematics both within and related to other disciples. They further argue that, "This fragmentation has divorced the subject from reality and from inquiry" (p. 851). The fragmentation of mathematical concepts does not provide opportunities for students to understand how these individual "pieces" fit together in the larger field of mathematics. Students experience mathematics as a series of rules and algorithms which must somehow be "memorized" and filed for future reference. Traditional instruction does not provide students with a relational understanding of the underlying mathematical concepts and processes, but instead provides a view of mathematics as an abstract, mechanical, and meaningless series of symbols and rules (Burns 8: Lash, 1988; Porter, 1989; Stodolosky, 1988). Traditional mathematics instruction is not conducive to the development of an inquiry orientation to learning and doing mathematics advocated by current reform efforts. A change is required in the way in which mathematics is organized and taught. A different type of mathematics instruction requires a change in teachers' beliefs about what it means to know and do mathematics (Romberg, 1988b; Spector 8: Phillips, 1989; Steff, 1988). This change in beliefs includes a different perspective of both students' and teachers' roles during instruction (NCTM, 1989, 1991). The proposed changes argue for a shift in orientation from a computational perspective to a 24 conceptual perspective. Teachers, not policy makers and researchers, are the ones who must put these reforms into practice (Cohen, 1990; NCTM, 1991; Sykes, 1986; Zeichner, 1992). Teacher education has a responsibility to help teachers think about the changes needed (e.g., teacher knowledge, beliefs, and dispositions) to transform the vision into reality. Teacher education has had a relatively brief history and has struggled to establish credibility among academe (see e.g., Conant, 1963; Lanier 8: Little, 1986). In 1823 the first American school for the preparation of teachers was established. This was a private academy that included in its program "(1) provision for a series of lectures on 'school keeping' and (2) admission of a limited number of children to be used as a class in which good teaching techniques could be demonstrated for prospective teachers" (Gibb, Karnes, 8: Wren 1970, p. 303). The first state normal school was opened in 1839 in Massachusetts. The curriculum consisted of "reading, writing, arithmetic, geography, grammar, spelling, composition, vocal music, drawing, physiology, algebra, geometry [original emphasis], philosophy, methods of teaching, and reading of the Scriptures" (Gibb, et al., 1970, p. 304). By the late 18005, the debate concerning the type of education needed for teachers began. Consistent with the subject listings above, mathematical training consisted of arithmetic with a bit of algebra and geometry. This is understandable if one is reminded that the arithmetic program for elementary schools focused on the fundamental operations with whole numbers and common fractions. Prior to 1950 there was general agreement that elementary schools provided arithmetic programs, not mathematics programs (DeVault 8: Weaver, 1970). What is taught in elementary schools certainly influences teacher preparation for prospective elementary teachers. 25 Gibb et al. (1970) describe the commonly held view regarding the type of education needed by elementary teachers and the impact on mathematical requirements for prospective elementary teachers in the 192051: The training of teachers of arithmetic was still regarded as not being of any concern to college or university departments of mathematics. . . left to the departments of education. . . . program consisted only of methods courses, few of which gave much attention to subject matter. . . the postwar uncertainty concerning the goals of arithmetic was one of the greatest forces against an improved program of arithmetic in the preparation of elementary teachers. . . a theory of "incidental learning" became popular. Thus, in many teacher education programs for elementary teachers, methodology overshadowed content. The prevailing philosophy held that the teachers should teach the child, not teach arithmetic (p. 318). During the 19405 there was an increase in content and methods requirements for prospective elementary teachers. World War II had helped to reveal severe limitations in mathematical abilities. Perhaps elementary teachers did not have sufficient mathematical backgrounds. One might predict that requirements for mathematical content would be greatly increased for prospective elementary teachers. However, in 1960, Ruddell, Dutton, and Reckzek (cited in Gibb et al., 1970) found that, although the amount of college training required for certification of elementary teachers had increased significantly, there had been practically no change in the amount of prescribed mathematics. Only twelve states required any specified mathematics, and this requirement was an inadequate minimum which consisted of a course in general mathematics or a methods course in arithmetic. . . . Approximately two—thirds 1Training of elementary teachers was a two year program during this time period. 26 of the institutions studied would admit prospective elementary teachers whose programs showed no high school mathematics. . . . A typical graduate from the four-year program for certification to teach in the elementary school had completed two years of high school mathematics, one 3-semester-unit course in general mathematics, and one 2-semester-unit course in methods of teaching mathematics (p. 328). Fisher (1967) conducted a study of catalog descriptions of course offerings and found that the amount of mathematics required for prospective elementary teachers graduating in 1960 was a mean of 1.97 semester hours. By 1965 the mean had increased to 4.15 semester hours. These kinds of limited mathematical requirements for prospective elementary teachers provide some insight about why elementary teachers struggled to understand the "new" math of the 19505 and 605 which focused so heavily on the structure of mathematics. A review of the history of teacher education for elementary teachers highlights the tremendous uphill battle facing reform efforts of the 19905. One need only review the current mathematical requirements for prospective elementary teachers in the state of Michigannnone-to illuminate the magnitude of the challenge. Subject Matter Knowledge Needed to Teach Until the late 1980's, research about subject matter knowledge was virtually ignored or at best considered as a contextual variable (see e.g., Schoenfeld, 1986; Shavelson 8: Stern, 1981; Shulman, 1986a). Shulman (1986a) referred to the absence of focus on subject matter as the "missing paradigm" of research (p. 7). Recently subject matter knowledge has emerged as a significant concern in teacher education and teacher assessment (Anderson, 1982; Ball 8: McDiarmid, 1990; Buchmann, 1984; Carnegie Forum on Education and the Economy, 1986; Floden 8: Buchmann, 1989; The 27 Holmes Group, 1986; Lampert, 1986; Leinhardt 8: Smith, 1987; National Center for Research on Teacher Education, 1988; Shulman 8: Sykes, 1986; Shulman, 1987; Wilson, Shulman 8: Richert, 1987). Researchers argue for a particular kind of subject matter knowledge. Shulman and Sykes (1986) define content knowledge as: . . . the amount and organization of knowledge per se in the mind of the teacher. . . . Properly to think about content knowledge requires that we go beyond knowledge of the facts or concepts of a domain. It requires understanding the structure of the subject matter in the manner defined by such scholars as Joseph Schwab and Jerome Bruner (p. 8). Schwab's (1964) definition of structure includes the substantive structure, which represents the relationship among the facts, ideas, and concepts, and the syntactic structure, which represent the ways in which the discipline creates and validates knowledge of that discipline. Shulman (1986b) also argues for subject matter knowledge for teachers that goes beyond mastery of facts and skills: Teachers must not only be capable of defining for students the accepted truths in a domain. They must also be able to explain why a particular proposition is deemed warranted, why it is worth knowing, and how it relates to other propositions, both within the discipline and without, both in theory and in practice (p. 9). like Shulman, Buchmann (in Floden 8: Buchmann, 1989) thinks that the subject matter knowledge needed by teachers must go beyond mastery of facts and skills of the discipline. The subject matter knowledge she describes requires: 28 . . . a knowledge of subject matter that includes an elaborated understanding of the various aspects of a content domain, so that teachers can recognize inconsistencies in pupil responses and can generate hypotheses about what connections pupils have made incorrectly—or appropriately, though deviating from the textbook. . . . knowledge about the subject as well as knowledge of the subject. . . . Teachers need to give pupils "tutored" uncertainty; that gift requires understanding of the bases and processes of knowledge, not merely its conclusions (p. 19, 20). Mathematics educators and members of the wider educational research community provide compelling arguments for providing prospective teachers with a different view of subject matter content (see e.g., Buchmann, 1984; Cohen, 1989; Cohen 8: Ball, 1990; Fennema 8: Franke, 1992; Grossman, 1987; Lampert, 1989a; McDiarmid, Ball, 8: Anderson, 1989; Quimby 8: Barnes, 1986; Shulman, 1986b; Grossman, Wilson, 8: Shulman, 1989). Post, Harel, Behr, and Lesh (1991) assert that elementary teachers need to know more mathematics and that a "firm grasp of underlying concepts is an important and necessary framework for the elementary teacher to possess" (p. 191). Most elementary teachers can solve the computational exercises presented in K-6 textbooks, but this is not sufficient to teach K-6 mathematics to children in ways that are consistent with reform recommendations (see e.g., Brown, Cooney, 8: Jones, 1990). Rather than arguing for greater (i.e., more) mathematics requirements for prospective teachers, Ball (1989) argues for a particular kind of mathematical understanding. She describes four dimensions of subject matter understanding: "(1) knowledge of the substance of mathematics, (2) knowledge about the nature and discourse of mathematics, (3) knowledge about mathematics in culture and society, and (4) capacity for pedagogical reasoning about mathematics" (p. 89). 29 Considerable subject matter knowledge is required if teachers are to develop interesting problem situations which actively engage students in "making conjectures, investigating and exploring ideas, discussing and questioning their own thinking and the thinking of others, validating results, and making convincing arguments" (NCTM, 1987, p.54). A teacher who has appropriate understanding of subject matter should be able to assess students' abilities to validate and/ or justify solutions for problems presented. Clearly this focus on content knowledge demands more from teachers than their simply being able to do the computational problems that presently exists in K- 8 mathematics curriculum. Various studies (e.g., Carlsen, 1987; Hoyle, 1988; Lampert, 1988; Steinberg et al., 1985) describe the importance of subject matter knowledge in teacher's decisions about instruction. Teachers should understand why problems can be solved in various ways. Teachers also need to be able to comprehend how various mathematical concepts relate to the larger field of mathematics (see e.g., Steinberg et al., 1985). If a teacher understands how individual concepts relate to one another then he/ she can provide appropriate links for students between prior knowledge and new concepts. Romberg (1976) supports the argument that elementary mathematics teachers need to have a strong mathematical background which includes not only isolated content but relationships among concepts: They [elementary teachers] must not only know the material well but to justify mathematical statements themselves so that they can ask students to validate and explain mathematical ideas (p. 134). The number of college-level mathematics courses does not necessarily equate to the kind of subject matter knowledge described in this section (see e.g., Ball, 30 1989). Increasing the number of university mathematics courses could, in fact, impede developing the kind of subject matter described. The common pedagogy of university coursework—teaching by telling, and perpetuating the portrayal of mathematics as a technical, static discipline (Davis 8: Hersh, 1981; Kline, 1977) does not help overcome the patterns imprinted from earlier experiences with mathematics in K-12 grades. While there are limited studies about changing preservice teachers' mathematical orientations, there is evidence that it can occur in work with practicing teachers. Findings from two research projects at Michigan State University, the General Mathematics Project (Madsen-Nason 8: Lanier, 1986) and the Middle Grades Mathematics Project (Lappan, 1987b) indicate that teachers can change from a computational to a conceptual problem solving orientation in the classroom. However, both projects concluded that the change evolves over a long period of time (two years minimum) and with a great deal of support (Lappan, 1987b). These studies offer encouragement but also provide evidence of the thoughtful attention required to help teachers think differently about learning and teaching mathematics. At least three areas of research in mathematics education—problem solving, community and discourse, and conceptual and procedural knowledge—are relevant to understanding the current mathematics education reform recommendations. Research in Mathematics Education Problem solving. A phrase that the teaching profession and general public frequently hear associated with discussions concerning mathematics is "problem solving." Caution must be exercised so as not to equate the frequent use of the term to a common understanding or interpretation of the term. Schoenfeld (1992) argues that problem solving is "one of the two most 31 overworked and least understood buzzwords of the 19805" (p. 336). Schroeder and Lester (1990) add additional support to Schoenfeld’s argument: "In recent years problem solving has been the most written-about and talked-about part of the mathematics curriculum and at the same time the least understood" (p. 31). Some people may define problem solving narrowly as "word" or "story" problems. Driscoll (1981) suggests that for many adults the phrase, problem solving, "triggers memories, often uncomfortable, of textbook word problems" (p. 101). He further argues that this interpretation, the interchangeableness of "problem solving" and "word problems," would be consistent with practices in many elementary mathematics classrooms. A limited view of problem solving is exacerbated by the placement of "problem solving" sections or as isolated topics in textbooks. When "buzz" words or phrases such as problem solving promote wide spread discussion among educators and the general public, textbook publishers react. One way for the textbook authors to respond is to scatter problem solving "sections" throughout the book typically at the end of a section or chapter. This type of placement encourages the notion that problem solving occurs separately from other aspects of mathematics. Perhaps, the publishers do not intend this notion of separation. Among the factors which might influence such placement include: (1) It does not involve changing the entire mathematics curriculum when viewed as an "add-on." Thus, it is quick, easy, and relatively inexpensive. (2) It meets the criteria identified on many textbook selectors' "must include" list. Having the words in the table of contents looks good. When contrasted with the Commission on Standards group, the problem solving view described above provides a limited notion of problem 32 solving. The Commission on Standards working group, grades 5-8 (N CTM, 1987), describes problem solving in the following manner: It [problem solving] is not a separate content area but is the process by which one learns and does mathematics. . . . Problem solving is the process whereby students experience the power and usefulness of mathematics in the world around them. Problem solving should be a process that actively engages students in making conjectures, investigating and exploring ideas, discussing and questioning their own thinking and the thinking of others, validating results, and making convincing arguments. Therefore, problem solving does not happen when students do a page of computations, when they "follow the example at the top of the page," or when all the word problems practice the algorithm presented on the preceding pages (p. 54). A wide range of definitions is associated with the phrase, "problem solving." Schroeder and Lester (1990) identify three approaches to problem-solving instruction: "(1) teaching about problem solving, (2) teaching for problem solving, and (3) teaching via problem solving" (p. 32). Teaching about focuses on the process itself. Schroeder and Lester caution that the danger of teaching about is that problem solving can become just another topic to be added to the mathematics curriculum. Teaching for focuses on applications of various mathematical concepts. The limitation of this strategy is that it is often taught in conjunction with a specific mathematical skill or topic and thus promotes a narrow view of problem solving. Schroeder and Lester argue that teaching via problem solving is more consistent with the NCTM Curriculum Standards because mathematical skills and concepts are embedded in problem situations and occurs in an "inquiry-oriented, problem solving atmosphere" (p. 34). Schroeder and Lester acknowledge the potential overlap of the three approaches and do not argue to eliminate the first two but rather to broaden 33 one's perspective to include all with a particular emphasis on teaching via problem solving. Schoenfeld (1985) conducted a problem solving study which explored high school students' abilities to draw from both deductive and empirical mathematical knowledge in attempts to solve geometric problems. Most of the students were unable to successfully make the appropriate connections between available deductive and empirical mathematical knowledge. Students appeared to "compartmentalize" their knowledge. Researchers argue that it is not what you know but how that knowledge is organized and the ways that problem solvers represent problems that characterize success as a problem solver (Chi, Glaser, 8: Rees, 1982; Glaser, 1984; Resnick, 1987b; Schoenfeld, 1985; Silver, 1987). Cpmmunity and discourse. As described earlier, there is general support in the mathematics education community that students actively construct knowledge rather than passively receive it. While this assertion focuses on the individual, there also is support for the interactive nature of constructing knowledge in some type of social context (see e.g., Bauersfeld, 1988; Cobb, 1989; Cobb, Wood, Yackel, Nichols, Wheatley, Trigatti, 8: Perlwitz, 1991a; Cobb, et al., 1991b; Lampert, 1988; 1989a; 1991; Resnick, 1988; Steffe, 1987; Wilcox, Schram, Lappan, 8: Lanier, 1991). Cobb (1989) asserts: Each child can be viewed as an active organizer of his or her personal mathematical experiences and as a member of a community or group who actively contributes to the group's continued regeneration of taken-for—granted ways of doing mathematics. . . . Children also learn mathematics as they attempt to fit their mathematical actions to the actions of others and thus to contribute to the construction of consensual domains (p. 34). 34 Cobb, et al. (1991a) describe the tension between focusing on the individual and focusing on the community: "When we focus on an individual student's sense-making activity we lose sight of the community and when we analyze communal knowledge individual sense—making slips from our view" (p. 101). Classroom community can be constructed in two sites—small groups and whole-class discussions. Small groups provide opportunities for students to (a) communicate about mathematical ideas; (b) talk to others to clarify understanding; (c) view multiple ways to approach and solve problem situations; (d) learn ways to work collaboratively; (e) develop independence as learners; and (f) take intellectual risks (Wilcox et al., 1991). Whole-class discussions provide opportunities for students to engage in experiences similar to those of small groups but provide a wider audience in which to clarify and challenge understandings; to reflect upon understandings; to make connections among various mathematical ideas; and to develop shared understandings among community members. Van Engen argues for the need to communicate during mathematics instruction: "People learn and understand mathematics through discussion [original emphasis] about mathematics with peers and knowledgeable teachers." (in Henderson, 1972, p. 19). Many mathematics educators and researchers are beginning to describe the value of verbalizing mathematical thinking during classroom discussions (e.g. Hoyle, 1985; Lampert, 1989b; Lappan 8: Schram, 1990; Putnam et al., 1990; Skemp, 1987). Putnam et al. (1990) describe the importance of talking about mathematics and encouraging students to verbalize their thinking: 35 From the discipline of mathematics, we have seen the importance of conjecturing and defending ideas in mathematics - activities that require extended public discourse (p. 138). Cobb, Yackel, and Wood (1992) offer additional support for the value of verbalizing one's thoughts: "It is by attempting to communicate that discrepancies in individual interpretations become apparent" (p. 17). Simon and Schifter (1991) support the argument that discussions among peers about mathematical ideas and understandings helps individuals to further clarify their thinking: As group meanings are negotiated, group members engage in making sense of and resolving disequilibrium caused by differences between their ideas and those of others. Thus, cognitive reorganization is promoted by these attempts at communication and cooperation (p. 310). Hoyle (1985) describes the "cognitive function" of "talk" as follows: Language facilitates reflection and internal regulation since difficulties in formulating the language to describe a situation may lead the speaker to modify her analysis of that situation (p. 206). The creation of a community subjects each participant's private world of understanding and may challenge the participant's currently held views and lead to the construction of more powerful views. Creating a mathematical community within a classroom takes place over time and requires creating a total environment where students will take risks to make conjectures, offer arguments in support of assertions, and assume the authority for deciding about the reasonableness of mathematical representations and solutions (Wilcox, et. al., 1991). 36 Qpngeptpal and procgural knowledge. Resnick and Ford (1981) argue that, "The relationship between computational skill and mathematical understanding is one of the oldest concerns in the psychology of mathematics" (p. 246). The argument of "meaningful" learning has been made by mathematics educators for many years (see e.g., Brownell, 1935; Dewey, 1965; Moore, 1903/ 1926; Van Engen, 1953). Early debates among educators focused on making a choice between computational skill- procedural knowledge and mathematical understanding—conceptual knowledge. As early as 1903, Moore (1903/1926) debated the issue of emphasizing drill and practice which portrays mathematics as isolated ideas and procedures versus meaningful learning which stresses mathematics as interrelated concepts and ideas. Psychological studies (e.g., Anderson, 1983; Bruner, 1960, Glaser, 1984) support the importance of the ways in which knowledge is organized and stored in memory. Mathematics educators now recognize the complexity of separating procedural knowledge and conceptual knowledge. Recent discussions among mathematics educators (e.g., Carpenter, 1986; Davis, 1986; Hiebert 8: Carpenter, 1992; Hiebert 8: Lefevre, 1986; Schoenfeld, 1986; Silver, 1986) attempt to understand the relationship between procedural knowledge and conceptual knowledge. Hiebert and Lefevre (1986) define conceptual knowledge as: . . . knowledge that is rich in relationships. . . . a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information. Relationships pervade the individual facts and propositions so that all pieces of information are linked to some network. In fact, a unit of conceptual knowledge cannot be an isolated piece of information; by definition it is a part of conceptual knowledge 37 only if the holder recognizes its relationship to other pieces of information (p. 3—4). As students engage in learning experiences, they individually generate internal representations for these experiences. Each of these representations is either connected to an existing network or a new system is begun. Hiebert and Carpenter (1992) describe the ways in which the learning experiences influence the creation of the networks and the relationship between the connections built and a student's interpretation of mathematical meaning: The richness of the information and material influences the richness of their [students] internal representations: If the mathematical activities in which students are engaged are overly restrictive, their internal representations are severely constrained, and consequently the networks that are built are bounded by these constraints. Connections between these restricted networks are difficult to establish. Students are forced to search for meaning within these relatively small bounded networks, and because meaning in mathematics often comes by relating ideas, facts, and procedures across a range of situations, a search for meaning within an overly restricted domain is bound to be deficient (p. 76). Hiebert and Lefevre (1986) define procedural knowledge as follows: One part is composed of the formal language, or symbol representation system, of mathematics. The other part consists of the algorithms, or rules, for completing mathematical tasks (p. 6). Results from statistical studies (e.g., National Assessment of Educational Progress, 1978; Second International Mathematics Study, 1987) and studies which incorporate clinical interviews (e.g., Erlwanger, 1973, Schoenfeld, 1985) support the conclusion that many students have only rote procedures for manipulating numbers and symbols and little conceptual understanding of 38 the underlying mathematical concepts and processes. Campione, Brown, 8: Connell (1988) suggest that the process of teaching algorithms before understanding may lead to acquisition of knowledge that cannot be applied flexibly. Early use of algorithms may interfere with an attempt to attach meaning at a later time. They argue that attention should be focused on metacognitive and contextual factors during the learning process. Bransford, Hasselbring, Barron, Kulewicz, Littlefield, and Coin (1988) identified several examples to support the assertion that students memorize facts and procedures without attaching meaning. They assert that "the facts and procedures must be transformed into conceptual tools" (p. 132). The relationship between conceptual and procedural knowledge provides power and flexibility in using mathematical knowledge. Hiebert and Lefevre identify three factors that limit the connections that are made between conceptual and procedural knowledge: "deficits in the knowledge base, difficulties of encoding relationships, and tendency to compartrnentalize knowledge" (p. 17). Deficits in the knowledge base refers to gaps or missing pieces of information to which other concepts and procedures need to be connected. It is logical that connections cannot be made if particular knowledge does not exist. For example, the relationship between percents and common fractions cannot be made if knowledge about common fractions does not exist. Hiebert and Lefevre describe difficulties of encoding relationships as particularly pertinent with young children. Relationships and connections that may seem obvious to adults may not be noticed or understood by children. The tendency to compartrnentalize knowledge is an interesting factor. Hiebert and Lefevre argue that it is possible to have well developed individual "compartments of knowledge" but few or no connectors to other 39 compartments. This tendency to isolate knowledge is exacerbated by the way mathematics is normally taught—tidy two-page lessons. Mathematics textbooks traditionally are designed in the two-page lesson format without reference to related big ideas or analysis (Tyson 8: Woodward, 1989). If connections are not made explicit, these lessons can be viewed as isolated bits and pieces of knowledge. The similarities and connections to prior knowledge may not be readily apparent to the learner. Context-specific knowledge has been described in many mathematical studies (see e.g., Carpenter, 1986; Hiebert 8: Wearne, 1986; Lawler, 1981; Schoenfeld, 1986; Silver, 1986; Tulving, 1983). Ball (1989) identifies "connectedness" as one of three specific criteria for teachers' "substantive knowledge" (p. 89). The connectedness she describes is the opposite of compartmentalized knowledge described above. A review of previous mathematics education reforms affords a historical perspective for this study and provides additional information about the kinds of mathematics experiences that contributed to shaping prospective teachers' views about mathematics and about teaching and learning mathematics. Historical Perspective 19505 amid 605: Modern mathema_ti&. In 1950 the National Science Foundation (NSF) was established by an act of Congress and was appropriated $225,000. The expressed purpose of NSF was to "develop a national policy for the promotion of basic research and education in the sciences" (Jones 8: Coxford, 1970, p. 74). In 1957, the news of Sputnik launched a flurry of activity on the national scene in mathematics education. The public demanded that schools "produce" mathematicians and scientists who could meet and surpass their Russian peers. 40 Substantial amounts of federal monies (e.g., NSF and US. Office of Education) were funneled into the development of curriculum projects (e.g., School Mathematics Study Group, University of Illinois Committee on School Mathematics, Comprehensive School Mathematics Project) to support change in school mathematics (see e.g., Fey, 1982; Wooten, 1965). Money also was provided to support summer institutes and fellowships for teacher training. The focus of the training of teachers was on increasing their mathematical knowledge. The new curricular projects focused on the structure of mathematics with an emphasis on languagez, the field properties,3 and symbol notation. The focus for many of these curricular projects was on the "best and brightest" students. Mathematics instruction during this era was commonly referred to as "new" math or "modern" math. This resulted from the argument that the mathematics curriculum was outdated and did not reflect the more modern view of the nature and role of mathematics (Jones 8: Coxford, 1970). The suggested changes in curriculum marked a major shift in the elementary grades. Prior to this time, arithmetic was taught in those grades. The introduction of new content shifted instruction in elementary school from an arithmetic program to a mathematics program4. Early on, attention was focused on the content taught in schools but as additional funding became available (e.g., The National Defense Education Act of 1958), other 2An example of the emphasis on language was the debate about the distinction between the use of number and numeral. 3Freld properties include the commutative (i.e., a+b=b+a) and associative properties (e.g., a+(b+C) = (a+b) +c); the multiplicative property of 1 (i.e., a x 1 = a); additive property of 0 (i.e., a + 0 a a). 4It is interesting to note that at least one Michiganian had some mathematical vision. In 1899, E. C. Goddard at a state teachers' meeting gave an address entitled "The Distribution of Mathematics through the Twelve Grades." "He recommended that mathematics, not merely arithmetic, be taught in the elementary school” (In Jones, 1970, p. 465). 41 aspects of the elementary schools were subject to experimentation (e.g., open classrooms, team teaching, nongraded primary schools, teacher aides). The Madison Project introduced a "discovery" approach to teaching mathematics which eventually led to mathematics laboratories (Jones 8: Coxford, 1970). Relatively few teachers were involved in setting project goals, objectives, or determining content. These decisions were made primarily by a select group of mathematicians and scientists. Instruction tended to be technical, formal and heavy with symbolism (Rising, 1977). The emphasis was on fostering conceptual understanding of the "structure" of mathematics as a discipline (Resnick 8: Ford, 1984). Instruction centered around students memorizing properties and definitions. It was also important to write mathematics correctly so symbol notation, for example, being able to produce the notation for sets (i.e., { l) became increasingly important.5 As a historical backdrop to a recent chapter, Calfee (1989) describes this era from the perspective of a graduate student: When I entered graduate school in the late 19505, the "teacher- proof" curriculum was in ascendance. This approach called for careful task analysis, precise specification of instructional objectives, and a tight linkage between teaching and testing- 5A5 a student during this era, I recall homework assignments that required us to fill a notebook page of I]. The teacher explained that this practice was important so we could make our math look nice. Many answers required using the set notation, [ l, in one form or another so this became an "important" skill. Other personal memories from this time include memorizing the field properties. I remember at the time being puzzled about the purpose for doing that. I was fascinated by all the symbolism and what it must represent and the teacher assured me that knowing the properties was a prerequisite to pursuing ”higher" level mathematics. The "new" math was exciting to me because it was more appealing than repeating the same "old stuff" that we had done for the previous five years. Our teacher was taking a modern math course and learning the math in the evenings. She struggled with the course and decided to put five of us in the back of the room to work ahead on our own so we could help the rest of the class and, I suspect, her understand this new "stuff". 42 proper engineering was the key—, and the teacher's role was to manage the system (p. 33). Discourse in the modern math class was limited. The University of Illinois Committee on School Mathematics (UICSM) was considered as one of the "founding fathers" of all the curriculum projects of the 19505. UICSM was created in an effort to correct the weakness of secondary school mathematics programs. Many of the K—12 curricular projects that followed incorporated aspects of UICSM. Max Beberman was the director of UICSM. In a lecture in 1958 he described various features of the program. It is interesting to note UICSM's position on verbalization of mathematical thinking: It is important to point out here that it is unnecessary to require a student to verbalize his discovery. . . . In fact, immediate verbalization has the obvious disadvantage of giving the game away to other students. . . (In Osborne 8: Crosswhite, 1970, p. 255). Not only was the discussion of mathematical ideas not encouraged, it was explicitly frowned upon. Criticism of modern math and the apparent results from that type of instruction abounded. Today, the phrase, new math or modern math, evokes distasteful expressions, unpleasant memories, and discussions among many educators and parents. Many teachers who were teaching during this period and other adults schooled during this time continue to harbor intensive negative images and feelings. A commonly held misconception was that the new math curriculum was fully implemented. Analysis and reflection of the "New Math" reform of the 50's and 60's indicate that the "new" curriculum was never fully implemented and resulted in little change in the elementary and middle school curriculum (see e.g., Fey, 1978; Powell, Farrah, 8: Cohen, 1985). 43 Several factors influenced the limited implementation of the new math curriculum. Many of the curricular materials were planned, produced, field tested and evaluated in a short period of time (Fey, 1978). Limited attention was given to teacher concerns and needs. For example, a new school year began and elementary and middle school teachers were greeted with "foreign" sounding terms5 and told that they would begin teaching "modern mathematics" the following week. The newly created curricular materials were designed primarily to be "self explanatory" and acquired labels such as "teacher proof" materials. The focus of teacher "training" was on content—the structure of the disciple of mathematics-with little or no attention to the "teaching" aspect. Textbooks appeared to adopt the ideas of the new math curriculums. Vocabulary of the "new math" (e.g., sets, commutative and associative properties, inequalities) was dispersed throughout the textbooks, but a closer examination of the texts indicated little significant change in actual content. The reform of the 505 and 605 resulted in cosmetic changes but the traditional, technical "how to" focus continued to permeate mathematics instruction. 19705: Individualized instrpction: and "back to the ba__si_C§." One of the goals of the modern math era was "to develop an understanding of the number system and its properties" (Dessart, 1981, p. 6). Computational drill and practice received much less attention than in previous years. Standardized test scores that measured computational skills dropped and the modern math curriculum was held responsible for the decline in scores. There was a general plea from the public for "back-to-the-basics" implying a 6Examples included modular arithmetic, set theory, commutative, associative, and transitive properties. 44 renewed focus on drill and practice. Goodlad (1984) characterizes the public's perception of this era as follows: The back-to-basics movement of the 19705 was fueled in part by the charge that the schools had been taken over by a generation of teachers stressing progressive beliefs and practices. That is, instead of exercising firm control in the classroom and concentrating on fundamental skills and subject matter, teachers had become overly permissive in regard to both student behavior and academic performance. The consequences, it was charged, were poor discipline and poor student achievement (p. 173). Goodlad continues by describing a portion of data from their study regarding teachers' educational beliefs and practices. The data did not support a significant distinction between teachers who were teaching prior to the 19605 and 705 and teachers who were teaching after the 19605 and 705. He concludes, "The rhetoric about what should be undoubtedly shifts more rapidly and strongly than either the beliefs or practices of teachers" (p. 174). The role of teachers during the back-to-basics movement became "less decision maker and more the implementer, maybe even technician, of mandates from above" (Cooney, 1987, p. 2). The teacher's role was to "drill" the basics into students' heads. The student's role was to memorize basic facts and practice, practice, practice. The focus of instruction was on computational proficiency. Timed tests and classroom competitions became common features of mathematics classrooms. The nature and need of society, the increasing knowledge concerning developmental psychology and learning theories7, and a general 7For example, Gagne's (1962) work related to task analysis—breaking down skills and concepts into basic units-promoted a hierarchical perspective about learning. 