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Math penpals as a context for learning to teach: a study of preservice teachers' learning Crespo, Sandra 1998

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MATH PENPALS AS A CONTEXT FOR LEARNING TO TEACH: A STUDY OF PRESERVICE TEACHERS' LEARNING by Sandra Crespo B.Ed., Pontificia Universidad Catolica Madre y Maestra, 1987 M.A., University of British Columbia, 1990 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in The Faculty of Graduate Studies Department of Curriculum Studies We accept this thesis as conforming to the required standards THE UNIVERSITY OF BRITISH C O L U M B I A March, 1998 © Sandra Crespo, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of CvRjlJtvLUM Studies The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract This study explores preservice teachers' learning to teach mathematics in the context of an innovative version of the mathematics methods courses typically offered at UBC. Thirteen preservice teachers engaged in a math letter writing exchange with Grade 4 students are the focus of this study. This math penpal experience was meant to provide a "laboratory setting" for preservice teachers to try out and investigate the ideas discussed during methods classes. Interactions with students, in turn, served as the focus of further class discussions and reflective journal writing. Two research questions were investigated: ( 1 ) What were preservice teachers learning through their math penpal investigations; and (2) What factors influenced their learning? Preservice teachers' written records (math letters, journals, and case reports) were the main sources of data used to address these questions. My perspectives as a participant, teacher, and researcher were used to guide and inform my analysis of this data. An analytical framework was constructed based on preservice teachers' "pedagogical puzzles" (issues and challenges faced and deliberated on). These puzzles related to their problem posing, interpreting, and responding practices. This framework was used to explore patterns and changes in preservice teachers' views and practices. Learning themes discussed include: learning to broaden goals and expectations of problems; learning to see and construct meaning from students' work, and learning to question and revise claims about students' mathematical attitudes and abilities; learning to recognize and interrogate hidden messages in their discourse, and learning to respond differently. Factors found to be associated with preservice teachers' learning include: (a) interactive experiences with students, (b) engagement in collaborative explorations of problems and comparable students' work, and (c) the opportunity to revisit and reinterpret their experiences with students in multiple occasions. Implications for the research and practice of mathematics teacher education are discussed in relation to: (a) preservice teachers' learning of mathematics and mathematical pedagogy, (b) learning in course-related field experiences, and (c) learning to reflect and write about teaching and learning. Table of Contents Abstract ii Table of Contents i i i List of Tables ix List of Figures x Acknowledgments xi Chapter 1 Introduction to the Study 1 I. The Problem 1 II. The Study 5 Goals of the Study 5 The Research Questions 6 Significance of the Study 6 III. Organization of the Chapters 7 Chapter 2 Framing the Study: Learning to Teach and the Pedagogy of Teacher Education 8 I. Visions of Good Mathematics Teaching 8 Rethinking the Nature and Goals of School Mathematics 8 Teaching Mathematics for Understanding 10 Worthwhile Tasks 11 Classroom Discourse 12 Learning Environment 13 Analysis of Teaching and Learning 14 II. Learning to Teach Mathematics for Understanding 14 What Teachers Need to Know 15 Knowledge of Mathematics and School Mathematics 16 Knowledge of Students as Learners of Mathematics 18 - iii -Knowledge of Mathematical Pedagogy 20 Challenges of Learning to Teach Mathematics 23 Preservice Teachers' Prior Knowledge, Beliefs, and Expectations 23 The Complex, Uncertain, and Situated Nature of Teachers' Knowledge 30 III. Changing the Pedagogy of Preservice Mathematics Education 32 Towards a Pedagogy of Inquiry and Investigation 33 Interweaving Knowledge of Mathematics and Pedagogy 33 Pedagogical Potential of Case Methods and Teacher Inquiry 35 Contexts and Tools for Inquiry and Investigation 38 Interactive Experiences with Students 38 Writing to Learn About Teaching and Learning Mathematics 44 Summary 47 Chapter 3 The Study 48 I. The Research Framework 48 Math Penpals as a Context for Preservice Teachers' Learning 48 Introduction to the Penpal Idea 48 The Context of Teacher Education Reform 49 Using Math Penpals as a Site for Investigation 51 Teacher Research: Researching from the Inside and the Outside 53 II. The Research Context 58 The Mathematics Methods Course 58 The Course Setting 58 The Course Design 58 Assignment to Field-Related Projects 60 The Penpal Group: Pairing and Grouping 62 Schedule and Timelines 64 The Main Assignments: Math Letters, Math Journal, and Case Report 68 The Participants 70 The Instructors 70 The Penpal Teacher 70 - iv -The Penpal Students 71 The Preservice Teachers 72 III. The Data Analysis 72 Sources of Data 72 Written Records: Penpal Letters, Math Journals, and Case Reports 73 Looking for Evidence of Learning 74 Analytic Strategy 76 Deciding What to Look For 76 Organizing the Data 80 Analytical Framework 80 Writing Strategy 81 Chapter 4 Analysis of Preservice Teachers' Learning Part I: Posing Mathematical Problems 84 I. Selecting Tasks 85 Choosing Problems Students Can Solve Easily 85 Realizing Need for Better Questions 90 Trying New Problem Posing Strategies 93 Posing New Kinds of Questions 96 II. Making Adaptations 103 Simplifying Problems 103 Removing Ambiguity and Providing Clues 106 Reconsidering How to Make Problems Interesting 111 III. In Summary: From Problems to Solve to Situations to Investigate 116 Chapter 5 Analysis of Preservice Teachers' Learning Part II: Interpreting Students' Mathematical Thinking 118 I. Interpreting Students' Work 120 Initial Expectations 120 - v -Focusing on General Features of Students' Work 122 Looking for Clues into Students' Understandings 124 Focusing on the Meaning of Students' Work 129 II. Drawing Inferences About Students 136 Making Claims About Students' Attitudes and Abilities 136 Revisiting and Reinterpreting Claims 139 III. In Summary: From Evaluative to Analytical Interpretations 143 Chapter 6 Analysis of Preservice Teachers' Learning Part III: Responding to Students' Work 147 I. Responses to Students' Correct Answers 149 Reacting to Students' Answers: Praising and Correcting 149 Praising More than the Right Answer 153 Responding with More than Praise 153 Problematizing Praise 155 II. Responses to Students' Incorrect Work 158 Considering "Not-Telling" as a Response to Students' Wrong Answers 158 Responding to Correct Students' Thinking: Supplying the Answer vs. Helping Students Correct Themselves 160 Responding to Understand Students' Thinking 170 Challenges of Using Inquiry-Oriented Responses 174 In Summary: From Evaluative to Investigative Responses 179 Chapter 7 Conclusions and Discussion of Research Questions 181 I. Research Question 1: What Were Preservice Teachers Learning? 181 Learning Related to Problem Posing 182 Learning About Challenges of Posing Mathematical Problems 182 - vi -Learning to Value "Problematic" Mathematics Problems 184 Learning to Broaden Goals and Expectations of Problems 186 Learning Related to Interpreting 188 Learning About Challenges of Interpreting Students' Mathematical Work 188 Learning to See and Construct Meaning From Students' Work 189 Learning to Question and Revise Claims About Students 191 Learning Related to Responding 193 Learning About the Challenges of Responding to Students' Work 194 Learning to Recognize and Interrogate Hidden Messages in Discourse 195 Learning to Respond Differently 197 II. Research Question 2: What Factors Were Influencing Preservice Teachers' Learning? 198 Inquiry Tools 199 Penpal Writing 199 Journal Writing 201 Case Writing 204 Pedagogical Interventions 206 Types of Problems and Posing Strategies Modeled in Class 207 Journal Prompts and Responses 209 Interactive Experiences with Students 212 Penpal Interactions 212 Personal Interactions 214 Chapter 8 Implications for Research and Practice 215 I. Learning of Mathematics and Mathematical Pedagogy 215 Structural Conditions which Facilitate Learning 216 Relationship Between Mathematics and Pedagogy 217 Focusing on Promoting Connected Understanding 219 II. Learning in Course-Related Field Experiences 223 Issues Related to Learning in the Context of Practical Experience 223 - vii -Issues of Interpreting Preservice Teachers' Learning 225 Issues of Transfer of Learning to Other Contexts 226 III. Learning to Reflect and Write About Teaching and Learning 229 Structural Conditions that Facilitate Reflective Writing 229 Issues Related to Learning in Individual and Collaborative Settings 229 Issue of Content and Form of Written Reflections 230 References 235 Appendix A - Information for Assignment to Field-Related Project, Letters to Parents and Preservice Teachers, Penpal Pairs 244 Appendix B - Course Outline 256 Appendix C - Collection of Math Problems and Journal Activities Assigned on Wednesday Classes 270 Appendix D - Collection of Journal Prompts Related to Math Penpal Project 293 Appendix E - Sample of Summary of Penpal and Journal Data 310 Appendix F - Samples of Preservice Teachers and Students Penpal Letters 314 - viii -List of Tables Table 1. Class schedule of weekly topics and activities 66 Table 2. Overview of research findings on preservice teachers' learning 182 Table 3. Overview of contributing factors to preservice teachers' learning 198 Table 4. The "posing-interpreting-responding" framework as a pedagogical tool 222 - IX -List of Figures Figure 1. Classroom layout and seating arrangement for the penpal group 64 - x -Acknowledgments These are the last words I will be writing in this document. These are the happiest words I have had to write, though they are not the easiest. Expressing on paper my gratitude to those who have helped me reach the completion of my graduate work and this dissertation is a challenge. There are not enough words nor paper to describe what their support has meant to me over the past few years. In the absence of a better medium, here are my words. I am forever grateful to my teaching partners who struggled with me, day in and day out, to understand and make sense of our preservice teachers' thinking and learning in our mathematics methods course. Teaching in your company has been a privilege. To the preservice teachers in our class and those who chose to participate in this study, I thank you for your generosity and willingness to share your work. To the Grade 4 student penpals and their teacher who so willingly shared with us their thoughts and their ideas about mathematics and its teaching and learning, I am eternally grateful. To my research committee, David Robitaille, Ann Anderson, and Tony Clarke, I am most grateful for their incredible patience, constant support, and unwavering belief in me. I thank each and every one of you for teaching me about writing, scholarship, and discipline. I thank Dr. Robitaille for introducing me to the world of mathematics education research and for letting me experiment and wander off while keeping me on track. I thank Ann for giving me the strength to finish this work with her unlimited enthusiasm and her always valuable insights. I thank Tony for giving me the courage to write and to enter the world of teacher education. I could not have done this work without each and every one of you! This study would not have been possible without the influence and support of many. Especially, without the help and encouragement of my very best friend, Cynthia Nicol who has been my long-time thinking partner over the years and very instrumental in helping me conceive and write my dissertation. I am most grateful to Cynthia for helping me become a better researcher, a better teacher, and a better person. She has helped me pose, interpret, and respond differently with her insightful questioning, listening, and responding. Eileen Phillips has also been an important person in my life. She introduced me to the world of elementary school mathematics and has taught me about teaching and learning to teach by her example. She has been a great source of inspiration to my work. I am also - xi-indebted to Kara Suzuka who has been a long-distance thinking partner over the years. Our many conversations during conferences and through e-mails about teaching and teacher education has made my work that much more interesting. I am also thankful to Jo Towers for her willingness to listen and provide me with needed encouragement in my darkest hours. I give my heartfelt thanks to Deborah Ball for always finding time to keep in touch and provide encouragement. Deborah's work has been my great source of inspiration. I especially thank her for posing the most interesting questions of my work, since they are pushing my thinking beyond this study. I thank Rita Silverman for helping me see what a master teacher educator looks like. I thank Klaus Hoechsmann for bringing mathematics back to my life. I thank Brent Davis for reading pieces of my work and getting my thinking gears in motion. Thanks to David Pimm for his interest and always timely encouragement. Thanks to Linda Peterat and Carl Leggo for making my thesis defense so memorable. I also want to thank the past and present faculty, graduate students, and staff in our department who have enriched my academic and personal life at UBC. Faculty members like Gaalen Erickson, Jim Gaskell, Ricki Goldman, Walter Szetela, and Doug Owens who have provided me with their generous advice and encouragement over the years. Graduate students like Sue Brigden, Renee Fountain, Thomas Garcia, Gary Hepburn, Gary Hoban, Heather Kelleher, Hari Koirala, Rama Menon, Ed Robeck, Alberto Rodriguez, and Joel Zapata who have shared with me the ups and downs of graduate life. I thank all of them for enriching my life with their generosity, wisdom, and friendship. I am forever grateful to our department's secretaries (Alex, Bonnie, Michelle, Saroj, Winnie) and technicians (Brian, Bob, Will) for making my life so much easier. To my wonderful family and friends from the Dominican Republic who see very little of me, I thank them for their unconditional love and support. To my parents, Nano and Any, I am eternally grateful for encouraging me to follow my dreams. To my brother Miky for doing mathematics with me over the e-mail. I also thank my sister-in-law Roxana and my beautiful nieces, Claudia and Daniela, for their many calls and pictures. I thank my uncle Eduardo for introducing me to the world of research and for being my role model. Finally, I am eternally indebted to my husband Reid for his willingness to do "simple" mathematics with me, look over my writing, and listen many times over to my same old stories. Most importantly, Reid's constant support and love gave me the strength to do this work. I thank him for encouraging me to continue to follow my dreams. - xii-Chapter 1 Introduction to the Study I. The Problem The mathematics education community has been promoting changes to the way mathematics is taught and learned in schools. In contrast to the image of speed, accuracy, and repetition commonly associated with school mathematics, the image of mathematics teaching and learning proposed by current reform documents (e.g., NCTM, 1989; NRC, 1990) highlights, instead, mathematical inquiry, problem solving, and communication as the mainstays of mathematics learning. Changes to the nature and substance of school mathematics have, in turn, substantially changed what teachers must do to enable students to learn this kind of mathematics. Rather than relying on demonstrations or emphasizing repetitive practice and memorization to help students learn, teachers are now expected to construct learning opportunities for students to make their own sense of mathematics and to adopt what some educators have called an inquiry approach to the teaching and learning of mathematics (e.g., Borasi, 1992; Lampert, 1990). These radical changes to the nature of teaching mathematics have in turn posed serious challenges for mathematics teacher educators in their attempts to help prospective teachers learn to teach mathematics in ways which are much different from their own school mathematics experiences. Having spent 12 to 14 years in schools, preservice teachers have vivid images about what mathematics is and what mathematics teaching looks like. Their past experiences, more often than not, have helped develop implicit and imitative pedagogical tendencies, such as habits of correcting, telling, and supplying the answers (Feiman-Nemser, 1983), and a limited knowledge of and about mathematics along with negative affective dispositions towards it (Ball, 1990a). Preservice teachers' past experiences with school mathematics, most educators agree, militates against their efforts to learn to teach reform-oriented mathematics (e.g.; Ball, 1988; Borko, Eisenhart, Brown, Underbill, Jones, & Agard, 1992; Smith, 1996). Yet, preservice teachers' prior knowledge and beliefs tend to remain unchallenged and unchanged throughout their teacher preparation program. Preservice teachers, therefore, have great difficulty learning to teach differently from the way they were taught during their school years. Furthermore, their school field experience, researchers say, is likely to reaffirm rather than challenge their ideas about knowing, teaching, and learning (Britzman, 1991; Feiman-Nemser & Buchman, 1987; Feiman-Nemser, 1983). Initial teacher education courses are in a unique position to help "break" the "apprenticeship of observation" cycle in the learning to teach process (Ball, 1990a, 1988). The separation of university-based courses from school settings makes it possible to create educational environments which are quite different from what prospective teachers may have experienced before and are likely to find in their field experience classrooms (Suzuka, 1996). The formal on-campus aspect of mathematics teacher preparation could be an important context for preservice teacher learning. This is an important consideration since education courses have tended to play a secondary role (to the field experience) in the preparation of teachers. The research literature, on the other hand, has made it clear that current designs and instructional strategies in teacher education courses, which rely on a "pedagogy of presentation"—instructional strategies which introduce and demonstrate innovative ideas, practices and techniques unproblematically—have not had much success in altering the traditional images prospective teachers bring to their teacher preparation programs. It has become evident that a different design and pedagogy is needed if initial teacher preparation is to have a greater impact on prospective teachers' mathematics teaching. As a response, leading mathematics teacher educators have called for a "pedagogy of investigation" which parallels the advocated changes to the pedagogy of school mathematics and relies on problem solving and reflective inquiry to "help prospective teachers learn what it is like to teach rather than learn how to teach. ... [and] to think and inquire about teaching rather than to learn answers about teaching" (Ball, Lampert, & Rosemberg, 1991, p. 7). Education courses can help introduce teacher candidates to the knowledge base of the profession and the responsibilities, attitudes, and commitments involved in teaching. Methods courses, in particular, offer an opportunity that is quite different from general foundation courses. They might be the only education courses preservice teachers encounter that deal with the philosophy and pedagogy of specific subjects, especially for those preparing to teach in elementary classrooms. But more importantly, "prospective teachers typically look forward to them because they expect that they will learn how to teach specific things. These are the 'practical' classes, unlike foundations or general education courses" (Ball, 1990a, p. 12). Yet preservice teachers' expectations of their methods courses tend to contrast with the changes teacher educators are trying to implement in such courses. Preservice teachers' expectations are, according to Ball (1990a), both too high and too low. Their expectations are unrealistic because they expect to master in the few weeks (10-12 weeks) generally available for such courses what takes years of thoughtful practice to achieve. Their expectations are also quite narrow, Ball (1990a) also says, in that preservice teachers "do not expect the course to challenge what they already know about teaching mathematics. They want to get better at what they know math teachers have to do: explain, show, and tell" (p. 12). Finding ways to help preservice teachers reconsider their long-held assumptions and to develop new ways of seeing, thinking, and acting are challenges teacher educators face. Some teacher educators have addressed this challenge by exploring innovative ways of structuring learning experiences in order to initiate prospective teachers into the practice of teaching as inquiry. Many have turned to case methods as a promising pedagogy to engage students in problem-based situations that simulate the world of classroom practice. Supporters of this approach argue that by discussing and constructing cases1 prospective teachers face some of the challenges involved in teaching and learning to teach, in particular the challenges of learning to "think like a teacher" (Lee Shuiman, 1992), that is, to inquire into and try to make sense of the complexities of classroom life. Case method approaches also parallel teacher-research pedagogies in advocating teacher inquiry2 as a means of improving one's teaching practice and understanding. Both of these pedagogical strategies seek to help experienced and novice teachers represent, study, and make sense of the complex world of practice. By doing so, teachers not only develop and grow professionally, they contribute to their profession by discussing, sharing, communicating, and documenting their knowledge. The pedagogical ideas of case-based approaches and the teacher researcher movement provide a particularly helpful way of thinking about the purposes and goals for initial teacher preparation courses and the kinds of learning opportunities that may help prospective teachers face the challenges of learning to teach mathematics for understanding. These alternative pedagogies draw on the idea that knowledge for beginning teaching needs to be "anchored"3 in rich educational contexts. Such educational environments provide prospective teachers with opportunities for in-depth exploration of the teaching-learning process. This is an important feature, because many classrooms, including university classrooms, provide little opportunity for sustained and deep thinking about content. The Cognition and Technology Group at Vanderbilt (1990) note that "most topics explored in class are ephemeral and change nearly every day" (p. 7). With such an inquiry-oriented pedagogy it may be possible to develop a cognitive apprenticeship 1 Cases are compelling stories highlighting an important problem, issue, or dilemma of teaching and learning. These stories represent a "case of something," an instance or illustration of theoretical, practical or normative principles (see Shuiman, 1986). 