45 dissatisfaction with mathematics instruction all influenced the "movement" toward individualized instruction in the late 60's and early 70's. Technological advances created additional needs for application of mathematics. Integration and society's demand for compensatory education created change and growth within the school population. This change and growth provided a wider range of abilities within individual classrooms. Reports from groups such as the Commission on Mathematics of the College Entrance Examination Board (CEEB) (Osborne and Crosswhite, 1970) laid the groundwork for "tracking" and an emphasis on differences among individuals. One of the commission's recommendations included: All students need not be taught at the same pace, in the same order, or to the same extent, or with the same emphasis (p. 262). The commission also stated that students with "similar interests and similar intellectual abilities" (e.g., college bound students) should be taught as a group. During this time period considerable attention also was given to the need for "gifted" education. A natural outgrowth of this emphasis on individual learners was a return to individualized instruction. Individualized instruction has faded in and out of education for many years. As early as 1888, Preston Search, a superintendent in Colorado, was implementing a form of individualized instruction (Gibb, 1970). A common definition for "individualized instruction" does not exist. Contracts, unipacs, flexible grouping, programmed instruction, self-pacing, self-selection of materials and activities, are all considered under the large umbrella of individualized instruction. A review of the Arithmetic Teacher and Mathematics Teacher during the 1970's established that numerous articles were written about various aspects of individualized instruction. The entire 46 January 1972 issue of Arithmetic Teacher was devoted to individualized instruction. Individualized instruction typically meant a classroom of students, each of whom was given some type of packet or carefully sequenced worksheets, working alone at their desks. As students completed their packets some routine was in place to "check" their answers. When students achieved a certain "mastery" (e.g., 80% correct), they moved on to the next "level". The teacher's role was to be available to answer questions as students sought help. If one visited one of these classrooms, it would not be uncommon to find the teacher seated at her desk with students lined up waiting their turn. It was possible that a teacher might instruct one student about how to multiply fractions and the next student about how to subtract decimals, and the next student about how to multiply fractions. In other words the teacher might repeat the same instructions several times during a single class period or might be telling students how to do a wide range of problems. The teacher's role was one of management: how to manage the paperwork—correcting and recording scores, assigning worksheets, giving and correcting tests; how to manage the classroom of students—no doubt restless from such tedious and repetitive work; how to manage one's sanity—coping with the frustration that must accompany repeating "how-to" instructions a dozen times or jumping among a wide range of mathematical topics and skills. Students' success in individualized programs was measured by the number of right answers and attaining mastery (i.e., a predetermined percentage) of carefully sequenced skills. Erlwanger's (1973) study of a group of students who had participated in Individually Prescribed Instruction (Lipson, Koburt, 8: Thomas, 1967) for several years illustrates the 47 consequences that can result from a focus on "right" and "wrong" answers rather than the processes students use to solve problems. The case studies of Benny and Matt provide evidence of children identified by teachers as "good" math students (i.e., they made good test scores and were progressing through the booklets at an appropriate rate) but who had serious misunderstandings and misconceptions concerning basic mathematical concepts. Benny and Matt also had unusual ideas concerning the rules and "answers" for mathematics problems. Magical arbitrary rules dominated their world of "mathematics." They learned quickly that the goal was to figure out the pattern of the correct answers as they appeared in the "answer key." The focus of what teachers needed to know to teach mathematics during this era was not related to subject matter knowledge but rather an emphasis on managerial skills. During this time period NSF existed on a skeletal budget. Many of the earlier NSF funded programs were severely cut back or eliminated. 1_9_&)s_and 905: Analysis and reform. During the 1980's national concern and attention once again focused on mathematics instruction. Results from studies and national reports such as, A Nation at Risk (The National Commission on Excellence in Education, 1983), Educating Americans for the 215t Century (National Science Foundation, 1983), A Nation Prepared: Teachers for the 215t Century (Carnegie Forum on Education and the Economy, 1986), Tomorrow '5 Teachers: A Report of the Holmes Group (The Holmes Group, 1986), and The underachieving curriculum (International Association for the Evaluation of Educational Achievement, 1987), prompted the public to demand changes in mathematics education. Professional organizations such as the National Council of Teachers of Mathematics, National Science Board, Board of Mathematical 48 Sciences, American Association for the Advancement of Science and state departments of public instruction including California, Texas, Wisconsin, and Oregon responded to the call for reform in school mathematics. The documents produced by these organizations reveal a similar description of the vision of mathematics education (American Association for the Advancement of Science, 1989; California State Department of Education, 1985; 1992; NCTM, 1989; 1991; NRC, 1989). The reform efforts of the 19605 and 19705 focused on curricular changes and were mandated from "up above". Cooney, (1987) characterizes these reform efforts as "paper reforms." He argues for the current reform efforts to be "people reforms": To me, reform should be considered a humanistic enterprise rather than a matter of paper reform. Reform needs to be based on human innovation and activity. . . . A humanistic orientation emphasizes the teacher as a decision maker who determines what mathematics students are capable of learning and what strategies are appropriate given the mathematical maturity of the students (p. 4). The spirit of the current reform documents (N CTM, 1989, 1991; NRC, 1989) provides a "vision" rather than a "recipe" of proposed changes in mathematics teaching and learning. Transforming the vision into reality requires thoughtful teachers and thus implies a "people reform." This study will focus on what students learn about the mathematical content in a mathematics course designed for prospective elementary teachers. In chapter 2, I will describe the purpose and design of the study. Chapter 3 will focus on a description and analysis of the mathematics course, Math 201. Chapters 4 and 5 are data analysis chapters. In chapter 4,1 will analyze and describe students’ views about mathematics, students' patterns of 49 reasoning and problem solving, and students’ ideas about selected number theory concepts and ways in which those changed. This chapter focuses on what students learned about the number theory ideas embedded in contexts similar to those explored in Math 201. In chapter 5, I examine what students were able to do with their knowledge and understanding about the number theory concepts explored in the Math 201 course. The focus of this chapter is on students' flexibility in using these number theory ideas in generalized and unfamiliar3 contexts. In the final chapter, I provide a framework that represents the complexity and interrelatedness of several factors that impact what students learn about mathematical content. Implications for findings from the study and future research questions also will be explored. 8"Unfarniliar' is defined as a context that is different from ones explored in the Math 201 class or one that is different from the way in which the mathematical idea is commonly encountered in traditional mathematics classes. CHAPTER 2 THE STUDY: PURPOSE, DESIGN, AND ANALYSIS Purpose This study investigated what a group of six students came to understand about the mathematical content embedded in a mathematics course, Math 201, for prospective elementary teachers. The design and goals of this particular section of Math 201 were closely related to the goals advocated by mathematics educators in the current reform efforts (e. g., NC'I'M, 1989, 1991 ; NRC, 1989). The main question informing this study was: What do prospective elementary teachers learn about mathematical content and reasoning in a conceptually oriented mathematics course? Subsidiary questions included: OIn what ways and to what extent do students understand the mathematical content presented in Math 201? 0What strategies and patterns of reasoning do students use when solving problems related to the mathematical content in Math 201? OIn what ways and to what extent do students apply their knowledge of this mathematical content to solve problems or create new mathematical knowledge? The research reported here is part of a longitudinal intervention study, the Elementary Mathematics Study (EMS)1, in an elementary teacher education 1EMS is a companion study to the larger Teacher Education and Learning to Teach Study of the National Center for Research on Teacher learning (NCRTL), Michigan State University. The 50 51 program, Academic Learningz, at Michigan State University. The Academic Learning Program emphasizes the development of thorough subject matter understanding as well as knowledge of how students learn in each subject area and how to teach each subject matter effectively. EMS is a longitudinal research project (1987-1992) studying the change in preservice and beginning teachers' perceptions and beliefs about mathematics, what it means to know mathematics, and how mathematics is learned (see Schram et al. 1988; Schram 8: Wilcox, 1988; Schram, Wilcox, Lappan, 8: Lanier, 1989; Lappan 8: Even, 1989; Wilcox et al., 1991; Schram, Wilcox, Lappan, 8: Lanier, 1992; Wilcox, Lanier, Schram, 8: Lappan, 1992). Perry Lanier3 directed the EMS project. Glenda Lappan4 was associate director of the project and principal designer and instructor for the sequence of mathematics courses. Ruhama EvenS participated in the conceptualization and development of the sequence of mathematics courses and the research instruments. Sandra Wilcox6 and I were researchers for the project and did not participate as instructors; however, we did participate in discussions about the conceptualization and development of the sequence of mathematics courses. The EMS project focused on knowledge and contextual constraints in implementing a conceptual approach to mathematics in elementary Center was formerly known as the National Center for Research on Teacher Education (NCRTE) (1985—1990), the Center was renamed in 1991. 2T'he Academic learning Program is one of five alternative teacher education programs at MSU. 3Perry Lanier is a professor in the Department of Teacher Education. 4Glenda Lappan is a professor in the Department of Mathematics. 5Ruhama Even was a graduate assistant/ curriculum developer on the EMS Project (1987—1990). She completed her doctorate in mathematics education at MSU and has returned to Israel where she joined the staff of the Weizmann Institute in Rehovot. 6Sandra Wilcox assistant professor, in the Department of Teacher Education at MSU, is a senior researcher in the NCRTL. 52 classrooms. The study investigated how the intervention of the mathematics sequence influenced the responses teacher candidates initially brought to questions such as: What is mathematics and what does it mean to know mathematics? What should children study and what are guiding principles for selecting curriculum in elementary school mathematics? How is mathematics learned and what are effective ways of building mathematical experiences for children? What is the teacher's role and what does she/ he need to know to teach elementary mathematics? What are mathematics classes like and what causes them to be this way? During the EMS project, the Academic Learning Program required prospective elementary teachers to take a sequence of three mathematics courses. These courses were designed, piloted, and taught from 1986-1989. During this same time period, many of the reform documents, including the NCTM Curriculum and Evaluation Standards (1989) and the N C TM Professional Teaching Standards (1991), were in various stages of development and drafting. Glenda Lappan, primary instructor and course designer, was an integral participant in crafting the NCTM documents. She chaired the grades 5-8 group of the Curriculum Standards and was the overall chairperson for the Professional Teaching Standards. Analysis of the EMS Project was one component from the body of literature on which the authors of the Professional Teaching Standards drew. Thus, it is not surprising that the mathematics courses and the NCTM documents shared many philosophical similarities and emphasis areas. In addition to the EMS, the Academic Learning cohort also participated in a study conducted by the National Center for Research on Teacher Education (NCRTE). Participation in the NCRTE study included periodic completion of questionnaires during their two year teacher education I nil 53 program. The EMS Project did not focus on the mathematical content students were learning. Thus, my study was intended to be a complementary study and was independent from the longitudinal study in that research questions and interview protocols were different7. Design There were 28 students in the cohort of prospective elementary teachers. 16 were randomly selected to participate in the EMS or NCRTE project leaving 12 from which to select the six for my study. The six were selected using "purposeful sampling" (Bogdan 8: Biklen, 1982). This research procedure [purposeful sampling] insures that a variety of types of subjects are included, but it does not tell you how many, nor in what proportion the types appear in the population. . . . You choose particular subjects to include because they are believed to facilitate the expansion of the developing theory (p. 67). I selected the six along the dimensions of age, university GPA's, university mathematics courses taken, and subject area sections for TE 200 and TE 2053. Cphort profile 7Human subjects approval was granted independently from the EMS Project. Participants were informed about the purpose of this study and they signed separate consent forms to participate in this dissertation study. Careful attention was given to provide confidentiality to participants and to ensure that responses were not shared with instructors during the evaluation period. 8The first two courses in the Academic Learning Program are re 200 and TB 205. TB 200, Learning of School Subjects, focuses on "theory and practice of the personal and social dimensions of teaching, including communication skills, interpersonal and group dynamics, and personal educational philosophy“ (MSU, 1987). TE 205, Curriculum for Academic Learning, focuses on "effects of curriculum on understanding of academic subjects. Political and cultural influences on curriculum. Teachers' use of curriculum" (MSU, 1987). The entire Academic Learning Cohort (65 students including secondary and elementary majors) take these courses together. They meet together for large group lectures and then the students select subject area sections for small group discussions that are led by Academic Learning faculty. The students in this study were enrolled in TE 200 during fall, 1987 and TE 205 during winter, 1988. 022 females, 6 males 0Age range: 20-47; modal age of 21 with 5 students 25 or older OUniversity GPA range (possible 4.0)9: 2.81 to 3.72; average 3.399 OPrevious university mathematics courseslo—all students had taken at least one University mathematics course Math 082/ 1043 Intermediate Algebra“: (1 student) Math 108 College Algebra and Trigonometry I (3 students) Math 109 College Algebra and Trigonometry II (2 students) Math 111 College Algebra and Trigonometry12 (6 students) Math 112 Calculus and Analytic Geometry I (7 students) Math 113 Calculus and Analytic Geometry 11 (2 students) Math 201 Mathematics Foundations for Elementary School Teachers13 (19 students) Math 214 Calculus and Analytic Geometry III (1 student) Math 215 Calculus and Analytic Geometry IV (1 student) GTE 200 and TE 205 subject area sections: Social Studies (10 students) Science (6 students) Language Arts (6 students) Math (6 students) The following are profiles for the six students in this study: 9Some researchers might argue that GPA is an important measure when assessing the quality of prospective teachers so I included it as a dimension. loAnother variable that is sometimes used in assessing the quality of prospective mathematics teachers is the number and kind of mathematics courses taken. 11Remedial—developmental-preparatory course. 12Students' enrollment in Math 108, 109 or 111 is determined by placement test scores. Math 108- 109 is a two quarter version of Math 111. 13The content of this Math 201 was different from the version described in this study. 55 Andrea“, 21 years old, University GPA 3.63, University math courses taken: 112 Honors (Calculus and Analytic Geometry 1); Social Studies subject area section Tamara, 20 years old, University GPA 3.26, University math courses taken: 201 (Mathematics Foundations for Elementary School Teachers); Language Arts subject area section Tim, 20 years old, University GPA 3.57, University math courses taken: 111 (College Algebra and Trigonometry), 201 (Mathematics Foundations for Elementary School Teachers); Science subject area section Jason, 21 years old, University GPA 292, University math courses taken: 111 (College Algebra and Trigonometry), 112 (Calculus and Analytic Geometry I), 113 (Calculus and Analytic Geometry 11), 214 (Calculus and Analytic Geometry III), 215 (Calculus and Analytic Geometry IV); Social Studies subject area section Kim, 32 years old, University GPA 2.81, University math courses taken: 082/1043 (Intermediate Algebra- remedial course), 201 (Mathematics Foundations for Elementary School Teachers); Math subject area section Linda, 21 years old, University GPA 3.69, University math courses taken: 108 (College Algebra and Trigonometry I), 109 (College Algebra and Trigonometry II), 201 (Mathematics Foundations for Elementary School Teachers); Science subject area section For the purposes of this study, only one section of the first course of the sequence was examined. The first course focused on an exploration of numbers and number theory and emphasized patterns, relationships, and multiple representations of problem situations. The course was divided into 14All participant names are pseudonyms. 56 two sections and also included an overview of the sequence of mathematics courses. The two primary sections focused on number theory and sequences. Observations of the entire course were made, but only the number theory section, consisting of ten, two-hour class sessions, was analyzed. Number theory represents a piece of mathematics that focuses on structure and the close interrelations of a set of math concepts, facts, and skills. Number theory takes mathematics itself as a context for study. The rationale for focusing on the number theory piece is described in chapter 3. The questions, outlined at the beginning of this chapter, focused the data collection. I wanted to know what students thought about number theory concepts prior to taking the Math 201 course. The interview conducted at the beginning and end of the course (i.e., A1 and A2) was structured to capture the knowledge and understanding that students brought to Math 201 related to number theory. Bogdan and Biklen (1982) argue that researchers should consider planning data collection sessions to address issues that emerge during observations and to help one think about the question: "What is it I do not yet know?" (Bogdan 8: Biklen, p. 149). I constructed a different interview (i.e., Interview B) after the Math 201 course began. The questions for Interview B were based upon Math 201 observations and addressed issues not raised in Interview A. Data sources Sources of data included four interviews (see Appendix B for complete interview protocol) that were audio-recorded and transcribed, a NCRTE questionnaire, an EMS questionnaire, twenty-one classroom observations audio—recorded and field notes, Math 201 assignments, quizzes, and tests. Interview A1 was conducted at the beginning of the Math 201 course, March, 1988; Interview B was conducted at the conclusion of the focus on number 57 theory topies, May, 1988; Interview A2 was conducted at the conclusion of the Math 201 course, June, 1988; and Interview C was conducted at the conclusion of the sequence of math courses, June, 1989. Interviews A1 and A2 consisted of the same set of questions, B was composed of a different set of questions, and C repeated selected questions from A and B. The data from interview C is discussed in conjunction with implications and significance of the study (see chapter 6) and is not included in the analysis described in chapters 3-5. W: An interview with the Math 201 instructor provided insight into the number theory concepts that would be highlighted during the course—factors; multiples; primes and composites. I created a concept map (see Figure 21) to highlight these and related number theory concepts (e.g., Fundamental Theorem of Arithmetic, least common multiple, greatest common factor). u us o 883:. x .o . t can! u 8 063.9 moi 9.5305 .3320 :58 to: £35; 2:853 Egan 3:5. .9ch 0:95 8.5853: been: .833 .0 3.689.”. 5.8... £25.... Maggi .0 E2022. .acoEavcau grease: 2:62.. l i 033:: 93:52 .8352 Figure 2.1: Number Theory Concept Map 59 I examined several mathematics textbooks designed for prospective elementary mathematics teachers and talked to other mathematics educators to stimulate my thinking as I began to generate a pool of tasks and questions that might provide insight into the ways students think about factors, multiples, primes and composites. The research questions that guided my Study suggested the need for a variety of questions including straight-forward questions about basic notions of factors, multiples, and primes and composites; application questions; and questions that embedded the mathematics in a variety of representations. A pilot interview was constructed. Pilot participants included prospective elementary teachers who were in their junior and senior years. I e>'<£amined the pilot interviews with attention to the clarity of questions, the q‘l-l ality of the answers given, and the time needed to complete the interview. I Shared copies of the interview with colleagues in mathematics and mathematics education. The interview was piloted and revised during Winter Term, 1988. After piloting the interview, a 3 x 3 matrix (see Table 2.1) was generated to reflect the mathematical topics and types of questions to be included in the reVised version of the interview. Factors, multiples, and primes and Q<>Inposites represented the particular mathematical content and procedure, applications, and representations described the types of questions. Questions were generated to represent a cross section of the matrix. The numbers in each cell indicate the number of questions for that cell in the version of the itlterview used in the study. 60 Questions were coded as procedures if they could be answered by retrieving a procedure/ formula, rule or definition from past mathematical experiences. An example of a Factor x Procedure ce1115 was: What is the greatest common factor of 8 and 10? How do you know? Is there any other way you could figure it out? Application questions required taking ideas about a particular concept or idea arid applying it to solve a problem. An example of a Factor x Application cell was~ 1523 .5 a factor of24 . 32 0 5? If yes, how do you know? If no, could you change it to become a factor of 24 ~32 . 5? Representation questions included questions that illustrated a range of ways- numerical, graphical, concrete materialsuto represent ideas about factors, II‘I-lsaltiples, primes and composites. Representation questions primarily f(>C‘used on alternative representations. "Alternative" was defined as a representation that did not usually appear in traditional instruction about the mathematical topic or idea. An example of a Factor x Representation cell was: Here are 24 square tiles. How many different rectangular squares can you make by using all the tiles? [After the person stops, ask]: Do you have all of the possible rectangular shapes? Explain. Many of the questions were related to more than one cell and were coded based upon the primary emphasis. A few questions seemed to have an equal e1'Irphasis and were categorized in multiple cells. For example, the question, The greatest common factor of 630 and 1716 is 6. What is their least common multiple? was coded as Factor x Application cell as well as Multiple x Application cell. \ 1F’See Appendix B for a complete coding of other interview questions. 61 Table 2.1: Interview Matrix Procedures 4 3 2 Factors Mtrlti Primes/ mtewiew A consisted of twenty-six questions related to the major number tlrxeory ideas—«factors, multiples, and primes/ composites-to be explored in Math 201. These number theory ideas were identified by the Math 201 I1='\.str'uctor (interview, 2 / 88). Interview A was constructed without the benefit of knowing the day- to~1, can be expressed as a product of primes, this representation is unique, apart from the order in which the factors occur (Burton, 1980). 3For example, 8 = 3+5; 36 . 31 +5; 100 = 97+3. 92 uncover the two original large primes“. Phillips (1991) describes what this process can entail: The difficulty of breaking these codes depends on the difficulty of factoring a composite number with 100 or more digits into prime factors, each having 50 or more digits. It is estimated that such a problem would require over a billion years on the largest imaginable supercomputer (p. 21). Thus cryptographers can easily create "codes" that are virtually impossible, even with the aid of powerful computers, to decipher5. An analogy that can be helpful in thinking about the power of this idea is DNA fingerprinting. It is fascinating to think that each person possesses a unique DNA "fingerprint", but the real power comes when one starts with the fingerprint and can trace it back to identify a unique person. Patterns are an intricate component to understanding and creating mathematics, yet it is the relationship of patterns to primes that creates part of the mystique surrounding the study of primes. Golos (1981) describes the tension: The most perplexing, frustrating and fascinating thing about the primes lies in their patterns, or worse, their lack of them. On the one hand, they do not seem to be randomly scattered throughout the counting numbers, and yet, no one has ever (in over 2,000 years of intensive study) found a precise rule to describe a pattern (p. 173). 4As of 1988, the largest prime found is 2216091 - 1; after several months of preparation it took a computer, performing 400 million calculations per second, three hours to determine this prime. This number has 65,050 digits (Phillips, 1991 ). 5For an elaboration on this coding process, see Buxton, 1985. 93 Patterns and relationships among patterns play a dominant role in understanding additive and multiplicative structures, two "conceptual fields‘" (Vergnaud, 1983) embedded in the study of number theory. Additive and Multiplicative Structures. Additive structures refers to thinking about building numbers by continuing to add the same quantity to a starting number. In particular, we can begin with one and continue to add one. Symbolically we would represent this as n --> n+1. Vergnaud (1988) identified several concepts involved in additive structures: "cardinal, measure, state, transformation, comparison, difference, inversion and directed number" (p. 35). Multiplicative structure refers to the notion of numbers generated as products of primes. This multiplicative structure is based on one as a unit and the Fundamental Theorem of Arithmetic which tells us all whole numbers other than one can be written in a unique way as a product of primes. Vergnaud (1988) defines multiplicative structures: The set of situations that involve the multiplication or the division of two numbers, or a combination of such operations. Most of these situations are in fact simple—proportion or multiple-proportion problems, in which two variables are proportional to each other (simple proportion), or one variable is proportional to several other independent variables (multiple proportion) (p. 35). Concepts related to the study of multiplicative structures include: fraction, ratio, rational number, linear and n-linear function7, dimensional analysis, 6Vergnaud defines conceptual field as "a set of problems and situations for the treatment of which concepts, procedures, and representations of different but narrowly interconnected types are necessary" (p. 127). 7Alinearfunctionisa function which canbedescribed byanequationoftheformysmx+b, where m and b are constants. The graph is a line with slope m and y-intercept b. 94 and vector space8 (Vergnaud, 1983). Vergnaud identifies three different types of problems within multiplicative structures: "(a) isomorphism of measures9, (b) product of measures1 0, and (c) multiple proportion other than product“" (p. 128). Vergnaud cites "equal sharing, constant price, uniform speed or constant average speed, constant density on a line, on a surface, or in a volume" as examples of isomorphism of measure situations (p. 129). "Area, volume, Cartesian product, and work" are among the examples he describes as product of measure problems (p. 134). The difference among numbers multiplicatively form one of the fundamental insights behind success in algebra. . . . Knowing the multiplicative biographies of the integers will add to success in many aspects of algebra (Pollak, 1987, p. 256). Number theory topics are often found in elementary school textbooks. An interview with the primary instructor for Math 201 provided her perspective about the value of exploring number theory concepts with prospective elementary teachers. Excerpts from the interview follow: 1: Why do you think it is important to include number theory in a course for prospective elementary teachers? Lappan: It is sound and important mathematically for kids to know and therefore teachers must have deep knowledge of the concepts and connections. It's the 8Vectors are quantities that have both magnitude and direction. 9"isomorphism of measures is a structure that consists of a simple direct proportion between two measure-spaces M1 and M2" (p. 129). 10"product of measures is a structure that consists of the Cartesian composition of two measure- spaces, M1 and M2, into a third, M3" (p. 134). 11"multiple proportion is a structure very similar to the product from the point of view of the arithmetic relationships: a measure-space M3 is proportional to two different independent measure-spaces M1 and M2. . . In multiple proportion, the magnitudes involved have their own intrinsic meaning, and none of them can be reduced to a product of the others" (p.138). 95 backbone of an awful lot of what kids have to understand to really come to grips with the notions of fractions but it also underlies a lot of mathematical thinking that builds as kids move from the world of addition and subtraction into the world of multiplicative structures. An understanding of number which is fundamental to everything these students are going to be dealing with in elementary and middle school, no matter whether you are talking about number as it shows up in applications, number as it shows up in handling data, or number in terms of building students' understanding of addition, subtraction, multiplication, and division. Fundamental to understanding numbers is the additive structure of numbers which is one of the first mathematical ideas that kids meet. Two is one more than one—one plus one is two, two plus one is three. This includes counting and all that early experience with putting together and taking apart numbers of objects. The area of number theory gives students another look at the structure of numbers but from a multiplicative point of view. What other numbers are embedded in this number in a multiplicative way? What are the divisors? What are the factors? What are the multiples of this number? As students think about factors and multiples, they are beginning to look at repeated multiplication which moves into exponentiation, which opens the door for all of the very powerful mathematics that is based on multiplication as the underlying structure of the numbers as opposed to addition. Exponential growth and decay and logarithmic functions flow from an understanding of repeated multiplication. These preservice teachers need deep understanding of factors and multiples to begin to understand operations on fractions. Fundamental to all of this is really understanding the operations of multiplication and division and their relationship to each other. 96 I wanted students to have great facility with breaking numbers apart, seeing the structure of a number, seeing its relationship to another number, coming to understand divisibility, coming to understand the notion of starting with a number, then multiplying by the counting numbers to generate the whole collection of multiples of that number. You have produced this infinite set of numbers that have something in common. They are all divisible by the number used to generate the set. We wanted to introduce the students to all of those subtle ways of looking at the structure of numbers, looking for patterns we wanted to help students develop flexibility with primes, composites, factors, multiples, prime factorization of a number, that whole set of mathematical ideas, and we wanted to do this through interesting problems situations. We tried to keep a problem solving atmosphere in the class as we explored these concepts (6/ 88). Previous Exgriences with Number Theogy Concepts. Most students are introduced to the number theory concepts / ideas explored in Math 201 during the middle school years. While it was impossible to go back in time and analyze the particular middle school experiences for the students in this study, an examination of three textbook series that were popular during the time that these students were in middle school (1978-1982) proved helpful (See Appendix C for an analysis of individual texts.). An analysis of the textbooks provided insight into some of the types of experiences these students might have had related to number theory concepts—factors, multiples, primes and composites. The following categories were.used for the analysis: Ways in which these number theory topics were introduced—factors including common factors and greatest common factors, multiples including common multiples and least common multiples, and primes and composites; connections made among these topics; 97 mathematical ideas and topics that followed each one. Common themes that emerged from this analysis were that the topics were treated in isolation and there were few, if any, connections made among any of the number theory t0pics. What follows is an illustration of the ways in which these topics were introduced in the middle school textbooks. A lesson about multiples follows a page emphasizing multiplication basic facts (e.g., 3 x 4:12, 8 x 7:56). The format of the lesson was a series of examples labeled as follows: Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27 Multiples of 4: 4, 8, 12, 16, 20, 24, 28 Common multiples: 12, 24 Least common multiple: 12 It is possible for a student to mimic the example provided and get 100% on the practice problems with little or no understanding about the concepts being used. The next lesson introduces multiples of 10, 100, 1000 but does not build on the organized listing used the previous day. Rather, the emphasis is on the number of zeros to use—determined by the number of zeros in the problem”. The next reference to multiples occurs 75 pages later in a lesson entitled, "Finding Factors and Multiples." The focus is on the 1—100 chart and patterns related to common multiples within that. There is no evidence that any work is done on the relationship between factors and multiples or any applications for either. The next lesson focuses on finding primes and composites and the language suddenly shifts, without explanation, from factors to divisors. Definitions are given for prime and composite and two statements are made—The numbers 0 and 1 are 12If you are multiplying by 1000, it has 3 zeros so you add 3 zeros to the number being multiplied. E.g., 60 x 1000 = 60,000. 98 neither prime nor composite. All numbers greater than 1 are either prime or composite. Within these middle school texts number theory concepts are not developed in a meaningful way. The lack of connections was quite prominent in all of the textbooks examined. In one text, multiples were introduced on p. 66 which was 24 pages before factors were mentioned. No applications were provided. The pictures on the page were a row of houses—a farm house and barn, a house that might be found in the suburbs, and an apartment building that might be found in the city. These pictures seemed totally irrelevant to thinking about multiples. This lesson was followed by an unrelated page of "problem solving" that concluded the chapter. Factors were introduced with a series of lists—factors, common factors, and greatest common factors. There was no comment about factor pairs, just a list of factors, and no applications were provided. Again, the pictures on the page were irrelevant and the next lesson was another "problem solving" page that was unrelated and concluded the chapter. Following the practice chapter test was a page entitled "Something Extra." The "something extra" was a lesson about prime numbers. There were three examples of prime numbers followed by a definition. Students were then required to copy a list of numbers from 1 to 50 and instructed to circle and cross off various multiples. The statement was made that "the circled numbers and all numbers not crossed off in the chart are prime numbers." No applications were provided. None of these topics appear anywhere else in the text. If these textbook examples are representative of the experiences these prospective teachers encountered related to number theory, it is likely that their perceptions of these concepts are primarily as bits and pieces of isolated facts and procedures. Recent studies about textbooks support the notion that 99 textbooks lessons are constructed in ways that promote a view of mathematics as isolated bits and pieces (see e.g., Tyson 8: Woodward, 1989). Stodolsky's study (1988) about teaching mathematics and more general studies (e.g., Goodlad, 1984; Powell et al., 1985) describe the low-level, routine tasks, repetitive lessons that occur in many classrooms at all grade levels. Porter (1989) describes a study about 5th grade teachers' focus on the four basic operations and computational skills with little attention to other topics. He also found that elementary mathematics teachers tended to focus on coverage rather than in-depth understanding of content. Doyle (1986) argues that classrooms are easier to manage when students engage in routine tasks. While the studies cited do not reflect the particular instruction in which this set of students engaged, they provide evidence of wide-spread themes that occur during typical instruction. Thus, it is likely that these prospective teachers' experiences related to these number theory concepts were embedded in a traditional instruction perspective. Number theory is not part of the standard high school curriculum so formal instruction related to number theory occurs during the middle school years. It is often during the middle school years that students develop the belief that school mathematics lacks meaning and is unrelated to their lives Meme 8: Hiebert, 1988). Many students are fully entrenched in a model of viewing mathematics as manipulating symbols. . . As students progress in school and begin to use symbols to represent the mathematical ideas, however, the symbols begin to live a life of their own without strong attachment to any conceptual network. Students manipulate the symbols by recalling and applying memorized rules. . . The student who attempts to make sense of the various manipulations is something of an anomaly. (p. 220). 