2 "Teacher as inquirer", "teacher researcher", and "reflective practitioner" are all terms used to describe the "vision of a teacher who questions his or her assumptions and is consciously thoughtful about goals, practices, students, and contexts" (Richardson, 1994, p. 6). 3 The Cognition and Technology Group at Vanderbilt (CTGV) (1990) have coined the term "anchored instruction" to refer to learning environments which permit sustained explorations of particular topics. model of learning to teach that initiates prospective teachers into the complex and uncertain world of teaching and learning in a safe and supportive manner. Through such an inquiry approach prospective teachers would not only learn important professional knowledge for teaching, they would also learn to think, act, reflect, and inquire as teachers without the immediacy or the pressure of the classroom setting. As a mathematics methods course instructor interested in designing meaningful learning experiences for prospective elementary teachers, I have been inspired by the literature on case-based pedagogies and the ideas and rationale for introducing teacher inquiry methods in teacher education courses. While these ideas are argued convincingly and make sense to me, some questions still remain: How can such an inquiry-oriented pedagogy be integrated within teacher preparation courses, in this case, a mathematics methods course? How can it help preservice teachers reconsider their preconceived notions and redirect them towards alternative ways of thinking and acting? It is with these questions in mind that the present study has been designed. II. The Study Goals of the Study This study was designed to explore the nature and substance of preservice elementary teachers' learning in the context of an inquiry-based mathematics education methods course. During the course preservice teachers engaged in an investigation project with school students. This investigation activity was an integral part of the course and engaged preservice elementary teachers in a math penpal exchange with fourth graders as a major context for individual and collective, mathematical and pedagogical investigations. These interactive experiences with students were meant to serve as a source for reflective inquiry for preservice teachers, both during class and in their journals. To bring their inquiries to closure, preservice teachers then wrote a case study about their learning experience at the end of the course. Provided with such an opportunity, what might preservice teachers learn? What would they make of such an experience? How would this experience contribute to their learning to teach mathematics? These were some of the questions I was interested in exploring when I started this research project. The main goals of this study, therefore, were to (a) understand and characterize preservice teachers' learning in this particular context, and to (b) get a sense of how such learning was encouraged and developed throughout the course. The Research Questions The design, analysis, and write up of this study have been more narrowly defined by the following research questions: • What were preservice teachers learning through their math penpal investigations? • What factors influenced preservice teachers' learning in this context? These seemingly simple questions have turned out to be much more complex than I had originally anticipated. Through my attempts at identifying and analyzing preservice teachers' learning, I have also had to address other sets of related and also very complex questions: What is learning? What counts as evidence of learning? How is learning to teach mathematics similar to, and different from, learning mathematics? How can I best share and represent what I have learned about these preservice teachers' learning? I share and discuss how I have dealt with some of these issues in the last section of Chapter 3. Significance of the Study Teacher education courses have tended to be the most overlooked and the least researched elements of preservice education. They have become the black box of teacher education, or as Wilson (1990) notes, a well-guarded secret. Although these courses tend to be harshly criticized (both by teacher educators and preservice teachers), very little is known about what goes on in these classes. This proposed study intends to add to this literature by providing insights into the content and pedagogy of an on-campus mathematics education methods course and by examining its influence on preservice teachers' learning to teach mathematics. This study also contributes to the literature on case methods because few studies have explored the use of this pedagogy in subject-specific courses. In addition, results of this study will be of interest to advocates of teacher research since preservice teachers in this study are introduced to "practical inquiry"4 (Richardson, 1994). Although teacher research is fervently advocated in the literature, its introduction to teachers remains an issue. While some educators support the introduction of teacher research as early as preservice education (i.e., Rudduck, 1992, Stenhouse, 1985), teacher research is not considered a fundamental aspect of many undergraduate teacher preparation courses or programs. This study delves into the process of preservice teachers' inquiry, and will contribute to our understanding of its role in helping prospective teachers learn to teach for understanding. III. Organization of the Chapters Chapter One presents an introduction and brief rationale and overview of the study. Chapter Two provides a review of the literature which has guided the present study and informed the design of the mathematics methods course which was the site for this research. Chapter Three provides further detail into the rationale and design of the mathematics methods course and its associated math penpal experience. This chapter also discusses the teacher-researcher perspective adopted for this research as well as the analytical and writing strategies employed. Chapters Four through Six are analysis chapters constructed as narratives to highlight preservice teachers' learning themes related to their posing, interpreting, and responding practices. Results of the study are discussed in Chapter Seven. Implications of results are provided in Chapter Eight. Practical inquiry is undertaken by practitioners in order to understand their contexts, practices, and students (Richardson, 1994, p. 7). Chapter 2 Framing the Study: Learning to Teach A n d The Pedagogy of Teacher Education This chapter synthesizes the relevant research literature associated with teaching and learning to teach mathematics in teacher preparation programs. This literature provided the theoretical lens for framing the research questions that guided this study and informed the design of the mathematics methods course in which this study takes place. I. Visions of Good Mathematics Teaching Rethinking the Nature and Goals of School Mathematics The study of mathematics, mathematician John Allen Paulos (1991) argues, provides a way of knowing and reasoning that is quite often missed by school mathematics, that is, the mathematics that is taught and learned in schools. What is important to learn from and about mathematics, he contends, is not "this formula or that theorem" but rather "the ability to look at a situation quantitatively, to note logical, probabilistic, and spatial relationships, and to muse mathematically" (p. 6). In other words, mathematical competence encompasses more than the mastery of specific content and the solving of particular types of problems. Mathematical competence, or "mathematical literacy" as Paulos (1988) would call it, includes the appreciation and the understanding of "mathematical modes of thought" (Borasi, 1992), that is the kind of thinking which are characteristic of the discipline of mathematics (e.g., modeling, abstracting, conjecturing, testing, making inferences, etc.), and being able to use these mathematical reasoning tools in a wide range of situations. School mathematics, critics say, has little resemblance to the disciplinary structure and activity of mathematics. The mathematics many students experience in school tends to mainly emphasize quickness, precision, and correctness as qualities to aspire to and to achieve through practice and memorization. These are perhaps not the explicit goals of school mathematics, but conventional mathematics instruction has consistently helped students develop distorted views and "dysfunctional" beliefs about mathematics (Borasi, 1990; Lampert, 1990). Lampert (1990) explains it as follows: Commonly, mathematics is associated with certainty: knowing it, with being able to get the right answer, quickly. These cultural assumptions are shaped by school experience, in which doing mathematics means following the rules laid down by the teacher; knowing mathematics means remembering and applying the correct rule when the teacher asks a question; and mathematical truth is determined when the answer is ratified by the teacher. Beliefs about how to do mathematics and what it means to know it in school are acquired through years of watching, listening, and practicing, (p. 32) (emphasis in original) These commonly held views and beliefs contrast and conflict with current views about the nature of mathematical knowledge (Davis & Hersh, 1981; Ernest, 1991) as changing, uncertain, and constantly revisable. Commonly held views about mathematics also contrast with current research on, and assumptions about, how students learn mathematics articulated in mathematics education reform documents (e.g., NRC, 1989; NRC, 1990; NCTM, 1989; NCTM, 1991) and volumes of research articles. There is, at least on paper, a wide-spread consensus among mathematics educators of the need "to bring the practice of knowing mathematics in school closer to what it means to know mathematics within the discipline" (Lampert, 1990, p. 29), and to narrow the gap between theories of learning and instruction in the mathematics classroom (Cobb, 1988). In reform documents, mathematics is promoted as "an ongoing product of human activity," "a dynamic and expanding system of connected principles and ideas constructed through exploration and investigation" (NCTM, 1991, p. 133). Mathematics is also being promoted as the language and science of patterns (Steen, 1990). "The patterns are regularities in numbers and data, in shapes, in graphs and in symbols, and the connections among them, [and] the science is exploration, conjecturing, testing, abstracting, generalizing, and extending these patterns by communities of practitioners" (Romagnano, 1994, p. 8). Understanding this kind of mathematics requires more than knowing the rules, procedures, and answers. Learning mathematics with understanding, therefore, requires more than listening, memorizing, and practicing. Teaching Mathematics for Understanding Changes to the nature and substance of school mathematics have, in turn, substantially changed what teachers must do to enable students to learn mathematics with understanding. Rather than relying on demonstrations, repetitive practice, and memorization to help students learn, teachers are now expected to construct learning opportunities for students to make their own sense of mathematics through reasoning and problem solving and to adopt what educators are calling an inquiry approach to the teaching and learning of mathematics (e.g., Borasi, 1992; Lampert, 1990). This inquiry approach to teaching and learning mathematics is also referred to in the literature as "teaching mathematics for understanding." Teaching for understanding, different from "teacher telling" or "direct instruction," is founded on an inquiry view of learning, a view of learning which yields the kinds of mathematical understanding that are now valued and promoted. Two different kinds of mathematical understanding— functional and structural understanding—have been noted by Hiebert and colleagues (1996) in the discussions about teaching mathematics for understanding. There is, however, unprecedented consensus among mathematics educators that mathematical understanding is constructed through students' active and reflective participation in individual and collective inquiry into significant mathematical questions and problems. The functional perspective of learning mathematics with understanding equates understanding with participating in a community of practice. Therefore, learning with understanding depends on and develops through the students' and teacher's collaborative engagement in mathematical activities in the classroom. The structural perspective, on the other hand, equates mathematical understanding with the students' internal representations and constructions of the subject. Understanding is considered to be the "by-product" or "residue" of mathematical activity and to be dependent on the student's - 1 0 -prior knowledge and experiences. The product or residue of structural understanding are the "insights," "strategies," and "dispositions" that students construct as a result of their involvement in mathematical activities (see Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier, & Wearne, 1996). Regardless of which of these two views of mathematical understanding one subscribes to, Hiebert and colleagues (1996) point out, an inquiry approach to learning and teaching mathematics supports the development of functional and structural mathematical understandings. Hiebert et al. (1996) propose that any instructional approach which helps students make the subject matter problematic, that is to inquire and wonder about questions and problems that elicit students' curiosity and sense-making, would succeed in helping students learn mathematics with understanding. The Professional Standards for Teaching Mathematics (NCTM, 1991) makes a similar though more elaborate proposal outlining four aspects of teaching—selecting tasks, orchestrating discourse, designing environments, and analyzing instruction—which are central to teaching mathematics for understanding and promoting students' inquiry of mathematics. Worthwhile Tasks. Selecting worthwhile mathematical tasks is considered to be central to teaching mathematics for understanding. The tasks teachers pose in their classrooms deserve important consideration because they open or close the students' opportunity for meaningful mathematics learning. They also convey implicit messages about the nature of mathematics: what it is and what it entails, and what is worth knowing and doing in mathematics. Worthwhile tasks, therefore, are chosen, not because they are "fun" or because they are suggested in mathematics textbooks. Worthwhile tasks are chosen because they have the potential to "engage students' intellect," "can be approached in more than one interesting way," and "stimulate students to make connections and develop a coherent framework for mathematical ideas" (NCTM, 1991, p. 25). Worthwhile tasks, however, are not necessarily one-of-a-kind, innovative, colorful, and complexly designed tasks. Furthermore, not everyone would need to agree on the "worthwhileness" of problems. Some times it is not the content of the task itself which makes it worthwhile but the way in which it is posed to students which enhances the task's potential to engage students on sound and significant mathematics. Even the most routine of mathematical activities can be constructed into a worthwhile mathematical experience when posed in such a way as to engage students in mathematical inquiry (see Schoenfeld, 1989; and Hiebert, et al., 1996). Hiebert and colleagues (1996), for example, discuss how the task of adding two-digit numbers can enthrall a second-grade class when students are allowed to generate and share multiple solutions and methods to a computational problem. Another example is found in Sullivan and Clarke (1991) who suggest different ways of converting drill-and-practice exercises like rounding decimals—typically posed as: "round 5.77 to the nearest tenth"—into more inviting and mathematically interesting tasks by posing a related but different kind of question, such as: "a number is rounded to 5.8 what might the number be? [and] what is the largest and smallest possible answer?" They also suggest that routine word problems which typically require students to apply a known procedure to, for example, find the volume of a particular shape, could be instead posed as a problem to investigate patterns and relationships, such as: "In how many different ways can you design a box-shaped building using exactly 24 cubes" (p. 17). Similarly, Brown and Walter (1983), suggest extensions to the problem solving tasks teachers typically propose by encouraging students to generate their own mathematics problems. They propose using their "what-if-not?" process for systematically changing the conditions and goals of previously solved problems in order to engage students in problem posing activities, an often forgotten and neglected aspect of mathematical inquiry. Classroom Discourse. A common and important feature of the problems and strategies for engaging students in worthwhile tasks, such as those highlighted above, is -12-their potential to promote mathematical discourse. Classroom discourse refers to the ways in which students and teachers explore and exchange mathematical ideas. Classroom discourse, therefore, is an important element in the teaching of mathematics for understanding. It introduces students to different forms (e.g., verbal, pictorial, symbolic) and tools (e.g., concrete models, calculators, group discussion, journal writing) for exploring and communicating mathematics. It also helps students develop fundamental values about knowledge and authority. "It is through