1 00 If these prospective teachers in this study experienced typical mathematical instruction similar to that outlined in the examined middle school textbooks, it is unlikely that they attempted to reason about the symbol manipulation or rules they memorized. Comments during student interviews related to previous mathematical experiences support the hypothesis that these students' mathematical experiences, including their experiences with number theory concepts, were consistent with the experiences suggested by the middle school textbooks analyzed. Math 201 Course Analysis In an effort to challenge prospective teachers' beliefs about mathematics developed over years of traditional instruction, Math 201 instructors argued that prospective teachers should experience mathematics in a learning environment that was similar to that advocated for students (Lappan 8: Even, 1989). Thus these future teachers would have first-hand experiences from which to draw when making curricular and instructional decisions for their future students. Since the instructors designed Math 201 experiences to model the type of instruction they encourage for children, three of the standards—tasks, discourse, and environment--from the "Standards for Teaching Mathematics" section of the NCTM Professional Teaching Standards (1991) will be used to analyze the Math 201 course. As noted in the Standards introduction, teaching is a complex process and all of these components- tasks, discourse, environment, and analysis—are fundamentally intertwined. While acknowledging the difficulty in isolating a given Standard, there are important criteria related to each that deserves examination. Thus, each will be highlighted in an individual section. For each Standard, four perspectives will be discussed: (1) the traditional perspective, vignettes and analysis 1 01 related to traditional mathematics instruction; (2) the reform perspective, arguments related to each Standard drawn from NCTM Professional Standards and other reform documents; (3) the Math 201 perspective, vignettes and analysis related to classroom observations and instructor interviews; and (4) the students' perspectives, excerpts from interview data related to each Standard. Tasks Traditional perspective. The vignette that follows is a composite account of a mathematics lesson that is representative of a classroom that might be using any of the middle school texts analyzed”. As argued earlier, these textbooks are representative of the kinds of experiences these students had related to number theory concepts. Ms. Rankin: What is 1 times 3? Sally: 3. Ms. Rankin: [writes 1 x 3 = 3 on the board] What is 2 times 3? Tom. Tom: 6. Ms. Rankin: [writes 2 x 3 = 6 on the board] What is 3 times 3? Tasha. Tasha: 9. Ms. Rankin: [writes 3 x 3 = 9 on the board.] What is 4 times 3? Jim. 13The composite is based upon middle school textbooks analyzed. See Appendix C. Jim: Ms. Rankin: Bill: Ms. Rankin: 102 12. [writes 4 x 3 = 12 on the board and says]: What is 5 times 3? Bill. 15. [Ms.Rankinwrites5x3=15onthe board. She then says as she writes on the board]: The numbers 3, 6, 9, 12, and 15 are multiples of 3. [Ms. Rankin writes multiple on the board.) The list of all multiples for a number is endless so we can continue our list of multiples of 3 by writing, [adds to the list on the board] 18, 21, 24, 27, . . . Now let's list the multiples of 4 [writes on board]: Multiples of 4: 4, 8, 12, 16, 20, 24, . . . 12 and 24 are common multiples of 3 and 4 but 12 is the least common multiple of 3 and 4. [writes least common multiple on the board]. Here are the steps to follow to find the least common multiple for numbers. [Ms. Rankin writes these steps on the board.] Analysis of task: What does a student need to know about multiples to complete this exercise? How does a student know if he has found enough multiplies? What do multiplies relate or connect to? This task requires little more than skip counting by different numbers. This task is not likely to engage students intellectually. There is nothing to reason about. This task exists in isolation. There is nothing about which to problem solve. Step 1: List the first 7 multiples of one number. List the first 7 4: 4, 8, 1_2, 16, 20, 2515, 28 6: 6, 1_2_, 18, 2:1, 30, 36, 42 multiples of the other number. Compare. Make sure that you have at least one number in each list that is the same. For example: Step 2: List the common multiples of the two numbers. 12, 24. Step 3: List the least common multiple of the two numbers. 12. 103 Mrs. Rankin: Okay class, let's try one together. 6 and 9. [Mrs. Rankin allows time for the students to do this one on their paper.] OK. Jill what did you get? Jill: 6: 6, 12, 18, 24, 30, 36, 42. 9: 9, 18, 27, 36, 45, 54, 63. 18, 36 are common multiples. And 18 is the least common multiple. Mrs. Rankin: Good work, Jill. Any questions? Does everyone understand? Now class, I would like for you to do problems 1-20 on page 142. [Page 142] Find the least common multiple for each pair of numbers. (1)2and8 (2)10and5 (3)6and9 (4)3and7 (5)8and5 (6)2and9 (7)6and8 (8)6and8 (9)4and10 (10)6and7 (11) 2 and 5 (12) 3 and 9 (13) 4 and 8 (l4) 5 and 10 (15) 7 and 8 (16) 3 and 6 (17) 4 and 7 (18) 2 and 3 (19) 3 and 4 (20) 5 and 15 Classroom teachers teaching mathematics from a traditional perspective rely on textbook authors to prescribe tasks for students to do. The "lesson for the day" is laid out in a two-page format in which the "new" concept is introduced by listing definitions, rules, and/ or formulas. A few examples of problems are provided and often a series of steps are given for students to follow or imitate to solve a similar set of problems. On the second page one finds a set of problems, similar to the examples provided on the previous page, that are progressively more difficult and often conclude with a challenging problem(s) for the "smart" students. 1 04 Reform perspective. The Professional Teaching Standards (N CTM, 1991) argue for a different rationale related to task selection: Tasks should: 0 integrate mathematical thinking with mathematical concepts or skills 0 capture students' curiosity 0 invite students to speculate and to pursue their hunches 0 nest skill development in the context of problem solving (NCTM, 1991, Transparency 96). Madsen-Nason and Lanier's (1986) work with the General Mathematies Project also argues the value of task selection. The study examines the ways in which a set of students are thinking about particular pieces of mathematics content but the mathematics is embedded in the tasks students have the opportunity to experience. Lanier (1986) argues that teachers should consider, "What is it that you want your students to learn, not what do you want your students to do?" as a major criteria for task selection. Doyle's (1983) study of academic work in elementary and secondary classrooms examines how the tasks (i.e., work) that students are asked to do provides the context for learning. Doyle argues that the kind of tasks (i.e., work) that students engage in influences students' beliefs about the subject area in which the tasks are embedded. Reform efforts imply that teachers have an important role in terms of task selection. Problems should be limited to one or two problem situations that actively engage students in problem solving. Tasks should communicate to students that mathematics is a dynamic and interesting discipline. The tasks that students explore should enable students to see the powerful connections among mathematical concepts and ideas. The task provides the 1 05 context for the mathematics students have an opportunity to learn in a mathematics class. Math 201 course mrspective. The locker problem described earlier in this chapter provides the vignette for this section. The interview that follows provides the Math 201 instructor's rationale for selecting the locker problem as a rich task for the Math 201 students to explore. Lappan: When I choose to do a problem like the locker problem I think about what kind of story shell will allow us to probe the mathematical ideas in it in a significant way. But to probe them in a context that the kids can imagine, so to a certain extent you're engaging their ability to visualize, to build mental representations. In this case we could even build physical representations, act out a problem. It's a situation where you can look at the notions of prime, factor, and multiple, and special characteristics of square numbers. But you can actually do it. You can see yourself in your mind doing this problem, or you can actually walk down a series of mock lockers and tactually do the problem. It's a story shell that has all of the richness of the interrelationships between factors, divisors, and primes. We discover that the square numbers have an odd number of factors, and all the other numbers seem to have an even number of factors. Those were things that the situation allowed you to probe, but to probe so that the story shell and the mathematical problems had an almost direct relationship to each other. The problem embedded all of the ideas with which we had worked into this one setting. In addition, the problem encouraged conjecturing, gathering evidence to see whether or not your conjectures held up, looking at patterns, devising a strategy for how you'd keep track of the simulation. How far should the simulation go? How far was convincing? Once you saw in your patterns that the locker numbers that were closed were one, four, nine, 16, 25, then you were stuck 106 with the problem of trying to figure out why that was happening. This takes you back to the structure of numbers (6/88). Tasks were selected for the Math 201 class to provide opportunities for students to encounter mathematics in a different way. Problems were selected to engage students in mathematical inquiry. Students were encouraged to explore the richness of the embedded mathematics and to discover the patterns and relationships among mathematical ideas. Students' persmctives. Students' comparisons between Math 201 and other math classes included references to the mathematical tasks they were asked to do. The students' description of tasks required in other math classes were of a technical and procedural nature while the tasks given in Math 201 were described as requiring work with ideas and concepts. 1113: This class is totally different than any other math course I've had. They [Math 201 instructors] require you to think a lot. . . In this class we have to think through a problem and not just plug and chug with formulas like I used to do in other math classes. (5/ 9 / 88, p. 5). Students' descriptions of other math classes were similar to Tim's—plugging numbers into given formulas, doing large numbers of similar problems from a ditto or textbook, memorizing large quantities of information and thinking was rarely required. As students described Math 201, a different image emerged-thinking, figuring out why something works, persisting, using a variety of strategies to solve problems. Jason and Andrea's comments are representative of the contrasting image painted by other students: lason: In this class [Math 201] we look at interesting type problems, it's not just a ditto sheet of "Find the least common multiple," it's "There are a thousand lockers 107 in this hallway and which ones are visited by which students and why. You really have to push yourself to think." (5/88, p.12). An rea: I've had a lot of math and I've seen a lot of these problems but I've never had to conceptually think about them. I've just known a formula and stuck numbers into the formula. In this class, I've really tried to push myself to think about why something works and it takes a lot longer at first, but after awhile you start to get the hang of thinking that way and it ends up being a lot more challenging and a lot more interesting than the way I've always done it before. . . Before, I used to do problem after problem after problem, and then make sure I knew how to do every kind of problem and I would be able to work a problem until I could fit it into a formula. . . . Now, I draw a lot of pictures, I think, and use a lot of trial and error, it's a little bit different kind of trial and error because I'm working with more concrete things than abstract. And I like to have my math paper, well I write all over them and then they're all messy. I'll try anything, and if it doesn't work, I'll just try again, that's how I usually do it now. (6/ 88, p. 13-14). Most of the students' references to mathematical tasks were described in the context of class activities, but references also occurred in the context of comparing the nature and purposes of tests. As Tamara compared Math 201 to other math classes she included the following remarks about testing: Tamara: The tests are different. It's not just plug in numbers and go to it, you have to reason your way, think about how you did it. Or sometimes it will be a variation of another problem and you have to explain all the reasons why you did something instead of just saying, 108 "this is my answer," you have to explain why you got that answer, explain your thinking. It's not like, oh, we're having this test just to get a grade but really we're trying to get a look at how much you know and understand and where to go next. (6/ 88, p.12). One could infer from Tamara's comments that the purpose of testing in other math classes was for evaluative purposes. She perceived that tests in Math 201 served a different function—assessing student understanding and informing instruction. Another student, Tim, also raised the issue of tests but in the context of comparing the process of preparing for tests in Math 201 to other math classes: Tim: I used to learn just by memorizing formulas and stuff because that's what got me good grades on the test. I've always done well in math, but I've always thought that I didn't know math either. Because I would know it for the test, and I'd get an A on the test or something and then next year if I had a friend or something that wanted help on a problem, I'd say, yeah, well, I've seen that before, but I couldn't do it. . . . Now, it's the night before the test, and I'll study for about 15 minutes. I don't have to cram or anything because we've actually done it in class and worked through it and had a chance to internalize it. (6 / 88, p. 15). The tasks that teachers select provide the grist for classroom interactions. The mathematical tasks that students engage in sends strong implicit and explicit messages about what it means to know mathematics and what it means to do mathematics. Tasks that fail to intellectually engage students imply that mathematics is a sterile and technical discipline. Meaningful tasks convey messages that mathematics is a dynamic, humanistic discipline. The tasks also provide the contexts for students to 1 09 explore mathematical concepts and ideas. Does the task provide an isolated view of concepts or does it promote an image of connectedness and relationships among mathematical concepts and representations? Making good decisions about task selection is critical to the possibilities for what students can and will learn during mathematics instruction. Discourse Traditional perspective. The vignette that follows is a composite account of a mathematics lesson that is representative of a classroom that might be using any of the middle school texts analyzed“. Ms. Rankin: Who can name 2 numbers which Analyst? 0‘ discourse: can be multiplied to make 20? Teacher '3 role. The teacher provided an example for Sally: 4 times 5. students to follow in order to complete the assignment; the teacher was considered as the Ms. Rankin: [writes 4 x 5 = 20 on the board] authority to d I .m Tom. whether or not students' answers were right or wrong; Tom: 2 times 1 0. no discussion or debate about why answers were "right" or "wrong"; the teacher Ms. Rankin: [writes 2 x 10 = 20 on the board] provided practice for the Tasha. students to do; the teacher did not ask any of the students to justify their Tasha: Negative 4 times negative 5. answem, The teacher does not encourage nor explore students' thinking when an Ms. Rankin: You can't use negative numbers to . . alternative answer is do these problems. Jim. suggested that does not fit the teacher‘s preconceived ways lim: 5 times 4. of thinking about the problem. 14T'he similarity to the previous traditional perspective vignette was deliberate. Typical traditional instruction basically uses the same generic template for instruction. Ms. Rankin: Bill: Ms. Rankin: 110 5 times 4 is the same as 4 x 5. Bill. 1 times 20. [Ms. Rankin writes I x 20 = 20 on the board. She then writes 1, 2, 4, 5, 10, 20.] A number is divisible by its factors. Two factors of 18 are 3 and 6. Here are the steps to follow to find all the factors of 18 and to list them in order. [Ms. Rankin writes these steps on the board] Step 1: Write 18 as the product of two whole numbers in as many ways as possible. 1x18, 2x9, 3x6, 6x3, 9x2, 18x1. Step 2: List the whole numbers in order. Show each only once. 1, 2, 3, 6, 9, 18. Mrs. Rankin: Okay class, let's try one together. 24. [Mrs. Rankin allows time for the students to do this one on their paper.] OK. Jill what did you get? Jill: 6x4=24,2x12=24,3x8=24,1x24=24. 1, 2, 3, 4, 6, 8, 12, 24. Mrs. Rankin: Good work, Jill. Any questions? Does everyone understand? Now class, I would like for you to do problems 1-20 on page 125. Student '5 role. Students were to follow the example that the teacher provided; students were to complete the assigned problems; students were to accept the teacher as the authority for determining right and wrong answers; students were not expected to reason about answers; students were to figure out what the teacher had in mind for answers and were not encouraged to offer alternative ideas. [Page 125] Find all the factors. List them in order. (1) l2 (8) 64 (15) 77 (2) 16 (3)27 (4)36 (5) 42 (9) 104 (10) 100 (11) 24 (12) 150 (16) 60 (17) 88 (18) 56 (19) 33 (6) 25 (7) 54 (13) 72 (14) 39 (20) 225 111 Discourse is limited in the traditional classroom. The teacher dominates the talk as he/ she introduces the new concept for the day and provides examples for the types of problems students will encounter on the homework and / or assignment. Students work independently to complete the assignment and the teacher and/ or textbook serve as the authority for determining correctness of problems. The teacher does not attempt to understand students' patterns of reasoning or problem solving strategies. The process is not valued; completion and percentage of correct problems provide the grist for rewards. Reform perspective. In the NCTM Teaching Standards (1991) discourse is described much differently: The discourse of a classroom—the ways of representing, thinking, talking agreeing, and disagreeing—is central to what students learn about mathematics as a domain of human inquiry with characteristic ways of knowing. Discourse is both the way ideas are exchanged and what the ideas entail. . . . In order for students to develop the ability to formulate problems, to explore, conjecture, and reason logically, to evaluate whether something makes sense, classroom discourse must be founded on mathematical evidence. . . . Students must talk, with one another as well as in response to the teacher. . . . When students make public conjectures and reason with others about mathematics, ideas and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community. . . . The teacher's role is to initiate and orchestrate this kind of discourse and to use it skillfully to foster student learning (p. 34). The traditional model of instruction in which the teacher acts as the "giver" of knowledge and the student as the "receiver" is no longer appropriate. 1 12 Knowledge is not a static product that can be moved from one person‘s mind and "placed" in another. Some type of processing must occur. Romberg (1983) borrows terms from Dewey to argue for a distinction between "knowledge" and "the record of knowledge". The former requires active student construction of meaning while the latter implies absorption of someone's work. Traditional mathematics instruction places emphasis on the absorption of the record of knowledge. Proposed reforms emphasize active student construction of knowledge. The authors of Everybody Counts (NRC, 1989) describe this change as a transition - "The teaching of mathematics is shifting from preoccupation with inculcating routine skills to developing broad-based mathematical power." (p.81). Genuine problem solving and mathematical power are terms that surface often in current mathematics education reform documents. The NCTM Curriculum and Evaluation Standards (1989) describe mathematical power as: Mathematical power denotes an individual's abilities to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods effectively to solve non routine problems. This notion is based on the recognition of mathematics as more than a collection of concepts and skills to be mastered; it includes methods of investigating and reasoning, means of communication, and notions of context. In addition, for each individual, mathematical power involves the development of personal self-confidence. (p. 5). The National Research Council (1989) argues that mathematical power "requires that students be able to discern relations, reason logically, and use a broad spectrum of mathematical methods to solve a wide variety of non routine problems." (p. 82). The use of the term "non routine" supports the 1 13 notion of unpredictableness of future mathematical problems that students may encounter. Thus it is important to make good curricular decisions but perhaps equally important are decisions related to task selection and ways to structure the mathematical environment to provide opportunities for students to develop mathematical power and to engage in genuine problem solving. Math 201 course mrspective. A partial transcript15 shows the collective efforts of four students solving the locker problem described in the task section. This vignette provides a flavor for the kind of discourse that occurred during Math 201 classes. We enter the conversation just as students have decided to check out the first several lockers: S1: So the ninth person goes to locker nine and opens it. 32: What about the factors involved? S3: Seven stayed open until the seventh person got there. Five stayed open. 51: These are primes. S2: Four is closed. 53: But 4 isn't prime. 15Taken from Wilcox et al. (1991). 52: S3: 81: 82: S1: S3: S1: S3: 114 So all primes stay open until that person changes the state: . . . So we know eventually all primes are closed except for one. . . One, four, and nine are open. These are perfect squares. Let's try 4 squared. Just do 16. But you couldn't do just 16 because you might have multiples you have to close or open prior. [to the teacher who has approached this group] We're going to conjecture that perfect squares are open. Why? You have a very good conjecture but why? What is peculiar about square numbers? What is there about the structure of numbers so that primes are closed and many composites are closed? [Teacher moves on to another group.] Primes get touched only by that person. But why are square numbers open? Well, the squares have two people passing over it. . . . Let's look at composite numbers. Collaboration and shared responsibility for learning are two norms that were established in Math 201. Teacher '5 role: asking probing questions, finding ways to extend students' thinking, asking questions based upon the different levels of understanding, posing questions that allowed students to build abstractions and generalizations. Student '3 role: questioning each other, reasoning about ideas, making suggestions about various strategies, trying to explain and validate their mathematical thinking and solutions. S4: 52: $3: 115 It [a composite] gets hit for each factor. Sixis2and3but4is2and 2,and9i53and3. Then why shouldn't composites be open as well? How about if we go back to what you said: 4 is 2 times 2. When you go over it with the first two it closes, when you go over with the second two it opens. With nine, the first three opens it, the second three closes it. The four students pursued this problem together for nearly 30 minutes. It took another set of guiding questions from the teacher to help them rethink what it would mean to have a repeated factor. The teacher's questions helped the students to look more closely at the structure of numbers: What distinguishes composite numbers from square numbers? Look at their structure and see if you can puzzle it out. . . . Try 36. Figure out who is going to touch 36. . . . Now try a number that is nonsquare. Eventually the students concluded that only square numbers have an odd number of factors. The "routine" of the Math 201 classes stood in sharp contrast to the routine of traditional mathematics classes. In the context of small groups, students were encouraged to make conjectures, validate assertions with convincing arguments, and communicate with others as they attempted to make sense of mathematical situations. 116 Students' persmctives. Students' perceptions about the role of the student and the teacher in a math class were embedded in their comparisons between Math 201 and other math classes they had experienced. Their descriptions of previous experiences were consistent with the traditional perspective described above and their descriptions of Math 201 provided evidence that the Math 201 experience was, in many ways, consistent with the reform vision. lason: Tamara: It‘s a lot more relaxed atmosphere, and there is more verbalization and interaction than other math classes I've had where we just sit there and the instructor is reading on an overhead at a thousand miles an hour and we're just cramming it all into writing. Then you have to go home and try to decipher your notes and you weren't actually learning anything unless you were just writing things down, because you didn't have time to process what you were writing. You were just taking it in long enough to get it on the page. It was like from his mouth, through your hand, onto your page and it didn't make any sense. (6/ 88, p.15). Well in other math classes, either you were right or you were wrong. If you were right, you were praised. If you were wrong, well you were shown why you were wrong and how to do it right, there was nothing really good that came out of it. (5/ 9/ 88, p5). In other math classes it always seemed like the teacher knew everything and you were always looking for the "right" answer. I mean, they gave you an equation and they definitely wanted an answer. They didn't really care how you thought about it. Maybe they did, but that's not the way they presented it. As long as you got the right answer, it didn't seem to matter about the way you were thinking. In this class, it seems like the 117 instructors, even the people in the class like to know and will say, "How are you understanding this problem?" or "What's the strategy you're using?" not "What did you get?" (6/ 88, p.11). Jason's comments about student participation compared active and passive student roles: lason: The math experience is more memorable when you've actively worked through it, rather than to be a passive recipient of someone telling you how to do it. (5/9/88, p.12). Students were encouraged to talk about the mathematics that they were doing and all agreed that mathematics discussions were helpful. Their reasons for valuing these discussions varied. Andrea valued talking about math as an aid to clearer understanding: Andrea: When you verbalize your thinking, it gets more, you understand it more yourself. If you try to explain it to other people, it makes you more clear in your own thinking. (5/ 9/ 88, p.13). Tim perceived talking about mathematics as a tool for thinking about the process of solving problems: Em: Talking about math is really helpful because it makes you think about why you're doing something and not just what you ended up with for an answer. She [instructor] never just asks for an answer. If she asks what you got for an answer, she always asks for an explanation. I think it's helpful for us to hear the others explain how they got something because sometimes there is more than one way to go about getting the right answer. (5/ 88, p.10). 118 Jason perceived discussions as helpful in hearing multiple solutions and approaches to solving problems: I ason: We help each other because one person may see a problem from one view and another sees it from a different view. One person understands an aspect of a problem and another person understands another aspect of it. You can put those together and come up with a solution. . . usually after we work in small groups, the teacher brings us back as a large group and we discuss different ideas and we see that even different groups approach problems differently. Part of the environment is that you see that there's more than one way to think about a problem. And there is not always necessarily one way to do it. (5/ 88, p. 11). Students' comments about the role of discussion about mathematics is consistent with other research (see e.g., Pimm, 1987; Hoyle, 1985). Pimm argues: Articulating aspects of a situation can help the speaker to clarify thoughts and meanings, and hence to achieve a greater understanding. . . One force of talking aloud is that it requires the use of words, whereas merely thinking to oneself allows words to be bypassed. It may be only when you discover a difficulty in expressing what you want to say that you realize that things are not quite as you thought. (p. 24-25). Students' comments about issues related to discourse in the Math 201 classes indicate a view that is consistent with the vision advocated by reformers. Students felt that the instructor valued the process they used for thinking about mathematical problems. The descriptions of Math 201 classes support the notion that the students were active learners and had many opportunities to work together and share ideas as they developed ways to think about and 1 19 solve problems. They were encouraged to make conjectures and provide supporting evidence to convince their peers of the merit of their arguments. Students came to value what they could learn from listening to one another. The discourse within the Math 201 class influenced what students learned about the mathematical ideas embedded in problem situations posed. Environment Traditignal firsgctive. The mathematical environment suggested by traditional mathematics instruction promotes an image of teaching mathematics as a technical (how to) course. Textbooks are often set up as a two-page lesson that focuses on a mathematical skill or concept. Often, there are implicit connections to a series of two—page lessons but students may view each lesson in isolation and do not automatically note the connections. Romberg and Carpenter (1986) describe the consequences of traditional mathematics instruction that promotes fragmentation of mathematical concepts and ideas: For schools, the consequences of this traditional view of mathematics are that mathematics is divorced from science and other disciplines and then separated into subjects such as arithmetic, algebra, geometry, trigonometry, and so on. Within each subject, ideas are selected, separated, and reformulated into a rational order. This is followed by subdividing each subject into topics, each topic into studies, each study into lessons, and each lesson into specific facts and skills. This fragmentation has divorced the subject from reality and from inquiry (p.851). The fragmentation of mathematical concepts does not provide opportunities for students to understand how these "pieces" fit together in the larger field of mathematics. Students experience mathematics as a series of rules and algorithms which must somehow be "memorized" and filed for future 120 reference. Traditional instruction does not provide students with a relational understanding of the underlying mathematical concepts and processes, but instead provides a view of mathematics as an abstract, mechanical, and meaningless series of symbols and rules. The vignette that follows is a composite account of a mathematics lesson that is representative of a classroom that might be using any of the texts that were analyzed“. Ms. Rankin: Ms. Rankin: Andrew: Ms. Rankin: Amanda: Ms. Rankin: Tom: The numbers 1, 2, 9, and 18 are factors of 18. Factors are also called divisors. Who can give me two other divisors of 18? Sally. 3 and 6. Good, Sally. Who can name the divisors of 4? Andrew. 1, 2, 4. Right. How about the divisors of 15? Amanda. 1, 15. 3 and 5 are also divisors of 15. What about 11? Tom. 1 and 11. 16See Appendix c for textbook analysis. 1 21 Ms. Rankin: Right. [Mrs. Rankin draws a chart on the board] Number Divisors 1 1 2 1, 2 3 1, 3 4 1, 2, 4 6 1, 2, 3, 6 11 1, 11 15 1, 3, 5, 15 18 1, 2, 3, 6, 9, 18 24 1, 2, 3, 4, 6, 8, 12, 24 Ms. Rankin: A prime number has only two divisors, 1 and the number itself. Who can name the prime numbers listed in our chart? Tom. Tom: 1, 2, 3, 11. Ms. Rankin: No, Tom, not 1. The numbers 0 and 1 are neither prime nor composite. Who can name a prime number that is not in our chart? Tasha. Tasha: 51? 122 Ms. Rankin: No, 3 is a divisor of 51, so 1, 3, 17, and 51 are all divisors of 51 so 51 is not a prime number. Who can name a prime number that is not on our chart? Kathy. Kathy: 13? Ms. Rankin: Right. Class, numbers that have more than two divisors are called composite numbers. Who can name a composite number that is found on our chart? Jim. Jim: 4, o, 15, 18, 24. Ms. Rankin: Right. Who can name a composite number that is not on our table? Bill. Bill: 20. Ms. Rankin: Good, Bill. Here are the steps to follow to tell whether or not a number is prime or composite. [Ms. Rankin writes these steps on the board] Step 1: List 211 the divisors of each number. Step 2: If the number has only 2 divisors, the number is primg. If the number has more than 2 divisors, then the number is W. 123 Ms. Rankin: Okay class, let's try one together. 28. [Mrs. Rankin allows time for the students to do this one on their paper.] OK. Jill what did you get? Jill: 1, 2, 4, 7, 14, 28. Ms. Rankin: Good work, Jill. Any questions? Does everyone understand? Now class, I would like for you to do problems 1-20 on page 132. [Page 132] List all the divisors of each number. Tell whether the number is prime or composite. (1) 12 (2) 16 (3) 27 (4)36 (5)42 (6) 25 (7) 54 (8)64 (9) 14 (10) 10 (11) 24 (12) 15 (13) 72 (14) 39 (15) 77 (16) 60 (17) 88 (18) 56 (19) 33 (20) 41 The environment that dominates traditional classroom instruction does not encourage students to work collectively or collaboratively. It does not support students inventing alternative ways to solve problems. The classroom is intended to be quiet except at the beginning of class when the teacher is introducing the "new" concept for the day or describing how to work the set of problems. Reform perspective. The NCTM Teaching Standards (1991) offers a different image for the type of learning environment in which students engage in mathematical inquiry. Excerpts from the learning environment standard follow: This standard focuses on key dimensions of a learning environment in which serious mathematical thinking can take place: a genuine respect for others' ideas, a valuing of reason 124 and sense-making, pacing and timing that allow students to puzzle and to think, and the forging of a social and intellectual community. Such a learning environment should help all students believe in themselves as successful mathematical thinkers (p. 57). Math 201 course perspective. The locker problem vignette provides insight into the kind of environment that existed in the Math 201 course. The environment in Math 201 encouraged students to work independently and collaboratively to make sense of the mathematics embedded in problem situations. The learning environment fostered an image of "safeness"- students felt comfortable taking risks by asking questions and offering conjectures as ideas were developing. Students were given time to pursue various ways of thinking about a problem and "allowed" to let unsolved problems "hang around" while ideas gelled. Students' perspectives. During interviews, many of the students often commented about the type of environment provided in the 201 class. Andrea and Tim's comments are representative: Andrea: In 201, the thinking is sometimes a little bit frustrating, but if you can't figure it out, you feel like it's okay, it's okay to not know what you're doing, it's okay to ask her [instructor] a question, it's really an open, open environment. (5/ 88, p.11). Tim: It's pretty much risk free, no one judges you for what you say. (5/ 9/ 88, p.4). As students described Math 201, four common themes emerged—Math 201 created opportunities for student engagement in mathematical drinking, making connections among mathematical concepts, promoting a feeling of 1 25 confidence that anyone can do mathematics, and changing views about mathematics. Engagement in mathematical thinking. Ja_spr1: It's [Math 201] different in that it's not all lecture as in all other math classes. In 201 we work in small groups and we actually work with real objects, it's not all done on the chalkboard or overhead. In other math classes they give you an assignment, come in the next day after it's due and the prof says, "Any questions?" and someone says that they didn't get the problem and they work it out up front and then you say, "Oh, I see," and then you go on. There's very little thinking involved. (6/ 88, p.14). Students were asked to do writing assignments in Math 201. For one of the assignments, students were to read an article17 and respond to questions. Tim chose to weave in some personal reflections and wrote the following: I have always done well in math, by doing what the teacher told me to do and plugging the right numbers into the right formula. I never knew why it worked, though. . . . Although I often get frustrated because I have to think so much, it makes me understand why I got the answer I did. It makes me think about the process of getting the answer and not just the product. (3/ 88, p. 1). References to "thinking" were common when students described the 201 class: Andrea: Time goes by really fast because you're always thinking. . . You have an opportunity to think about things that you just never really thought about why, you just memorized it. (5/ 88, p.11). 17de Walle, 8: Holbrook (1987). Patterns, thinking, and problem solving. Arithmetic Teacher, 34(8), 6—12. 