the interactions in the classroom that students learn what mathematical activities are acceptable, which ones need to be explained or justified, and which explanations or justifications are acceptable" (Lappan & Theule-Lubienski, 1994, p. 251). Teachers play a crucial role in shaping classroom discourse through the problems they pose, through the questions they do and do not pursue, through the way mathematical answers and solutions are accepted and validated, and through the tools and models of representation and communication they provide and introduce to their students. Traditional patterns of teacher-student interactions whereby the teacher looks and asks for the correct answers convey very different messages from a classroom where possible answers are shared, investigated, and justified (regardless of correctness) and ultimately assessed and accepted based on mathematical reasoning and evidence, rather than the authority of the teacher or textbook. Establishing a classroom discourse that focuses "on exploring mathematical ideas, not just on reporting answers", the NCTM (1991) suggests, requires teachers to value, encourage, and accept a variety of tools and means of communication (including drawings, diagrams, invented symbols, and analogies) in the mathematics classroom. Learning Environment. The learning environment teachers and students construct affects the seriousness with which they (the students and the teacher) engage in mathematical inquiry, how and whether they listen to each other's ideas, and their willingness to take intellectual risks and to collaboratively work towards making sense of mathematics. Creating a conducive learning environment, therefore, is paramount to -13-helping students learn mathematics with understanding. The ways in which teachers organize the classroom environment—the physical (e.g., group, individual, whole class activities) as well as the intellectual and social climate—serve to communicate to students the values and norms of the classroom community in particular, and the mathematics community in general. Analysis of Teaching and Learning. Teaching for understanding is ultimately affected by the teachers' disposition and capacity to analyze the effects of his or her efforts (Ball, 1991a). The systematic study and examination of teaching and learning is the trademark of "reflective practice" and "teacher inquiry" which are fundamental to teaching for understanding. It is through the continual examination of the tasks, discourse, and learning environments, and through the analysis of students' learning that teachers inform and transform their own teaching practices. The mathematics education reform vision, therefore, considers the inquiry into one's own teaching practices and students' learning to be fundamental to practicing good teaching and the continual improvement of teaching mathematics (NCTM, 1991). Ball (1991a) explains: Teaching involves making plausible conjectures—about what will work well, what will promote desired goals, and what will help particular students—and revising these conjectures in light of the outcomes. The more teachers see, hear, and sense and the more they bring new perspectives to bear in interpreting their observations, the more adept they become in developing their own teaching. The analysis standard discusses ways to study and learn from one's own teaching. (Ball, 1991a, p. 20) II. Learning to Teach Mathematics for Understanding The previous discussion served to highlight the mathematics education community's vision of teaching mathematics for understanding. It also provided an overview of what this kind of teaching might mean and entail for students and teachers. The discussion now turns towards exploring what such ambitious visions for mathematics teaching and - 14-learning require and demand of teachers, and what it might take to help prospective teachers learn to teach mathematics for understanding. What Teachers Need to Know Fundamentally, [teachers] would need to know how to get a large group of students gathered in a small place interested and engaged in doing intellectually challenging work. One would need to know how to manage a rather complex set of interactions, both between teachers and students as a whole class and among students as they work together in problem solving groups. One would need to be able to figure out how students think about mathematical phenomena and to respond to that thinking in ways that are both supportive and challenging. One would need to know how to listen to students and how to organize the classroom so that students could express their thinking and listen to one another with respect. One would need to know where the mathematics teaching and learning process was headed, not in a linear sense of one topic following another, but in the global sense of a network of big ideas and the relationships among those ideas and between ideas, and facts, and procedures. One would need to know a variety of ways to represent big ideas to students drawing on concrete, pictorial, verbal, and contextual as well as abstract modalities. One would need to know how to assess student understanding and represent that assessment in terms that students, parents, and administrators would understand and accept. (Lampert, 1988, p. 163-164) Teaching mathematics for understanding, mathematics educators agree, requires a rich, deep, and broad understanding of mathematics and its pedagogy. Research studies into teachers' knowledge (e.g., Shuiman, 1986; Wilson, Shuiman, & Richert, 1987) and the knowledge structures of novice and expert teachers (e.g., Borko and Livingston, 1989; Leinhart & Smith, 1985) have documented three areas of specialized knowledge needed for teaching—knowledge of the subject they teach (subject matter knowledge), pedagogical knowledge specific to the subject (pedagogical content knowledge), and knowledge of the curriculum content and materials of specific topics (curricular knowledge). Consistent with this research literature, the NCTM (1991) also proposes that in order for teachers to teach mathematics for understanding and incorporate inquiry-- 1 5 -oriented practices in their classrooms (select worthwhile tasks, effectively orchestrate classroom discourse, design constructive learning environments, and analyze teaching and learning) they need expertise in at least three interrelated domains of teacher knowledge: knowledge of mathematics and school mathematics, knowledge of students as learners of mathematics, and knowledge of the pedagogy of mathematics. Knowledge of Mathematics and School Mathematics. Having a strong understanding of mathematics is essential to teaching mathematics for understanding. Teachers' knowledge of mathematics is known to affect teaching practice in terms of what and how teachers choose to teach. Teachers' knowledge and conceptions of mathematics, therefore, serve to "shape their choice of worthwhile mathematical tasks, the kinds of learning environments they create, and the discourse in their classrooms" (NCTM, 1991, p. 132). The mathematical knowledge needed in order to teach mathematics for understanding, the Standards say, includes more than the understanding of specific concepts and procedures. It also includes knowledge of the discourse of mathematics and "a deep understanding of the mathematics of the school curriculum and how it fits within the discipline of mathematics" (NCTM, 1991, p. 134). Deborah Ball's writings (e.g., 1990b, 1990c, 1991b) about the mathematical understandings needed to teach and learn to teach mathematics have illuminated the complex structure and character underlying the knowledge of mathematics required for teaching. Drawing from Shulman's (1986) work on the structure of teachers' subject matter knowledge (based on Schwab's model of disciplinary structures)—substantive and syntactic knowledge—and her research studies into preservice teachers' knowledge of mathematics, Ball (1991b) has outlined a framework of the subject matter knowledge needed for teaching mathematics. She asserts that teachers need substantive and syntactic knowledge of mathematics, what she calls knowledge of and about mathematics. The substantive knowledge of mathematics, Ball (1991b) says, is what most people would recognize as mathematical knowledge. It is the knowledge of mathematical principles, procedures, concepts and their relationships. Teachers' substantive knowledge, -16-however, requires more than understanding and being able to do mathematics for oneself. While for many people, their knowledge of mathematics tends to be rule-bound, teachers' knowledge needs to be conceptually based. In other words, teachers need to know the underlying principles and meanings of, for example, how and why the "invert and multiply" rule works in the division of fractions and how this principle relates to other mathematical concepts and principles, such as its connection to the meanings of division and division of whole numbers. Therefore, in order to help others understand mathematics, Ball (1991b) says, teachers' substantive knowledge needs to be explicit: Tacit knowledge, whatever its role in independent mathematical activity, is inadequate for teaching. In order to help someone else understand and do mathematics, being able to "do it" oneself is not sufficient. A necessary level of knowledge for teaching involves being able to talk about mathematics, not just describing the steps for following an algorithm, but also about the judgments made and the meanings and reasons for certain relationships and procedures. Explicit knowledge of mathematics entails more than saying the words of mathematical statements or formulas; rather, it must include language that goes beyond the surface mathematical representation. Explicit knowledge involves reasons and relationships: being able to explain why, as well as being able to relate particular ideas or procedures to others within mathematics. (Ball, 1991b, p. 17) Besides the explicitness criteria, teachers' knowledge of mathematics also needs to be connected (Ball, 1990c, 1991b). This means that teachers need to be able to see mathematics as a "network of big ideas," as Lampert (1988) says, and to see the larger picture of connections between and among particular principles and procedures and the larger mathematical concepts and ideas. To teach mathematics, Ball (1991b) says, teachers need to know that "mathematical knowledge is not a collection of separate topics, nor a laundry list of rules and definitions" (p. 18). They need to know, for example, that "addition is fundamentally connected to multiplication, algebra is a first cousin of arithmetic, and the measurement of irregular shapes is akin to integration in calculus" (Ball, 1991b, p. 18). -17-In addition to understanding the substance of mathematics, knowledge about the nature of mathematics or "syntactic knowledge" is also a very important part of knowing mathematics and school mathematics. Syntactic knowledge of mathematics or the knowledge about mathematics refers to the understanding about the nature of mathematical knowledge and activity—what is involved in doing mathematics, and the ways in which the validity of mathematical claims are determined. Knowledge about mathematics is not often associated with mathematical knowledge and therefore not often the explicit focus of mathematics instruction. Ideas about the nature of mathematical thinking and activity, however, are communicated through mathematics teaching practices and develop implicitly through years of schooling. Teachers' understandings about the nature of mathematics have been shown to influence the choices they make regarding the norms for knowing, doing, and communicating mathematics in their classrooms (Ball, 1991b). Hence, the critical importance of this kind of knowledge in teaching mathematics for understanding. Knowledge of Students as Learners of Mathematics. Teaching mathematics for understanding also requires extensive knowledge of diverse learners, how they learn particular mathematics topics, and of students' informal ways of thinking about mathematics. It is this pedagogical content knowledge related to students' mathematics learning which "enables teachers to build an environment in which students may learn mathematics with appropriate support and acceptance" (NCTM, 1991, p. 144). Teachers may learn about students' learning of mathematics from their own experiences as learners of mathematics, from the growing research literature on students' mathematical thinking, and their own experiences teaching students. Knowing students as learners of mathematics, the NCTM (1991) suggests, requires teachers to "develop habits of mind that include becoming active researchers in their own classrooms as well as users and interpreters of research as it relates to their everyday teaching" (p. 144-145). Teachers' knowledge and beliefs about students as learners of mathematics is believed to exert a powerful influence on teachers' instructional practices, on the learning -18-experiences they design for their students, and their analysis of students' learning. The effect of teachers' understandings of students on their instructional decisions and their interactions with students have been widely documented in studies of teacher expectations (see Good, 1987). There are also a few studies which have focused on extending teachers' knowledge of students' mathematical thinking both in practicing teachers (e.g., Cognitively Guided Instruction or CGI studies by Carpenter and his colleagues—Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; also, Cobb, Wood, & Yackel, 1990) and in preservice teachers (e.g.; D'Ambrosio & Campos, 1992). These studies have found that becoming sensitive to students' ways of thinking and understandings encouraged teachers and preservice teachers to engage in inquiry-oriented practices which foster the teaching and learning of mathematics for understanding. In a series of research studies led by Thomas Carpenter at the University of Wisconsin, researchers have found that teachers' instructional decisions substantially change as a result of their developing knowledge about students' strategies for solving addition and subtraction problems. In one study they compared the instructional decisions and teaching practices of teachers in experimental and control groups. Teachers in the experimental group had access to research-based knowledge of students' mathematical understandings of addition-subtraction problems. Carpenter and colleagues found that experimental teachers "spent more time listening to children explain the mental processes they used while solving word problems and expected and accepted a larger variety of problem-solving strategies from children than did the control teachers" (Fennema & Franke, 1992, p. 155). In another study, this team of researchers studied changes in the beliefs and instruction of 21 primary grade teachers over a 4-year period (see Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996). They introduced teachers to their research-based model of children's mathematical thinking during a teacher development program and observed changes to their beliefs and practices through classroom observations and interviews. They reported that students' achievement gains in these teachers' classrooms - 19-were substantial. They also reported changes in the teachers' teaching practices which can be connected to the increase in their students' mathematical performance. Changes to their teaching practices included: (a) providing more opportunity for students to grapple with concepts and engage in problem solving, (b) encouraging students to share their thinking and valuing their contributions, and (c) adapted instruction to the problem-solving abilities of their students (Fennema, et al., 1996). Similarly, Cobb and colleagues (1990) report that teachers in their studies began to change their instructional practices as a result of collaborating with them in the analysis of children's mathematical learning within the setting of the classroom. They report that teachers in their studies began to renegotiate classroom norms to allow students to express their mathematical thinking, to seek assistance when they encountered difficulties interpreting their students' mathematical solutions, and to appreciate the relevance of detailed knowledge of children's mathematical cognition to their practice (Cobb, Wood, & Yackel , 1990). D'Ambrosio and Campos' (1992) study investigated the development of preservice teachers' knowledge of children's mathematical thinking as they were engaged in the research of students' understanding of fractions. They studied five preservice teachers who agreed to participate in a research experience which required them to: (1) become acquainted with existing research literature on students' understandings of fractions and methods of collecting data of students' thinking, (2) to design and administer a data gathering instrument, (3) to revise the instrument and i f necessary administer it to different students, and (4) write a report of the results. D'Ambrosio and Campos (1992) report that their preservice teachers' research experience led them to become "inquisitive of children's understanding of a topic, [and] more knowledgeable of and more skilled in ways to gain insight into a child's formal knowledge as well as informal or intuitive understanding of the topic" (D'Ambrosio & Campos, p. 227). Knowledge of Mathematical Pedagogy. Knowing mathematics pedagogy is also fundamental to practicing good teaching. This knowledge, as discussed in the Standards, -20-is a mixture of understandings related to pedagogical-content and curricular knowledge. It refers to the knowledge of powerful instructional representations, tools, materials, and strategies which have the potential to make the subject matter understandable to students. It is this knowledge, the NCTM (1991) suggests, which enables teachers to ask central pedagogical questions about the subjects they teach and that leads them to use powerful pedagogical representations, strategies, and actions. Teachers, therefore, need to be able to ask the following fundamental questions of each of the curricular topics of a discipline: What are core concepts, skills, and attitudes which this topic has the potential of conveying to students? ... What are the aspects of this topic that are most difficult to understand for students? What is the greatest intrinsic interest? What analogies, metaphors, examples, similes, demonstrations, simulations, manipulations, or the like, are most effective in communicating the appropriate understandings or attitudes of this topic to students to particular backgrounds and prerequisites? What students' preconceptions are likely to get in the way of learning? (Shulman & Sykes, 1986 cited in Carter, 1990, p. 305) Carter (1990) also suggests (citing Tamir, 1988) that additional aspects of teachers' knowledge of subject specific pedagogy include: "a teachers' knowledge of students' interest and motivation to learn particular topics within a discipline [and] a teachers' understanding of how to make outside-school settings (e.g., museums and laboratories) quality learning environments" for students (p. 305). This knowledge, the Standards (1991) suggest, allows teachers to employ alternative forms to the conventional "show, tell, and do model" of instruction and to create learning environments that support students' mathematical inquiry and understanding. This pedagogical content knowledge is considered to be unique to the profession of teaching and therefore relatively undeveloped in novice teachers (Brown & Borko, 1992). Furthermore, this knowledge, Ball (1991b) says, interacts with teachers' subject matter knowledge and beliefs in complex and nonlinear ways which adds to the challenge of understanding how this knowledge is constructed. -21-Research studies contrasting experts' and novices' teaching practices have reported that beginning teachers tend to have great difficulties coming up with powerful means of representing the subject matter to students and that their efforts are more time-consuming and less efficient than those of expert and more experienced teachers. These differences, Brown and Borko (1992) note, "can be explained by the assumption that novices' schemata for pedagogical content knowledge are less elaborate, interconnected, and accessible than those of experts" (p. 221). Expert teachers, Brown and Borko (1992) point out, also have difficulties expanding their pedagogical content knowledge. Making the transition to thinking pedagogically about the subject matter, therefore, is known to be one of the most difficult aspects of learning to teach (Brown & Borko, 1992). Learning to make pedagogical representations of the subject is unquestionably a very important aspect of teaching mathematics for understanding yet it is the most difficult knowledge for teachers and preservice teachers to construct. Ball's (1991b) analysis and contrast of the teaching practices of three different teachers—having differing understandings and beliefs and who were teaching the same mathematical topic (multiplication) at the same grade level—highlight the complex interactions between teachers' instructional representations and their knowledge and beliefs of mathematics, students, teaching, learning, and school contexts. Ball (1991b), for example, notes that while using inquiry-based practices for teaching mathematics depends heavily on the teachers' subject matter knowledge, having a strong and conceptually based understanding of mathematics does not necessarily guarantee that the teacher would draw upon this understanding in their teaching. Teacher educators have, therefore, concluded that attempts to help prospective and practicing teachers construct and extend their pedagogical content knowledge should not focus on one single part of the knowledge equation. Feiman-Nemser and Buchmann (1989) after following six preservice teachers through their teacher preparation program and examining the development of their pedagogical thinking also suggest that the alignment of person, program, and setting may not be enough to help prospective -22-teachers make the transition to thinking pedagogically about the subject matter. In other words, pedagogical content knowledge, while fundamental to teaching mathematics for understanding, is a complex and challenging knowledge for novice and expert teachers to construct, and for teacher educators to promote in teacher preparation programs. Challenges of Learning to Teach Mathematics Much of the learning-to-teach literature has focused on and has helped define the multiple sets of knowledge teachers need and draw from in order to teach. The more recent learning-to-teach literature has focused on the challenges of teaching and learning to teach reform-oriented mathematics. Two main sources of difficulties associated with learning to teach mathematics for understanding are worth noting and are discussed next. These are: (1) preservice teachers' prior knowledge, beliefs, and expectations, and (2) the complex, uncertain, and situated nature of teachers' knowledge. Preservice Teachers' Prior Knowledge, Beliefs, and Expectations. Preservice teachers enter teacher preparation programs having spent many years in school classrooms. They accumulate a wealth of experiences, understandings, and beliefs about mathematics, students, teaching, learning, and classrooms which inform and shape their learning to teach during courses and field experiences. These early experiences with mathematics teaching and learning often act as obstacles and constraints to preservice teachers' learning and hamper their attempts to teach reform-oriented mathematics. The "Teacher Education and Learning to Teach Study" conducted by the National Center for Research on Teacher Education (NCRTE) at Michigan State University in the late 1980's was one of the early studies to bring to the fore the importance of focusing the research and practice of teacher education on prospective teachers' entering knowledge, beliefs, and prior experiences. Deborah Ball's study mentioned earlier was a part of the NCRTE larger study and focused on the mathematical understandings of prospective elementary and secondary school teachers. Based on questionnaires from 252 (217 elementary, 35 secondary) -23-teacher candidates and interview data from a small portion (19 candidates: 10 secondary and 9 elementary) of the main sample, Ball (1990b) concluded that the mathematical understandings prospective teachers construct in their mathematics courses in schools and university are inadequate and insufficient for teaching elementary and secondary school mathematics. She reported, for example, that while many preservice teachers were able to correctly perform mathematical procedures, few of them understood the meaning underlying their procedures. Furthermore, in the topic of division of fractions (see Ball, 1990c), she found that very few secondary teacher candidates and none of the elementary candidates were able to generate appropriate representations for division statements such as l 3 /4 + V2. Similar findings have been reported in studies of preservice teachers' understandings in diverse mathematical topics and at the elementary and secondary school levels (e.g., Even, 1993, functions; Koirala, 1995, probability; Simon, 1993; division). These studies support Ball's conclusions that prospective teachers' prior experiences with mathematics in schools and university courses help them develop inadequate understandings (rule bound and sparsely connected) of mathematics for teaching. Based on her findings, Ball (1990b) suggested that teacher education programs cannot assume that teacher candidates' prior mathematical courses in elementary and secondary school or college courses provide them with adequate subject matter knowledge for teaching mathematics for understanding. These findings are not surprising given that preservice teachers have been students in the kinds of mathematics classrooms educators are trying to reform. Preservice teachers' experiences in these classrooms, Ball (1990b) suggests, also give them ideas about mathematics which influence their learning of mathematics. In these conventional classrooms, preservice teachers learn, as Ball (1990b) found, that: "doing mathematics means following set of procedures step-by-step to arrive at answers; knowing mathematics means knowing 'how to do it'; and mathematics is a largely arbitrary collection of facts and rules" (Ball, 1990b, p. 460). -24-Their classroom experiences also help preservice teachers develop ideas about students, teaching, learning, and schooling. Years of listening and observing in classrooms provide them with a specialized "apprenticeship of observation" (Lortie, 1975). "Watching teachers and paying attention to their own experiences, they develop ideas about the teachers' role, form beliefs about 'what works' in teaching math, and acquire a repertoire of strategies and scripts for teaching specific content" (Ball , 1988, p. 40). These experiences, as Smith (1996) notes, help preservice teachers perpetuate the practice of "teaching mathematics by telling" which is so pervasive in mathematics classrooms. The aforementioned N C R T E study also found that undergraduates often enter teaching with a limited view of their roles as teachers, tend to see teaching as a matter of telling, and see learning as a result of memorizing and practicing (see Ba l l , 1990b; McDiarmid , B a l l , & Anderson, 1989). Synthesizing the research literature on the views and beliefs of teachers and preservice teachers, Smith (1996) also says that practicing and beginning teachers hold a core set of beliefs about mathematics teaching and learning which are "internally consistent and mutually reinforcing" and lend support to the view and practice of "teaching mathematics by telling" (p. 391). Smith (1996) describes these views and commitments as follows: Mathematics content. School mathematics is viewed as a fixed set of facts and procedures for computing numerical and symbolic expressions to find determinant "answers." Through middle school, the focus is on arithmetic of natural and rational numbers; the high school years shift the focus to computation with algebraic expressions. School mathematics is defined by the content of mathematics textbooks and is largely invariant. Textbooks contain problems that students must learn to solve, and each problem— whether stated in numerical, symbolic, or verbal terms—is associated with a particular solution procedure. Teaching mathematics. Given that view of content, teachers' central task is to provide clear, step-by-step demonstrations of each procedure, restate step in response to student questions, provide adequate opportunities for students to practice the procedures, and offer specific corrective support when necessary. If students do not master a procedure, teachers should -25-repeat their demonstration. Teachers should also provide recurrent opportunities for students to refresh and strengthen their mastery of previously taught content. Learning mathematics. Students learn by listening to teachers' demonstrations, attending carefully to their modeling actions, and practicing the steps in the procedures until they can complete them without substantial effort. Solving problems is a matter of recalling and applying the procedures appropriate for the given problem type. Because of the heavy demands of memory, success in mathematics may well depend on students' effort or innate ability. Mathematics authority. The answers to all mathematics problems are known and found in textbooks. Teachers who control and interpret texts are the intermediate authorities for students on matters of mathematical truth. (Smith, 1996, pp. 390-391) Their prior experiences with school mathematics also help prospective teachers get ideas about what they need to learn in order to teach. They come to think and believe, as Britzman (1991) says, that "experience is the best teacher," that "teachers are self-made," and "product of their experiences" (p. 7). Prospective teachers, therefore, enter teacher education programs expecting to learn from practical experiences the know-how of teaching mathematics. They do not expect, as Ball (1990a) points out, to be challenged about what they think they already know about mathematics teaching and learning, and they expect to get better at what they think teachers do—explain, show, and tell. These views and commitments are not coherently nor explicitly articulated by teachers and preservice teachers, rather they are the product of their experiences in mathematics classrooms prior to entering university and teacher preparation programs. These beliefs and views are generally taken for granted and referenced without questioning. These beliefs, and expectations, however, become evident in preservice teachers' teaching practice and discourse in actual, observed, or simulated teaching situations. Wilson (1990), for example, reports that for preservice teachers in her class watching a videotape of a teacher who encourages students to set, explore, and solve their own problems, "fifteen minutes of students arguing over the meaning of one graph seems -26-an eternity to my students" (p. 206). Wilson (1990) also reports that her preservice teachers typical reactions to such video episodes are to state as fact that "the discussion went on for too long," and that "there is not enough time for all that talk" (p. 206). Teacher educators who work with practicing and prospective teachers often write and attest to similar scenarios in their classrooms. Holt-Reynolds (1992) studied preservice teachers' "personal history-based beliefs" (or "lay theories"5 ) about good teaching practice and explored their interactions with the values and principles about teaching and learning preservice teachers encountered during their teacher preparation course work. She suggests that preservice teachers' views and beliefs of good teaching are based on their analysis and conclusions of their experiences as students. These beliefs, she found out, "work behind the scenes as invisible, often tacitly known criteria for evaluating the potential efficacy of ideas, theories, and strategies of instruction they encounter as they formally study teaching" (p. 343). She reported that preservice teachers in her study tended to question the premises and values introduced in their courses and not their lay beliefs. They evaluated the course material by referencing their experiences as students which they used as prototypes for their generalized premises about teaching and learning. Preservice teachers' prior knowledge and experiences are powerful yet limiting resources for learning to teach. These early commitments are very important to consider in the design of contexts and opportunities for learning to teach mathematics for understanding. These are important because as noted in the previous section, teachers' knowledge is intricately connected to teaching practice. Furthermore, research into teachers' beliefs and conceptions of mathematics, teaching, and learning (see Thompson, 1992) have also shown their relationship to teaching practice. Studies of preservice teachers' entering knowledge and beliefs have convinced teacher educators that the problem of learning to teach demands more than providing prospective teachers with opportunities to add to their knowledge base. The task of teacher preparation courses has, According to Holt-Reynolds (1992) drawing from Vygotsky's work "lay theories" are beliefs developed naturally over time without the influence of instruction. - 2 7 -therefore, been defined as one of affecting preservice teachers' entering knowledge and beliefs in ways which prepare them to teach for understanding. This requires providing learning opportunities for preservice teachers to change and revise prior and unexamined knowledge and beliefs related to mathematics teaching and learning. Engaging preservice teachers' prior experiences, knowledge, and beliefs in productive ways, teacher educators have learned, is not a simple matter. Research studies of education courses attempting to challenge preservice teachers' knowledge and beliefs report having very limited success doing so (see Ball 1990a; Lappan & Theule-Lubienski, 1992; McDiarmid 1990; Simon, 1994). Furthermore, research studies also report that preservice teachers' initial views and ideas also change little and often are reinforced during their field experiences (see Feiman-Nemser & Buchmann, 1989). Experiences in innovative mathematics and mathematics methods courses, Lappan and Theule-Lubienski (1992, citing a longitudinal study conducted by NCRTE) report, can be powerful interventions into preservice teachers' disciplinary knowledge and dispositions to engage in mathematical inquiry or sense-making. They also report that although these courses affected preservice teachers' views of mathematics for themselves, preservice teachers' ideas about how children should learn mathematics and how they should go about teaching it remained intact. Ball (1990a), McDiarmid (1990), and Simon (1994) also make similar claims. Simon (1994, citing Simon and Shifter, 1991) reports that "teachers can learn a great deal about how mathematics is learned in the context of their own learning of mathematics when it is accompanied by reflection on their learning" (p. 80). However, he has also observed that "learning mathematics in a context in which the overarching goal is learning to teach may not cause much disequilibrium" (p. 90). McDiarmid (1990) and Ball (1990a) also found that the experiences in their respective courses succeeded at encouraging preservice teachers to construct new images for teaching and learning mathematics and to reinterpret their own past experiences with mathematics. They also found that the -28-course's experiences were also discouraging and unsettling to preservice teachers' confidence in their abilities to teach and learn to teach reform-oriented mathematics. These studies highlight the importance and the difficulties of influencing preservice teachers' prior knowledge and beliefs related to mathematics teaching and learning. They also highlight the challenges of designing learning experiences which influence and connect preservice teachers' mathematical and pedagogical understandings. Ball (1990a) attributes some of the difficulties affecting such long held views and beliefs to the short amount of time preservice teachers and teacher educators typically can spend on course's activities and experiences. "Ten weeks, four hours a week, is a minuscule, almost trivial, amount of time to contemplate the agenda I have set" (Ball, 1990a, p. 15). Another challenge Smith (1996) suggests, is that "teaching by telling" is an attractive model for teaching because even though good telling cannot guarantee that students will learn, "it narrows the scope of the content to manageable proportions, clearly defines what the central acts of teaching are and what counts as evidence of student learning, and provides a structure for daily classroom life" (p. 393). Similarly, Ball (1994) notes, there are powerful disincentives to teaching mathematics for understanding: Evidence that students may not understand is not always intriguing, for it can be quite uncomfortable. One major source of teachers' sense of efficacy and satisfaction is the sense that they can help students learn. And when we do not ask students to voice their ideas, we run less risk of finding out what they do and do not know. ... But things are more complicated still. If student understanding becomes more problematic, one's own understandings are soon more uncertain as well. And this is at least as unsettling. After all, teachers are "supposed to" know what they are teaching. Confronting one's own uncertainties in understanding can make a teacher feel inadequate and ashamed. That the mathematics reforms are aimed at helping students understand content in usable and powerful ways is part of the appeal for teachers whose own mathematical histories did not offer them such opportunities. Still, in pursuing such goals, deep anxieties about one's effectiveness and knowledge are likely to surface. (Ball, 1994, pp. 10-11) - 2 9 -The Complex, Uncertain, and Situated Nature of Teachers' Knowledge. The complexity of teachers' knowledge is evident in discussions about what teachers need to know. Teachers' knowledge is complex not only because of the underlying structures of each of its domains but also because of their complex interactions and interplay with one another. Teaching, however, requires more than transforming one's knowledge into usable pedagogical forms and actions. Teaching also involves considerable skill. It involves skills like "listening to one child while watching thirty others, using one's voice as a tool, 'reading' and interpreting the reactions and understandings of others who may communicate differently from the teacher, keeping a wide range of details in mind, posing appropriate questions" (Ball, 1994, p. 8). This, as Ball (1994) says, means that: The kinds of things that play a role in practicing as a teacher of groups of children is more complex than revealed by our usual lists of what teachers need to know. A host of personal qualities matter: patience, curiosity, generosity in listening to and caring about other human beings, confidence, trust, imagination. There is caring about seeing the world from another's perspective, as well as enjoying the humor, sympathizing with the confusion, caring about the frustration and shame of others. And there are things like tolerance for uncertainty, willingness to take risks, and patience with confusion and mess. The personal resources which teaching demands are not so often discussed, and even less often nurtured. Is the kind of patience that teaching requires something that can be learned? Can empathy grow? If these kinds of resources and qualities are central to teaching, then we need ways of thinking about what might be ways of cultivating and nurturing their development. (Ball, 1994, pp. 8-9) The inherent uncertainty and contextualized nature of teachers' knowledge is also highlighted in the way in which some teacher educators describe teaching and teachers' thinking. Lampert and Clark (1990), for example, describe teaching as "a complex act requiring the moment-by-moment adjustments of plans to fit continually changing uncertain conditions" (p. 21). Lampert (1985), has also characterized the work of teaching as coping with dilemmas that arise out of daily practice. To manage these problems of practice, researchers say, teachers draw from and integrate knowledge of multiple sources including "practical knowledge" (images, stories, or cases of past -30-classroom events). Furthermore, because "knowledge in teaching is both incomplete and contested," Ball (1994) points out, "teachers are continually in a position of interpreting conflicting evidence and making choices and judgments," that is, "they must figure out new things as they teach" (p. 9). Learning to reason and construct new knowledge from the data of one's teaching experiences, Ball (1994) argues, is another crucial dimension of learning to teach mathematics for understanding. However, this kind of learning tends to be left to the students' own resources and initiative