126 l5: This class [201] is really helpful because it makes you think about why you're doing something and not just what you ended up with for an answer. . . .The problems we do in here [Math 201] require you to think a lot. You can get really frustrated if you worry about it too much, but normally if you think about it for a while and come back to it, you can figure it out. You think a lot, that's the main thing (5/ 88, p.5). Connections among mathematical topics. Students' comments often reflected frustration regarding the isolation of concepts and topics in other math courses: K_i_rr_t_: Math 201 has everything that I didn't have when I was in school. And I think now, that's why I'm so frustrated. The pieces were never put together for me. Each day seemed like a completely different lesson. (6/ 88, p. 19). Jason described Math 201 as broadening his view of mathematics: Lam: 201 helped me to see that you don't have so much of this compartmentalized mathematics, where you've got geometry, you've got algebra, you've got this and all these little isolated things, but in 201 they all can work together so it's helped me have a broader view of mathematics. (6/ 88, p.15). Anyone can do mathematics. Students came to Math 201 with the notion that only a "select" subset of the population was capable of succeeding in mathematics. This is consistent with Schoenfeld's study (1985) that students believe that only geniuses can do well in mathematics. Comments during interviews towards the end of the course indicated a change in that perception. The following 1 27 comments illustrate change both about themselves as learners and the population more generally: 1.3.5331 And this class makes you feel like you can actually do math. . . Plus, you're not thinking that there is somebody out there who is just a genius that handed out this formula to me, but that I can actually figure it out myself if I'm pushed in the right direction. (6/ 88, p.15). ::3 I've realized that everybody can do mathematics, they just have to think it through, but if you work through a problem and don't get frustrated and give up, then you can do it. . . In this class [Math 201] we're learning that everything that you do has a reason for it, it's not just a bunch of numbers and rules, but there are ways that you can find out why you have a formula and how all this stuff works. It's not just something that someone higher up told you and you should follow it, we can do it ourselves, we can reason through it. (5/ 88, p.3 and 10). .17] This course gives you a more of a "you can do it" type attitude. In other classes, the instructor tells you that a mathematician like Gauss came up with some great formula and he must have been such a genius, but they [other math instructors] never say, but you can do it too. (6/ 88, p.14). Andrea: I think it opens up a world of math to anybody. (5/ 88, p.11). I think that one thing I've learned is that, anybody can do well in math, it doesn't take a genius, which I used to think before. (6/ 88, p.14). 1 28 Views amut mathematics. During interviews at the end of the course students often compared the 201 course to other math classes and described changes in the way in which they thought about mathematics, what it means to do mathematics, and what it means to know mathematics. The following excerpts are illustrative and provide a flavor for students' overall perceptions of Math 201: Andrea: I've tried to describe this class to people. The course is challenging and it'll change the way you think about math. (5/88, p.11). 1159p: The environment in here [Math 201] encourages us to think through problems. . . we're not just waiting for the teacher to show us how to do a problem and then mimic that or just kind of memorize the steps of doing the problem (5 / 88, p. 11). Tim volunteered the following comments at the end of an interview: Ti : You know, I've always done well in math classes but I seem unable to help other people with anything in math that's very complicated. H 00 Why do you think that is? Tim: I guess Ijust learn it long enough to test so I don't really know it. (3/88, p.?). Kim's responses to an item on the EMS questionnaire18 when she entered the teacher education program and at the end of the first course captured one aspect of change in Kim's thinking about what it means to know mathematics. The question began with the phrase, "Knowing mathematics 18The first EMS questionnaire was administered fall, 1987 when students entered the Academic learning Program. 1 29 means." Four choices were provided and students were asked to choose the statement with which they most agreed and the one with which they least agreed. At the beginning of the course, for most agreed, Kim chose "being able to do rapid mathematical calculations without the aid of a calculator" and for least agreed, she chose "being able to make connections in the mathematical ideas that arise from different situations." At the end of the course, Kim reversed her choices. The students' perspective about the environment in Math 201 provided evidence that students were encouraged to engage in mathematical thinking and to make connections among the mathematical ideas they explored. Students felt that the environment was safe and promoted a feeling that they could do mathematics and make sense of mathematical ideas for themselves. Changes in students' view about what it means to know and do mathematics suggest that the environment in the Math 201 course offered opportunities that encouraged and supported students to make such changes. Summary The prospective elementary teachers who entered the Math 201 course had logged many years of mathematics learning in traditional mathematics classes. Students thought they knew the "routine" of a mathematies class- check yesterday's homework, go over the new topic of the day, and practice doing problems that are similar to ones worked by the teacher. From a traditional perspective the student's role is to work independently and quietly; memorize certain rules and procedures; imitate the methods and steps prescribed by the teacher or textbook; and expect the teacher or textbook to serve as the authority for knowing. This type of instruction reinforces students' view of mathematics as a static, technical, how-to discipline. 130 The "routine" of the Math 201 classes stood in sharp contrast to the routine that students had experienced in previous mathematics classes. The Math 201 course had as a basic goal demonstrating the feasibility of creating in new teachers a more conceptual level of knowledge about mathematics and the teaching and learning of mathematics. The teacher's role, the student's role, the tasks selected, and the social organization and discourse in this Math 201 course provided students with different messages about mathematics, learning, and teaching. Rather than telling students what to do and how to do it, the instructor expected students to work out problems and ideas on their own and with their peers. The instructor expected students to talk and to justify their answers, not just remember what she had told them. Math 201 experiences engaged students in making sense of mathematical situations. The environment was constructed in such a way that these prospective teachers experienced mathematics much as their own students might. Rather than being somewhat invisible in mathematies class, in the Math 201 class each student was expected to contribute to the development of ideas. The Math 201 context required students to engage in doing mathematics- analyzing, inventing, proving, and applying. The Math 201 class provided the context for what students learned about the mathematical ideas-number theory concepts—embedded in problem situations posed. In the next chapter, I will analyze students' views about mathematics, students' patterns of reasoning and strategies that they used, and students' ideas about number theory concepts and how those changed. Chapter 4 focuses on what students learned about the number theory ideas embedded in contexts similar to those explored in Math 201. CHAPTER 4 LEARNING MATHEMATICS TO TEACH: WHAT STUDENTS LEARNED ABOUT PARTICULAR NUMBER THEORY CONTENT, RELATIONSHIPS, MATHEMATICAL WAYS OF THINKING AND PROBLEM SOLVING The analysis in this chapter and the next draw on three interviews conducted with students before, during and at the end of Math 201. These interviews were designed to explore what students learned about particular number theory content, relationships, mathematical ways of thinking and problem solving. The interviews also probed students' views about mathematics and how they perceived their own progress in mathematics in the Math 201 intervention environment. The intent was also to track changes in those views and understandings across the ten-week course. The interviews were structured to capture what students learned about particular concepts—factors, multiples, primes and composites—embedded in the course as well as the relationships among those concepts. Many of the interview tasks were posed in a problem context different from what they encountered during the course to probe students' depth and breadth of understanding and their flexibility in using knowledge about these concepts1 . The two research questions addressed in this chapter are: In what ways and to what extent do students understand the mathematical content presented in Math 201? What strategies and patterns of reasoning do students use when solving problems related to the mathematical content in Math 201? Many studies have documented the pervasive pattern of U. S. mathematics classes. Overwhelrningly, teachers press rules and memorization, doing most of the talking while students absorb information 1For a more detailed analysis and description of interview design, see clmpter 2. 131 1 32 passively. Number and computation dominate, and students spend little time investigating topics such as geometry or probability. Rarely do mathematics classes focus on conceptual understanding, on mathematical reasoning, on problem solving, or connections (e.g. Davis 8: Hersh, 1981 ; Goodlad, 1984; Resnick, 1987a; Stodolsky, 1988; Welch, 1978). The deeply rooted ideas about teaching and learning mathematics that prospective teachers bring to their professional education courses are shaped largely by their own experiences in mathematics classes (see, e. g., Ball, 1988a; Schram et al., 1988; Thompson, 1984). One goal of this sequence of mathematics courses was to provide experiences that would challenge those deeply held beliefs and understandings about mathematics and mathematics classes. Given the opportunity to experience mathematics in an environment where students are encouraged to become mathematical risk-takers, where students and teacher form a community of learners engaged with one another in inquiry, where the criterion for what seems reasonable is determined by students and teacher working together, what do students learn about the mathematical content? Addressing this question is not a simple matter. What students learn about the mathematical content is deeply intertwined with their ways of thinking about mathematics and their patterns of reasoning. Teacher education students, like all learners, come to learning to teach with knowledge, beliefs, and experiences that they use to interpret and construct their own meaning as they interact with various learning opportunities in a classroom (see e.g., Putnam et al., 1990; Resnick, 1987b; Romberg 8: Carpenter, 1986; and Shuell, 1986). This chapter focuses on six teacher education students' learning of mathematics. Mathematical knowing is an interaction between substantive mathematical ideas and patterns of thinking and reasoning. The analysis 1 33 described in this chapter focuses on students' views about mathematics, students' patterns of reasoning and strategies that they used, and students' ideas about number theory concepts and how those changed. The first part of the chapter will be devoted to students' views about mathematics and patterns of reasoning and the strategies they used to solve problems posed during the interviews. The second part of the chapter will focus on what students learned about the mathematical content. The analysis of mathematical content is discussed using two categories-[a] knowledge about basic number theory concepts and [b] flexibility in using that knowledge. In the first category, I examine three concept domains—factors, multiples, and primes and composites. In the second category, I examine structure of number and relationships. This chapter focuses on what students learned about the number theory ideas embedded in contexts similar to those explored in Math 201. Students' flexibility in using that number theory knowledge in generalized and unfamiliar contexts will be discussed in the chapter that follows. Students' Views about Mathematics Examination of the six students' interview transcripts from across the term provided insight into their views about mathematics. There was a marked difference between one of the students, Andrea, and the other five students in terms of their ways of thinking about mathematics and their expectations as they entered Math 201. Andrea entered Math 201 with the desire and expectation that her previous views about mathematics would be challenged and altered. The remaining five students entered the course with the desire and expectation that their experiences in Math 201 would fit within their existing framework of what it means to know and do mathematics. 134 Five of the students in this study came to Math 201 expecting it to be like other math classes that they had experienced. For these students, being good at math meant being able to search one's memory and retrieve an appropriate rule or formula. They entered the course expecting the instructor to provide formulas to use, examples to follow, and problems to "practice." They expected that the 201 mathematical experiences would fit their beliefs about what it meant to know and do mathematics. Following are excerpts from these students' interviews that illustrate some of the ways their beliefs about mathematics emerged: Tamara: In math you were either right or wrong. . . they [math Linda: instructors] would ask questions where there is only one right answer and one way to go about finding it. . . you were always looking for the answer. . . I've forgotten a lot of math, a lot of the math that I've had in my past has just been memorizing. We haven't had to work through the problems, it's just "Here is the formula, do it.". . . the teacher knew everything and you were always looking for the right answer. They give you an equation and they definitely want an answer. They didn't really care how you thought about it. . . (5/ 88,6/ 88). This class makes me think. . . there isn't that emphasis on an exact correct answer. I think she [Math 201 instructor] just wants to know how you're thinking. All my other math classes, they want their right answer or you get it wrong. . . This is not your typical math class. It's a class in getting you to really think about math. . . not just rote memorization or just doing the equations, you know one after the other like the other math classes that I've had. . . the other math teachers I've had were more concerned about me I ason: 135 getting the right answer, just follow the steps the teacher gave us (5/ 88, p. 20). I expected Math 201 to be like my other math classes, but it's a lot different from other math classes where they give you about 60 or 70 problems during the week, but you're only using three or four formulas and you just keep plugging the numbers in and coming out with an answer. All of these [Math 201] problems are related but they are different too, you have to change your thinking a little bit for each problem, it's not just because the numbers are different like in other math classes. . . I thought Math 201 would be like the other math classes where the teacher gives you the formulas and you plug in the numbers. . . I never really thought that I could actually do math, I always thought there was somebody out there who is just a genius that handed out this formula to me. Now, I realize that I can actually figure it myself if I'm pushed in the right direction. . . I used to learn just by memorizing formulas and stuff because that's what got me good grades on the test. . . I expected Math 201 to be more of the plug and chug through formulas and stuff, I didn't expect to have to think (5/ 88, 6/ 88). I expected Math 201 to be like other math classes where you wait for the teacher to show you how to do it and then mimic that or just kind of memorize the steps of doing the problem (5/ 88, p. 19). By the end of the course, all of the students recognized that Math 201 was different from other math courses they had experienced. Prior experiences had reinforced the notion that success in mathematics was dependent upon technical expertise. Kim's description of high school math represented similar comments offered by the other students. 136 Kim: High school math was procedures -This is the rule, this is what you do, and if you follow this rule all the time, you'll get the right answer. And that's why this class is so frustrating for me. I need to change. I think the hardest thing for me is to admit I don't know how to think about math. (5/ 88, p.9). Recognition of differences between what it meant to know and do mathematics in previous mathematics courses and Math 201 and a desire to change one's beliefs was not enough to enable students to think and act differently. During an interview at the end of the course, Kim continued to voice a desire to change the way she thought about mathematics: This math is just such a different way of thinking for me. I've always used the trial and error method or just "tell me what to do" and you can't do that in this class. I have trouble seeing the patterns and going far enough to be able to reach the generalization...I just have to get used to thinking about math in a different way. No one is going to tell me what to do next. (6/ 88, p.20). Andrea entered Math 201 wanting to think about mathematics differently from her previous mathematical experiences but was not sure what that entailed or how to accomplish it. Evidence supporting this notion emerged as Andrea described some of the goals she had for Math 201: Andrea: I knew that I had a lot of inhibitions about math. I knew that my thinking about math was really limited. I know that I can do it and I know that I can think like that but I've just never had a chance to, and I knew that it would be hard for me at first to get myself to think and work that way, but I wanted to change the way I was doing math problems, because I don't want to have my students doing that. I wanted to change the way 137 I approached problems and the way I've learned math. . . . Well, the focus for me was just trying to get myself to work on problems in a way that I could make sense out of them. . . Just to get myself out of the mold or the way I was thinking about math before was the most important thing to me (6/ 88, p.14). Interviewer: You said that one of your goals was to start thinking about math differently. Do you remember when or why you decided that was something you really wanted to do? Andrea: Well, it started, I think in TE 2002. We read an article about Benny3 and when I was kid, I did math like that [independently] and that started me to think. . . that was a great article, because I never knew there was another way to think about math until we started talking about that article in 200. . . I realized that I had thought the whole time that I was really good in math. . . And after I read that, then I started thinking, maybe I don't know that much, and then when we started doing these interviews at the beginning of this class, I knew that I needed to make some changes. Interviewer: I'm curious, you read this article in TE 200 so you began to think about math differently, what made you think that this particular math class would help you to do that when none of the other math classes you had before had done that? 2Siee chapter 2 for description of TE 200. 3Benny's conception of nrles and answers in IPl mathematics (Erlwanger, 1973). 138 Andrea: I knew there was no way they [Academic Learning Program] were going to give us a class where you had to do math the way we did before so I knew that it would be different. I didn't know how, because I didn't know what the difference was yet, I really didn't have any idea what was going to really happen once I got in there, just that it was going to be different and I wanted to think about math differently (6/ 88, p.16). Comments such as these provide insight into why Andrea often approached problems from a reasoning perspective during the first interview and continued to move in that direction throughout other interviews. At the conclusion of an interview four weeks into the course, Andrea volunteered additional thoughts related to her desire to think about math differently: Andrea: Math has always been really easy for me. I whizzed through everything even college calculus but since I've been in Academic Learning. I realize how much I need to learn about things I thought I already knew. I'm glad we're doing this math sequence. I want to think about math in a different way. I just get frustrated that it takes me so long. I've always been able to figure out math so quickly but I never thought about what I was doing or why, I just came up with right answers. (4/ 8/ 88, p. 24). Students' comments about mathematical experiences they had prior to Math 201 were consistent with the traditional perspective—technical, how-to orientation—described in the previous chapter. By the end of Math 201 students recognized and articulated differences between Math 201 and prior mathematics courses. In Math 201, students were expected to think about mathematics differently, the process one used to solve a problem was valued, students were expected to create formulas or figure out alternative ways to 139 solve a problem, the teacher did not tell students how to do the problems, students were expected to search for patterns and generalizations. Students' views about mathematics also influenced the strategies they used to solve problems during interviews. Schoenfeld (1985) supports this assertion: One's "world view" (or set of beliefs, or epistemology) determines the kind of reasoning (and meta-reasoning) one does (p. 368). Patterns of Reasoning and Strategies The strategies students use to solve mathematical problems are important as a lens through which to view students' drinking about mathematics. Lampert (1991) described the value of examining strategies: It is the strategies, rather than the answers, that are the site of the mathematical thinking, and it is these strategies that reveal the assumptions a student is making about how mathematics works (p. 129). Participants used a variety of strategies to solve problems presented during interviews. Interview data were coded to examine the strategies“. The categories, derived from multiple readings of interview transcripts, were used to code the interviews systematically. Strategies used by students seemed to fall into two categories, an emphasis on reasoning and an emphasis on technical strategies. Reasoning strategies are strategies that include reflection about the method used to solve a problem. Student responses that were coded as a reasoning strategy included: 0 Using a procedure/ formula/ rule and reasoning why it worked; 4For a detailed analysis, see chapter 2 140 0 reasoning based on mathematical symbolism; 0 searching for an example to serve as a counter-example and reasoning why it worked; 0 using an example to test or verify a solution and reasoning why it worked; 0 using an example in a trial/ error manner and reasoning why it worked; 0 talking through a proposed solution and then changing or verifying the solution; 0 using a pattern and reasoning why it worked; 0 comparing to a similar problem; 0 exploring various possibilities related to concepts within a problem. The following is an example of "using a procedure/ formula/ rule and reasoning why it worked (MF/M/ R)": I: What is the greatest common factor for 24 and 72? Tim: 24, because 24 is the largest number that divides evenly into both numbers, and if I do a prime factorization then I'll get 2 to the third times 3 which is 24, and for 72, I would get 2 to the third times 3 squared, and I need to take two to the third times 3 out of both and you get 24. Technical strategies are strategies that emphasize a "how to" technique. Student responses that were coded as a technical strategy included: 0 Retrieving a procedure/ formula/ rule from memory to manipulate symbols; 0 using an example to test or verify solution; 0 using an example in a random trial / error manner; imitating a perceived pattern; 0 viewing a problem in isolation; 0 generating a correct solution intuitively but unable to articulate why it worked; 1 41 0 responding "don't know" or did not attempt a solution5. The following is an example of "retrieving a procedure/ formula/ rule from memory to manipulate symbols (MP/M)": I: How many factors does 28 0 35 have? Kim: 13. I: And why do you think it is 13? Kim: Because if you add the exponents 8 and 5, you get 13. (5/ 88, p.5). A description and example for the other individual coding categories are included in Appendix A. The first part of this section will describe strategies used by individual students categorized as a technical or reasoning emphasis and changes in these across the term. Examination of strategies used by students during the interviews revealed changes within individual students. There also were some similar tendencies across students. For the five students who entered Math 201 expecting to find a math course that matched their previous mathematical experiences, the use of technical strategies was prominent. These students were content to find a formula, plug in the numbers, and arrive at an answer. They were easily frustrated when pushed to explain or think about "why" something worked. Strategies used by these students during the first interview often included attempts to retrieve a formula or procedure from memory and a desire to manipulate symbols. By the end of the course a comparison of the total number of reasoning strategies to the total number of technical ones revealed changes within all of the students. 5m: was coded as technical if the student did not try to think about what they knew or to reason about the problem. 1 42 Three of the five students, Kim, Jason and Tim, made significant shifts from a predominantly more technical to a more reasoning orientation (see Figures 4.1, 4.2, and 4.3) Another student, Tamara, had a slight shift- decreasing the number of technical strategies and increasing the number of reasoning ones (see Figure 4.4) Linda's total number of technical strategies increased slightly while the total number of reasoning strategies remained fairly constant (see Figure 4.5) In the next section, the changes for each of the students will be described and supported by excerpts from interviews. Kipp. Kim exhibited the most dramatic change in total number of strategies used in a given category. During the first interview, Kim's strategies were predominantly technical (88.57%). Analysis of strategies Kim used during the interview at the end of the course indicated a strong emphasis on reasoning strategies (75%). 143 Technical Reasoning I 100-- IE] Percentage of Coded Responses Figure 4.1. Kim: Percentages of Technical / Reasoning Strategies During the first interview, Kim searched her memory for a procedure to use and was content with almost anything she was able to retrieve even if it was unrelated to the given problem. Often, Kim's nonverbal behavior (e.g., a heavy sigh) seemed to connote relief when she was able to write a formula or remember a procedure. There was little evidence to indicate concern about whether or not the solution itself was reasonable. The following excerpt is illustrative: Kim: All I can remember to keep GCF [Greatest common factor] and LCM [Least common multiple] straight is that one day on an 144 overhead sheet on GCF and LCM, I realized that the GCF is always lower than the LCM so I just try to remember that to keep the two straight. (5/ 88, p.7). Kim seemed to focus on trying to find "tricks" or ways to help her remember rather than understanding the particular concept. The exception to the technical emphasis surfaced during the series of three "real-world" context problems (#21-23) in which she not only reasoned through the problem situations but provided correct solutions as well. Kim‘s response to the question about the pan of brownies illustrates the contrast: Interviewer: A pan of brownies is to be cut so that each member of the family can have the same number of brownies. How many brownies should there be if four members of the family will definitely be present and they may or may not be joined by (a) one other member? (b) two other members? (#22) Kim: I can do this one. [Kim draws a rectangular shape and draws lines to show 4 x 5.] Because all four people could have five pieces or if the fifth person showed up, they could all have four pieces. If they are joined by two others, then I would cut it into 24 [draws another rectangle and draws lines to show 6 x 4.]. So if four were there they'd have six pieces, but if six were there they'd have four each. (3 / 88, p. 15). During the B interview Kim began to show signs of moving toward using more reasoning type strategies. She sometimes responded, "No, I can't do it," but when asked probing questions she proceeded to reason through the problem quite well. An illustration follows: I: How many numbers between 1 and 1000 would be divisible by 6 and 9? Kim: Kim: Kim: Kim: 145 I have no idea. Do you have any way of talking about how you might go about solving the problem even if you were going to just grind it out? If I was going to grind out, I would start with the number 18. Why 18? 18 is the first number that 6 and 9 are divisible by. And I would just take the pattern of the 6 and 9 multiples until I found the next one. I know it wouldn't be 24 and it wouldn't be 27, so probably 36 would be the next one. And then maybe go in intervals of 18 until I get to 1000. Why would you go intervals of 18? Because that's the first number that I came to, and I think it's going to work in a pattern all the way to 1000. (5/ 88, p.7) [reasoning strategy]. Kim's expanded response was a long way from 'T have no idea" but she was not confident of her ability to just think through a problem, and she continued to look for a polished procedure even though she could reason quite well when asked probing questions. When the B interview was conducted, Kim was just beginning to think about mathematics differently, so the questions that still caused her difficulty were the ones that required more flexibility in using ideas and knowledge. Individual knowledge pieces were still quite fragile and she was not yet able to make connections among the mathematical ideas. 1 46 By the end of the course Kim still approached problems looking for a "procedure" but she tried to ask herself about the reasonableness of solutions. She talked to herself and said things such as, "Now, does that make sense?" "Now that makes absolutely no sense." "No, that can't be right, it doesn't make sense." She also was more persistent as she attempted to come up with a solution. In reflective comments Kim described ways in which she had changed: I'm not content anymore to just plug numbers into a formula, I want to know why that formula and why it works. I used to read a problem and if I didn't immediately know a solution, I gave up. Now I think about questions that I can ask myself to help me think about what I know about a problem and I feel like given time, I have a chance to be able to solve it. (6/ 88, p. 18). From the beginning of the course to the end, Kim's patterns of reasoning and strategies used shifted dramatically from a technical emphasis (88.6%) to a reasoning emphasis (75%). Jpgpp. Jason's patterns of reasoning also indicated a marked change from a technical to a reasoning emphasis. More than half (59.46%) of Jason's responses during the interview at the beginning of the course emphasized technical strategies. By the end of the course, reasoning strategies dominated his responses (78.79%) (see Figure 4.2). 147 Technical Reasoning I 100"- “868 70" F 80'[ L 60‘ Percent eolCoded R 01 O l l Interview A1 Interview B Interview A2 3/88 5/88 6188 modem Figure 4.2. Jason: Percentages of Technical/ Reasoning Strategies The following examples reflect the shift in strategies that Jason used during the two interviews: I: What is the greatest common factor of n and n+1? (#Sc) Jason: I don't think you can know. I: And why don't you think you can know? Jason: Because you don't know what "n" is so you don't know what n+1 is either. (3/ 88, p.3). [technical]. Jason: Jason: 148 [same question A; interview] You can't have one. Why not? Because your second expression has plus [his emphasis] one. If you're dealing with multiplication, you could do what I did here [points to 5b] to find the greatest common factor. But when you're adding, it's different. For example, if n was 13, then n+1 would be 14 and there's nothing in common. It doesn't matter what n represents because n+1 is just one more and there's nothing in common. (6/ 88, p.3). [reasoning strategy]. A comparison of Jason's responses to a different question show a similar contrast. Jason: Jason: The greatest common factor of 630 and 1716 is 6. What is their least common multiple? Explain your answer. (#24) [uses calculator] 630 times 1716 divided by 6 is [shows me the calculator—180,180]. How did you decide to do that? Using my system, that procedure from before [points to his work on problem ?]. First I tried a simpler problem that I knew the answer for. Now let me make sure. [uses calculator] Well that's [180,180] definitely a multiple but I don't know if it's the least, it seems pretty big. 1 49 I: How could you‘find out? Jason: I'm not sure. (3/ 88, p.18). [technical strategy]. 1: [same question A2 interview] Jason: [He uses the calculator and figures 105 x 1716 and gets 180,180. On his paper he had divided 630 by 6 to get the 105.] I want to check something. [He flipped back to another problem in which he wrote]: 4 6 (2)-2 @-3 [He used the calculator again and said]: The GCF is 6 so I divided 6 out of 630 which is 105 and 105 times 1716 is 180,180. In this other problem, I took out the GCF from one number and multiplied the rest together. Now I see. I went a round about way. I could have multiplied 630 by 1716 and then divided by 6 to get the same thing. [Long pause and then on his paper he wrote]: LCM(A,B) = ALE GCF That makes sense. If I divide the GCF out of one number, it's the same as dividing it out of the whole thing after I multiply the two numbers together. (6 / 88, p.16). [reasoning strategy]. When Jason entered Math 201, he was content to apply remembered rules and formulas without questioning the reasonableness of his choices. At the end of the course, he continued to apply rules and formulas but usually 150 reasoned why they worked or seemed to be appropriate. He also reasoned about mathematical notation and symbolism. IE!- Examination of Tim's responses evidenced a similar shift from an emphasis of technical strategies (54.55%) to reasoning ones (75.76%). During the first interview, Tim's responses reflected that he utilized almost the same number of technical (54.55%) as reasoning (45.45%) strategies. By the end of the course, the percentage of reasoning strategies (75.76%) greatly outnumbered the technical ones (24.24%) (see Figure 4.3). Technical Reasoning - 100+ TIM 90—1— 230-— 70-1— 60- 50" 40- Percentage of Coded Responses 30- 1 9 ' axes 5/88 6/83 Figure 4.3. Tim: Percentages of Technical/ Reasoning Strategies 1 51 A comparison of Tim's responses to the same question during Interview A1 and A2 illustrates a shift from technical to reasoning strategies: Interviewer: A number is a multiple of 7 and another number is also a multiple of 7, is the sum of the two numbers a multiple of 7? Tim: Can I work this out? Interviewer: Sure. Tim: I guess it would be because I multiplied 7 times 2 and got 14 and 7 times 5 and got 35, and I added them together and got 49 which is a multiple of 7. (3/ 88, p. 10). [same question 6 / 88] Tim: Sure, all you're doing is adding different chunks of 7 together so you'll always get another multiple of 7 (6/ 88, p. 11). Tamara. Tamara's coded interview responses indicated less dramatic changes in the total number of reasoning (12.9% to 23.33%) or technical (87.1% to 76.67%) strategies (see Figure 4.4). 152 Technical Reasoning I 90'- Percentage of Coded Responses Figure 4.4. Tamara: Percentages of Technical/ Reasoning Strategies Some of Tamara's responses to the same questions were almost identical during Interview A1 and A2. The following example illustrates: Interviewer: What is the greatest common factor of n and n+1? Tamara: 11. Here's a n [points to n] and here's a n [points to n in the expression n+1] (3/ 88, p. 3). [same question 6/ 88] each quid way that C011 153 Tamara: I'd say n because they both have n in them (6/ 88, p.3). For Tamara, there was a small change in the percentages of strategies in each category. During the first interview, Tamara gave up if she could not quickly obtain a solution to a problem. She was less likely to respond that way during interviews towards the end of the course. Tamara perceived a change in the way she approached problems for which she did not have immediate answers. Her response to the problem of finding the greatest common factor for 2a2 and 8ab illustrates: Tamara: I'm not sure. [pause] I could break it down into factors and see what they each share [writes 2 . a . a and 8 . a . b]. They each share a two and they each share an a so I'll say 2a (6/ 88, p.3). The following comments support her awareness of this change in herself: Tamara: I used to give up if I couldn't do a problem immediately. It's different now, just being able to break a problem down and try to see something there instead of just saying, "This is so overwhelming, what am I going to do?" You just keep trying and you ask questions or try and look at it in a different way or see if there is some kind of pattern. (6/ 88, p. 19). In other comments related to herself as a learner in Math 201, Tamara expressed a continuing need to have someone tell her "the answer" and that she needed more "practice." At the end of the Math 201 course, Tamara seemed to be looking for a more traditional mode of instruction. Tamara thought that Math 201 was hard and she was ambivalent about whether or not the changes in instruction were beneficial to her. These excerpts illustrate: Tamara: Discussion is big is this [Math 201] class. I know it's good to discuss but I‘ve never done it in math so it's hard for me 154 to do it and that's frustrating to know that I should be doing it but, I can't do it. Well, I can, but it's hard (6 / 88, p. 16). Tamara: I need more practice in math because for me, I don't know if it's being scared of math or being scared that I won't be able to do it. . . .I still think math is hard. At least now I'm not totally scared of it like when I just hear the word, now I can at least look at the problem and try and figure out something, see if there is some kind of pattern or something. . . . (6/ 88, p. 18, 19). Interviewer: Anything, else you can think about that might be helpful for me to know about you as a learner in math? Tamara: I like to take my time. I don't like to rush. I like lot and lot of practice. I mean even if it is worksheet after worksheet or problem after problem, I need to it over and over again (6/ 88, p. 20). Linda. Linda's strategies changed very little from the beginning of the course to the end (technical 86.21% to 87.1%; reasoning 13.79% to 12.9%). She continued to want to rely on her memory and a technical, ”how to" orientation (see Figure 4.5). 155 100-- Technical 5 Reasoning lntewiew B Interwew A2 5/88 6/88 mtemiellls Figure 4.5. Linda: Percentages of Technical/ Reasoning Strategies During the first interview she frequently made references to memory: Linda: Let's see if I can remember that. . . I'm not sure exactly. If I remember correctly. . . I could kick myself for not really remembering this stuff. . . I can't remember. . . (3/ 88). Throughout the interview, Linda looked to the interviewer for approval or confirmation. Linda: Is that right? . . . Is it really okay to use a calculator? . . . Can I draw a picture? Can you tell me, did I do this right? (3 / 88). 1 56 At the conclusion of the interview after I turned off the recorder, Linda expressed concern about the number of questions she got "right" or "wrong." She said, "I don't feel that I have a logical mind like some of my friends. I've done okay in math because I study hard." (3/ 88). This seems consistent with her view of mathematics as a fixed body of knowledge, the expectation that she should be told what to do, the assumption that a good memory was critical, and that authority for knowing emanates from an "expert" (e.g. teacher). During the interview at the end of the course, Linda's primary strategies continued to emphasize a technical orientation—retrieving formulas from memory or selecting an example, often only one, to prove or disprove a conjecture. If nothing came immediately to mind, she responded, "I don't know that one." Andi-pa; As described earlier in this chapter, Andrea's views about mathematics as she entered Math 201 were very different from the other students. Unlike the other five students in this study, Andrea entered the Math 201 class wanting to think about mathematics differently. She thought mathematics could make sense. She wanted to take the "mystery" out of math. She continued to expand and build upon these views as the course progressed. Andrea used more reasoning (61.76%) than technical (38.24%) strategies (see Figure 4.6) initially and the range increased significantly from the first interview (61.76% to 38.24%) to the one at the end of the course (93.94% to 6.06%). 157 Technical Reasoning I 100.. go of Coded Responses 03 O l l Percenta to O I l 'igure 4.6. Andrea: Percentages of Technical/ Reasoning Strategies During the first interview many of Andrea's responses evidenced a iasoning disposition. Often, as she talked through a solution she would say, iut let me think why." Her response to one of the multiple questions (#20) illustrative: I: A number is a multiple of 7 and another number is also a multiple of 7. Is the sum of the two numbers a multiple of 7? Andrea: Yes. I: How do you know? Andrea: Andrea: 158 Because, whatever number you have, it's going to be divisible by seven just because each number has 7 as one of its factors. If you add them together, they're still going to have 7 as their factor. And is the product of those two numbers a multiple of 7? Let me think. [pause, writes on paper 7 x 14 and then uses her calculator] I think yes, but let me think why. If you're going to multiply two numbers that are divisible by 7, [writes 7 ( x y) you can always factor out the 7. Whatever is in there, it doesn't matter because you're always going to be able to factor out the 7. (3 / 88, p.13). [reasoning strategy]. Vhen Andrea was unable to figure out a problem, she often talked about why 'arious solutions would not work or why they did not make sense: Andrea: The greatest common factor of 630 and 1716 is 6. What is their least common multiple? Explain your answer. [using her calculator] No, that wouldn't work. I multiplied, I don't know why I did this, I'm thinking backwards. I multiplied 1716 by 6 and then divided it by 630, but that's thinking backwards. [pause, uses calculator again] That won't work either. [pause, uses calculator] Whatever the number is, [uses calculator] I'm not sure about this question. [uses calculator] I multiplied this by 10, so this 0 on the end tells me [pause] it doesn't tell me that either, I changed my mind. I thought if I multiplied them by 6 again that 159 then you get another thing, but that's not right either. Nothing I try makes sense. (3/ 88, p.15). During the first interview, Andrea was not always able to talk explicitly about the "why" but referred to her "gut feelings" which were often correct: Andrea: Andrea: Andrea: Andrea: This is a graph that shows the greatest common factors of 6 and the natural numbers, 1-12... (#7). I don't really understand this graph. I would just finish the pattern here which is what I would do if I was just going to do it. . . I don't really understand this graph so I'll just do this [imitates the pattern of dots] which would be my most natural instinct. Can you talk about any patterns that you see? I guess, I don't really understand this at all. What if you were to compare this graph to a similar one that shows the greatest common factors of 5 and the natural numbers 1-12. Will it be different? I would think so. This would go straight across up till you get to five and then you go up like that [sketches the graph correctly]. I'm not sure but this is my gut feeling, you would have it go straight across until you got here [points to 5]. And why do you think it does that? Because when you think of 6, it's divisible by 2 and by 3, and then 1 and 6, but 5 isn't divisible by any other number except for 1 and 5. . . But, I'm not exactly sure how it works. I can see the 3 but I just don't see why. . . 160 I don't really know how this works, it's just how I feel about it. (3/ 88, p.5). During the first interview Andrea often was content to rely on instincts and feelings to justify solutions. During the B interview she continued to want to rely on memory or how it "felt" but when pushed to explain responses she was usually able to reason through her solutions. During the A2 interview, it seemed important to Andrea to have a reasonable explanation and she generally offered these explanations without probes. The contrast between Andrea's responses to a least common multiple question (#15) during A1 and A2 interviews is illustrative: I: What is the least common multiple for 36 and 63? Andrea: [During the A1 interview Andrea used the calculator and tried several different procedures] No, can't be that. No, I don't think that would work either. [pause] 252. [probe] I just kept multiplying 63 and then dividing the answer by 36 to see if it came out even and the lowest number that I got was 252 and that just feels right to me so 252 (3/ 88, p. 7). During the A2 interview she did the prime factorization of the two numbers and used that to find the least common multiple. After she worked this out on paper [9-7 9-4 9(7- 4)] she said: 252. I remember how to do this and I remember I figured out why it worked, but I'm trying to confirm now why it works. Well, it would have to have a 9 in it and then the 7 and the 4. The reason it's the lowest is because the 9 is the one thing that they have in common and then you have to multiply these two [points to 7 and 4] because to get a multiple of a number you have to have all the factors. It's going to be 9 times 7 times 4. You have to have the factors that both of the numbers have but 161 if you put 9 twice, it won't be the lowest common multiple (6/ 88, p. 8). Reflective comments by Andrea regarding changes in the ways she approached or solved problems provided additional insight into strategies that she used to solve problems during interviews. Andrea: Math 201 is different. It's a lot different than what I've had before. I think I feel like I'm starting from the very beginning. I just feel like I've discovered a lot of new things about math that I never knew before, never thought of before. I guess it's hard to adjust yourself to the change, but I think now I can approach some problems and at least have a better way of attacking them than I did when I first started the class. I: Could you say some more about that? Why do you think that's true? Andrea: Before I had a formula to attack it with, a formula that I often didn't even understand. And now I'm doing concrete - a table or a graph or something, then you generalize from there, you don't start out with a generalization. (5/ 9/ 88, p.14). Andrea was aware that she sometimes reverted to previous approaches to solving problems. After we had completed an interview that occurred more than half-way through the course, Andrea said: Andrea: I still sometimes approach a new problem like I used to do math and then I catch myself and think, "I can reason this out if I just look at it and think about it - think about what I know about the problem. It's difficult to overcome the old way because I've had so many years of doing it the other way. (5/ 9/ 88, p.15). 1 62 By the end of the course, Andrea felt that she had moved closer to making the transition in her thinking that she desired: Andrea: I've been trying to get myself to work on problems in a way that I could make sense out of them. . . Just to get myself out of the mold or the way I was thinking about math before was the most important thing to me, and I think I have for the most part. (6/ 88, p.14). From the beginning of Math 201 to the end of the course, patterns of thinking rnd problem solving changed for each of the six students. For all of the students, except Andrea, a technical emphasis dominated their solutions vhen they entered the course. By the end of the course, Kim, Jason, and Tim's strategies were predominantly reasoning ones. Analysis of Tamara and .inda's strategies revealed much less change from the beginning to the end of he course. As previously described, Andrea entered Math 201 with views bout mathematics that were markedly different from the other five students. 1 the first interview, Andrea used more reasoning than technical strategies. .ndrea had the entire ten-week term to nurture and strengthen the patterns f reasoning and problem solving that she brought to Math 201. Students' atterns of reasoning influenced their success as problem solvers. Resnick 987b) argues that Becoming a good mathematical problem solver - becoming a good thinker in any domain - may be as much a matter of acquiring the habits and dispositions of interpretation and sense- making as of acquiring any particular set of skill, strategies, or knowledge (p. 58). ie process of changing mathematical ways of thinking and dispositions is mplex and occurs over time. A single mathematics course, for a ten-week riod, is not enough to undo years of mathematics learning; however, a 1 63 mathematics course such as Math 201 can begin to challenge some students' ways of thinking and patterns of reasoning. In the next section, I will discuss what students learned about the number theory concepts embedded in the Math 201 experiences. In the next chapter I will examine their flexibility in using this knowledge. In the previous section I focused on the patterns of reasoning and the strategies that students used to solve interview problems. The strategies used by students have implications for what students learned about the content in Math 201; thus the discussion of strategies continues as the focus shifts to highlight what students learned about the content in Math 201. Mathematical content This mathematics course, Math 201, was a ten-week course. The part of the course this portion of the analysis focuses on is the first four and one-half weeks which included exploration and study about the following number theory topics: factors, multiples, and primes and composites5. Knowledge about Number Theory Concepts Fem Included in the section of "factors" are elementary factoring ideas, lotions related to divisibility, the relationship of factors and factoring to multiplication and division, and ideas related to greatest common factor GCF). Figure 4.7 shows the interview questions in this section. lee chapter 3 for a discussion about the rationale for focusing on number theory in a athematics course for prospective elementary teachers. ,0) 164 This is an example of two numbers that have this relationship, 4 is a factor of 12. What is a factor? What is the greatest common factor of 8 and 10? Explain why you think your answer is correct. Is there any other way you could figure out the greatest common factor of 8 and 10? What is the greatest common factor of a) 24 and 72 Here are 24 square tiles. How many different rectangular shapes can you make by using all the tiles? I would like for you to cut out paper models from the grid paper to represent each different shape. Probe: Are you sure that you have all of the possible rectangular shapes? Explain. The following graph shows the greatest common factors of 6 and the natural numbers 1 through 12. [INSERT GRAPH] (a) Extend the pattern through 18. Talk about any patterns that you observe. (13) Compare this graph to a similar one that shows the greatest common factors of 5 and the numbers 1—12. Is it different? Why do you think this occurs? :3) Decide whether or not each number is a prime or composite. Explain how you decided. c) 237 d) 239 An interior decorator would like a wallpaper pattern that would fit exactly in a room with walls 8 feet high and also fit exactly under window sills that are 30 inches from the floor. What is the height of the largest pattern that can be considered? Figure 4.7. Factors Interview Questions Figure 4.7 shows the type of strategy used and indicates whether or not students gave a correct or incorrect response to interview questions related to factors. 165 Factors A1A2 Jason R Reasoning strategies T Technical strategies Gray indicates correct solution Stripes indicate calculation wrong, but reasoning correct Figure 4.8. Participants' Strategies for Factors Questions All six students came into the course with some ideas about factors. Questions #3, 4, 5a, and 6 referred to basic factor concepts. Half or more of the six students responded correctly to all four of these questions at the beginning of the course and all six students gave correct solutions to the questions at the end of the course. Examination of the strategies employed by these students for this set of questions offered insight into students' understanding of factors. Certain questions seemed to lend themselves to particular strategies both within an individual and across the group. In the category of factors, all but two students used a memorized procedure to respond to questions 4 and 5a (see Figure 4.8). The difference was whether or not the student reasoned about 166 why the procedure or solution made sense. Students could provide correct solutions to many of the problems in this set by retrieving appropriate procedures or formulas, thus, it is not surprising that memory-related strategies dominated. Many of the students explicitly referred to memories from earlier math courses or talked about familiar strategies such as using factor trees7 to find the greatest common factor. Tamara's response during the first interview illustrates this: Tamara: Tamara: Tamara: Can you think of another way you could figure out the greatest common factor of 8 and 10? (#4) By doing a factor tree. Could you do that on your paper? Sure, [pause while she draws factor trees for 8 and 10] oh boy, I forgot how to do this. What I would do is [pause] circle the one that they have in common or is it all the ones they have in common and then the one that is [pause] greatest is the one [GCF], I think that's right. Are you comfortable with that? I think so, [pause] yes, I checked it. (3/ 88, p.2). [technical strategy] \ factor tree consists of the composite number to be factored and "branches” from the composite imber to factor pairs until all of the numbers in the row are primes. 24 6x A A Zx3x2x2 1 67 During the A2 interview Tamara continued to rely on a procedure but appeared more confident in using it. Tamara: Take the prime factorization of each of the numbers and then take the lowest power that they each share. I: Could you show me what you mean? Tamara: Okay. [writes as she talks] For 8 you would have two ’ times two times two and then for ten you would have two times five and the only factor that they share is the two [circles it in each prime factorization]. (6/ 88, p.2). [technical strategy] Sometimes the remembered rules were incorrect as evidenced by Andrea's response to determining whether or not 237 was prime or composite (#10c): Andrea: 237, that's prime. I remember some rule about adding the last two digits and if it's divisible by three or four or something like that, it isn't prime, so this must be prime. (3/ 88, p.5). [technical strategy] Andrea combined divisibility rules for three and four8 and despite being incorrect she seemed content with the result. Although all six students seemed to have some knowledge about the ideas related to factors, they were not clear about what they knew or what they could do with these pieces. No more than half of the students were able to provide correct solutions for the questions that required students to apply their knowledge about factors in this section (#7, 10c, 10d, 23) during the first interview (see Figure 4.8). 81f the sum of the digits is divisible by 3, then the number is divisible by 3. If the last two digits are divisible by 4 then the number is divisible by 4. 1 68 Responses to the wallpaper problem (#23) illustrate this well as four of the six students were incorrect and struggled to understand the problem. The four students who correctly answered the same problem during the interview at the end of the course quickly defined the problem as one related to finding the greatest common factor. Tim‘s response represents the correct solutions: Tim: [On his paper, Tim converts the eight feet to 96 inches and then finds the prime factorization for 30 and 96 and says:] Six inches because six is the greatest common factor of the two numbers. I: And why would you want to use the greatest common factor of the two numbers? Tim: Because you want the largest pattern that works for both numbers so that would be the same as the greatest common factor. (6/ 88, p.12). [reasoning strategy] The overall number of correct responses in the factor section increased from the first interview to the last one. Students' knowledge of factors seemed to be growing; however, that knowledge remained somewhat fragile for several of the students. Kim's response during the interview at the end of the course (A2) to the question regarding whether or not 239 was prime or composite (#10) illustrates this. Kim began in the same way as at the beginning of the course, by testing for divisibility9 by 2 and 3, but as she explained her reasoning she began to form some hypotheses and attempted to expand her understanding of the relationship of factors to one another and the structure of a number: 9A number is divisible by 2 if the number is even-the last digit is 0, 2, 4, 6, or 8; the number is divisible by 3 if the sum of the digits is divisible by three. 169 Kim: Well I know it's not divisible by 2 because it is odd, and I know it's not divisible by 3. I checked all the numbers up to 9—you know there should be an easier way to do this. I'm thinking that, and I know you can't tell me this, but if 239 isn't divisible by 2, then it's not going to be divisible by anything that 2 is a factor of. Does that make sense? I: What do you think? Kim: Yes, because that means that 2 would have to be a factor of that number and if it's not, then that rules out 2, 4, 6, and 8, so it would just be the odd numbers and I've checked through 9, I just checked 11 and 13, and neither one of those worked. Would that also work the same for 3? Then that would rule out 3 and 9 and 15 so really all I would have to check is 2 and 3 and 5 and 7 and 11 and 13. (6/ 88, p6). [reasoning strategy] Kim's voice and inflection reflected uncertainty. She did not appear to recognize that she had just listed the prime numbers less than 15 and to determine primeness, one needs only to "test" prime numbers. Instead, she was trying to reason about the results of her longer procedure of testing every number. Although this procedure had been used by the instructor and by other students, for Kim this seemed a new and uncertain idea. By the end of the course, Kim was beginning to use reasoning to help her think about problems that required application of factor ideas, but she was not always aware of the generalizability of some more fragile notions. This was true for other students as well. By the end of the course, there was evidence of growth in all six students' knowledge and understanding of factors. Half of the students gave correct responses to all of the questions in this section. A fourth student gave only one incorrect response. The remaining two students gave correct 1 70 responses to the basic questions related to factors but continued to struggle with some of the application problems related to factors. Students were able to be successful responding to these questions using both reasoning and technical strategies. One could draw upon isolated skills and offer correct solutions to this set of questions. However, from the first interview to the last one, the percentage of reasoning strategies used by students increased from 38.7% (12 out of 31) to 66.7% (28 out of 42). Knowledge about factors remained somewhat fragile but students seemed inclined to reason more about their responses. Multiples Another content category under major concepts was multiples. Basic ideas about multiples and least common multiple (LCM) were explored. Figure 4.9 shows the questions in this section. 13) This is an example of two numbers that have this relationship: 8 is a multiple of 4. What is a multiple? 14) What is the least common multiple of 4 and 6? Explain why you think your answer is correct. Do you know any other way you could figure out the least common multiple of 4 and 6? 15) What is the least common multiple of (a) 36 and 63 18) Show the student rectangular models for multiples of 4 and 6. [INSERT RECI'. MODEL] (a) Can you use these models to show common multiples for 4 and 6? (b) [LABEL THE MODELS SO THE SEQUENCE OF NUMBERS IS CLEAR.) In the group of models showing multiples of 6, every second one is a common multiple of 4 and 6. In the group of models for multiples of 4 every third one is a common multiple of 4 and 6. Explain why this pattern occurs? 21) Draw a picture and label the dimensions of rooms whose floors can be tiled completely, without cutting any tiles, using tiles 9 inches square. 22) Apanofbrowniesistobecut sothateach memberofthefamilycanhavethesamenumberof brownies. How many brownies should there be if four members of the family will definitely be present and they ngy or may not be joined by (a) one other member? (b) two other members? Figure 4.9. Multiples Interview Questions 1 71 Figure 4.10 shows the type of strategy used and whether or not students gave a correct or incorrect response to interview questions related to multiples. R Reasoning strategies T Technical strategies Gray indicates correct solution Figure 4.10. Participants' Strategies for Multiples Question Overall the students entered the course knowing many of the basic ideas about multiples that were explored in the questions in this section. Two of the students responded correctly to all of the questions in this section. Only one student, Kim, seemed baffled by multiples. She struggled with the problems and eventually treated them as she would if they were questions about factors—she simply replaced "multiple" with "factor" in these questions and responded accordingly. It was interesting, however, that when she encountered multiples in a real-world context (e.g. #21 & 22) she did not 172 interchange "factor" for "multiple" and she was able to provide correct solutions. Some of the application problems that required students to apply knowledge about multiples (e.g., #18, 21, 22) proved to be stumbling blocks for some of the students during the initial interview. The exception, was the following problem (#22): A pan of brownies is to be cut so that each member of the family can have the same number of brownies. How many brownies should there be if four members of the family will definitely be present and they may or may not be joined by (a) one other member? (b) two other members? All of the students gave correct solutions to the brownie problem. One plausible explanation for this anomaly is that the context-sharing food among family members—drew upon familiar experiences and enabled students to reason about the situation rather than viewing the question as a ”math problem." There was a wide range of responses to the other multiple questions. Similarly, as in the factor section, students could provide correct solutions to many of the problems in this set of questions by retrieving remembered procedures (see Figure 4.10). The choice of strategy did not seem to play much of a role in this set of questions. The following excerpts from responses to the question about rectangular models for multiples of 4 and 6 (#18) are illustrative: I: You said that the common multiples were 12, 24, and 36. When I look at the multiples of six I notice that every second one is a common multiple and when I look at the multiples of four, I notice that every third one is the common multiple? Do you have any ideas about why it occurs in that pattern? 173 Tamara: I don't know. [pause] Six would have to be a factor of the number, [pause] no, that doesn't work. Four times three and then times another three. Four times three is 12 times another three, but that's not 24 though. No, I really don't know. (3/ 88, p. 12). [technical strategy] In contrast, Andrea responded immediately as follows: Andrea: Because of the 12. 12 is their lowest common multiple and every time you add 12 on, you're going to have another common multiple. Four would have three and six would have two because six times two is 12 and four times three is 12. (3/ 88, p. 12). [reasoning strategy] During the interview at the end of the course, only two of the questions in this section were missed by any of the students. Three students (Andrea, Tim, and Jason) gave correct solutions to all of the questions. These students entered the course able to think about multiple problems quite easily and their knowledge and understanding remained fairly stable. At the conclusion of the course Kim was able to distinguish between factors and multiples but her knowledge and understanding remained fragile. Linda and Tamara continued to struggle with application problems like the room dimensions (#21) and the pattern of multiples (#18). Since many of these questions could be solved by using familiar ideas about multiples, the particular type of strategy did not seem to influence success in solving the problems in this section. Primes and commsites The questions in this section focused on determining primeness of numbers and providing a representation for primes and composites. Two of 1 74 the questions referred to numbers that are relatively small (i.e., less than 13), two pertained to larger numbers (i.e., greater than 200) and two probed expressions that included variables (see Figure 4.11). 10) Decide whether or not each number is a prime or composite. Explain how you decided. (a) 12 (b) 7 (c) 237 (d) 239 (e) 4a2 (0 7a2 11) Draw a picture or model to represent a prime and a composite number. Probe: Can you generalize for any prime or composite number? Figure 4.11. Primes and Composites Interview Questions Tamara R Reasoning strategies T Technical strategies Gray indieates correct solution Figure 4.12. Participants' Strategies for Primes and Composites Questions 1 75 Figure 4.12 shows the type of strategy used and whether or not students gave a correct or incorrect response to the interview questions related to primes and composites. During the first interview, all but one of the students, Jason, responded correctly to the two smaller number questions. Jason was unable to answer correctly because he was not familiar with the terms, "prime" and "composite". Most of the students used a memorized procedure or rule to respond to questions in the primes and composites category (see Figure 4.12). Success was greater for students who pondered the reasonableness of procedures selected and/ or their problem solutions. Students were limited in the number and types of strategies that they used to determine primeness of the larger numbers, 237 and 239 (10c and d). The most popular approaches included the application of remembered divisibility rules and an "inspection" method. Tamara's response illustrates the inspection method: ”239 is prime because I can't think of anything right off hand that would divide into it." Even when students, such as Tim, gave a correct response, their "proof" seemed to rely on limited evidence: Tim: I would say that 239 is prime because I can't think of anything that divides evenly into it. I: What numbers were you trying? Tim: I know 2 and 5 because of divisibility rules so I divided by 3, 9, and 7. (3/ 88, p.4). [technical strategy] Tim responded to this question quickly and confidently. After testing only three numbers he seemed to be satisfied that the number was prime. 1 76 In the interview at the end of the course, some of the students seemed to have developed other strategies to use when confronted with larger numbers as evidenced by Tim's response to the same question (#10d). Tim: 239 is prime because I tried to divide it by all the prime numbers less than 15. I: Why did you stop at 15? Tim: Because if you go higher than the square root, you start repeating factor pairs from earlier numbers. I: Why did you just try prime numbers? Tim: Because if it's a composite number, it is divisible by at least one prime number smaller than it. (6/ 88, p.5). [reasoning strategy] Jason also used the square root notion, but he seemed to use it more as a procedure than as a means for thinking about the structure of the numbers: Jason: Well, I'd use the line of squares, or whatever that thing is called. 15 is right around the square root, close to the square root so I'd check all the numbers, 2, 3, 4, 5, 6 up to 15. (6/ 88, p.6). [technical] Linda was the only student who was unable to correctly determine that 239 is prime. Her approach seemed rather random: Linda: For these next two [237 and 239], I would just try a couple of numbers by trial and error to see if I can get that number to go in there evenly. I: What would be the first number that you would try? 177 Linda: I would try 21, just for the heck of it. I figure you are most likely going to have an odd number going into it because, the endings of these numbers are odd. Like I said, I'm just trying them. I: How will you know when you are finished? Linda: Well, one thing that I might do is try to find a prime number that would be as close to it as possible or a number that maybe was like 236 that could be broken down into certain numbers, you know two by whatever or four by whatever. And then I would maybe look at numbers that would be very close to it, and just try it out and see if I could find some kind of comparison. (5/ 88, p.7). [technical strategy] Variables that appeared in generalized expressions were a stumbling block for some of the students. During the first interview, two students, Kim and Tamara, correctly determined primeness for the first four numbers but the variables and symbolism of a multiplicative structure created problems for them. For both 4112 and 7112, they seemed to ignore the a2 and looked only at the 4 and 7 to determine overall primeness. Kim said, "4 is composite, so this [4:12] is composite and 7 is prime so this [7a2] is prime." Tamara's response was virtually the same. During the second interview Tamara's response did not change. Kim made the same decisions but questioned her thinking: Kim: This one [7a2] is prime but [pause] this [points to a2] is what throws me, because I know there's two a's there, but I also know 7 is prime so I'll say it's prime. (6/ 88, p.9). 1 78 Kim's response indicated that she had limited understanding of the power of prime factors or the Fundamental Theorem of Arithmetic”. Additional evidence for this assertion is provided by the following interview excerpts: I: It has been said that prime factors are the "building blocks of arithmetic." What does this mean to you? (#814). Kim: That's because that would be the starting point for everything. I: Could you say more about what you mean by that? Kim: Not really. (5/ 88, p.4). [Response to a different question] I: What is the Fundamental Theorem of Arithmetic? Kim: I don't know that one. (5/ 88, p. 8). The combination of a number and a variable also created uncertainty for Linda. She gave a correct response but noted that she was unsure: Linda: I'm not sure about these two numbers [4112 and 7a2]. This [4:12] would be a composite number because four can be broken up into two times two. I guess I would have to say that this one [7a2] is composite [pause] but seven is prime. I'm really not sure but I'll say composite. (6 / 88, p.7). [technical strategy] Other students, such as Andrea, seemed comfortable with all of these questions during both interviews. The two problems involving variables did not create problems for Andrea as they did for Kim and Linda. In fact, in 10The Fundamental Theorem of Arithmetic is that every whole number greater than 1 can be written as a product of primes in a unique way. 179 Andrea's response to 7a2, she qualified it by saying, "7a2 would be composite unless a is 1 and then it would be prime." Her responses to the questions about the prime factors as "building blocks of arithmetic" evidenced a very different understanding as well: Andrea: They're like basic numbers. . . it doesn't have any other factors in it. And every number you can start with a prime number and just keep building up. Like, I'm trying to say this the way I think, all the composite numbers have other factors. Below them there are prime numbers, when you keep factoring down you end up with the powers of prime numbers. And once you get to the prime numbers, you can't go below it, and so prime numbers make all composite numbers. They make all numbers in some way or another. (4/ 88, p.11). Like Kim, Andrea did not recall the label "Fundamental Theorem of Arithmetic," but Andrea's responses indicated some degree of understanding of the concept. None of the students remembered the Fundamental Theorem of Arithmetic as such. Some of the students used it as if they understood but when their responses were probed, it became evident that they did not remember or understand. Tim and Linda's responses illustrate this contrast: Linda: Linda: [part of response to #Blc] . . . I would think that the Fundamental Theorem of Arithmetic, I mean, this would be your basic corner stone of everything. It seems like everything, you know, maybe this would go at the top of my concept map. And what is the Fundamental Theorem of Arithmetic? Good question, I don't know. (5/ 88, p.6). 180 Tim: [part of response to #Blc] . . . And with the Fundamental Theorem of Arithmetic, I would say that goes in with mathematical reasoning. I: What is the Fundamental Theorem of Arithmetic? Tim: I'm not sure, yes, it would be associated with everything else. It's encompassed under, I'm really not familiar with the term, I'm not even sure, I don't remember it. (5/ 88, p. 2). Others understood some of the underlying concepts but did not label them as such. I: Some people refer to prime factors as the building blocks of arithmetic. Why might that be appropriate? Tim: Because all numbers are going to end up with prime numbers. You figure them out, by breaking the numbers down into the prime factorization and going from there. For example 6 and 42. They don't look all that much alike, but if you break them down, 6 is 3 x 2 and 42 is 3 x 2 x 7, so you break them down into more familiar numbers which are the prime numbers and you can see the similarities. (5/ 88, p.11) Similarly as in the set of questions in the factors and multiples sections, students could provide correct solutions to this set of questions by retrieving a formula or procedure or remembered rule. While students increased the percentage of reasoning strategies used (24% to 79%) to correct responses to these questions, it was not a necessary condition for success. It does support the assertion that students were continuing to change the ways they 1 81 approached mathematical problems from a more technical orientation to a reasoning one. Summary By the end of the course, five of the six students seemed to have a working knowledge of factors and multiples. They recognized some of the basic ideas related to factors and multiples and were able to apply that knowledge to related problems that were similar to problems explored in Math 201. For the interview at the end of the course, two of the students, Jason and Tim, gave correct responses to all of the questions. Determining primeness of larger numbers remained a challenge for Andrea but other areas seemed stable. For these three students, their knowledge and understanding about factors and multiples deepened throughout the course. From the beginning of the course to the end of the course, Kim clarified many ideas about factors and she developed some new notions about multiples, but variables continued to pose a stumbling block. Linda and Tamara's knowledge and understanding about factors and multiples seemed quite fragile. They were facile with basic factor and multiple questions but struggled with related application problems. By the end of the course, the number of correct responses did not change substantially but in all three sections, students were more inclined to use reasoning rather than technical strategies to respond to interview questions. In the next chapter I will examine questions related to what students were able to do with their knowledge and understanding about number theory concepts embedded in the Math 201 course content. What did these students learn about relationships between factors and multiples? What did they learn about the structure of number? What did they learn about the 182 "power" of these ideas? The focus of the chapter is on students' flexibility in using number theory ideas in generalized and unfamiliar contexts. CHAPTER 5 LEARNING MATHEMATICS TO TEACH: RECOGNIZING AND APPRECIATING THE POWER OF VARIOUS MATHEMATICAL IDEAS As argued in the previous chapter, what students learn about the mathematical content is deeply intertwined with their ways of thinking about mathematics and their patterns of reasoning. The previous chapter focused on students' ideas about number theory concepts embedded in contexts similar to those explored in Math 201 and how those changed. I argued that these prospective teachers were continuing to change the ways they approached mathematical problems from a more technical orientation to a reasoning one. They recognized some of the basic ideas related to factors and multiples and were able to apply that knowledge to related problems that were similar to problems explored in Math 201. In this chapter, the strategies and patterns of reasoning that these prospective teachers used to solve problems continues to be examined as I investigate questions related to what prospective teachers were able to do with their knowledge and understanding about number theory concepts embedded in the Math 201 course content. What did these prospective teachers learn about relationships between factors and multiples? What did they learn about the structure of number? What did they learn about the "power" of these ideas? The research question that is examined in this chapter is: In what ways and to what extent do students I prospective teachers] apply their knowledge of this mathematical content to solve problems or create new mathematical knowledge? In thinking about some of these issues it is helpful to me to use an analogy to a road map. As I describe this analogy, some aspects will relate to 183 1 84 the learner while others relate to the teacher. This is appropriate since the students were learners in Math 201 but they also were prospective teachers. If one thinks about a road map, one has some understandings about the ways in which road maps are typically constructed and the meaning that conveys. There are cities located on the map and usually a number of ways or routes to reach a particular city. Depending upon what area the road map represents, one may know different things in greater or lesser detail. Sometimes the area is totally unfamiliar and thus one is dependent upon prior general knowledge to figure out or make sense of the map. Other times it may be quite familiar and one may have traveled across many of the roads so one may have other information and experiences on which to draw. The frequency with which roads are traveled varies and some roads exist that are unknown to some travelers. New roads are often constructed and repairs are ongoing. One also can think about a type of mathematical road map where the "cities" represent mathematical concepts. There are various routes to travel from one mathematical concept to another. Even though one can only physically be in one place at a time, if one is familiar with the area, one can use that familiarity to help one make other decisions. Just as cities on a regular map vary in size and importance so do these mathematical concepts. The "terrain" along the various routes chosen also may vary and the more one knows about the terrain the more informed the decisions can be. As a teacher, the more detailed the mathematical road map can be the more flexibility the teacher has in allowing students to choose different routes and in being able to make sense out of where students are going and how they are getting there. The teacher can also assist in the construction of new roads. 