in most teacher preparation programs. Yet the pitfalls of novices learning to teach from their teaching experience have been widely documented (e.g., Feiman-Nemser & Buchmann, 1987; 1989). Although first hand experience offers rich possibilities for helping beginning teachers learn to teach, it can also be a source of problems. "Familiarity with classrooms and teachers," Feiman-Nemser and Buchmann (1987) point out, "may prevent beginners from searching beyond what they already know and from questioning the practices they see" (p. 256). Their research on the subject has led them to conclude that "in teacher preparation, experience is a trusted though not always a reliable teacher" (Feiman-Nemser & Buchmann, 1987, p. 256). From the above discussion, it is not hard to see that representing and constructing teachers' knowledge in teacher preparation programs are challenging undertakings. Furthermore, teachers' knowledge is not so easily transferred from situation to situation and therefore defies prescription and reliance on pre-specified steps and procedures. The classroom setting, long considered the main context for learning to teach, is not necessarily a conducive environment for learning to teach for understanding. Learning to teach, teacher educators have come to believe, requires opportunities to construct teaching knowledge in multiple contexts and opportunities to learn through reflective inquiry. Experience, teacher educators have concluded, can be a powerful context for learning to teach only when approached with a stance of reflective inquiry. Reflection, therefore, is considered to be central to the process of learning to teach for understanding. -31-III. Changing the Pedagogy of Preservice Mathematics Education The faith in the educational powers of first-hand experience is apparent in the very structure and curricula of teacher preparation. Research into the cognitive and socialization effects of field experiences, however, have challenged the notion of experience as the best teacher (Feiman-Nemser & Buchmann, 1989). As a result, the teacher education field experience is currently seen as a mixed blessing. On the one hand, the practice setting provides powerful and memorable experiential learning. Yet on the other hand, the familiarity of classrooms serves to confirm more than to challenge prospective teachers' limited conceptions and beliefs about teaching and learning. The realization that the practice setting may not necessarily be the best place to learn to teach for understanding has revived interest in the educative role and potential of teacher preparation courses. Teacher education courses, on the other hand, have been shown to be a weak intervention in preservice teacher education. Studies from the "teacher socialization" perspective highlight the potency of preservice teachers' early apprenticeship of observation in limiting the impact of the on-campus and field-based experiences offered in teacher preparation programs. In this literature there are many "examples of students interpreting the messages of teacher education courses in ways that reinforce the perspectives and dispositions they bring to the program, even when these interpretations involve a distortion of the intentions of teacher educators" (Zeichner & Gore, 1990, p. 337). Teacher educators have, therefore, grown concerned with the structure, content, and pedagogy of teacher education programs. The research literature, as Brown and Borko (1992) say, have provided evidence that current structures and designs are not working. It is, however, widely recognized that very little is known about the specific content and pedagogy used in the various courses and field experiences of teacher preparation -32-programs and their contribution to preservice teachers' learning. Nevertheless, discussions about teacher education reform (e.g., The Holmes Group, 1987; NCTM, 1991) denounce the typical structural arrangement that tends to separate teacher education courses (foundation and methods) and field experiences. In addition, teacher education's widely practiced "pedagogy of presentation"—that is, instructional strategies that introduce and demonstrate innovative ideas, practices, and techniques unproblematically—has also become problematic and insufficient to help prospective teachers learn to teach mathematics for understanding. Towards a Pedagogy of Inquiry and Investigation in Teacher Preparation Research on the process of learning to teach has given teacher educators good reasons for revising the content, practices, and settings of teacher preparation courses and field experiences. Some teacher educators have begun to design and systematically investigate pedagogical interventions to help prospective teachers learn to teach mathematics for understanding. There is a growing and more explicit articulation of goals, content, and instructional strategies for the kinds of learning experiences teacher preparation programs should be offering to prospective teachers. These are largely informed by: (a) the goals of teaching and learning mathematics highlighted in reform documents (e.g., NCTM Standards, 1989, 1991); (b) analogies between learning mathematics and learning to teach mathematics; and (c) the learning-to-teach research literature. As a result, various practices are currently advocated as means to improve the teaching and learning to teach in teacher preparation programs. Some of these practices are reviewed next to provide a rationale for the design and study of innovative learning-to-teach experiences in teacher preparation programs, in general, and in mathematics education methods courses, in particular. Interweaving Knowledge of Mathematics and Pedagogy. Knowledge of subject matter and of subject-specific pedagogy are known to be essential to good teaching. Yet prospective teachers are known to enter teacher preparation having incomplete and inadequate subject matter understandings for teaching. They also enter teacher -33-preparation having limited ideas and views about what mathematics teaching and learning looks like and entails. Learning to teach, therefore, requires opportunities to re-construct conceptually sound, elaborate, and connected knowledge of mathematics and its pedagogy. Teacher education programs have tended to offer such opportunities in disconnected ways and in separate settings. In other words, mathematics courses, pedagogical content courses, and field experiences tend to address either, but seldom both, of these knowledge domains. Teaching and learning to teach requires weaving together mathematical and pedagogical understandings. Knowledge of students and of mathematical pedagogy is understandably underdeveloped in beginning teachers. It requires the ability to transform subject matter knowledge into powerful pedagogical representations which are responsive and adaptive to diverse learners. This ability (often called "pedagogical reasoning"5 in the literature) is known to be the most difficult aspect of learning to teach. Although little is known as to how this pedagogical content knowledge is constructed, research findings suggest preservice teachers seldom make these connections on their own or in contexts which separate mathematics content from its pedagogy. It, therefore, makes sense to design teacher education learning experiences to provide opportunities for preservice teachers to learn and connect both mathematical content and mathematical pedagogy. This, idea, however, is far from being widely shared among teacher educators, or to be reflected in teacher preparation programs. Some teacher educators have observed the difficulties of constructing pedagogical content knowledge in a course with a focus on mathematical content (e.g., Lappan and Theule-Lubienski, 1992). Others report having difficulties helping experienced and prospective teachers construct mathematical knowledge in courses with an explicit focus on mathematical pedagogy (e.g., Simon, 1994). 5 See Shulman, 1987; Wilson, Shulman, & Richert, 1987. -34-Furthermore, there are important challenges to adopting such an integrated approach in teacher education courses. These include: the small amount of time generally available for course work and the difficulties of covering the breadth of the mathematical topics teachers need to be prepared to teach. Another source of difficulty is the customary separation of the university course setting from the classroom setting. This makes it a challenge to establish connections between the topics studied at the university and the practice setting. This means that it would be useful to explore alternative designs for the on-campus teacher education learning experiences in order to establish connections between course work and the school setting. Pedagogical Potential of Case Methods and Teacher Inquiry. Preservice teachers' knowledge of mathematics is known to be incomplete and conceptually shaky. Teaching is also a complex and uncertain activity requiring situational decision-making and responsiveness to continually changing and unpredictable classroom conditions and interactions. This means teachers need to be able to teach with incomplete and uncertain knowledge. Learning to teach is also considered to be a lifelong process. Preservice teachers cannot be expected to learn everything they need to know in the two or three years they spend in their teacher preparation programs. An alternative way of thinking about initial teacher preparation is as a place where "it may be possible to learn in two or three years the kind of practice which then leads to another twenty years of learning" (Hawkins, 1973, p. 7). It seems, therefore, important to orient teacher education pedagogy towards developing a stance of reflection and inquiry towards learning mathematics and learning to teach mathematics. Two different pedagogies—case methods and teacher inquiry—have been heralded in the teacher education literature as promising approaches to encourage practicing and prospective teachers to construct habits of thinking about teaching and learning which foster learning from one's own teaching and learning experiences. In beginning teacher preparation, case-based pedagogies are often associated with course-based teacher education and tend to engage preservice teachers in collaborative discussions and inquiry -35-related to a particular episode or "case" of somebody else's teaching (in narrative or video form) (see Judith Shuiman, 1992a; Wassermann, 1993). Teacher inquiry methods (teacher research, action research, practical inquiry) in teacher preparation programs, on the other hand, tend to engage preservice teachers in individual inquiry on their teaching practice while they are in their practicum field experiences (see Rudduck, 1992; Stenhouse, 1985). In case-based pedagogies, the focus on a common "case" makes it possible to engage preservice teachers in group discussions and deliberations about a particular episode of classroom practice. Furthermore, the case is permanently recorded and therefore can be analyzed in multiple occasions. This allows preservice teachers to engage, vicariously, in the situational decision-making process that classroom practice demands and to collaboratively devise plausible solutions and plans for future action. Case discussions, however, do not provide opportunities to carry out any of the devised plans for action, to see the consequences of the group's deliberations and action plans, or opportunities to examine the subsequent dilemmas that may arise from trying to implement one's plans. Yet ultimately, teacher educators expect preservice teachers to be able to reflect and analyze their own teaching practices along with the consequences of their decisions and actions. Teacher inquiries, on the other hand, provide opportunities for preservice teachers to reflect on their own teaching experiences. Usually some form of recording is used (e.g., video or audio recording, teacher journal, students' work) in order to facilitate later reflection on the classroom events and interactions. The pressures of preparing for the next day's class, however, make it difficult for preservice teachers to choose to reflect on past classroom events in order to design future classroom practice. Furthermore, the time constraints and the immediacy demands of classroom practice tend to encourage preservice teachers to frame problems of practice as problems of classroom management. Preservice teachers, therefore, do not tend to focus their reflections on issues of content and pedagogy while they are teaching (see Kagan, 1992; and Grossman, 1992). This is -36-also complicated by the fact that in the practicum setting there is little time to consult and deliberate with other colleagues. The individualistic nature of what happens in each classroom also makes collaboration an unlikely resource for teacher inquiry. The above discussion points to the potential influence that the structure and the context of learning experiences have on the process and content of preservice teachers' inquiries. Richert (1992), for instance, compared four different arrangements of two different approaches (portfolios and reflective partners) commonly used to facilitate teacher reflection and inquiry. She found out that the content of the reflections of the beginning teachers in her Master's level course varied according to the conditions under which the reflection occurred. Depending on whether they had reflected with or without the aid of either a partner or a portfolio, the focus of their reflections ranged from personal concerns (no partner and no portfolio), to the content of instruction (no partner, and portfolio), to the social and interactive aspects of teaching and learning (a partner and no portfolio), to matters of general pedagogy (partner and portfolio). Richert (1992) concluded that different structures affect the form and content of teachers' reflections. Another study by Levin (1995) also revealed the influence that structural conditions have on students' reflective engagement and activity. She found that adding group discussions to a case-based learning experience substantially contributed to the participants' understanding. Her study explored differences in the quality, form, and content of teachers' learning as a result of their experiences with a written case of teaching and learning. She compared teachers' learning under two different instructional conditions, one group analyzed the case individually through reading and writing; and another group analyzed the case through reading, writing, and discussing it. Levin (1995) reported that "only reading and writing about a case appeared to provide little stimulus for teachers to elaborate their understanding or increase their perspective on the issues in the case" (p. 75). She also asserted that the conflicts that arose during the teachers' discussions of the case were an important source of changes in teachers' thinking regardless of whether they were listeners or talkers. -37-Interestingly, Levin (1995) also reported that the teachers in her study did not draw from their previous case discussion when discussing a similar case a month later. She suggested that such learning experiences should be offered over an extended period of time if they are to have a longer and lasting influence in preservice teachers' thinking and teaching practice. This again highlights the need to attend to the conditions under which preservice teachers reflect and inquire in teacher preparation programs. In addition to structural considerations, there are also cognitive factors that need to be considered in the design of learning experiences for preservice teachers. Calderhead (1992), for example, notes that preservice teachers have different ideas and expectations about professional practice and their professional development. Many preservice teachers, Calderhead (1992) says, "expect to be told how to teach..., others expect to learn from their own trial and error, others expect to model their practice on a teacher familiar to them, a few regard learning to teach as completely unproblematic" (p. 141). Richert (1992) also points out that there are also cognitive barriers to reflective practice. She reminds us, as Sumara (1989) also does, that reflecting on one's teaching actions is often a very painful and humbling experience which many would rather avoid. Reflective practice, therefore, is not an easy activity to ask preservice teachers and teachers to do in private, let alone in public. The point here is that different structural arrangements have different effects on preservice teachers' inquiry and reflection. It is, however, also clear that there is much to learn about the conditions and tools which facilitate inquiry and reflection in teacher preparation courses and field experiences. This means that exploring different ways in which to organize learning experiences to engage preservice teachers in reflective inquiry—particularly on their own teaching practices and in collaboration with others—is an important area of inquiry for the practice and research of teacher education. Contexts and Tools for Reflective Inquiry and Investigation Interactive Experiences with Students. One of the central tasks of teacher education programs is to provide opportunities for preservice teachers to learn pedagogical content -38-knowledge, that is learning to see the subject matter pedagogically—from the teachers' perspective and through the students' eyes. Helping preservice teachers construct this knowledge, however, has proven to be a difficult undertaking in both teacher education courses and field experiences. Although practical experiences with students abound during teacher education field experiences, research has shown that pedagogical content knowledge is not simply constructed by virtue of experience alone. Grossman (1989), for instance, says that "although much pedagogical knowledge has been characterized as common sense, knowledge is not hanging, ripe, and fully formed, in the classroom waiting to be plucked by inexperienced teachers" (p. 205). Research has also revealed that subject matter knowledge is essential in the development of pedagogical content knowledge (Ball, 1991b). Learning about students as learners and mathematical pedagogy, therefore, may require opportunities to learn in contexts which integrate subject matter and its pedagogy in ways which are similar and related to the practice setting. The educational powers and potential of learning in "authentic" contexts have been widely recognized in the education community. What one learns, researchers in the "situated cognition" field assert, is intricately connected with how it is learned. Knowledge is said to be situated and inseparable from the activity, context, and culture in which it is developed and used (see Brown, Collins, & Duguid, 1989). Although relatively little is known about how people learn and construct meaning from their experiences, active engagement in authentic activity is considered to be essential for learning to think and act in a community of practice. Learning, therefore, is seen as a process of "enculturation" or "cognitive apprenticeship" into the practices and modes of thinking of a particular field whether it is an academic discipline, trade, or profession (Brown, Collins, & Duguid, 1989). The field experience has tended to be the main place for learning general and subject-specific pedagogy during teacher education programs. The educative powers of the field experience, however, are no longer taken for granted. Feiman-Nemser and Buchmann's -39-(1987) study of six preservice teachers during their two years in teacher preparation showed that preservice teachers' familiarity with classrooms makes the field experience a problematic context for learning to teach. Familiarity with classrooms, Feiman-Nemser and Buchmann (1986) point out, makes it difficult for preservice teachers to see beyond what they see and experience in the classroom setting which in turn also prevents them from envisioning what teaching could be like. Classroom familiarity, therefore, tends to prevent preservice teachers from problematizing current teaching practices and from conceptualizing alternative ways of teaching and learning. Teacher education programs, Feiman-Nemser and Buchmann (1986) assert, further aggravate the problem of familiarity of classrooms by markedly separating learning the theoretical aspects of teaching in professional course work and learning the practical aspects in field experiences. Another source of difficulty for getting over the familiarity of the classroom setting, Feiman-Nemser and Buchmann (1986) also suggest, is that "classrooms are not set up for teaching teachers" (p. 71). This means preservice teachers need to learn to make sense and learn from their teaching experiences throughout their teacher preparation programs, not only during field experiences. The process of learning pedagogical content knowledge in the practice setting is complicated by the fact that preservice teachers have had lengthy personal experiences in classrooms. This means that they enter teacher preparation with a variety of knowledge and beliefs about teaching and learning. These understandings, as discussed earlier, tend to be incomplete and inappropriate for teaching and learning to teach for understanding. Furthermore, their own experiences as students do not provide preservice teachers with opportunities to learn about students who are different from themselves or to develop pedagogical strategies to help students learn particular subject matter concepts and procedures. Prior experience as students, therefore, is a powerful though insufficient and often misleading source of knowledge for learning to teach subject matter to diverse groups of students. -40-The task of providing opportunities for preservice teachers to learn about students as learners of mathematics and about mathematical pedagogy in teacher preparation programs is, therefore, not a simple one. The classroom setting is the authentic place for learning to teach. Yet learning to teach in the context of field experiences has not been very successful in helping preservice teachers learn to teach for understanding. Furthermore, the remoteness of teacher education courses from the world of practice makes it very difficult for preservice teachers to relate to the knowledge imparted in those courses and to use or recognize when to use it in the practice setting. Blurring the boundaries between teacher education course work and field experiences seems, therefore, like a good pedagogical move for teacher preparation programs. In addition to the extended field experience in teacher education programs, various other opportunities are provided to preservice teachers to observe and work with a group of students and their teacher during "early field experiences" or "short practicums." These early field experiences, however, have not always been associated with specific courses. Field experiences which are connected to a specific introductory or methods course are understandably far less common as there are time and structural constraints to the typical design of education courses. It is, for example, not always possible to arrange course-related field experiences over a long period of time nor with a large group of preservice teachers in a classroom. Most teacher educators, therefore, attempt to connect their course material with the practice setting through vicarious experience with classroom videos or written cases, and through various forms of firsthand experience observing, interviewing, and teaching students in a short and sometimes single session. Connecting course work in some way with the practice setting is an important element in current teacher education pedagogies. Many teacher educators have come to believe that course-related field experiences can be a powerful context for engaging preservice teachers in the inquiry and investigation of mathematics teaching and learning. There is, however, no conclusive research to support nor to challenge this claim. According to Carter and Sanders (1996), in their review of the literature on field-based -41-pedagogies, there is "little solid evidence concerning the impact of field experiences in general or of specific intervention strategies" (p. 575). There are however strong indications from the theoretical and research literature of teacher learning that the practice setting can become a powerful context for learning to teach. Lampert (1985), in discussing the differences between how research and teaching knowledge are constructed, points out that teachers construct knowledge in the context of attempting to resolve problems of their practice. Similarly, Schon (1983), who studied the construction of knowledge in professional domains, describes how practitioners learn in the context of the practice setting. He highlights the role that surprises in their practice play in stimulating practitioners' sense making and learning of their practice: Ordinary people and professional practitioners often think about what they are doing, sometimes even while doing it. Stimulated by surprise, they turn thought back on action and on the knowing which is implicit in action. ... There is some puzzling, or troubling, or interesting phenomenon with which the individual is trying to deal. As he tries to make sense of it, he also reflects on the understandings which have been implicit in his action, understandings which he surfaces criticize, restructure, and embody it in further action. (Schon, 1983, p. 50) Reports from studies of practicing teachers learning in the context of professional development programs also provide evidence of the potential of field-related experiences in promoting reflective practice. The studies of Carpenter and colleagues (1989) cited earlier (see p. 26), Cobb and his colleagues (1990), and D'Ambrosio and Campos (1992) all provide evidence that opportunities to attend to students' thinking can be a transformational experience for practicing and beginning teachers. The opportunities for learning provided in these professional development projects support and lend credence to the idea of linking course work with field experiences. The above discussion suggests that in order to make field experiences productive contexts for learning to teach for understanding, they need to offer opportunities for preservice teachers to investigate, interpret, and discuss problems of practice. Such field--42-based opportunities can also be incorporated during preservice teachers' on-campus course work. Course-related field experiences can therefore be more than artifacts to help preservice teachers see the relevancy of the course's content (McDiarmid, 1990) and (or) to help preservice teachers put into practice what they are learning (Feiman-Nemser & Buchmann, 1989). Furthermore, course-related field experiences do not need to take place at the school setting nor do they need to take the form of actual classroom practice in order to become occasions for thinking and deliberating about issues of teaching and learning. In fact, field experiences which differ from the classroom setting may help make unfamiliar the familiar context of teaching and learning and therefore facilitate the process of reflection. Such surrogate field experiences can also provide preservice teachers the necessary distance from everyday practice to allow them to begin to reflect on and inquire into their own teaching practices and their students' learning. Another way in which course-related field experiences may aid preservice teachers' learning is by helping to surface preservice teachers' prior and implicit knowledge and beliefs about teaching and students' learning which, as Schon notes, is essential to the reflective process. During field-related experiences, preservice teachers reveal their understandings and convictions, not only through what they say and write about their experiences, but also through their actions. Preservice teachers' actions in field-related contexts is not always an accessible source of information or reflection in teacher education courses, particularly for the course instructor. Yet, as McDiarmid (1990) indicates, changes in preservice teachers' beliefs can only be reliably tested in the context of what prospective teachers do in their classrooms. He also says that, as a course instructor, he can only "infer [that preservice teachers' beliefs have changed] from the discourse in my classroom and from [his] students' written work" (p. 17). Having opportunities to act as a teacher during their teacher education course work can, therefore, help reveal gaps between preservice teachers' intentions and actions which can then become accessible for further inquiry and reflection. This is a particularly important point, since there are times, as Sumara (1989) points out, when we can become -43-aware of what we think by examining our actions. This is so, Sumara (1989) says, because our actions are a representation of what we think. This point is eloquently stated and further elaborated in an excerpt Sumara (1989) quotes from the book Ascent of the Mountain, Flight of the Dove. It reads: It is not true that faith, creed, convictions come first and then action. It is rather true that we are already acting before we are clear about our ultimate convictions. More importantly still: our actions, reflected on, reveal what it is we really care about, more accurately than our words or aspirations about what we would like to care about. We do not know what our deepest views and root concerns may be until we see them bloom into action. Action is the starting place of inquiry. Action reveals being. Action is our most reliable mode of philosophizing. In action we declare our cosmology, our politics, our identity. Who am I? I am what I do. (Novak, 1971, cited in Sumara, 1989, p. xiii) Writing to Learn About Teaching and Learning Mathematics. Journal writing has become a popular instructional tool in teacher preparation programs to promote preservice teachers' reflection and inquiry as they learn to teach. Practicing teachers who engage in teacher research in their own classrooms use journals as a tool for recording and analyzing data from their classroom experiences and their students' learning. Writing in journals makes it possible for teachers to distance themselves from the world of practice and the immediacy of action (Van Manen, 1990). This allows them to better understand their own practice and the context in which occurs. Journal writing also makes it possible for teachers to share data and interpretations from their own classrooms with other teachers and engage in reflective inquiry in collaboration with others (Altrichter, Posch, & Somekh, 1993). Journal writing has also become a popular tool for helping school students learn mathematics with understanding. Aside from the traditional role of writing for recording and reporting purposes, educators are also advocating the use of writing as a learning and teaching tool in mathematics and mathematics education classrooms. Supporters of writing as a tool for learning highlight the communicative and the reflective functions of -44-writing as its major contributions to the learning process (see Borasi & Rose, 1989). Writing in math classes, Burns (1995) says, "adds an important and valuable dimension to learning by doing. Writing encourages students to examine their ideas and reflect on what they have learned" (p. 13). Writing, according to various mathematics educators, benefits students' learning by slowing down the thinking process and by helping students clarify and become aware of their own thinking (see Buerk, 1990; Countryman, 1992; Rose, 1990). Furthermore, writing is a useful pedagogical tool for encouraging all students to engage in mathematical inquiry, particularly those who do not tend to participate in more public classroom activities (NCTM, 1989). Rose (1990), who studied the benefits of journal writing in one of her college mathematics classes, concluded that writing supports mathematics learning in multiple ways and has great benefits for the student as writer, for the teacher as reader, and for the classroom environment. She writes: When students write, they can be encouraged to express and reflect upon their feelings, subject matter, processes, and views of mathematics, and consequently cope with negative emotions, learn new content and skills, ask questions, and reconceive their beliefs about mathematics. As teachers read the writing, they are exposed to individual needs, common difficulties, and feedback on the course; with this information, teachers can become more responsive to short-term adjustments in the course and long-term improvements in teaching. As students and teachers engage in a dialogue, the interaction can produce a more personal, cooperative, and active learning environment. (Rose, 1990, p. 71) Similarly, journal writing in preservice mathematics teacher education courses and field experiences is also believed to benefit preservice teachers' learning in multiple ways. Writing activities can take many forms (e.g., autobiographies, case writing, essays, portfolios) and they have multiple uses and purposes—such as, eliciting past experiences, analysis of teaching and learning experiences, or assessment of material covered in course work. Journal writing, in particular, is a tool teacher educators use to encourage preservice teachers to interpret and reflect on their own teaching and learning -45-experiences. Writing, as Bean and Zulich (1989) suggest, can help preservice teachers "generate their own questions in a course, explore hunches and hypothesis, and begin to perceive the multiplicity of views inherent in human experience" (p. 44). Writing is also helpful to both preservice teachers and teacher educators in helping make explicit preservice teachers' thinking (Bean & Zulich, 1989; Hoover, 1994). There are, however, several problems associated with the use of writing as a teaching and learning tool. Students' resistance to writing, time constraints, and the overuse of this tool by instructors tend to undermine the potential of writing activities (see Anderson, 1992). There are also concerns associated with the focus and the structure of journal writing which have been shown to influence the content and quality of reflections. Richert's (1992) study mentioned earlier, for instance, reported on the difficulties of moving preservice teachers' reflections from the personal and emotional aspects of learning to teach towards matters of general and subject-specific pedagogy. It seemed, however, that having access to tangible records of what happened in their classroom and a dialogue partner served to help the teachers in her study recollect and focus their reflections onto pedagogical concerns of their teaching practices. Anderson (1992) also shared his concerns with the content and form of preservice teachers' writing in the Reading and Language Arts methods courses he teaches at the university. He notes that most of his preservice teachers' writing takes the form of reporting and summarizing, and very few of their entries show analysis, synthesis, deliberation, or reflection. "Although this reporting might be of minimal benefit to the student," Anderson (1992) asserts, "it is highly questionable whether the time spent in this type of exercise is justified" (p. 307). He also notes the ethical dilemmas he has faced when responding to some of the overly critical and prejudiced remarks preservice teachers make in their journals. Grading journal writing, Anderson (1992) also says is problematic as it invites students to "write for the teacher" rather than use it as a learning tool. Yet grading based on completion alone with disregard for quality of content can also discourage and limit the benefits of serious engagement in journal writing. -46-Summary In this chapter I have reviewed the arguments and theories associated with the reform vision of teaching and learning mathematics for understanding. Relevant literature on learning to teach—what teachers need to know and the challenges of learning to teach— was also reviewed. This literature provides insights into the goals for teacher preparation and the problems associated with helping prospective teachers learn to teach for understanding. Preservice teachers' prior knowledge and beliefs and the uncertain and situated nature of teachers' knowledge were highlighted for their relevance to the design of learning opportunities in teacher education courses. This was then followed by a review of the literature on current innovations in teacher education pedagogies which are pertinent to teacher education courses, particularly for the methods course setting. This review provides the rationale behind the design and study of preservice teachers' learning in a mathematics methods course incorporating a field-related experience. This is now followed in the next chapter with the details about the design of this study's mathematics education methods course and the research strategies used to examine preservice teachers' learning experiences. -47-Chapter 3 The Study This is a study of preservice elementary school teachers learning to teach mathematics in the context of an innovative mathematics methods course which I, in collaboration with other colleagues, designed, taught, and researched. The main focus of this study is on one of the field-related experiences associated with the course, namely the math penpal investigation project. The purpose of this study is not to evaluate this interactive experience with students but rather to describe and explore preservice teachers' learning experiences with it. This chapter reports on the design of both the course and the research study associated with it. Then, the strategies and issues of collecting and analyzing data are discussed. But first, I discuss the rationale for using math penpals as a context for investigating teaching and learning in a mathematics methods course. This is then followed by a rationale for using a teacher-researcher perspective in the study of preservice teachers' learning. I. The Research Framework Math Penpals as a Context for Preservice Teachers' Learning The idea of designing a course around the notion of investigating teaching and learning did not come overnight, but rather it was a slow realization over a period of two years of experience teaching and researching preservice teacher education. This latest course built on the earlier designs of my (and my colleagues') previous mathematics methods courses and incorporated the ideas from the current teacher education and mathematics teacher education literature. Introduction to the Penpal Idea. In January of 1994, the year prior to the present study, I was approached by my "cooperating penpal teacher" with the idea of engaging her students and my preservice teachers in a math penpal exchange. This invitation was a welcome extension to my regular weekly visits to this teacher's mathematics class. At - 4 8 -that time I had been a volunteer in her class since September 1992. My cooperating penpal teacher was a full-time teacher at a school near the University and, at the time, a part-time graduate student in our department. She had become interested in the idea of promoting her students' mathematical communication and had read about a penpal exchange between students and preservice teachers in a journal article (see Fennel, 1991). She wanted to provide her students with an authentic reason for communicating mathematically with an audience other than the teacher. While her interest was in exploring the development of her students' mathematical communication, I became interested in exploring the potential benefits of this activity for preservice teachers' learning. The penpal idea was immediately attractive to me and that is why I accepted the invitation, but I did not know how my preservice teachers would respond to such an invitation. In particular, I was concerned that they would not be interested in adding an extra commitment to their already heavy course load. There were a number of reasons which made it impossible for me to incorporate the penpal exchange as a regular feature of my course that year. One of them was the mismatch of the number of students in this teacher's class and the number of preservice teachers in my methods course. Another was the timing of the invitation which had afforded me very little time to organize and design the activity. I decided, therefore, to present the penpal invitation to my preservice teachers as a voluntary, non-credit, and extra-curricular activity which they would do in their own time. To my surprise, 26 of my 38 preservice teachers that year volunteered. The Context of Teacher Education Reform. As a beginning teacher educator I have turned to the research literature for guidance and direction in my teaching of preservice mathematics education courses. The NCTM Professional Standards (1991) have been a useful resource in the design and conceptualization of a pedagogy which relies on investigation, rather than presentation of mathematical ideas and practices. This, and the emerging literature about the use of cases and case methods as a pedagogy for teacher - 4 9 -education courses has influenced my thinking about mathematics methods courses as sites for investigating the teaching and learning