1 85 Imagine planning a trip to another city, for example, Boston. Various options related to planning the trip may be investigated. There are many "routes" to choose from and within each of those choices, other decisions must be made. For example, what form of transportation—plane, bus, car, train? If a person chooses air, she1 must then select an airline and many factors influence that choice—cost, time schedules, frequent flyer mileage, length of stay, previous experience with that airline. Then she needs a place to stay, food to eat, things to do. . . . Each of these decisions will affect a person's experience in getting to Boston. Even if two people choose the same route, travel together and make the same "stops," the interpretations of the experience will vary. Once one arrives in the "city" the experiences also will vary - experience in Boston as a first timer will probably be quite different from someone who has lived there or who visits frequently or knows someone who lives there. Each person will return home with different impressions and interpretations of their trip to Boston and of the city itself. As a first timer to Boston one may ask questions that may seem unusual to a native Bostonian. As a resident, one may take for granted things that seem extraordinary to a "fresh" eye. But the newcomer's observations and questions may make the "old timer" take another look at familiar territory through a different lens. A similar phenomenon can occur in the mathematics classroom when a student "visits" a concept for the first time, she may ask fundamental questions that may appear almost trivial to a student who thinks she knows this area. As the newcomer works hard to make sense and raise questions about the topic, the student more familiar 1For ease in reading, I have chosen to use the pronoun she rather than he/ she. 186 with the idea may be forced to take a different look at what she thinks she knows about that idea. A trip to one city may enrich the way one visits a second city. One remembers things one saw in the first city and perhaps it enables one to view things in the second city in a different manner. For example, if a person has visited some of the fraction "cities," and knows something about them, that understanding can enhance her thinking as she visits probability "cities." After a person has visited several cities she begins to develop some notions about "cities" (i.e., mathematical ideas). She begins to expect certain things that seem common to many mathematical cities-mathematics makes sense, patterns often occur; all problems are not quickly solved. To continue our thinking about this analogy, imagine moving to an unfamiliar city. During the first few weeks, if a person knows one way to get to a particular place, she feels fortunate. Later, she may know how to get to a number of different places but have no idea of the relationship among the routes leading to the individual places so her ability to maneuver around the area may be quite limited. But once she makes the decision to live in that city, she will try to make sense out of "trips" and begin to develop a mental "map" of the area. This mental map provides greater flexibility for traveling. One begins to realize that roads are laid out in a logical mannerz, so just knowing the general direction (e.g., west) provides useful information and invites exploration without a great risk of "getting lost." The evolving mental map also enables one to make sense out of references by others to places in the surrounding areas. It is as if once one begins understanding the general layout and connections one can begin to organize knowledge about the area in a different way. 2This is not true in some geographic locations, but is often true in many areas of the U. S. 1 87 As a mental map of the area develops, one also can become more knowledgeable about the terrain of the area. Some roads are better than others; some routes are more efficient; some are more scenic and the choices one makes are dependent upon the criteria for getting there. If a person encounters unexpected construction/ traffic problems, she can change direction if she is aware of alternative routes. Thus knowledge of the area can become quite flexible. It can be organized in a way that allows one to draw on it in a number of different ways and situations. I think a similar means of organizing one's mathematical understandings can provide flexibility and enable one to draw on that knowledge when appropriate. When one considers mathematical road maps, the journey to the "cities" or mathematical concepts may vary and there can be many variations in experiences along the way as well as when one reaches a destination. Students can share the same experience in a mathematics course but interpret the mathematical ideas differently as they draw from different experiences and prior knowledge. The road map analogy will continue to guide our "journey" as I return to the analysis of what students learned about number theory concepts during the Math 201 course and, in particular, what students were able to do with their knowledge and understanding. Flexibility in Using Number Theory Ideas "Mathematical power" is a term often used by current math reformers. The National Research Council (1989) argue that mathematical power "requires that students be able to discern relations, reason logically, and use a broad spectrum of mathematical methods to solve a wide variety of nonroutine problems" (p.82). Students construct new knowledge and add to existing ideas within some type of organizational framework. The ways in which one connects knowledge and ideas influences the context in which it Bi at nu 1 88 might be retrieved for later use and application (see e.g., Confrey, 1990; Hiebert & Lefevre, 1986). Initial connections can be quite fragile. Organization and storage of knowledge becomes critical to accessing appropriate information in unfamiliar contexts (Silver, 1981; Garofalo & Lester, 1985; Lesh, 1985; Schoenfeld, 1992). The interview questions categorized in this section were designed to push students to employ mathematical "power" and flexibility in using individual pieces of knowledge related to the number theory concepts explored in the course. Two major areas related to this kind of power were explored: (1) structure of number and (2) relationships. I discuss each in turn below. Strumre of number How are numbers constructed? What is the scaffolding on which the whole numbers can be built? It can be additive which is the way in which children meet the whole numbers in the early grades. This means that the numbers can be thought of as generated by continuing to add the same quantity to a starting number. In particular, we can begin with one and continue to add one. Symbolically we would represent this as n -> n+1. Or as children proceed through the grades they need the notion of numbers generated as products of primes. This multiplicative structure is based on one as a unit and the Fundamental Theorem of Arithmetic which tells us all whole numbers other than one can be written in a unique way as a product of primes. Different situations call for an interpretation of whole numbers in each of these ways. Understanding how numbers are built (i.e., composition of numbers) is important but an even greater flexibility occurs when one is able to think about the decomposition of numbers as well. Ideas related to the structure of numbers are fundamental to understanding many mathematical patterns and 1 89 relationships. The ability to recognize structure of numbers' principles and ideas in a variety of contexts also is valuable and increases one's flexibility in understanding and using numbers. Multiplicative and additive structures in a generalized context. The structure of numbers category includes how one builds numbers (e.g., the distinction between the multiplicative structure of a number and an additive structure—the distinction between p131 . p282. . . pnan and n -> n+1)3. Ideas related to the multiplicative structure of a number include understanding and interpreting the symbolism of exponents; understanding and interpreting the concept of division and remainders. Researchers (e.g., Bell, 1979; Kaput, 1987a, 1987b; Romberg, 1983) have argued the value of knowledge about generalization and symbolization. Figure 5.1 shows the interview questions in this category. 5b) What is the greatest common factor (GCF) of 2a2 and 8ab 5c) What is the greatest common factor of n and n + 1 10) Decide whether or not each number is a prime or composite. Explain how you decided. (e) 4a2 (0 7a2 15b) What is the least common multiple of 2a2 and 8ab 15c) What is the least common multiple of n and n + 1? BB) Findallthe factors of2 . 3-5- 7. B4) How many factors does 28 . 35 have? BS) If p and q are primes, how many factors does pk 0 qm have? Figure 5.1. Multiplicative and Additive Structure Interview Questions 3p1 a1 - pzaZ. . . pnan stands for the product of n distinct primes each of which may be raised to a whole number power greater than or equal to one. _M&&Cg 1 90 Figure 5.2 shows the type of strategy used and indicates whether or not students gave a correct or incorrect solution to interview questions related to structure of number. 5b Additive in A1 A2 Jason R Andrea Tamara R Reasoning strategies T Technical strategies Gray indicates correct solution Stripes indicate calculation wrong, but reasoning correct Figure 5.2. Participants' Strategies for Multiplicative and Additive Structure Questions The first and last question in the A interviews was: "What can you tell me about the number 16?" This open-ended question was intended as a conversation starter about numbers. How do students think about numbers? How do they think about the composition and decomposition of a number? 16 was chosen because it lends itself to a wide range of possibilities including 1 91 squareness, many multiplicative and additive combinations, and parity‘. Omission of any of these did not indicate that students were incapable of thinking about those ideas but provided insight into initial ways that they think about numbers. During the first interview (A1), some of the students had only one or two ways to describe the number 16. For example they described multiplication and division combinations (e.g., eight sets of two, four sets of four, divisible by two and eight). Others added one or two other ideas including the notion that 16 represents a square number, it can be derived from other arithmetic combinations such as subtraction, it has a position in the counting numbers, or is an example of an even number. In contrast, Andrea came into the course with flexible ways to think about numbers. When she talked about the number 16, in addition to the ideas expressed by other students, she included both additive and multiplicative strategies; she used fractions (e. g., 15 1/2 + 1/ 2); she described 16 in relation to objects such as 16 apples; she drew pictures and described 16 squares or a particular group of 16 things; and she used language that included terms like factors and primes. By the end of the course all of the students seemed to have increased the range of ways to think about the number, 16. Kim's response illustrates well the expanded notions students used to describe the number 16: Kim: Sixteen singles; its factor pairs are 16 and 1, 2 and 8, 4 and 4. It is the least common multiple of 16 and 8, it's an even number, it's the greatest common factor (GCF) of 16 and 32, it's a square number. (6/ 88, pl). 4T‘he oddness or evenness of an integer. 1 92 The range of responses to an open-ended question like describing the number 16 is not necessarily indicative of all of the ways in which students could think about the question. However, the responses may raise questions that push our thinking about possible ways in which one organizes knowledge, makes decisions about which knowledge to access, as well as the flexibility with which one uses knowledge related to the question. Student responses also can provide insight into how students responded to more context specific questions related to the structure of number. In this section there are four sets of questions that probe students' recognition and understanding of the distinction between the multiplicative and additive structure of numbers in generalized contexts. The context for #5b and c5 is finding the greatest common factor between two expressions involving variables and for #15b and c6, it is finding the least common multiple between two expressions involving variables. As evidenced by responses for problems #10e and f7 (see Figure 5.2), variables created tension for some of the students. During the first interview most of the students were able to respond correctly to the greatest common factor and least common multiple questions involving numbers (refer to Figures 5.1 and 5.3). Only one student, Jason, gave an incorrect response for the greatest common factor of 2a2 and 8ab (question 5b). He said that he needed to know what number the variables represented to determine the greatest common factor. Kim and Linda gave 5#5b) What is the GCF of 2a2 and Bob? (5c) What is the GCF of n and n+1? [structure of numbers, multiplicative and additive structure]. 5#15b) What is the LCM of 2a2 and 8ab? (15c) What is the LCM of n and n+1? [structure of numbers, multiplicative and additive structure]. 7#10) Decide whether or not each number is a prime or composite. (e) 4:12; (1’) 7a2 . [structure of numbers, multiplicative and additive structure] 1 93 correct responses but each questioned their thinking. Kim's response illustrates this: Kim: It would be 2a. This [points to a] is where you lose me. 2112 is 2a x 2a [she thought the 2 was squared as well]8, then 8ab couldbebroken downto2x2x2xaxb, well, waita minute, maybe there isn't one. I was thinking [repeats the expanded form]. But now I don't think that that could be done. [pause] Well, yes it could because you could divide each side by 2a. I seem to remember something about that from factoring in algebra so my answer is 2a. (3/ 88, p.4). [technical] The distinction between the multiplicative structure in 2:22 and 8ab (questions 5b and 15b) and the additive structure in n and n+1 (questions 5c and 15c) was not apparent to some of the students. Three of the students responded to the additive structure represented in problem #Sc as if it represented the same type of multiplicative expression as problem #Sb. Each thought the greatest common factor was n.9 Linda explicitly pointed out the sameness: Linda: Well, along the same lines of reasoning, it would have to be u. (3/ 88, p. 4). [technical] Tim recognized the distinction between the multiplicative and additive structures but thought the two numbers did not have a greatest common factor. Apparently he did not consider that one could be the greatest common factor“): 31f 2a2 represented2ax2aasKimargued,theGCl'-‘wouldbe4aandnot2a. 91f #Sc had been multiplicative, for example n and 3n, rather than additive, n and n+1, Linda and Kim's responses would have been correct. 100m hypothesis might be that students who use the procedure of breaking down numbers into their prime factors and then identifying common factors to find the GCF may confuse the fact that one is neither prime nor composite number and thus do not consider that the GCF can be 1. If] wasconsidered tobeprime,allnumberscould notbeexpressedasauniqueproductofprimes (e.g., 10couldbe2x50r2x5x1 or2x5x1 x1 andsoon. 194 Tim: There is no common factor. I: And how do you know that? Tim: Because, you can't divide n + 1 by n and get an even number. (3/ 88, p.2). Andrea's response to this question indicated a different understanding: Andrea: There's not one. No, wait, it would be one. I: And why do you think that? Andrea: Because, it would be like two numbers in a sequence. There wouldn't be a factor that you could pull out of them that could be the same besides one. ( 3/ 88, p.3). [reasoning] During the interview at the end of the course, all of the students responded correctly to question #Sb. Kim hesitantly responded 2a but became more confident of her answer as she explained why: Kim: 2a. 1: And why do you think 2a? Kim: Well the 2, I'm not having a problem with, it's the a. Let me think. [pause] Because they both have one a and ifI factored it out, that [2a2] would factor out as 2 x a x a and that [8ab] would factoroutastZxeaxb. Soifyou take a Zn out of both, that's the biggest that you would be able to get. (6 / 88, p.2). [reasoning] Kim's understanding about the exponent notation had also changed. In contrast to the first interview when she interpreted 2a2 as (2a)2 (i.e., 2a x 2a), 195 she now realized that this notation indicates that only the a is squared (i. e., 2 xaxal Andrea again was the only student to respond correctly to the greatest common factor problem of n and n+1 (#5c). Kim and Linda continued to respond as if it represented a multiplicative structure. Each was explicit about the sameness in their response: Kim: N because I can factor out an n in both of these. (6/ 88, p.3). Linda: It would be 71. They share a common factor, n. (6/ 88, p.3). Tim, Jason, and Tamara each recognized the distinction between the additive and multiplicative structure but were unable to find a correct solution. Tim and Jason both thought that there would be no greatest common factor. They did not seem to consider one as a possible GCF. Tamara initially responded 11 but changed her mind as she explained her reasoning. Rather than trying to elaborate and make sense of her reasoning about the distinction between additive and multiplicative expressions, she tried searching her memory from high school math to help her respond: Tamara: I'd say n. I: And why would you say n? Tamara: Because they both have n in them, but this one is addition so I'm not real sure about that. Here [points to 2a2 and 8ab], when we are dealing with factorization, that's factors. And this, [points to n and n+1] is addition and factors are multiplication. [pause] I'm trying to remember from high school. [pause] It 196 just seems familiar that we don't do anything. I just can't remember what I'm supposed to do with these kind [emphasis author's]. I: So what is the greatest common factor? Tamara: They don't have one (5/ 88, p. 3). [first part reasoning, last part technical] In the context of finding the least common multiple (LCM), during the first interview two students gave correct responses to questions #15b and #15c11. Andrea, Tim and Jason all recognized the distinction between a multiplicative and an additive structure. Andrea's response was correct for both questions. Jason applied a "system"12 that he developed finding the LCM of 36 and 63. When confronted with finding the LCM of n and n+1, he said that he did not know because he needed to know what it represented. He recognized the distinction between multiplication and addition but seemed unsure what to do with that information: Jason: Once again you need to know n. I: I'm curious that you didn't need to know a or b to solve this problem [#15b], but you need to know n to solve this one [#15c]. Why do you need to know u? Jason: This is multiplication, a here [2112] is the same as the a there [8ab], a factor to multiply. But here In and n+1] if n was five, you've got a five and a six so the LCM is 30 but 11 #15b) What is the LCM of 2a2 and Sub? (15c) What is the LCM of n and n+1? [structure of numbers, multiplicative and additive structure] 12Using several number examples to experiment, Jason eventually generalized his ideas into a formula A x B divided by the GCF and called it his "system". In essence he multiplied the two numbers together and then divided by the largest common factor. 197 if n is another number, you'd get a different solution. (3/ 88, p. 9). [reasoning] In response to question #15b (LCM 2a2 and 8ab), Tim multiplied the two expressions and seemed to disregard the common factor. He applied the same idea of multiplying the two expressions to find the LCM of n and n+1. As mentioned earlier, Kim did not distinguish between factors and multiples during the first interview. She responded to these questions (#15b and c) as if the questions were about finding greatest common factors, thus, she duplicated her answers for problems #5b and C”. Linda attempted to apply a remembered rule to solve both of these questions: Linda: It would be 8. I have to figure out how to do this. It would probably be So. I don't know exactly the rule for this, but if you start off adding this up, I don't know if it goes a to the fourth or a to the sixth. I'm not exactly sure on the rule. And I don't know if you'd include the b since it's not originally there. (3/ 88, p.10). [technical] Tamara felt that there would not be a least common multiple for 2a2 and 8ab(#15b) because "2a squared doesn't have a b." In response to n and n+1, she said, "I don't know." At the end of the course, all students responded correctly in determining the LCM for 2a2 and Bob. Linda remained unsure and used the previous problem to help her think about this problem: Linda: I get confused on these [points to the variables]. To tell you the truth, I know the least common multiple would be eight, because the smallest number here is eight and two is eventually going to get to eight. I'm not sure when you start off with 2a squared, I don't know if you double this to get a to the fourth or not. But, if you're 13#5b) What is the GCF of m2 and 8ab? (5c) What is the GCF of n and n+1? [structure of numbers, multiplicative and additive structure] 198 taking the LCM between both of the numbers, it would definitely have to be eight a squared b somehow because you need something that is going to go into both of the numbers equally. If this has an a and b and this one has an a squared, you definitely have to have an a squared and a b, but [pause] I don't think this would be the right answer but that's what I'm going to say. (5/ 88, p. 8). [technical] When Linda looked at n and n+1, she said, "I would have the same problem here, exactly the same. . . I don't know what it would be." During the first interview, Jason said that he could not solve this problem without knong what n represented. By the end of the course, Jason was able to reason about finding the LCM of n and n+1 without knowing what n represented: Jason: These numbers are consecutive, they would be right next to each other, so you've got an odd and an even number, so I guess if it's like two and three, it would be [pause] it would be the two numbers multiplied by each other. I: And why would it be that? Jason: Well, I guess, you know when you multiply them together, the number is going to be a multiple of both of those numbers because you just multiplied them. And it's nothing less, because the numbers are right next to each other. (5/ 88, p.9). [reasoning] At the end of the course, Tamara's response to finding the LCM of n and n+1 was significantly different from her response (i.e., "I don't know.") during the first interview: Tamara: That would be n times It plus one so it would be n squared plus n. 199 I: And why would it be that? Tamara: Because if you break them down, they don't share any factors so you have to multiply them by each other and then you get n squared plus n. (5/ 88, p.9). [reasoning] Some of the students who were able to solve greatest common factor and least common multiple problems when specific numbers were given were unable to solve similar problems that included a variable which represents a more generalized form. Solving generalized problems requires one to analyze specific number examples and apply what one knows more broadly. It raises the level of the question beyond a response to a specific number problem and promotes a more generalized way of thinking about a mathematical idea. It also requires students to recognize that the generalized form shares a similar structure to the specific form. In this particular set of questions the distinction between additive and multiplicative structure was important. Again, recognizing and appreciating the power of that distinction proved to be a discriminating factor. For the set of questions in the B interview in which students were to identify all possible factors by examining the prime factorization (B3-5)14, five of the students seemed to just manipulate the numbers and/ or exponents in an effort to find a solution. Kim's work is one example. As Kim worked on finding the factors for 2. 3.5.7 (#B3), she became confused about whether or not to include 105 as a factor. After a great deal of work and talking about the problem she multiplied the prime factors to get the actual number: 210. Then 1433) Find all the factors of 2- 3.5. 7. (34) How many factors does 28 - 35 have? (as) If p and q are primes, how many factors does pk - qm have? [structure of numbers, multiplicative and additive structures] 200 she started over by taking the square root of 210 and checked each number from 2 to 12. She recognized that she could use a "procedure" to arrive at the solution. Having the prime factors seemed unimportant to her other than to enable her to multiply them to find the "number" itself. When probed about how the square root "rule" related to this problem she replied, "It's something we learned in class that you're supposed to do." None of the students had a systematic way to determine all the factors (e.g., generating all possible combinations of prime factors). The next question was How many factors does 23 .35 have? (B4). Kim saw a "quick" readily available procedure and grabbed it - she added the exponents, said, "13" and was content with that response. Three other students initially added the exponents but decided that "13" was not large enough so perhaps they should multiply. In contrast to the other students, Andrea reasoned through this set of problems (B3-5). Her solution to B4 was partially flawed because she counted the number itself twice - in the beginning and in the 8 x 5 match ups thus she always had one too many factors. The process she worked through to solve this problem was interesting. She made notes on her paper and reasoned by starting with one and the number itself. On her paper she wrote: 1 n 2 3 28 . 35 8 5 combinations of the 2 Next she talked about each factor by itself and said, "In this case eight possibilities for two and five for the three so +8+5 and then the matches so 8 x 5. She wrote: 28~8 combinations-—21 , 22, 23, 24, 25, 25, 27, 23 35—5 combinations-—31 , 32, 33, 34, 35 201 All the matches—2 . 31;2 . 329 2- 33; etc. For the next question, "If p and q are primes, how many factors does pk- qm have?", she generalized her thinking from the previous problem and wrote: Ln 2 factors k +m kxm = 2+k+m+km Again she counted the number itself twice but otherwise her reasoning was valid. In this group of interview questions whether or not students recognized the common characteristics and relationships among the set of problems proved to be an important factor in students' level of success in finding solutions to this set of questions. Students who approached these questions as a related set, that is, were able to identify the shared features, were more successful than students who approached each problem as a distinct and isolated question. A continuing theme that proved discriminating among successful responses was recognizing and appreciating the power of what one knows. In this set of questions it was important to recognize and appreciate what one knows given the prime factorization of a number including the relationship between the structure of a number and its prime factorization. Symbolismuunderstanding exponents and their relationship to the structure of numbers was another important factor in solving some of the questions in this set. Another recurring theme—being able to recognize the generalized form (e.g., pk . qm) and how to apply what one learned from a specific example to a more generalized format also was valuable. In the structure of number category, four of the six students relied on technical strategies during the first interview (see Figure 5.2). They 202 experienced limited success. Unlike the questions analyzed in the previous chapter, this set of questions required students to use their knowledge in flexible ways and in unfamiliar contexts. Tim's emphasis towards using reasoning strategies changed little as did the number of correct responses. Tamara provided correct solutions to more than half the questions in this set by utilizing predominantly technical strategies. Andrea continued to be an anomaly. She primarily employed reasoning strategies and responded correctly to all of the questions in this set during both interviews. By the end of the course, half of the students continued to struggle with questions that involved variables or required students to recognize and appreciate the distinction between the multiplicative and additive structure of numbers. For these students, even in a familiar context of factors and multiples, knowledge needed to think about the structure of numbers was fragile. Andrea was the only student who responded correctly to all of the questions in this section in both interviews. The remaining students seemed unable to recognize that often they possessed the knowledge needed to solve the problems in this set of questions, but perhaps did not appreciate the power of the ideas they possessed. They did not appear to be able to recognize mathematical ideas that would be useful in some of these unfamiliar contexts. Returning to the city and map analogy, as one continues to add to one's knowledge about specific roads in an area, it is helpful if one begins to look for relationships among roads and the ways in which the roads fit together into a larger map of the entire area. Other important features about the layout of a city include knowing multiple names for the same road, awareness of areas where there are frequent one-way streets. Having the knowledge is important but knowing when and how that knowledge might 203 be useful is equally important. Another important factor in gaining flexibility in traveling within a city is to begin to identify the main roads/ arteries of that particular city. All roads do not represent equal access to other areas. Some roads dead end or wind through residential areas while other roads provide access to major business and life-line areas. Recognizing and appreciating the power of this knowledge is critical to obtaining flexibility in maneuvering around a particular city. Recognizing and appreciating the power of various mathematical ideas also is critical to achieving mathematical power and flexibility in using one's knowledge about mathematical ideas. Structure of numbers in unfamiliar contexts. The questions in this section are posed in contexts that differ from the way in which students commonly encounter these number theory ideas (e.g., what is given and what students are asked to find is different - the notion of reversibility). Many of the questions in the B interview pushed students to use mathematical knowledge in unfamiliar contexts (i.e., not encountered in the Math 201 class“). Many of the students struggled to make sense of these questions and often gave incorrect solutions. There can be many types of relationships within a given concept domain. One special relationship is being able to think about mathematical ideas in multiple directions, reversibility. One example would be to work from a number to its prime factorization (e.g., 100 = 2- 20505), the "reverse" is to work from the prime factorization to understand a number. Students encounter mathematical problems related to a particular idea in a particular context or format. Changing the way the problem is framed may provide an unfamiliar context. The interview questions in this section 15For further elaboration about questions in Interview B and unfamiliar contexts, see chapter 2, methodology chapter. 204 were posed in the more unfamiliar contexts. Also included in this section are questions that may appear unfamiliar because of the representation used. For example, posing the question, 'Ts 40 a factor of 720?" might be a more familiar context for students than framing the questions as "Is 23 . 5 a factor of 24 . 32 . 5 ? " The purpose of the questions was not to "trick" students but to focus on the relationship between the structure of the two numbers. A question posed in this format raises the level of the question beyond a response that focuses on a procedure and promotes a more generalized way of thinking about structure of number more broadly. Figure 5.3 shows the interview questions in this section. 205 8) 2 and 4 are factors of a number. Is 8 a factor of that number? Explain. 9) h23-5ataetorot24-32-5? If yes, how do you know? If no, what could you change to make it a factor? 18b) Show the student rectangular models for multiples of 4 and 6. [label the models so the sequence of numbers is clear.) In the group of models showing multiples of 6, every second one is a common multiple of 4 and 6. In the group of models for multiples of 4 every third one is a common multiple of 4 and 6. Explain why this pattern occurs? 19) 1:32-32 o53amu1tip1eot22 o3-5? I-Iowdoyouknow? If no, what could you change to make it a multiple of 22 0 3 0 5? 20) A number is a multiple of 7. Another number is also a multiple of 7. (a) Is the sum of the two numbers a multiple of 7? How do you know? (b) Is the product of the two numbers a multiple of 7? How do you know? 82) A number is less than 200 and has more factors than any other number less than 200. What is the number? B6) What is the smallest number which is divisible by 2, 3, 4 and 5? B7) What is the smallest number which leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6 regectively? B8) What is the smallest number which is divisible by 2, 3, 4, 5, and 6 and leaves a remainder of 1 when divided by 7? B9) How many numbers between 1 and 1000 are divisible by 6 and 9? 812) Without multiplying the number out can you tell how many zeros are on the end of this number: 24 . 34 . 53 . 732 Figure 5.3. Structure of N umbers in Unfamiliar Context Interview Questions Figure 5.4 shows the type of strategy used and indicates whether or not students gave a correct or incorrect solution to interview questions that highlight the structure of numbers in unfamiliar contexts. R Reasoning strategies T Technical strategies Gray indicates correct solution Stripes indicate calculation wrong, but reasoning correct Figure 5.4. Participants' Strategies for Structure of Numbers in Unfamiliar Contexts Questions One particular question focused on understanding the structure of a number when given an exponential representation“: Without multiplying the number out, can you tell me how many zeros are on the end of this number 24 .34 ~53 . 73? Tell me why you think your answer is correct. (B interview #12, structure of numbers, unfamiliar contexts). 16T'he discussion about this question first appears in chapter 2. It is repeated here to highlight a different point. 207 This question required students to use knowledge about the structure of numbers in an unfamiliar context. A more familiar context that students might use this same knowledge is to look at a number such as 80 and identify numbers that divide it evenly without actually dividing the number. Most students would immediately list 2, 5, and 10 because the number 80 ends in a zero. They may or may not recognize that there is a relationship between the three factors. Reversing the way in which the question was posed, proved difficult for most of the students. Only one student, Andrea, responded with a correct solution to the number of zeros question and even after she constructed a reasonable argument to support the solution, she lacked confidence that the solution was correct. I would have predicted that at least four of the six would have been able to respond with a correct solution. The answer seemed so obvious to me—such a few symbols can reveal a great deal about such a large number without actually figuring out the particular number represented. I listed some of the things I knew about this number based upon the symbol notation given: the number is even; there are 400 factors of the number and all of the factors can be identified without dividing,17 the numbers that are not factors also can be identified; there are only 4 primes that are factors-2, 3, 5, 7; and of course, the number of zeros on the end of the number18 can be assessed. I carefully examined both the question and the students' responses in an attempt to get clearer about the differences among students' responses and the mathematics that might be required to figure out the question. 17For example, these numbers are all factors: 1, 2, 3, 4 (2 x 2), 5, 6 (2 x 3), 7, 8 (2 x 2 x 2), 9 (3 x 3), 10(2x5),12(2x2x3),14(2x7),15(3x5),l6(2x2x2x2), 18(2x3x3),20(2x2x5),21(3x7),24 (2x2x2x3),. . .thenumberitself. 18A number that ends in zero is divisible by 10 (2 x 5). In the given expanded notation there are threesetsof'ZxS" thus there wouldbeexactly3zerosontheend ofthisnumberwhenitis multiplied out. 208 Why might this be an important and interesting problem or question? It embeds fundamental knowledge about our number system-base ten, place value notions-ideas related to how our system works and thus has implications for understanding the structure of numbers within our numeration system. There are many layers in this problem including symbolism that stands for a mathematical idea—a shorthand mathematical notation19 that focuses on the multiplicative structure. Questions raised by this problem include: What are some of the mathematical ideas that might enable a person to think about this problem? Prime factorization-including symbolism and the ideas represented by it; the power of the Fundamental Theorem of Arithmetic and the notion of uniqueness; factors, divisibility; relationship to operations on numbers- multiplication and division; primes, composites; structure of numbers—how one builds numbers including the distinction between the multiplicative and additive structure of a number (e.g., n and 3n or n2 compared to n and n+1). Each of these mathematical ideas had been explored, to some extent, during the Math 201 course. Another question raised by this problem is what enables students to recognize the power of the mathematics they know and to be able to draw upon that knowledge when appropriate? Careful examination of two of the students-Andrea and Tim- responses pushed my thinking about these questions. On the surface these two students appeared to be quite similar-they had a background of being successful in mathematics including this course and they each had taken 19'Ihenotation24 034- 53 073 representsandeanbeexpandedt02x2x2x2x3x3x3x3x5x5x5 x7x7x.... 209 mathematics beyond the requirements for prospective elementary teachers20 including part of the calculus sequence. There did not seem to be much difference between their individual understanding of the "mathematical pieces" related to this problem as evidenced by their success in responding to other interview A questions related to this question. It appeared to me that each should be able to solve this problem quite easily. So, if there is so much similarity, what made the difference? To explore this question, I examine Andrea and Tim's responses to the question: Without multiplying the number out, can you tell me how many zeros are on the end of this number 24 . 