of mathematics. In the broadest sense case-based pedagogies rely on pedagogically problematic situations—presented in the form of cases—that serve as contexts around which to organize good discussions and explorations into classroom teaching and learning. These cases, or pedagogical problems, could take the form of a ficticious or real narrative or story about a teacher's dilemma (referred to as written cases); or they could take the form of a video clip of a classroom episode which raises important issues and questions (referred to as video cases). The idea is to provide a vicarious teaching-learning experience through which to analyze, problem solve, and reflect on one's own ideas about teaching and learning. Extensions to the original case method ideas were also flourishing in the research literature. "Cases," as described in the literature, were becoming more elaborate and often included "artifacts" (e.g., teacher's journals, interview transcripts, students' workings, etc.) that embellished and supplemented the cases. Some teacher educators were also using their class sessions as "cases" for discussion and analysis with the participants (See Wineburg, 1991). Others had also begun to ask their practicing teachers and preservice teachers in their courses not only to analyze someone else's case, but also to construct and analyze their own cases about their classroom dilemmas (See J. Shuiman, 1992b). The work of Deborah Ball and Magdalene Lampert (see Lampert & Ball, 1990; Ball, Lampert, & Rosemberg, 1991; and Lampert, Heaton, & Ball, 1992) and their design of a virtual (hypermedia) classroom which served as a context for individual and collective investigations in their education courses was instrumental in the design of the penpal investigation project. These educators had collected data (video, audio, and written) from their Grade 3 and Grade 5 classrooms and had made it accessible to others in a computer (hypermedia) environment to use as contexts for learning about teaching. They considered this virtual classroom as a case-like environment through which one could -50-engage individually or collaboratively in investigations about the teaching and learning of mathematics. In the summer of 19941 attended a week-long institute organized by these teacher educators and had the opportunity of participating in the investigations of these virtual classrooms. It was during this institute that I began to see Math Penpals as a case-like context for learning about teaching and learning mathematics. I also began to see parallels and similarities between Ball's and Lampert's notion of a "pedagogy of investigation" and the teaching methods advocated by the mathematics education community. Through such a pedagogy, preservice teachers would be learning about teaching in a similar fashion to the way they would be expected to teach mathematics to students. That is, to learn by making the subject problematic (Hiebert et al., 1996) rather than through mastery, repetition, and memorization. I also made some important connections and discoveries during this intensive week-long institute which greatly influenced my later design of the math penpal experience. One thing I thought about throughout that week was how much I would like to have access to that kind of data, but from my own classroom, in order to examine, reflect on, and learn more about, my teaching practices. I could see how such an experience (investigating one's own practice) would introduce preservice teachers to an important way of learning and continuing to learn about their teaching and their students' learning. Another important revelation came while examining the vast amount of data available in Ball's and Lampert's hypermedia environment. I found that access to the written records of the students' workings and the teachers' journals was surprisingly engaging and revealing, while the viewing of a videotape of 45 minutes of class was both time consuming and tiring. This served to ease my concerns about the usefulness and appeal of analyzing and working with written data. Using Math Penpals as a Site for Investigation. The previous volunteer penpal experience proved to be very successful in terms of the interest and enthusiasm it brought about in the students and the preservice teachers, and in terms of the mathematical -51-communication that developed through their penpal exchanges. This experience, I noticed, had given my preservice teachers the opportunity to experiment and try out the problems and ideas we had discussed in our classes. I also observed that these interactions with students had encouraged preservice teachers to question some of their assumptions about mathematics and its teaching and learning. I was, therefore, reassured of the potential pedagogical benefits of this experience for my preservice teachers. Writing, I could also see, slowed down the interactive process, thus allowing preservice teachers the time to carefully consider, either by themselves or with others, what to ask or how to respond to students' questions. The written medium also provided an instant record of the interaction which could be analyzed and shared on the spot or revisited later for further reflection. I also realized, however, that one drawback of the activity was that my preservice penpals did not have much support from me as their teacher nor their peers while working with their students since they worked individually and on their own time. For me, this was a missed opportunity as I did not integrate what my preservice teachers were doing with their penpals into our regular classroom activities (since not everyone was participating in the penpal exchange with the students). I realized that opportunities and support for reflection and analysis of their experiences were missing from this penpal activity and that this oversight undoubtedly limited the quality and potential impact of this experience. The literature on case methods and my own experiences investigating Ball's and Lampert's virtual classrooms helped me recognize the importance and the role of such support and to think of ways of including such opportunities within the penpal activity for my future preservice teachers. The penpal investigation project described in this study is the result of much deliberation about how to make penpal exchanges between students and preservice teachers an integral part of the methods courses I teach, rather than an additional activity that preservice teachers do on their own time. The current math penpal investigation project was supported in terms of time, group discussions, and course work allotted during our mathematics methods course. For example, readings, class activities, and -52-assignments were selected and organized to support preservice teachers' penpal investigations. Teacher Research: Researching from the Inside and the Outside In this study I played the dual role of teacher and researcher. I join the increasing number of researchers and teachers who use their classrooms as a site for inquiry into their teaching practices and their students' learning (e.g., Ball, 1990a; Lampert, 1990, 1985; Romagnano, 1994; Wilson, 1995; Wong, 1995). Along with my two collaborating teacher educators I engaged in what Richardson (1994) calls "practical inquiry" while conducting "formal research" on my preservice teachers' learning. This teacher-researcher strategy provided me the opportunity to study preservice teachers' learning from both an insider's and an outsider's perspective. These two complementary perspectives made it possible for me to study and address questions relevant to both research and practice. From the inside, as one of the teachers in the mathematics education methods course, I was able to design and manipulate the learning environment in which to study preservice teachers' learning. I became familiar with the context of my students' learning, not only through observing it, but also through my active participation and interaction in it. My insights and puzzlings as a teacher about my students' learning informed and aided the research process: for example, the decisions I made about sources of data to use, about the questions I chose to investigate, and about the analysis of the data. I, therefore, agree with other researchers who claim that studying learning from a teacher's perspective provides an important lens through which to understand students' learning (Bissex, 1987; Cobb & Steffe, 1983; Cochran-Smith & Lytle, 1993). I also join Richardson (1994) who argues that practitioners' and researchers' practical inquiry into their practice is foundational to formal research which aims to inform and improve practice. -53-In addition to the teacher's perspective I was also able to gain further insider's insight by participating in the math penpal activity along with my preservice teachers. I too corresponded with two Grade 4 student penpals. I exchanged the same number of letters and met my student penpals as many times as the preservice teachers did. I also kept a journal where I wrote my reflections and deliberations about my penpal exchanges which I often shared with preservice teachers during class. This perspective was very valuable to me both as a teacher and as a researcher. It provided me with another source of information from which to make pedagogical decisions and interventions during the course. It also gave me better access to and rapport with preservice teachers participating in the penpal investigation project, as I could also step into the participant's role. As a researcher, this perspective was another resource I had at my disposal for understanding my students' experiences and the pedagogical potential and the pitfalls of the math penpal activity as I had designed it. As a researcher I also brought an outsider's perspective to the practice setting. I was conducting a "teaching experiment" (Cobb & Steffe, 1983) largely designed and informed by the research literature on preservice teachers' learning and the literature on interpretive methods and standards of research. My experiment, however, was also grounded in my practice as a teacher educator. The questions and goals I, as a researcher, had set out to investigate, were also questions I, as a teacher, was grappling with every day of my teaching. My dual perspectives, therefore, complemented one another more often than they conflicted in the practice setting. The outsider's perspective also played a critical role in that it allowed me to distance myself from the particular context of my classroom and to have the time and the access to interpretive tools which were unavailable to me at the time of teaching. It helped me develop what Wilson (1995) calls "peripheral vision" in order to seek alternative explanations and to challenge and test my initial hunches and conclusions. Therefore I, like Lampert (1991), have used the language and tools of educational scholarship to -54-examine, extend, and communicate what I, as a teacher, have come to understand about my students' learning. Playing this dual teacher-researcher role while compatible and advantageous in many respects (as noted above) is not without its challenges. Wong (1995), for example, discussed his difficulties managing both roles in terms of their conflicting purposes. Wong (1995) reported experiencing conflict in his classroom when as a researcher he felt his teaching interventions were interfering with his research of the students' thinking. As a researcher he wanted to understand and get to the bottom of his students' thinking, yet as a teacher he felt responsible and pressured to intervene and help the students overcome their difficulties. Wong (1995) explained it as follows: "I felt a distinct tension between trying to be systematic and thorough and trying to be responsive and compassionate" (p. 25). Wilson (1995), on the other hand, in a rejoinder to Wong's article, argues from her experience as a teacher-researcher of her practice that the two roles, rather than conflict, complement and collaborate with each other. She argues that Wong's conceptions of research and of teaching could be the reason why he experienced such tension between the two roles. She says that Wong's conflict suggests a view that "researchers are interested in what's happening in students' minds; [and that] teachers may have such interests but they are, first and foremost, morally obligated to 'change' those minds" (p. 20). Wilson (1995) disagrees with this "black-and white contrast" and says that "in his portrait of both teaching, and research, Wong focuses on questioning as a tool of research and ignores the fact that it is equally a tool of teaching" (p. 20). She explains: When I decide to do research on my teaching, I don't enter the classroom one part teacher, one part researcher. I'm Suzanne, moved at once to help students learn and intensely curious about teaching and learning. In the room and in relation with my students, I am teaching. I am also collecting information (journals, videotapes, interview transcripts, fieldnotes, students' work) that can be used in subsequent analyses. Al l of this work is driven by the same questions: What might it take to help students learn social studies in meaningful ways? Are my students learning?... As a -55-teacher, I have always cared about these questions. As a teacher-researcher, I care about them still. In fact, I think learning to do research made me a better teacher. (Wilson, 1995, p. 20) I, like Wilson (1995), did not experience the conflict Wong discusses. Even though the focus of both his and my research was on our students' learning (rather than our teaching actions and deliberations), I found that my intentions as a teacher and as a researcher were quite often complementary to each other in the practice setting. This is not to say that I did not struggle in my attempts to figure out what my students were thinking and saying. I found this to be problematic both as a teacher and a researcher. Therefore, understanding what preservice teachers were saying, writing, thinking, and learning during my methods course was my goal both as a teacher and as a researcher. Furthermore, my deliberations as a teacher about my preservice teachers' thinking and learning informed and helped focus the questions I asked and pursued as a researcher. On the other hand, I, like Baumann (1996), found that my two roles sometimes competed with each other in terms of time and task constraints. Baumann (1996) related that he found that "on occasion, the roles and duties associated with teaching conflicted with the plans I had for gathering, analyzing, and reflecting on research data" (p. 31). This was also true in my case. My teaching duties (e.g., preparing class activities and assignments, reading and responding to journals) and participation in the penpal exchange with two students competed heavily with the time I could devote to research tasks (e.g., preparing and conducting interviews, arranging videotaping of our classes, photocopying written records, reflective writing in my research journal). Like Baumann (1996), I found myself putting my teaching commitments ahead of my research agenda. There were, for example, a few unrecorded classes, unwritten teaching journal entries, ill-prepared interviews, and postponed photocopying tasks. Our weekly instructors' meetings also reflected the primacy of teaching over research as our discussions and deliberations generally dealt with the more immediate tasks of teaching rather than with issues related to our research. -56-There were, however, a few occasions in which our research agenda may have played a more predominant role in our decisions and actions. A vivid example that comes to mind is the occasion on which some preservice teachers requested that we give them the option of preparing a unit plan instead of their final "case study" assignment. There were many good pedagogical reasons for us not to comply with this request. On the other hand, altering our pre-specified course design midway through our research study would have had serious repercussions. I, therefore, wonder if we would have been more willing to entertain our preservice teachers' request and concerns about having more flexibility and choice in the kinds of experiences they were engaged in during our course if a research study was not associated with the course. In addition, I also found that my insider's perspective as a teacher sometimes interfered with my later, more systematic, attempts to engage in the analysis and interpretation of the data I had collected in the practice setting. This was true even when my researcher role became more prominent once, and long after, the course was over. While I, as a researcher, had set out to explore what and how my preservice teachers learned through their engagement in a course-related field experience, I, the teacher, became overly critical of my attempts to support their learning. I therefore spent a lot of energy and time reflecting on what I, the teacher, could have or should have done that I did not do at the time I was teaching. While this is an important and very useful activity to engage in, it nonetheless often detracted me from the main focus of my research study. Another source of tension between my two roles arose during the process of re-reading my preservice teachers' work. Some of the comments our preservice teachers made about what they had found to be useful, troublesome, and unhelpful during our course were at times quite difficult for me, the teacher, to read; particularly when the comments were full of anger and blame. These angry comments from my students often blinded and derailed my attempts to analyze the data. Reliving such difficult moments of my teaching, therefore, made the data analysis an uneasy and often unappealing prospect for me, the researcher, to engage in. Only the passing of time made it possible for me to -57-put such "intense" data in perspective and become sufficiently detached from it to be able to analyze it. II. The Research Context The Mathematics Methods Course The Course Setting. This study took place within the context of a mathematics teacher education course called "Curriculum and Instruction in Mathematics," MAED 320, more commonly known as the "elementary mathematics methods course" for preservice teachers. This course is offered every year during the second half of the winter term (January through March) at the University of British Columbia. This is a required course for all preservice elementary teachers (Grades 1-7) enrolled in the two-year teacher education program in the Faculty of Education. Multiple sections of this course are offered during the same time period, therefore, they are not taught by a single instructor. The two-year teacher preparation program offers a combination of university-based courses and school-based practica in the following sequence: (1) foundational courses in education for the first half of the first year; (2) followed by methods courses in all subject areas during the second half of the first year of the program; and (3) a 13-week school practicum during the first term of the second year; (4) which is then followed by more university-based courses focusing on the social context of schooling and a chosen subject-area of specialization. Typically there are no long term school-based experiences scheduled concurrently or connected with the university-based course work. There is, however, a short 2-week practicum scheduled midway through the methods courses which is not associated with any of the methods courses, but rather is a program-wide experience. The Course Design. An innovative version of the elementary mathematics methods course was the site for my research study. This innovative course was a collaborative -58-venture among 4 university instructors, 34 preservice teachers enrolled in the course, 3 teachers and their 30 Grade 4 and 50 Grade 6/7 school students. The course ran from January through March of 1995. However, the planning and designing of the course began in the summer of the previous year when a fellow graduate student and I expressed interest in using the MAED 320 as a site for our respective doctoral research studies. It was at that time, around May 1994, that I teamed up with two collaborative partners, a mathematics education assistant professor (at the time) and a fellow graduate student, to participate in the design, teaching, and research of one section of the MAED 320 course. We had each taught this course at least twice before and collaborated with other instructors; however the innovations we had planned for this course made it a more complex and ambitious undertaking than any of the other courses we had previously