34 . 53 . 737 Tell me why you think your answer is correct (#B12). Andrea: I don't think there would be any zeros on the end of the number. I: Why do you think that? Andrea: Because all the numbers that are two or some power already have some even number on the end of them. Three's end up with some other number. You'd have to have a 10 in there to get zeros. I: What's your answer to the number of zeros on the end of this number? 20Prospective teachers are required to take Math 201. Prerequisite for this course is to take the placement example and place into college algebra (Math 108) or complete Math 082—an equivalent of high school algebra. 210 Andrea: Well, you've got 2 x 5 in there. I think we'd have three zeros. And why do you think that? Andrea: Because you have five to the third three times, and two to the fourth, four times. But you can only match them up to make a 10, three times because there's only three powers of five and then 2 x 5 is 10, so you'd end up getting those three zeros at the end. There wouldn't be any other zeros then? Andrea: No. [pause] I don't think so. I wish I could find out. Andrea had given a correct response and constructed a reasonable argument (5/ 88, p.10). to support the response but she was not confident of her solution. Tim's initial response was 16 zeros but as he explained his reasoning he became less sure and decided that there might be no zeros: Tim: 16. I: And why would you say 16? Tim: Because I added these exponents together and I came up with 14 and then I multiplied the numbers on the inside and I came up with 210, which would be some number to the second, which would be between ten squared [102] and ten to the third [103]. And it's closer to ten squared [102] so I gave it an exponent of two, which means that this whole total of the exponents would be 16. . . And the more zeros it has, if it has two zeros at the end then it's divisible by 100, if it has three zeros, it's divisible by 1000, and so on. I'm not sure if this is going to end up in 16 zeros, just 211 because I'm not even sure it is going to end up in a zero. (5 / 88, p.10). Tim appeared to be using a combination of strategies. Somewhere in his memory was something about adding exponents”. He recognized that the problem represented factors to be multiplied so he multiplied the base numbers (i.e., 2, 3, 5, 7) and obtained 210. He manipulated numbers, symbols and remembered "tricks" to arrive at his solution of 16 zeros. As he attempted to justify this solution of 16 zeros, he seemed to draw upon some remembered pieces of information related to the structure of numbers (e.g., the exponent used with powers of ten coincides with the number of zeros when one expands the number—107 represents 10,000,000). Tim demonstrated throughout the interviews that he had many rules, formulas, and procedures stored in his memory. In some instances he retrieved and used appropriate procedures, in others he seemed to randomly apply remembered procedures to enable him to manipulate numbers and symbols. Examples to support this assertion are described throughout chapters 4, 5, and 6 It seemed that Andrea thought about this problem differently from Tim. Tim immediately started to manipulate the symbols while Andrea seemed to start by thinking about what she knew about the relationship between the endings of numbers (i.e., the last digit of a number) and products of numbers. Puzzling about this difference led to a comparison of Andrea and Tim's responses to other questions which provided insight into their views of mathematics and patterns of reasoning and strategies used to solve other problems. 21When multiplying exponential factors of the same base, a short-cut is to add the exponents (e.g., 43 x45 = 43). 21 2 Returning to our road map analogy, driving around randomly trying to find a location is similar to a person who approaches a mathematics problem by manipulating symbols. This orientation is contrasted to someone who surveys the area (e.g., landmarks, familiar road names) and/ or uses a map of the area and attempts to make sense of a current location and the relationship to a projected location. In mathematics, one examines what one knows about a mathematical situation and the relationship or possible connections to other known concepts and ideas. The end results may sometimes be the same (i.e., the desired location is reached); however, the latter strategy offers consistency and predictability while the former involves luck. Andrea approached problems by thinking about what she knew about the problem and how that related to other mathematical knowledge and understanding. She reasoned about ideas and solutions. Tim was more likely to approach problems by manipulating symbols or searching his memory for an algorithm that might be useful. Unlike Andrea, Tim did not seem to approach mathematics expecting it to be reasonable or sensible. Examination of strategies employed by many of the students for questions in this section reveals the use of examples in one form or another. This was especially true for the questions in the A interviews (#8, 9, 18b, 19, 20)22. Selecting specific examples often is a useful problem-solving strategy, but the power of this strategy lies in being able to generalize from the set of specific examples to the set of all possible numbers. 22#8)2and4arefactorsofanumber. Is8afactorofthatnumber?(9)1323 05afactorof24 .32- 5? (18) In the group of models showing multiples of )6, every second one is a common multiple of 4 and 6. In the group of models for multiples of 4 every third one is a common multiple of 4 and 6. Explain whythispatternoccurs. (19)Is2 . 32 .53 amultiple ot22 o3-5? (20)Anumberisa multiple of 7. Another number is also a multiple of 7. Is the sum of the two numbers a multiple of 7? the product? [structure of numbers, unfamiliar contexts. 21 3 There are research studies that focus on the use of examples both as strategies by students and as a pedagogy strategy for teaching (see e.g., Anderson, Greeno, Kline, & Neves, 1981 ; Chi 6r Bassok, 1989; Eylon 8r Helfman, 1982; Reed, Dempter, 8: Ettinger, 1985; Sweller & Cooper, 1985; Van- Lehn, 1986). Examples are widely used in mathematics instruction to "teach" how to solve problems. The studies cited above support the notion that students experience difficulties solving problems that do not precisely match previously learned examples. Thus, students who rely on examples, without thoughtful analysis of the examples, often experience difficulty generalizing and transferring related understanding. In a physics study about knowledge acquisition and problem solving, Chi and Bassok (1989) found that: In general, all students liked to rely on examples in their initial attempts to solve problems. . . . We have thus characterized the poor23 students' use of examples as searching for a solution or a template from which they could map the to-be-solved problem so that they could generate a solution. . . . the good students used examples truly as a reference. . . . they were using the example to retrieve a particular piece of information. Thus, the good students must already have a plan for a solution in mind. . . . good students often prefaced their example-referencing episodes with the announcement of a specific goal (p. 276). The use of examples by students in my study seems consistent with studies cited above. During both interviews in my study, all of the students used specific examples to structure their response to the question, 2 and 4 are factors of a number. Is 8 a factor of that number? Explain (#8). The way they used the examples varied. During the first interview, two of the students, Kim and Linda, responded "yes" and their explanations indicated that they 23Chi and Bassok's criteria for categorizing students as "poor" and "good" students was directly related to the success rate in producing correct solutions to problems. 21 4 based their decision on knowing that the factors of 8 are 1, 2, 4, 8 thus 8 also must be a factor of the number. Two other students, Tim and Jason, used a single example to determine a judgment. Two students, Andrea and Tamara, used a couple of examples and counter examples and then generalized to think about multiples of 8. Andrea's response follows: Andrea: It would depend on whether it's higher than 4 or not, because it could be 4. If it's 4 then it isn't a factor. I guess it depends [pause], I guess it depends on whether the number, well, like I could think of like 12 where 2 and 4 are factors but 8 isn't a factor. [pause] So the number would have to be a multiple of 8. (3/ 88, p.5). At the end of the course, Tim joined Andrea and Tamara in giving correct responses. All three of these students initially used examples and counter examples and then generalized their drinking to conclude that the number had to be a multiple of 8. The only way Linda could decide was to "test" each example and she was unable to generalize from the collection of tested examples. Some of the students seemed to think about 2 and 4 separately and did not consider that 2 is a factor of 4 and thus the number would need to be composed of 23 in order to have 2, 4, and 8 as factors. The focus seemed to be on finding a counter example rather than considering the structure of the numbers. There was one exception during the interview at the end of the course, when Jason's response included some thoughts about structure: Jason: Yes, except for the number 4. 1: Why wouldn't it work for 4? Jason: Well 8 is not a factor of 4, but 2 and 4 are. I: And tell me why would it work for all the others? 215 Jason: Because any number that has a factor of 4 automatically has a factor of 2. I'm trying to think of how to explain it. Any number that's got a factor of 2 and 3 is going to have a factor of 6. [He checks this conjecture by writing down several examples, 6, 30, 12, 36, and 18.] (3/ 88, p.12). The example pair that Jason chose, 2 and 3, are relatively prime24 and thus his hypothesis was true. He seemed to assume that this hypothesis was true for all pairs of numbers and did not consider how sharing a common factor might influence his conjecture. If, for example, he had considered the pair, 4 and 6, he would have realized the limitations of his conjecture. Questions #9 and 1925 are similar in that a number is represented by its prime factorization written with exponents and students are asked to compare that number to another number to determine whether or not it is either a factor or a multiple of another number. During the first interview, three of the students multiplied the numbers out and then divided before answering the question. Tamara's response to #19 illustrates: I: Is this number [points to 2 . 32 . 53 written on card] a multiple of this number [22 . 3 . 5]? [#19] Tamara: N o. I: And how do you know? Tamara: I used a calculator. IfI hadn't used a calculator I probably would have said, yes, just because they both had a two, three and five. But by using the calculator it doesn't come out even. 24Two numbers are considered to be relatively prime if their greatest common factor is one. 25(9) Is 23 - 5 a factor of 24 . 32 . 5? (19) 132 - 32 - 53 a multiple of 22 .3 . 5? [structureofnurrbers, unfamiliar contexts] Tamara: Tamara: Tamara: 216 And how did you use the calculator? I multiplied the first set of numbers and got 1125 and I multiplied out the second set and I got 60 and I divided 1125 by 60 and got 18.75. So how did you know that it wasn't a multiple? Because it's not a whole number. So what could you change to make this a multiple of that? I really don't know. (3/ 88, p.12). Students who chose to multiply out the particular numbers rather than reasoning more generally about the structure of the numbers related to their bases and exponents, were limited in what they could articulate about this problem. During the interview at the end of the course, all three of the students who multiplied each of the prime factorizations and then divided, were able to look at the bases and exponents to determine whether or not the numbers were factors or multiples of the second number. Tamara's response to #19 is illustrative: Tamara: IsthisnumberlpointstoZ . 32-53writtenoncardla multiple of this number [22 . 3 . 5]? [#19] No. Why not? 217 Tamara: There are two factors of 2 in this one [points to 22 . 3 . 5] and only one in this [points to 2 . 32 . 53] so it can't be a multiple. I: Could you write down what the number would need to look like in order to be sure that it was a multiple of this number [points to 22 . 3 . 5]? Tamara: Two squared times three squared times five cubed [writes 22 . 32 . 53]. I: And how do you know that that number would be a multiple? Tamara: Because all of these numbers would be represented or contained within this number. (6/ 88, p. 13). The problems that required students to utilize the power of ideas they had worked with proved to be stumbling blocks for many of the students. For example, Kim easily identified factors of numbers but when she had information about relationships among factors of a number (#8)26 she could not integrate the ideas and reason about the solution. She was limited to thinking about the individual factors of that number. In contrast, Andrea approached this set of questions trying to think about what she knew about the problem and ways to connect concepts. For example, she used her understanding about factors to respond to a multiple question: I: A number is a multiple of 7 and another number is also a multiple of 7. Is the sum of those two numbers 26(#8) 2 and 4 are factors of a number. Is 8 a factor of that number? [structure of numbers, unfamiliar contexts] Andrea: Andrea: Andrea: 218 a multiple of 7? (#20 structure of numbers, unfamiliar contexts) Yes, it would have to be. And why would it have to be? You're not changing any of the factors of the numbers, so it's going to have [wrote 7 + (2 x 7)], that's really 7 x 3. Multiplication is just a longer form of addition. It's like saying 7+7+7, you know. Whatever, 7+7+7+7+7..., you just keep on adding 7's. Would the product of the two numbers be a multiple of 7? Sure. It still will have the factors of 7 in it. (6/ 88, p.14). Jason was not confident that he had found a correct solution but he was the only student who had a correct response to 82 - A number is less than 200 and has more factors than any other number less than 200. What is the number? : Jason: Jason: [after watching Jason work on his paper for a long time]: What are the numbers that you're writing down there? Factors of 18, 16, 12 [He circled 18]. Why did you circle 18? Well, I'm trying to look at twenty which has six factors and I'm trying to see if any of these numbers have a larger number of factors. I guess I'll say 180. And how did you get 180? 219 Jason: 18 and 12 are tied for the most factors under 20 and I'm just guessing 18 because it's a larger number. I: And how did you get from 18 to 180? Jason: I was just looking at a simpler situation and trying to get a number close to 200. I'm not sure if it really helped me any, but I can't think of any other way to do it without listing out all the factors of all of them. (5/ 88, p.4). Examination of Jason's explanation revealed that he considered some aspects of the structure of a number relative to the number of factors through investigation of a simpler situation but he did not seem to analyze why 18 and 12 had the most factors so he was unable to apply that knowledge to a larger number. He did not seem to have a rationale for multiplying the 18 by 10. Two students said that they did not know how to think about this question. The other students attempted the problem in various ways. Kim felt that it would be a number divisible by two so she chose 198, the largest even number less than 200. Tim chose 120 because 1 x 2 x 3 x 4 x 5 is 120. Tim's reasoning was headed towards thinking about the structure of numbers, but he did not consider multiples of the same base such as 22, 32 and he included non-primes (i.e., 4). Andrea tried to think about what she knew about the problem and noted that it was not a prime number; she did not think it was divisible by five but probably was an even number. She puzzled about the problem for quite a while and finally said, "No, I can't think right now about this one." (5/ 88, p.4). A number is less than 200 and has more factors than any other number less than 200. What is the number? was an interesting question in that it pushed students to stretch their thinking beyond ideas discussed in class. The students had explored many mathematical ideas that would have enabled 220 them to solve this problem. For example, the students knew the relationship between prime factorization and number of factors. If their knowledge was flexible and they could reverse their thinking from finding factors to building a number from primes, then they could analyze the problem. The factorization of 200 which is 23 x 52 gives an idea of the sizes of primes and powers to try. 2 x 3 x 5 = 30 and 302 = 900 so squaring each prime produces a product that is too large. Can we square some of them? 22 x 32 x 5 = 180 looks promising. Can one raise the 2 to the third power? 23 x 32 x 5 is too big and 23 x 3 x 5 has fewer factors than 180. 23 x 33 is too big and has only 16 factors. What becomes important is bounding the possibilities. Perhaps students did not realize that they knew many mathematical ideas that could help them solve this problem. Perhaps they did not recognize the power of the mathematical ideas that they knew. For example, knowing the Fundamental Theorem of Arithmetic (FTA) provides students with powerful knowledge about the structure of any number. However, in order to generalize and apply what they knew they have to first recognize the power of that knowledge. Questions B6-827 were intended to be a set of problems but three of the students, Linda, Tamara, and Kim, approached each of these problems separately and made no apparent attempt to use information figured out in the previous question to inform their thinking about related problems. All three gave correct responses to B5. Disregard for information obtained in previous problems and treating each problem separately seemed to be a pattern for some of the students, particularly these three (Linda, Tamara, 27(136) What is the smallest number which is divisible by 2,3,4 and 5? (137) What is the smallest number which leaves a remainder of 1 when divided by 2,3,4,5, and 6 respectively? (88) What is the smallest number which is divisible by 23,4,5, and 6 and leaves a remainder of 1 when divided by 7? [structure of numbers, unfamiliar context] 221 Kim), throughout this interview. At the completion of this set of questions, the interviewer probed concerning the possible relationship among the set of questions. Linda said that questions B7 and B8 were "remainder problems" and that was "totally different" from the B6 question. Tamara agreed that she saw no connection. Kim viewed this as a general difficulty in her way of thinking about mathematics: Kim: It probably would have been real easy if I would have thought about that. But that's my whole problem, it's easier for me to work it through with pencil and paper than it is to think about relationships like that. Now, I can see the relationship between #6 8r 7 but I just don't know where the relationship is between those two and #8. I just don't see it. (5/ 88, p.5). Two of the other three students, Jason and Andrea, continued to build from one problem to another as they worked through questions B6-8. Andrea's responses were representative of their thinking. She reasoned through B6 and then applied what she discovered to answer B7 8: 8: I: What would be the smallest number that is divisible by 2, 3, 4, 5, and 6, and leaves a remainder of 1 when divided by 7? (88). Andrea: It could be 60 or 120 or 180. I'll just divide those and see. It's not 60 so I'll try 120. [used calculator]. That's it, 120. I: Could you tell me how you came up with the numbers 60, 120, or 180? Andrea: Well, if 60 is divisible by these [points to previous problem B7], then the next one would be 120, which is just adding another 60 on and so on. And in order to 222 have a reminder of 1 when dividing by 7, I need to add 1 more. (5/ 88, p.8). Jason omitted one of the factors of 2 and calculated the least common multiple as 30 rather than 60 and used the incorrect LCM and responded 31 to B7. His reasoning was appropriate but his calculations were incorrect. Tim used what he figure out in #B6 to respond to #B7 but said he had "no idea" how to think about #B8. A common textbook format for questions related to least common multiples is to require students to determine the least common multiple for two or more numbers. Question B9 was a slightly different version and did not include the language of least common multiple: How many numbers between 1 and 1000 are divisible by 6 and 9? Only one student, Linda, was unable to provide a correct response to this question. She said that she would randomly test some numbers and then look for some patterns. The other students recognized that they needed to find the LCM of 6 and 9 and then divided 1000 by the LCM to determine the total number. During the interview at the end of the course, Tim and Tamara responded correctly to all of the questions and the strategies they utilized shifted to predominantly reasoning ones. During the second interview Kim responded correctly when she used reasoning strategies. The questions in the B interview proved difficult for most of the students but much greater success resulted when students used reasoning strategies than when they relied on more technical approaches. Most of the students' knowledge and understanding about individual number theory concepts grew but often they were unable to recognize and appreciate the power of what they knew. Perhaps they needed more time for some of these ideas to incubate and mature. Andrea seemed to be the exception. She came to the course with 223 some understanding about many of the individual number theory concepts. As the course progressed she became more facile synthesizing and applying her understanding of these number theory ideas. Similarly to the multiplicative and additive structures in a generalized context section, students possessed many of the mathematical ideas needed to solve the questions in this section, but they did not seem to be able to recognize and appreciate the power of these ideas and to use the knowledge in flexible ways. Relationships Another category related to flexibility in using ideas is relationships among concept domains. I am defining concept domain as a group of closely related concepts and ideas. In this study the concept domains related to interview questions include: factors; multiples; primes and composites. The questions that are categorized in this section have solutions that require using mathematical ideas known in one concept domain to think about ideas in another concept domain. At least two concept domains are explicitly stated in each of the questions in this category. Figure 5.5 shows the interview questions for relationships among concept domains. 12) The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 19, B, 29. Which ofthese primes would you have to consider as possible factors of 437 to determine whether 437 is prime or composite? 24) The greatest common factor of 630 and 1716 is 6. What is their least common multiple? Explain your answer. 25) The greatest common factor of two numbers a and b is 9, their least common multiple is 108, and a < b < 1m. Find a and b. Explain your thinking. 810) A student claims that the greatest common factor of two numbers is always less than their least common multiple. Is the student correct? Explain your answer. Figure 5.5. Relationships Interview Questions 224 Figure 5.6 shows the students' responses to interview questions related to relationships among the major individual concept domains studied- factors, multiples, primes and composites. 12 24 A1 A2 A1 A2 A1 Linda Andrea Tamara R Reasoning strategies T Technical strategies Gray indicates correct solution Figure 5.6. Participants' Strategies for Relationships Questions This set of questions proved difficult for most of the students. With the exception of Tamara, students who used reasoning strategies in the relationships category were more successful than those who relied on technical strategies (see Figure 5.6). Andrea was the only student to use reasoning strategies in responding to all of the questions in this set and she was the only student who provided a correct solution for all of the questions during the interview at the end of the course. 225 During the interview at the beginning of the course, Jason was the only student to offer a correct solution to the question that provided the greatest common factor of two numbers and required students to determine the LCM (#24)28. He played around with the calculator and some smaller numbers (e.g., 4 and 6) that he could easily determine the GCF and LCM. He said: According to my "system," I should be able to multiply 630 x 1716 and then divide by 6. [He does this on the calculator] 180,180. Now let's make sure. [Using the calculator, he divides 180,180 by 630 and then divides 180,180 by 1716] It is a multiple. I don't know if it's the least one but it's definitely a multiple. (3/ 88, p. 17). By using a smaller and simpler problem, Jason was able to develop a procedure to use for larger numbers but he was not confident that he had found the least common multiple and he was unable to explain why his "system" worked. Jason was unable to recognize all of the connections among related ideas, in particular, the relationship between factors and multiples29 and the structure of numbers. Two of the students, Kim and Andrea recognized and acknowledged that they did not know how to do this problem. Kim immediately responded, "I don't know." Andrea tried several things with her calculator and kept saying, "That's not right, that doesn't work." She multiplied one number by 4 or 5 and then divided but never determined a solution that satisfied her. She tried to write a formula but discarded it as well. 28(#24)T'he(3CFoftwonurnbersaandbis9,theirLCMis108,anda,1, can be expressed as a product of primes, this representaion is unique, apart from the order in which the factors occur (Burton, 1980). Focuses on the power of this mathematical idea. Understanding of the notion of uniqueness and relationship to multiplicative structure of numbers. 16) Describe the learning environment in Math 201. Cam Answer; Answers will vary. Can provide insight about the ways in which students interpreted the environment in Math 201. 17) During 201 class we've spent some time working in groups. Think about your experiences while working in your group. In what ways was group work helpful/ not helpful for you? Caisson: Anower; Answers will vary. Focuses on student's understanding and interpretation of value of group work and relationship to learning mathematics. 18) In 201 you are encouraged to talk about the mathematics you are doing. Sometimes this is in the form of large group discussion, 312 sometimes small groups. You are also encouraged to make conjectures and then to agree or disagree with someone's conjecture. How do you feel about this as a learner of mathematics? What about in terms of teaching mathematics? Caisson: Answer; Answers will vary. Focuses on student's understanding and interpretation of discussing mathematical thinking and the relationship to learning mathematics. 19) A friend of your is planning to take this section of Math 201 this summer. How would you describe the course to him/ her? Cam Answer; Answers will vary. In this type of context, students are encouraged to describe the Math 201 class honestly and in ways that might be helpful to prospective students. Can provide insight into what student observes and values about Math 201. 20) We are mid-way through the term. Think about your view and expectations prior to beginning to 201 and your experiences thus far in Math 201. In what ways have your views or perspectives changed and/ or remained the same? What would you like to occur during the remainder of the course? Calm Answer; Answers will vary. Focuses on student's reflection about the course. APPENDIX C MIDDLE SCHOOL TEXTBOOK ANALYSIS Three popular sixth grade textbooks were reviewed that were used when the majority of the students in this study were in middle school (1978- 1982). The analysis focused on the sections related to the number theory ideas from Math 201. It is during the middle school years that these ideas are introduced and deve10ped. These number theory topics usually are not part of the secondary curriculum. This examination of texts provided insight into some of the types of experiences these students might have had related to these number theory topics. The following criteria was used for the analysis: Identify way in which these topics were introduced-factors including common factors and greatest common factors, multiples including common multiples and least common multiples, and primes and composites; explore connections made among these tapics; identify mathematical ideas/ topics that followed each primary topic. Text A (Grade 6): more. "A number is divisible by its factor." This is the way factor is defined. It is not clear that they mean whole numbers because technically any number is divisible by any other number, it just may not be a whole number quotient. They do use "whole numbers" in the boxed steps: "Write 20 as the product of two whole numbers in as many ways as possible." The exercises that follow are of two types—one ask to "Find the missing factor." 1 x 5 = 75 and and the other says, "Find all the factors. List them in order." Numbers used range from 10 to 225. 313 31 4 Multiples. Once again "steps" to use to find LCM are boxed and an example following the steps is provided. The exercises that follow are just practice (40 problems) using the steps. This lesson ends the chapter. Primes and composites. There is a side box (considered "enrichment" labelled "PRIME NUMBERS" There is a definition of a prime number, statement that 1 is not a prime number, and no mention of composite. The task is to list all the prime numbers between 1 and 100. It would be fairly easy to include numbers that are not prime but on first appearance seem to be prime, e.g., 51. Connectionslrelationships among topics. The lesson that follows "finding factors" is "greatest common factors" followed by "least common multiples." This lesson completes the chapter. At the end of this chapter is an "enrichment" lesson on "prime factors". The next chapter begins with fractions and mixed numerals. The next mention of greatest common factors is 20 pages later. It's entitled "simplest name" but there is no discussion about why finding GCF is helpful to arriving at the "simplest name" for a fraction. The directions above the exercises that follow are "Find the greatest common factor" and the next one says, "Write the simplest name." Least common multiple is used in a similar way in the lesson that follows this one. Students are to use least common multiple to find the least common denominator for two fractions. There is never a connection between factors and multiples. Prime numbers are not mentioned again in the text. Text B (Grade 6): Factors. The term "factor" is first used 38 pages after multiples are introduced. The context is "finding missing factors." Example 6 x n = 348. 44 pages later, factors and multiples are used on the same page in a lesson entitled, "Finding Factors and Multiples." Focus is on the 1-100 chart and patterns within that. No work done on the relationship between factors and multiples or any 31 5 applications for either. This is followed by a lesson on finding primes and composites and the language shifts from factors to divisors with no explanation made. Multiples. p.27 Second lesson in chapter on multiplying whole numbers. It follows a page on multiplication basic facts. An example provided and then the exercises require students to repeat what was done in the example as they march through 15 problems. In the example they use . . . to indicate that there are more multiples. In the teacher text's margin it says, "Point out that the list of all multiples of a number is endless." A student could mimic the example provided and not understand very much about multiples, common multiples or least common multiples] The next lesson introduces multiples of 10, 100, and 1000 but not in the same organized manner. The emphasis is on the number of zeros to use. The next reference to multiples occurs 75 pages later in a lesson entitled, "Finding Factors and Multiples." Focus is on the 1-100 chart and patterns within that. No work done on the relationship between factors and multiples or any applications for either. This is followed by a lesson on finding primes and composites and the language shifts from factors to divisors with no explanation or connection. Primes and commsites. p. 104. [Definitions are given for prime, composite and two statements are made: The numbers 0 and 1 are neither prime nor composite. All numbers greater than 1 are either prime or composite. The other activity is the Sieve of Eratosthenes. This lesson is followed by a lesson that shows the use of factor trees to find prime factors. No explanation of when you would ever need to use this information. This concludes the chapter. The next chapter is "meaning of fractions and mixed numbers." None of these ideas are mentioned again in this text. 31 6 ConnectionsZrelationships among topics. There is no explicit connection made among any of these topics / ideas. Text C (Grade 6): I‘m-s. Page 90 (24 pages after multiples first introduced). Two numbers are used in the example—24 and 18. Each one is shown with its divisors and then the factors are listed out followed by a list of common factors and then the greatest common factor is given without explanation. No comment about factor pairs just factors. Exercises require students to find factors, common factors, and/ or greatest common factors. No applications are provided. The pictures on the page are jigsaw puzzle pieces. In the teacher's guide, it is suggested to divide the class into two teams, "the Jigsaws" and "the Puzzlers." The children number off. The teacher writes a number on the board and any child who thinks he is a number that is a factor, stands up. If correct, the team receives one point. If incorrect, the team loses one point. This lesson is followed by a problem solving page that is totally unrelated and this completes the chapter. Multiples. Introduced 24 pages before factors. Once again more statements- here is a list of multiples of 3 and here is a list of multiples of 4. These are common multiples of 3 and 4. This is the least common multiple of 3 and 4. Followed by exercises that require listing multiples, cormnon multiples, and / or least common multiple. No applications are provided. The pictures are distracting. They show a series of houses, one that looks like it is in the country with a barn, another "suburban" type home, and finally what appear to be "city" apartment buildings. This lesson is followed by a "problem solving" page that is totally unrelated. Before any of the story problems are given, there is a boxed list of four steps for "problem solving"understand the 31 7 problem, make a plan, do the arithmetic, and give the answer. This lesson completes the chapter. Primes and composites. Page 96 entitled "Something Extra" 3 examples of prime numbers followed by a definition. Students are then required to c0py a list of numbers from 1 to 50 and instructed to circle and cross off various multiples. "The circled numbers and all numbers not crossed off in the chart are prime numbers." N 0 applications are provided. This lesson follows the practice chapter test and precedes a page of "reviewing needed skills". Connectionslrelationships among topics. None of the topics appear anywhere else in the text. There is no connection made among any of these topics / ideas. LIST OF REFERENCES REFERENCES American Association for the Advancement of Science. (1989). Science for all Americans. Washington, DC: Author. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press. Anderson, J. R., Greeno, J. G., Kline, P. J., 8: Neves, D. M. (1981). Acquisition of problem-solving skill. In J. R. Anderson (Ed.), Cognitive skills and their acquisition (pp. 191-230). Hillsdale, NJ: Lawrence Erlbaum Associates. Anderson, R. D. (1982) Arithmetic in the computer/ calculator age. Washington, DC: National Academy of Sciences. Ball, D. L. (1988a). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation. Michigan State University, East Lansing. Ball, D. L. (1988b). Unlearning to teach mathematics, For the Learning of Mathematics, 8 (1), 40—48. Ball, D. L. (1989). Teaching mathematics for understanding: What do teachers need to know about the subject matter? In Competing visions of teacher knowledge: Proceedings from an NCRTE seminar for education policy makers: February 24-26, 1989. Volume 1: Academic Subjects. (Conference Series 89-1, pp. 79-100). East Lansing: Michigan State University, National Center for Research on Teacher Education. 318 319 Ball, D. L., 8: McDiarmid, G. W. (1990). The subject matter preparation of teachers. In W. R. Houston (Ed.), Handbook of research on teacher education (pp. 437-449). New York: Macmillan. Barbeau, E. J. (1989, September). Mathematics for the public. Paper presented at the meeting of the International Commission of Mathematical Instruction, Leeds University, England. Barrett, G., 8: Goebel, J. (1990). The impact of graphing calculators on the teaching and learning of mathematics. In T. J. Cooney 8: C. R. Hirsch (Eds), Teaching and learning mathematics in the 1990s (pp. 205-211). Reston, VA: NCTM. Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In T. Cooney 8: D. Grouws (Eds), Effective mathematics teaching (pp. 27-46). Reston, VA: National Council of Teacher of Mathematics; Hillsdale, NJ: Lawrence Erlbaum Associates. Bell, A. W. (1979). The learning of process aspects of mathematics. Educational Studies in Mathematics, 10, 361-387. Bereiter, C. (1985). Towards a solution of the learning paradox. Review of Educational Research, 55, 201-226. Bernstein, A. (1988, September). Where the jobs are is where the skills aren't. Business Week, pp. 104-108. Bogdan, R. C., 8: Biklen, S. K. (1982). Qualitative research for education: An introduction to theory and methods. Boston: Allyn 8: Bacon. Book, O, Byers, J., 8: Freeman, D. (1983). Student expectations and teacher education traditions with which we can and cannot live. Journal of Teacher Education, 34 (1), 9—13. 320 Bransford, J., Hasselbring, T., Barron, B., Kulewicz, S., Littlefield, J., 8: Coin, L. (1988). Uses of macro—contexts to facilitate mathematical drinking. In R. I. Charles 8: E. A. Silver (Eds), The teaching and assessing of mathematical problem solving (pp. 125-147). Hillsdale, NJ: Lawrence Erlbaum Associates. Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H. G. Steiner (Ed.), Theory of mathematics education (pp. 110-119). Occasional paper 54. Bielefeld, Germany: University of Bielefeld, Institute fur Didaktik der Mathematik. Brown, C. (1985). A study of the socialization to teaching of a beginning mathematics teacher. Unpublished doctoral dissertation, University of Georgia, Athens. Brown, S. I., Cooney, T. J., 8: Jones, D. (1990). Mathematics teacher education. In W. R. Houston (Ed.), Handbook of Research on Teacher Education (pp. 639—656). New York: Macmillan. Brownell, W. A. (1935). Psychological considerations in the learning and the teaching of arithmetic. In The teaching of arithmetic, 10th Yearbook of the NCTM. New York: Bureau of Publications. Teachers College. Columbia University. Bruner, J. S. (1960). The process of education. New York: Vintage Books. Buchmann, M. (1984). The priority of knowledge and understanding in teaching. In L. Katz 8: J. Raths (Eds), Advances in teacher education (Vol. 1, pp. 29-50). Norwood, NJ: Ablex. Burns, M. (1986). Teaching "what to do" in mathematics vs. teaching "what to do and why." Educational Leadership, 43, (7), 34-38. 321 Burns, R. B., 8: Lash, A. A. (1988). Nine seventh-grade teachers' knowledge and planning of problem-solving instruction. Elementary School Journal, 88. 369—386. Burton, D. M. (1980). Elementary Number Theory. Boston: Allyn 8: Bacon. Bush, W. (1983). Preservice secondary mathematics teachers' knowledge about teaching mathematies and decision-making during teacher training (Doctoral Dissertation, University of Georgia, 1982). Dissertation Abstracts Internations, 43, 2264A. Buxton, L. (1985). Mathematics for Everyone. New York: Schocken Books. Calfee, R C. (1989). Those who can explain, teach. . . In L. Weis, P. Altbach, G. Kelly, H Petrie, 8: S. Slaughter (Eds), Crisis in teaching: Perspectives on current reforms (pp. 33-68). Albany, NY: SUNY Press. California State Department of Education. (1985). Mathematics framework for California public schools. Sacramento, CA: Author. California State Department of Education. (1992). Mathematics framework for California public schools. Sacramento, CA: Author. Campione, J. C., Brown, A. L., 8: Connell, M. L. (1988). Metacognition: On the importance of understanding what you are doing. In R. 1. Charles 8: E. A. Silver (Eds), The teaching and assessing of mathematical problem solving (pp. 93-114). Hillsdale, NJ: Lawrence Erlbaum Associates. Carlsen, W. S. (1987, March). Why do you ask? Effects of science teacher subject matter knowledge on teacher questioning and classroom discourse. Paper presented at the annual meeting of the American Educational Research Association, New Orleans. Carnegie Forum on Education and the Economy. (1986). A nation prepared: Teachers for the 21 st century. New York: The Forum. 322 Carpenter, T. (1986). Conceptual knowledge as a foundation for procedural knowledge: Implications from research on the initial learning of arithmetic. In J. Hiebert (Ed), Conceptual and procedural knowledge in mathematics: The case of mathematics (pp. 113-132). Hillsdale, NJ: Lawrence Erlbaum Associates. Carpenter, T., Fennema, E., Peterson, P., 8: Carey, D. (1989). Teachers' pedagogical content knowledge of students' problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19, 385-401. Chi, M. T. H., Glaser, R., 8: Rees, E. (1982). Expertise in problem solving. In R. Steinberg (Ed.). Advances in the psychology of human intelligence (pp. 7- 75). Hilsdale, NJ: Erlbaum. Chi, M. T. H., 8: Bassok, M. (1989). Learning from examples via self- explanations. In L. B. Resnick (Ed), Knowing, learning, and instruction, (pp. 251-282), Hillsdale, NJ: Lawrence Erlbaum Associates. Clements, H. (1989). What do teachers need to be? In Competing visions of teacher knowledge: Proceedings from an NCRTE seminar for education policy makers: February 24-26, 1989. Volume 1: Academic Subjects. (Conference Series 89-1, pp. 123-157). East Lansing: Michigan State University, National Center for Research on Teacher Education. Cobb, P. (1989). Experiential, cognitive, and anthropological perspectives in mathematics education. For the Learning of Mathematics, 9(2), 32-42. Cobb, P. (1990). Multiple perspectives. In L. P. Steffe 8: T. Wood (Eds), Transforming children's mathematics education: International perspectives (pp. 200-215). Hillsdale, NJ: Lawrence Erlbaum Associates. 323 Cobb, P., Yackel, E., 8: Wood, T. (1988). Curriculum and Development: Psychological and Anthropological Perspectives. In E. Fennema, T. Carpenter, 8: S. J. Lamon (Eds), Integrating Research on Teaching and Learning Mathematics Education, National Center for Research in Mathematical Sciences Education. Papers from the First Wisconsin Symposium for Research on Teaching and Learning mathematics, Madison. Cobb, R, Wood, T., Yackel, E., Nichols, J., Wheatley, G., Trigatti, B., 8: Perlwitz, M. (1991 a). Assessment of a problem-centered second grade mathematics project. Journal for Research in Mathematics Education, 22, 3-29. Cobb, R, Wood, T., Yackel, E., Nichols, J., Wheatley, G. Trigatti, B. 8: Perlwitz. M. (1991b). Problem-centered mathematics project. Journal for Research in Mathematics Education, 22(1), 3-29. Cobb, P., Yackel, E., 8: Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33. Cohen, D. K. (1989). Teaching practice: Plus ca change. . . In P. W. Jackson (Ed), Contributing to educational change: Perspectives on research and practice (pp. 27-84). Berkeley, CA: McCutchan. Cohen, D. K (1990). Policy and practice: The classroom impact of state and federal policy. Unpublished manuscript, Michigan State University, East Lansing. Cohen, D. K., 8: Ball, D. L. (1990). Relations between policy and practice: An overview. Educational Evaluation and Policy Analysis, 12, 347-353. College Entrance Examination Board. (1959). Program for college preparatory mathematics. In the Report of the Commission on Mathematics. New York: Author. 324 Collins, A., Brown, J. S., 8: Newman, S. E. (1989). Cognitive apprenticeship: Teaching the crafts of reading, writing, and mathematics. In L. B. Resnick (Ed), Knowing, learning and instruction (pp. 453-494). Hillsdale, NJ: Lawrence Erlbaum Associates. Conant, J. B. (1963). The education of American teachers. New York: McGraw-Hill. Conference Board of Mathematical Sciences. (1975). Overview and analysis of school mathematics, grades K-12. Washington, DC: Board of Mathematical Sciences. Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. In C. B. Cazden (Ed), Review of research in education, (pp. 3-56). Washington, DC: American Educational Research Association. Cooney, T. J. (1985). A beginning teacher's view of problem solving. Journal for Research in Mathematics Education, 16, 324-336. Cooney, T. J. (1987, October). The issue of reform: What have we learned from yesteryear? Paper presented at conference sponsored by the Mathematical Science Education Board and the Center for Academic Inter institutional Program, Los Angeles. Davis, R. (1986). Conceptual and procedural knowledge in mathematics: Summary analysis. In J. Hiebert (Ed), Conceptual and procedural knowledge: The case of mathematics (pp. 265-300). Hillsdale, NJ: Erlbaum. Davis, P. J., 8: Hersh, R. (1981). The mathematical experience. Boston: Houghton Mifflin. Davis, P. J., 8: Hersh, R. (1986). Descartes' dream. New York: Harcourt, Brace Jovanovich. 325 de Lange, J. (1987). Mathematics, insight and meaning. The Netherlands: University of Utrecht. Demana, F., 8: Waits, B. K. (1990). The impact of graphing calculators on the teaching and learning of mathematics. In T. J. Cooney 8: C. R. Hirsch (Eds), Teaching and learning mathematics in the 19905 (pp. 212- 222). Reston, VA: NCT'M. Dessart, D. J. (1981). Curriculum. In E. Fennema (Ed), Mathematics education research: Implications for the 80's (pp. 1-21). Alexandria, VA: Association for Supervision and Curriculum Development. DeVault, M. V., 8: Weaver, J. F. (1970). Designing a contemporary elementary school mathematics program: 1952-Present. In NCTM T'hirty-second Yearbook, A history of mathematics education in the United States and Canada (pp. 133-154). Washington, DC: NCTM. de Walle, J. A. 8: Holbrook, H. (1987). Patterns, thinking, and problem solving. Arithmetic Teacher, 34(8), 6-12. Dewey, J. (1904). The relation of theory to practice in education. In R. Archambault (Ed), John Dewey on education (pp. 313-338). Chicago: University of Chicago Press. Dewey, J. (1965). The relation of theory to practice in education. In M. L. Borrowman (Ed), Teacher education in America: A documentary history (pp. 140-171). New York: Teachers College Press. (Original work published 1904). Dougherty, B. J. (1990). Influences of teacher cognitive] conceptual levels on problem-solving instruction. In G. Gooker et al. (Eds), Proceedings of the Fourteenth International Conference for the Psychology of Mathematics Education (pp. 119-126). Oaxtepec, Mexico: International Group for the Psychology of Mathematics Education. 326 Doyle, W. (1983). Academic Work. Review of Educational Research 53(2), 159-199. Doyle, W. (1986). Content representation in teachers' definitions of academic work. Journal of Curriculum Studies, 18, 365-379. Driscoll, M. (1981). Research within reach: Elementary school mathematics. Washington, DC: National Institute of Education. Duckworth, E. (1987). "The having of wonderful ideas" and other essays on teaching and learning. New York: Teachers College Press. Duckworth, E. (1991). Twenty-four, forty-two, and I love you: Keeping it complex. Harvard Educational Review, 61(1), 1-24. Dufour-Janvier, B., Bednay, N., 8: Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed), Problems of representation in the teaching and learning of mathematics (p. 109-122). Hillsdale, NJ: Lawrence Erlbaum Associates. Dunkin, P., 8: Barnes, K. (1986). Research on teaching in higher education. In M. C. Wittrock (Ed), Handbook of research on teaching (3rd ed, pp. 754- 777). New York: Macmillan. Erikson, F. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Ed) Handbook of research on teaching, (pp. 119-161). New York: Macmillan. Erlwanger, S. (1973). Benny's conception of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1(2), 7-26. Ernest, P. (1988, July). The impact of beliefs on the teaching of mathematics. Paper prepared for International Congress on Mathematics Education VI, Budapest, Hungary. 327 Eylon, B., 8: Helfman, J. (1982, February). Analogical and deductive problem- solving in physics. Paper presented at AERA meeting, New York. Feiman-Nemser, S. (1983). Learning to teach. In L. Shulman 8: G. Sykes (Eds), Handbook of teaching and policy (pp. 150-170). New York: Longman. Feiman-Nemser, S. McDiarmid, G. W., Melnick, S. L., 8: Parker, M. (1989). Changing beginning teachers' conceptions: A description of an introductory teacher education course. (Research report 89-1). East Lansing: Michigan State University, National Center for Research on Teacher Education. Fennema, E., 8: M. L. Franke, (1992). Teachers' knowledge and its impact. In D. A. Grouws (Ed), Handbook of research on mathematics teaching and learning, (147-164). Hillsdale, NJ: Lawrence Erlbaum. Fey, J. (1978). Change in mathematics education since the later 1950's: Ideas and realisation, U. S. A. Educational Studies in Mathematics, 9, 339-353. Fey, J. (1982). Mathematics education. In H. Mitzel (Ed), Encyclopedia of educational research (Vol. 3, 5th ed, pp. 1166-1182). New York: Free Press. Fey, J. (1989). School algebra for the year 2000. In S. Wagner 8: C. Kieran (Eds). Research agenda in mathematics education: Research issues in the learning and teaching of algebra (pp. 200—214). Reston, VA: National Council of Teachers of Mathematics, and Hillsdale, NJ: Erlbaum. Finkelstein, B. (1982). Technicians, mandarins, and witnesses: Searching for professional understanding. Journal of Teacher Education, 33(3), 25-27. Fisher, J. J. (1967). The extent of implementation of CUPM level I recommendations, Arithmetic Teacher 14, 194-97. 328 Floden, R. E., 8: Buchmann, M. (1989). Philosophical inquiry in teacher education. (Issue paper 8%). East Lansing: Michigan State University, National Center for Research on Teacher Education. Fotiu, R., Freeman, D., 8: West, B. (1985). Undergraduate follow-up study Spring, 1985 (Program Evaluation Series No. 11). East Lansing: Michigan State University, College of Education, Office of Program Evaluation. Frederikson, N. (1984). Implications of cognitive theory for instruction in problem solving. Review of Educational Research, 54, 363-407. Freeman, D., 8: Kalaian, H. A. (1989). Profiles of students completing teacher education programs at M. S. U.: Fall, 1986 through Spring, 1988 (Program Evaluation Series No. 25). East Lansing: Michigan State University, College of Education, Office of Program Evaluation. Gage, N. L. (1978). The scientific basis of the art of teaching. New York: Teachers College Press. Gage, N. L. (1985). Hard gains in the soft sciences: The case of pedagogy. Bloomington, IN: Phi Delta Kappan. Gagne, R. M. (1962). The acquisition of knowledge. Psychological Review, 69, 355-65. Garofalo, J., 8: Lester, F. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16, 163-176. Gibb, E. G. (1970). Through the years: Individualized instruction in mathematics. Arithmetic Teacher, 396-401. Gibb, E. G., Karnes, H. T., 8: Wren, F. L. (1970). The modern era: 1945-Present NCT'M Thirty-second Yearbook, A history of mathematics education in the United States and Canada. (pp. 327-352) Washington, DC: NCTM. 329 Glaser, R. (1984). Education and thinking: The role of knowledge. American Psychologist, 39, 93-104. Golos, E. B. (1981). Patterns in mathematics. Boston, Massachusetts: Prindle, Weber 8: Schmidt Goodlad, J. (1984). A place called school: Prospects for the future. New York: McGraw Hill. Grossman, P. (1987). A tale of two teachers; The role of subject matter orientation in teaching. Paper presented at the 1987 annual meeting of the American Educational Research Association, Washington, DC. Grossman, P. G., Wilson, S. M., 8: Shulman, L. S. (1989). Teachers of substance: The subject matter knowledge of teachers. In M. Reynolds (Ed), The knowledge base for beginning teachers (pp. 23-36). New York: Pergamon. Head, J. O., 8: Sutton, C. R (1985). language, understanding, and commitment. In L. H. T. West 8: A. L. Pines (Eds), Cognitive structure and conceptual change. Orlando, FL: Academic Press. Hembree, R. 8: Dessart, D. J. (1992). Research on calculators in mathematics education. In J. T. Fey 8: C. R. Hirsch (Eds), Calculators in mathematics education (pp. 23-32). Reston, VA: NCTM. Henderson, G. L. (1972). Individualized instruction: Sweet in theory, sour in practice. Arithmetic Teacher, 17-22. Hersh, R. (1986). Some proposals for reviving the philosophy of mathematics. In T. Tymazko (Ed), New directions in the philosophy of mathematics (pp. 9-28). Boston: Birkhauser. 330 Hiebert, J., 8: Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum Associates. Hiebert, J., 8: Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed), Conceptual and procedural knowledge in mathematics: The case of mathematics (pp. 199- 224). Hillsdale, NJ: Lawrence Erlbaum Associates. Hiebert, J., 8: Carpenter, T. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed) Handbook of research on mathematics teaching and learning. (pp. 65-97). New York: Macmillan Publishing Company. Hoffman, K. (1989, March). The science of patterns: A practical philosophy of mathematics education. Paper presented to the Special Interest Group for Research in Mathematics Education at the 1989 Annual Meeting of the American Educational Research Association, San Francisco. The Holmes Group. (1986). Tomorrow's teachers: A report of the Holmes Group. East Lansing, MI: The Group. Hoyle, C. (1985). What is the point of group discussion? Educational Studies in Mathematics 16(2), 205-214. Hoyle, C. (1988). From fragmentation to synthesis: An integrated approach to research on the teaching of mathematics. In D. A. Grouws 8: T. J. Cooney (Eds), Research agenda in mathematics education: Perspectives on research on effective mathematics teaching (pp. 143-168). Reston, VA: National Council of Teachers of Mathematics, and Hillsdale, NJ: Erlbaum. International Association for the Evaluation of Educational Achievement. (1987). The underachieving curriculum: A national report on the second international mathematics study. Champaign, I1: Stipes Publishing. 331 Jackson, P. W. (1990). Looking for trouble: On the place of the ordinary in educational studies. In E. W. Eisner 8: A. Peshkin (Eds), Qualitative inquiry in education: The continuing debate (pp. 153-166). New York: Teachers College Press. Jones, P. S. (1970). Present-day issues and forces. In A history of mathematics education in the United States and Canada (pp. 453-466). Washington, D. C.: NCTM Jones, P. S., 8: Coxford, A. (1970). Reform, "revolution," reaction: 1945- Present. In N CTM A history of mathematics education in the United States and Canada. (pp. 67-92). Washington, D. C.: NCTM. Kaput, J. J. (1987a). Representation systems and mathematics. In C. Janvier (Ed), Problems of representation in the teaching and learning of mathematics (pp. 19-26). Hillsdale, NJ: Erlbaum. Kaput, J. J. (1987b). Towards a theory of symbol use in mathematics. In C. Janvier (Ed), Problems of representation in the teaching and learning of mathematics (pp. 159-195). Hillsdale, NJ: Erlbaum. Kennedy, M. M. (1990). A survey of recent literature on teachers' subject matter knowledge (Issue Paper 90-3). East Lansing: Michigan State University, National Center for Research on Teacher Education. Kilpatrick, J. (1987a). What constructivism might be in mathematics education. In J. C. Bergeron, N. Herscovics, 8: C. Kieran (Eds), Proceedings of the 1 1th International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 3-27). Montreal: International Group for the Psychology of Mathematics Education. Kilpatrick, J. (1987b). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed). Cognitive Science and mathematics education (pp. 123-148). Hillsdale, NJ: Lawrence Erlbaum Associates. 332 Kliebard, H. M. (1985). What happened to American Schooling in the first part of the twentieth century? In E. Eisner (Ed), Learning and teaching the ways of knowing. Chicago: The University of Chicago Press. Kline, M. (1977). Why the professor can't teach: Mathematics and the dilemma of university education. New York: St. Martin's Press. Krulik, S., 8: Rudnick, J. A. (1986). Problem solving. In R. Lodholz (Ed), A change in emphasis. Creve Coeur, Missouri: Parkway School District. Kuhn, D., 8: Phelps, E. (1982). The development of problem-solving strategies. In H. Reese (Ed), Advances in child development and behavior (Vol. 17, pp. 1-44). New York: Academic Press. Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction , 3, pp. 305-342. Lampert, M. (1988). What can research on teacher education tell us about improving the quality of mathematics education? Teaching and Teacher Education, 4, 157-170. Lampert, M. (1989a). The teacher's role in reinventing the meaning of mathematical knowing in the classroom. Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 433-480). DeKalb, IL: Northern Illinois University. Lampert, M. (1989b). Choosing and using mathematical tools in classroom discourse. In J. Brophy (Ed), Advances in research on teaching: Teaching for meaningful understanding. Greenwich, CT: JAI Press. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63. 333 Lampert, M. (1991). Connecting mathematical teaching and learning. In E. Fennema, T. P. Carpenter, 8: S. J. Lamon (Eds), Integrating research on teaching and learning mathematics. Albany, NY: State University of New York Press. Lanier, J. E., 8: little, J. W. (1986). Research on teacher education. In M. C. Wittrock (Ed), Handbook of research on teaching, (pp. 527-569). New York: Macmillan. Lanier, P. E. (1986, January). Learning to teach mathematics conceptually. East Lansing: Michigan State University, Colloquium Series. Lappan, G. (1987a). The middle grades: Where we are—a look to the future- how we can get there. East Lansing: Michigan State University. Lappan, G. (1987b). The challenge: Good mathematics-taught well (Final Report to the National Science Foundation grant #MDR 8318218). East Lansing: Michigan State University, Middle Grades Mathematics Project. Lappan, G., 8: Even, R. (1989). Learning to teach: Constructing meaningful understanding of mathematical content. (Craft Paper 89-3). East Lansing: Michigan State University, National Center for Research on Teacher Education. Lappan, G., 8: Schram, P. (1990). Communication and reasoning: Critical dimensions of sense making in mathematics. In P. R. Trafton 8: A. P. Shulte (Eds), New Directions for Elementary School Mathematics (pp. 14- 30). Reston, VA: NCTM. Lawler, R. W. (1981 ). The progressive construction of mind. Cognitive Science, 5, 1-30. Leinhardt, G., 8: Smith, D. A. (1987). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 77, 247-271. 334 Lesh, R. (1985). Conceptual analyses of problem solving performance. In E. A. Silver (Ed), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 309-330). Hillsdale, NJ: Lawrence Erlbaum Associates. Lipson, J. I., E. Koburt, 8: B. Thomas (1967). Individually Prescribed Instruction (IPI) Mathematics. Pittsburgh: Learning Research and Development Center. Lortie, D. C. (1975). Schoolteacher: A sociological study. Chicago: University of Chicago Press (paperback edition). Madsen-Nason, A., 8: Lanier, P. (1986). Pamela Kaye's general math class: From a computational to a conceptual orientation (Research Series No. 172). East Lansing: Michigan State University, Institute for Research on Teaching. Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2-8. Mathematical Association of America. (1991). A call for change: Recommendations for the mathematical preparation of teachers of mathematics. Washington, DC: Author. Mathematical Sciences Education Board. (1990). Reshaping school mathematics: A philosophy and framework for curriculum. Washington: National Academy Press. McDiarmid, G. W., Ball, D. L., 8: Anderson, C. W. (1989). Why staying one chapter ahead doesn't really work: Subject-specific pedagogy. In M. Reynolds (Ed), The knowledge base for beginning teachers. Elmsford, NY: Pergamon. Michigan State University (1987, September). Descriptions of Courses. (Vol. 81, No. 2). East Lansing: Michigan State University Publication. 335 Miles, M. B., 8: Huberman, A. M. (1984). Qualitative data analysis. Beverly Hills, CA: Sage Publications. Moore, E. H. (1903/ 1926). On the foundations of mathematics. In R. Schorling (Ed), A general survey of progress in the last 25 years, 1st Yearbook of the NCT'M. Naisbitt, J. (1982). Megatrends: Ten new directions transforming our lives. New York: Warner Books. National Assessment of Educational Progress. (1978). Mathematics objectives second assessment. Denver, Colo: Education Commission of the States. National Center for Research on Teacher Education. (1988). Teacher education and learning to teach: A research agenda. Journal of Teacher Education, 32(6), 27-32. The National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform. Washington, DC: U. S. Department of Education. National Council of Teachers of Mathematics. (1987). Draft Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (1991). Professional standards for teaching. Reston, VA: Author. National Research Council. (1989). Everybody counts: A report on the future of mathematics education. Washington, DC: National Academy Press. 336 National Science Foundation. National Science Board Commission on Precollege Education in Mathematics, Science, and Technology. (1983). Educating Americans for the 21 st century: A report to the American people and the National Science Board. Washington, DC: Author. Oakes, J. (1985). Keeping track: How school structures inequality. New Haven, CT: Yale University Press. Osborne, A. R., 8: Crosswhite, F. J. (1970). Reform, revolution, and reaction. In NCTM A history of mathematics education in the United States and Canada. (pp. 235-300). Washington, D. C.: NCTM Peshkin, A. (1988). In search of subjectivity—one's own. Educational Researcher, (October), 17-21. Peterson, P. (1988). Teaching for higher order thinking in mathematics: The challenge for the next decade. In D. A. Grouws 8: T. J. Cooney (Eds), Perspectives on research on effective mathematics teaching (pp. 2-26). Hillsdale, NJ: Erlbaum. Peterson, P. , Swing, S. R., Stark, K. D., 8: Waas, G. A. (1984). Students' cognitions and time on task during mathematics instruction. American Educational Research Journal, 21, 487-515. Peterson, P., Fennema, E., Carpenter, T., 8: Loef, M. (1989). Teachers' pedagogical content beliefs in mathematics. Cognition and Instruction 6(1), 1-40. Phillips, E. (1991). Curriculum and evaluation standards for school mathematics addenda series, grades 5-8: Patterns and functions. Reston, Virginia: National Council of Teachers of Mathematics. Pimm, D. (1987). Speaking mathematically: Communication in mathematics classroom. London: Routledge and Kegan Paul. 337 Pollak, H. O. (1970). Applications of mathematics. In E. G. Begle (Ed), The Sixty-ninth Yearbook of the National Society for the Study of Education (pp. 311-334). Chicago: University of Chicago Press. Pollak, H. O. (1987). Cognitive science and mathematics education: A mathematician's perspective. In A. H. Schoenfeld (Ed), Cognitive science and mathematics education (pp. 253-264). Hillsdale, NJ: Lawrence Erlbaum Associates. Porter, A. (1989). A curriculum out of balance: The case of elementary school mathematics. Educational Researcher, 18(5), 9-15. Post, T. R, Harel, G., Behr, M. J., 8: Lesh, R. (1991). Intermediate teachers' knowledge of rational number concepts. In E. Fennema, T. P. Carpenter, 8: S. J. Lamon (Eds), Integrating research on teaching and learning mathematics (pp. 177-198). Albany, NY: SUNY Press. Powell,, A. G., Farrar, E., 8: Cohen, D. K (1985). The shopping mall high school. Boston: Houghton Mifflin. Putnam, R., Lampert, M., 8: Peterson, P. (1990). Alternative perspectives on knowing mathematics in elementary schools. In C. B. Cazden (Ed), Review of research in education (pp. 57-150). Washington, DC: American Educational Research Association. Quimby, M., 8: Barnes, D. (1986). Elementary teachers need strong academic credentials. Action in Teacher Education, 8(3), 25-31. Reed, S. K., Dempter, A., 8: Ettinger, M. (1985). Usefulness of analogous solutions for solving algebra word problems. Journal of Experimental Psychology: Learning, Memory 8: Cognition, 11, 106-125. Resnick, L. B., Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale, NJ: Erlbaum. 338 Resnick, L. B. (1987a). learning in and out of school. Educational Researcher, 16 (9), 13-20. Resnick, L. B. (1987b). Education and learning to think. Washington, DC: National Academy Press. Resnick, L. B. (1988). Treating mathematics as an ill-structured discipline. In R. 1. Charles 8: E. A. Silver (Eds), The teaching and assessing of mathematical problem solving (pp. 32-60). Hillsdale, NJ: Lawrence Erlbaum Associates. Resnick, LB, 8: Ford, W. W. (1984). The psychology of mathematics for instruction. Hillsdale, NJ: Lawrence Erlbaum Associates. Resnick, R. L. (1983). Toward a cognitive theory of instruction. In S. Paris, G. Olson, 8: H. Stevenson (Eds), Learning and motivation in the classroom (pp. 5-38). Hillsdale, NJ: Erlbaum. Rising, G. R. (1977). Which way mathematics education? New York State Mathematics Teachers Journal, 28, 5-17. Romberg, T. (1976). Individually guided mathematics. Reading: Addison- Wesley. Romberg, T. (1983). A common curriculum for mathematics. In G. D. Fenstermacher 8: J. I. Goodlad (Eds), Individual differences and the common curriculum (pp. 121-159). Chicago: National Society for the Study of Education. Romberg, T. A. (1987). The domain knowledge assessment strategy (Working Paper 87-1, Report from the School Mathematics Monitoring Center). Madison: Wisconsin Center for Education Research. 339 Romberg, T. (1988a). Can teachers be professional? In D. Grouws, T., Cooney, T., 8: D. Jones (Eds), Perspectives on research on effective mathematics teaching (pp. 224-244). Reston, VA: National Council of Teachers of Mathematics. Romberg, T. (1988b, September). Principles for an elementary mathematics program for the 1990s Prepared for the California Invitational Symposium on Elementary Mathematics Education. Romberg, T. A. (1992). Assessing mathematics competence and achievement. In H. Berlak, F. Newmann, E. Adams, D. Archbald, T. Burgess, J. Raven, T. Romberg (Eds), Toward a new science of educational testing and assessment. Albany, NY: State University of New York Press. Romberg, T., 8: Carpenter, T. P. (1986). Research on teaching and learning mathematics. In M. C. Whittrock (Ed), Handbook of research on teaching (pp. 850-873). New York: Macmillian. Sarason, S. B., 8: Klaber, M. (1986). The school as a social situation. Annual Review of Psychology, 36, 115-140. Schatzman, L. 8: Strauss, A. L. (1973). Field research: Strategies for a natural sociology. Englewood Cliffs, NJ: Prentice-Hall. Schoenfeld, A. H. (1985). Metacognition and Epistemological issues. In E. A. Silver (Ed) Teaching and learning mathematical problem solving: Multiple research perspectives. (pp. 361-379). Hillsdale, NJ: Lawrence Erlbaum Associates. Schoenfeld, A. H. (1986). On having and using geometric knowledge. In J. Hiebert (Ed), Conceptual and procedural knowledge in mathematics: The case of mathematics (pp. 225-264). Hillsdale, NJ: Lawrence Erlbaum Associates. 340 Schoenfeld, A. H. (March 1987). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. Paper presented at the OERI/LRDC Conference on Informal Reasoning and Education. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed), Handbook of research on mathematics teaching and learning (pp. 334-370). New York: Macmillan. Schram, P., Wilcox, S., Lanier, P., 8: Lappan, G. (1988). Changing mathematical conceptions of preservice teachers: A content and pedagogical intervention (Research Report 88-4). East Lansing: Michigan State University, National Center for Research on Teacher Education. Schram, P. 8: Wilcox, S. (1988). Changing preservice teachers' conceptions of mathematics learning. In M. Behr, C. Lacampagne, M. Montague Wheeler (Eds), Proceedings of the Tenth Annual Meeting PME-NA, DeKalb, IL. Schram, P., Wilcox, S., Lappan, G., 8: Lanier, P. (1989). Changing preservice teachers' beliefs about mathematics education. In C.A. Maher, G. Goldin, R. B. Davis (Eds), Proceedings of the Eleventh Annual Meeting PME-NA, New Brunswick, NJ. Schram, P., Wilcox, S., Lappan, G., 8: Lanier, P. (1992). Curriculum and teaching in the classrooms of novice teachers. Paper presented at National Council of Teachers of Mathematics Annual Conference, Nashville, Tenn. Schroeder, T. L., 8: Lester, F. K. (1990). Developing understanding in mathematics via problem solving. In P. R. Trafton 8: A. P. Shulte (Eds), New Directions for Elementary School Mathematics (pp. 31-42). Reston, VA: NCTM. Schwab, J. (1964). Problems, topics, and issues. In S. Elam (Ed), Education and the structure of knowledge (Fifth Annual Phi Delta Kappa Symposium on Educational Research), Chicago: Rand McN ally. 341 Shavelson, R. J., 8: Stern, P. (1981). Research on teachers' pedagogical thoughts, judgments, decisions and behavior. Review of Educational Research, 51 (4), 455-498. Shaw, K. (1989). Contrasts of teacher ideal and actual beliefs about mathematics understanding: Three case studies. Unpublished doctoral dissertation, University of Georgia, Athens. Shuell, T. J. (1986). Cognitive conceptions of learning. Review of Educational Research, 56, 41 1-436. Shulman, L. S. (1986a). Paradigms and research programs in the study of teaching: A contemporary perspective. In M. C. Wittrock (Ed), Handbook of research on teaching (3rd ed, pp. 3-36). New York: Macmillan. Shulman, L. S. (1986b). Those who understand: Knowledge growth in teaching. Educational Researcher, 15 (2), 4-14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1-22. Shulman, L. 5., 8: Sykes, G. (1986, March). A national board for teaching? In search of a bold standard (Paper commissioned for the Task Force on Teaching as a Profession, Carnegie Forum on Education and the Economy. Silver, E. A. (1981). Recall of mathematical problem information: Solving related problems. Journal for Research in Mathematics Education, 12, 54- 64. Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert (Ed), Conceptual and procedural knowledge in mathematics: The case of mathematics (pp. 181-198). Hillsdale, NJ: Lawrence Erlbaum Associates. 342 Silver, E. A. (1987). Foundations of cognitive theory and research for mathematics problem-solving instruction. In A. H. Schoenfeld (Ed). Cognitive science and mathematics education (pp. 33-60). Hillsdale, NJ: Lawrence Erlbaum Associates. Simon, M. A., 8: D. Schifter. (1991). Towards a constructivist perspective: An intervention study of mathematics teacher development. Educational Studies in Mathematics, 22, 309-331. Skemp, R. R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 26 (3), 9-15. Skemp, R. R (1987). The psychology of learning mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates. Smith, K. J. (1987). The nature of mathematics. Monterey, CA: Brooks/ Cole Publishing. Sowder J. T. (1989). Further opportunities for research. In J. T. Sowder (Ed). Setting a research agenda. Vol. 5 (pp. 32-38). Hillsdale, NJ: Lawrence Erlbaum Associates. Spector, B. S., 8: Phillips, E. R. (1989). Excellence in graduate education for mathematics and science teachers: A sciematics approach. School Science and Mathematics, 89, 40-48. Stanic, G. M., 8: Kilpatrick, J. (1988). Historical perspectives on problem solving in the mathematics curriculum. In R. 1. Charles 8: E. A. Silver (Eds), The teaching and assessing of mathematical problem solving (pp. 1- 22). Hillsdale, NJ: Lawrence Erlbaum Associates. Steen, L. A. (1986, November). Forces for change in the mathematics curriculum. Los Angeles: UCLA, the Mathematical Sciences Education Board and the Center for Academic Inter institutional Programs at UCLA. 343 Steen, L. A. (1990). Pattern. In L. A. Steen (Ed.),On the shoulders of giants: New approaches to numeracy (pp. 1-10). Washington, DC: National Academy Press. Steffe, L. (1987, April). Principles of mathematical curriculum design in early childhood teacher education. Paper presented at the annual meeting of the American Educational Research Association, Washington, D. C. Steffe, L. (1988). Children's construction of number sequences and multiplying schemes. In J. Hiebert 8: M. Behr (Eds), Number concepts and operations in the middle grades (pp. 119-140). Reston, VA: National Council of Teachers of Mathematics. Steinberg, R., Haymore, J., 8: Marks, R. (1985, April). Teachers' knowledge and content structuring in mathematics. Paper presented at the annual meeting of the American Educational Research Association, Chicago. Stodolsky, S. S. (1988). The subject matters: Classroom activity in math and social studies. Chicago: University of Chicago Press. Sweller, J., 8: Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 59-89. Sykes, G. (1986). Introduction. Elementary School Journal, 86, 365-367. Tabachnick, B., Popkewitz, T., 8: Zeichner, K. (1979-80). Teacher education and the professional perspectives of student teachers. Interchange, 10(4), 12-29. Thompson, A. (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127. 344 Thompson, A. (1985). Teachers' conceptions of mathematics and the teaching of problem solving. In E. A. Silver (Ed), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 281- 294). I-Iillsdale, NJ: Lawrence Erlbaum. Tirosh, D., 8: Graeber, A. (1990). Evoking cognitive conflict to explore preservice teachers' thinking about division. Journal for Research in Mathematics Education, 21, 90-108. Toffler, A. (1985). The adaptive corporation. New York: McGraw-Hill. Trow, M. (1977). The second transformation of American secondary education. In]. Karabel 8: A. Halsey (Eds), Power and ideology in education. New York: Oxford University Press. Tulving, E. (1983). Elements of episodic memory. New York: Oxford University Press. Turner, R. L. (1975). An overview of research in teacher education. In K. Ryan (Ed), Teacher education (74th yearbook of the National Society for the Study of Education, Part 2, pp. 87-11). Chicago: University of Chicago Press. Tyson, H., 8: Woodward, A. (1989). Why students aren't learning very much from textbooks. Educational Leadership, 3(11), 14-17. Van Engen, H. (1953). The formation of concepts. In H. F. Fehr (Ed). The learning of mathematics, its theory and practice, 21 st yearbook of the NCTM, Washington, D. C.: NCT'M. Van-Lehn, K. (1986). Arithmetic procedures are induced from examples. In J. Hiebert (Ed), Conceptual and procedural knowledge: The case of mathematics (pp. 133-179). Hillsdale, NJ: Lawrence Erlbaum Associates. 345 Vergnaud, G. (1983). Multiplicative structures. In R. Lesh 8: M. Landau (Eds), Acquisition of mathematics concepts and processes. New York: Academic Press. Vergnaud, G. (1988). Frameworks and facts in the psychology of mathematics education. In A. Hirst 8: K. Hirst (Eds), Proceedings of the sixth international congress on mathematical education. Budapest, Hungary: Malev. von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed), Problems of representation in the teaching and learning of mathematics (pp. 3-17). Hillsdale, NJ: Lawrence Erlbaum. Warren, D. (1982). What went wrong with the foundations and other off- center questions. Journal of Teacher Education, 33(3), 28-30. Wearne, D. 8: Hiebert, J. (1988). Constructing and using meaning for mathematical symbols: The case of decimal fractions. In]. Hiebert 8: M. Behr (Eds), Number Concepts and Operations in the Middle Grades (pp. 220-235). New Jersey: Lawrence Erlbaum. Welch, W. (1978). Science education in urbanville: A case study. In R. Stake 8: J. Easley (Eds), Case studies in science education (pp. 5.1-5.33). Urbana, IL: University of Illinois. West, L. H., Fensham, P. J., 8: Garrard, J. E. (1985). Describing the cognitive structures of learners. In L. H. West 8: A. L. Pines (Eds), Cognitive structure and conceptual change (pp. 29-49). Orlando, Florida: Academic Press. Wilcox, S., Schram, P., Lappan, G., 8: Lanier, P. (1991). For the learning of mathematics. The role of learning community in changing preservice teachers' knowledge and beliefs about mathematics education. 11(3) pp. 31- 39. 346 Wilcox, S., Lanier, P., Schram, P., 8: Lappan, G. (1992). Influencing beginning teachers' practice in mathematics education: Confronting constraints of knowledge, beliefs, and context (Research Report 92-1). East Lansing: Michigan State University, National Center for Research on Teaching Learning. Willoughby, S. S. (1990). Mathematics education for a changing world. Alexandria, VA: Association for Supervision and Curriculum Development. Wilson, 5. M. 8: Ball, D. L. (1991) Changing visions and changing practices: Patchworks in learning to teach mathematics for understanding (Research Report 91-2). East Lansing: Michigan State University, National Center for Research on Teacher Education. Wilson, S. M., Shulman, L., 8: Richert, A. (1987). "150 different ways" of knowing: Representations of knowledge in teaching. In J. Calderhead (Ed), Exploring teachers' thinking (pp. 104-124). London: Cassell. Wooten, W. (1965). SMSG: The making of a curriculum. New Haven, CT: Yale University Press. Zeichner, K. M. (1985). The ecology of field experience: Toward an understanding of the role of field experiences on teacher development. Journal of Research and Development in Teacher Education, 18, 44-52. Zeichner, K. M. (1992). Connecting genuine teacher development to the struggle for social justice (Issue Paper 92-1). East Lansing: Michigan State University, National Center for Research on Teacher Learning. Zeichner, K., Tabachnick, B., 8: Densmore, K. (1987). Individual, institutional, and cultural influences on the development of teachers' craft knowledge. In J. Calderhead (Ed), Exploring teachers' thinking (pp. 21-59). Eastbourne, England: Cassell. “llllll‘lllllllllllllll