taught by ourselves. We also sought the advice of a professor from the Mathematics Department who had recently retired and had expressed interest in our course. Al l of us worked collaboratively in designing a course which would engage preservice teachers in inquiry about mathematics and mathematical pedagogy as outlined in Chapter 2. We selected and organized classroom activities, readings, and assignments with this goal in mind. A major component of our course was the opportunity to interact with school students during the course. These interactions were meant to provide another (though a major) context for inquiry and investigation. This particular context, we felt, was an important addition to the experiences normally offered to preservice teachers during university-based courses. Such a context would not only simulate the practice setting, it would also allow for sustained investigations with the same students. In this way, the ideas explored during regular classes could be explored and connected with classroom practice. In addition, the experiences with students could in turn also serve as the focus for discussion during regular classes. These experiences were also meant to be a source for individual deliberation and reflection which preservice teachers were encouraged and expected to explore and document in their weekly math journals. The records of their interactions -59-with the students and their reflective journal entries would in turn serve as the data from which to construct a more in-depth analysis or case study of their learning experiences which they were to submit at the end of the course. The course was largely structured around these interactive experiences with school students. Two field-related projects were offered to the preservice teachers involving interactive experiences with elementary school students. One project, called "collaborative inquiry," was designed to engage preservice teachers in interactive teaching sessions with a small group of students in a Grade 6/7 class. The other project was called "penpal investigation" and it is the focus of this study. In this latter project preservice teachers engaged in a math letter writing exchange with one or two students in a Grade 4 class throughout the course. They also had two opportunities to visit with their students during this period of time. Assignment to Field-Related Projects. When we designed the course and the two main projects associated with it we did not know, until early December of 1994, the actual number of preservice teachers that were enrolled in our class. From previous years, we knew that these methods courses tend to have an enrollment of 30-40 preservice teachers. We therefore designed the two projects with a particular ideal size in mind. The collaborative inquiry project required an even number of preservice teachers (as they were to work in pairs) and could not handle a large group (partly due to the space available in one school classroom). The penpal project, on the other hand, was able to handle a larger number of preservice teachers (as many preservice teachers as there were students in the penpal class). The logical separation into two equal-sized groups was, therefore, problematic for either high or low enrollment in the course for both of the course's projects. The plan for the penpal group was to have each preservice teacher correspond with at least one student and some with two students if necessary. The possibility of having more preservice teachers than student penpals would present a serious design problem to the penpal -60-project. This, however, was considered as a remote possibility due to the availability of the two projects. When we learned that 34 preservice teachers were enrolled in our course we began to envision the composition of the two groups as follows: 10 preservice teachers in the collaborative inquiry group and 24 preservice teachers in the penpal group. We then began to consider the issue of how to assign the preservice teachers to each group. We deliberated on two options: random assignment and self-assignment. Neither of these gave us the flexibility we needed to deal with the needs and limitations of the two projects. It is important to note that at some point we also considered and quickly abandoned the idea of involving everyone in both projects. The short length of the course and the logistics of the course and the research made that an unappealing option. We decided to do a purposeful assignment to projects by considering several factors. An important consideration was the preservice teachers' interest in and preference for one of the projects and the reasons for their choice. We also took into consideration the grade level each preservice teacher was assigned to teach in their practicum and the background information they provided us related to their prior mathematical experiences (such as number and type of courses taken and their attitudes towards mathematics and teaching mathematics). The last consideration taken into account was whether each project had a sufficient number of preservice teachers who had volunteered to participate in the research of the course. During the first week of class we collected information to help us make the decision about project assignment: preservice teachers' background information, their level of. research participation, and stated preference for either project (see the content of these forms in Appendix A). Even though most preservice teachers opted for the collaborative inquiry project the groups were distributed as follows: 14 were assigned to the collaborative inquiry group (3 male and 11 female students) and 20 to the penpal investigation group (6 male and 14 female students). -61-Of the 20 preservice teachers in the penpal project 11 indicated a preference for this project, 2 indicated they could be assigned to either project, 2 did not indicate their preference, and 5 indicated their preference for the collaborative project. Of the 11 preservice teachers who had selected to work on the penpal investigation project, 8 had also elected to volunteer their work to the research aspect of the project (this is discussed more fully later). Therefore, volunteering to participate in the research study did not become a consideration for assignment to a project. Those who did not select the penpal project as their preferred choice, although initially disappointed, did not contest having been assigned to the penpal project. The Penpal Project: Pairing and Grouping. I originally intended to randomly pair preservice teachers with students regardless of their research participation. During a meeting with my research advisory committee we discussed whether volunteering preservice teachers should be assigned student penpals who were good writers. Although I could see reasons for why it could be useful to research the "best possible scenario," I did not think such purposeful pairing was appropriate or necessary. I did not want to "privilege" those who volunteered to participate in the research aspect of the activity by pairing them with the best writers. I also believed that the students' written work would provide opportunity for reflection and investigation not only for what students did write but also for what they did not write. I did, however, comply with my cooperating penpal teacher's request to pair a few specific students, whom she considered to be extremely capable in mathematics and writing, with preservice penpals participating in our research projects. I did this because this teacher had also made plans to look at the Grade 4 students' data for her own research interests. Another consideration for the pairing of the penpals was that the students outnumbered the preservice teachers. Therefore, I asked preservice teachers in the penpal group to consider corresponding with two penpals in order for every student in the penpal class to be included in the activity. Seven preservice teachers volunteered to work with two student penpals. Interestingly, three of these preservice teachers had originally -62-requested to be in the collaborative group. This gave us an indication of their acceptance and willingness to engage in their assigned project. Those preservice teachers who volunteered to be penpals to two students requested that a boy and a girl be assigned to them. One of them also requested to work with an ESL student. These requests were met in the pairing process. In addition, following the advice of my cooperating penpal teacher, I also made sure that the two students differed in their mathematical and communication abilities. Since the penpal class was composed of more girls than boys (19 girls and 11 boys) I did two draws, first the girls and then the boys. The matching was done by drawing the names of one student and one preservice teacher at the same time. Once all the preservice teachers had one assigned student, those who had requested two penpals were redrawn to match with the remaining boy penpals. To complete the draw I matched myself with two penpals and paired a fellow graduate student (who volunteered) with one student. After the names were drawn and pairs and triads were made, I provided my cooperating penpal teacher with a list of the 14 pairs and 8 triads along with the level of participation of each preservice teacher (see Appendix A) for her consideration and suggestions (8 students were eventually switched). Although preservice teachers only worked with one or two student penpals of their own they also had the opportunity to learn from the experiences of other preservice teachers and their student penpals. One way for preservice teachers to learn from and about other students was through having access to everybody else's letters. The students' and the preservice teachers' letters were all photocopied and kept in a common file for everyone to use as a resource. The class discussions organized throughout the course also provided further opportunities for all the preservice penpals to share and learn from each others' experiences with students. This form of collaborative investigation was also encouraged and promoted by assigning preservice teachers to work in groups for the duration of the penpal activity. - 6 3 -Preservice teachers in the penpal group were organized into 5 groups of 4 preservice teachers so that they could deliberate and work together on their reading and writing of letters. In assigning preservice teachers to their clusters I ensured that those who had volunteered to be videotaped sat together and were within camera range. I also made sure that at each table there were 5 or 6 student letters from a similar mixture of boys and girls. The seating arrangement of the preservice teachers at each group table and the number and gender of the student penpals assigned to them are shown in Figure 1. Notice that only those participating in this study are identified with their pseudonyms. (Susa l 5 penpal letters from 3 girls, 2 boys 14arri)i Resource Table File box with copies of letters 5 penpal letters from 3 girls, 2 boys e^ma)e 6 penpal letters from 4 girls, 2 boys 6 penpal letters from 3 girls, 3 boys 6 student penpals 4 girls, 2 boys 0. Figure 1. Classroom layout and seating arrangement for the penpal group. Schedule and Timelines. The course was scheduled twice a week for 1.5 hours every Wednesday and Friday morning. It ran for 11 weeks from January 4 to March 31, 1995. We conducted regular class activities at the Wednesday meetings. Friday classes were -64-generally reserved for the interactive work with students in both collaborative inquiry and penpal investigation groups. During the Wednesday classes we used various contexts to engage the preservice teachers in class activities and discussions related to mathematics education reform ideas. These discussions were organized around preservice teachers' experiences both as observers and as participants in various mathematical-pedagogical activities. The class problems and activities were structured around the mathematical theme of multiplicative thinking (reasoning about proportions, multiplication and division, percents, and relative size). This was partly our attempt to deal with both the depth and breadth of the mathematics curriculum. It was also meant to serve as a unifying concept for three mathematical topic domains of the British Columbia elementary mathematics curriculum (Ministry of Education, 1987): number sense and operations, size and shape, and data and chance. During the Friday classes, the class was separated into two groups. Those in the collaborative inquiry group met at the nearby school with the participating Grades 6/7 students. The penpal investigation group met in our regular on-campus classroom to receive and respond to their Grade 4 students' math letters. Although these two groups worked on separate projects during the Friday classes they were reunited during the Wednesday classes. On Wednesdays we often structured time for and encouraged preservice teachers to share their experiences, insights, and puzzlings from their work with students. The following table provides a more detailed overview of the course's weekly topics and activities. Further details on the course's interactive projects and assignments can be found in the course outline reprinted in Appendix B. -65-Table 1. Class schedule of weekly topics and activities Week Wednesday Friday 1 Jan 4,6 Introduction to the course and the research study. Preservice teachers work on three mathematical investigations. They rotate through three stations: (1) exploring body ratios, (2) comparing volume of cubes and pyramids, and (3) investigating mathematical probabilities. Introduction to idea of investigating students' thinking. Preservice teachers observed instructors work with four Grade 5 students. First, interviewing students on their views about mathematics and conceptions of fractions. Then, facilitating discussion on the "Three Hungry Monsters" Problem. 2 Jan 11, 13 Focusing on mathematical discourse. Preservice teachers solve and discuss the "Horse Problem." Whole class discussion about the experience. Also discussion about the role of problems in promoting discourse. A collection of problems are provided for preservice teachers to solve and consider posing to their students. Working with Students. Penpal Group receives the 1 st letter from students. They respond to students' i problems and pose their selected mathematical problems. 3 Jan 18, 20 Making sense of students' thinking. Whole class discussions around the Constance Kamii video clip (students' invented algorithms) and students' written workings on fractions. Working with Students. Receiving and responding to the students 2nd letter. Discussion about issues of communicating and writing in i mathematics. 4 Jan 25, 27 Investigating Area and Perimeter. Preservice teachers work in groups on the "Squares are In" set of six area and perimeter problems. Working with Students. Responding to the students' 3rd letter. Working with table partners to select and consider adaptating one of the "Squares are In" problems to pose as a common problem (group problem) to the students. 5 Feb 1,3 Making mathematical connections with other subjects. "Fish bowl" activity with half of the class acting as students and the other half as observers. Worked on two problems: "Foot Measuring" and "Packaging" (perimeter-area-volume) in the contexts of language arts and social studies. Working with Students. Responding to students' 4th letter. Working with table partners to send a group problem to students based on the "Foot Measuring" and "Packaging" problems. Whole class discussion about adaptations to problems and students' responses to common problems including the "Squares are In" problems. -66-6 Feb 8 Mid-term Exam: Case Study. Also, an advanced Organizer is provided for three mathematical investigations to be covered after they return from their practicum. Spring Break and Teacher Education 2-week practicum break. 7 March 1,3 Explorations into number theory and mathematical paradoxes. Solving and discussing the "Kanga and Pooh Bear" problem. Each instructor works with a group of 10-12 preservice teachers. Penpal Group receives students' 5th letter. Working with Students. Penpal visit. Preservice teachers meet with their penpals at their school. They conduct interviews and work on selected i math activities. 8 March 8, 10 Preparing for Math Fair. Five different investigations are planned. Preservice teachers work with one instructor (two other instructors are recruited to assist) to prepare their investigations on one of the following: fractals, topology, tessellation, logo, and mathmagic. Working with Students. Math Fair. Preservice teachers try out chosen mathematical investigation with | three different groups of students. Groups are composed of Grade 4 penpal 1 students and Grade 6/7 collaborative inquiry students. Preservice teachers hand in their response to students' 5th letter. 9 March 15, 17 Designing Good Problems. Using everyday life and popular culture as sources for mathematical problems. Investigations and discussions involved: (1) shrinking and enlarging pictures, (2) Comparing sizes and pricing of food and beverage containers, and (3) collecting and analyzing data with smarties (multi-coloured candy coated chocolate bits). Working with Students. Receiving and responding to students 6th and final letter. Each table constructs a problem for the whole class j based on the smarties investigation. 10 March 22,24 Teaching for understanding: Decimal operations. Discussion about various representations, tools, and contexts for exploring x and + of decimals. Project Consultation. Consultation time to help prepare final assignment of a case study about their learning experiences through their interactive projects. 11 March 29, 31 Teaching for understanding: Place value and powers of ten. Revisiting ideas discussed earlier about engaging students in meaningful mathematical explorations. Project Consultation. Consulting time for the preparation of final assignment. -67-To close this section it is important to highlight and expand on a few items from the above table. The penpal project began with the student penpals writing the first letter. These were received by the preservice teacher penpals during our second week of class. This exchange continued every week until our ninth week of class. Six letters were sent and received by both parties. The student penpals included one directed writing about their opinions about the use of calculators and computers in the mathematics classroom. Preservice teachers included three directed group problems which were also common to the collaborative inquiry group. The penpal group also had the opportunity to meet with their students on two occasions. First, a penpal meeting was hosted by the fourth graders at their school. The second visit was hosted by the preservice teachers. Students from both projects visited our university class and worked in pre-arranged groups (4 or 5) on a set of mathematics activities in what we called a "Math Fair" or stations format facilitated by two or three preservice teachers. These letters and meetings provided preservice teachers with opportunities for trying out and exploring questions and ideas related to mathematics, and the teaching and learning of mathematics, which were interesting and puzzling to them. They provided opportunities for gathering data on a problem and issue of interest through two different mediums, that is, through written and oral interactions. Preservice teachers kept a file with copies of the letters they sent and the ones they received. Videotapes, photographs, and/or audiotapes were used to make records of the meetings. These were meant to aid preservice teachers' later reflections in their journals. The Main Assignments: Math Letters, Math Journal, and Case Report. There were a total of 11 weeks and a total of 19 sessions in the course. Seven of these sessions were spent on the interactive experiences with school students. Three activities were associated with these interactions: (1) the interactions with students mainly in the form of math letters with two personal (face-to-face) interactions, (2) the weekly math journal, and (3) the final case report about the experience. Only the latter two were part of preservice teachers evaluation throughout the course. These two combined constituted -68-75% of the evaluation. The remaining 25% were assigned to (1) a math autobiography essay, and (2) a mid-term (case analysis) exam (See course outline in Appendix A for the assignments list and the distribution of marks). Preservice teachers wrote and received six math letters. They also sent and received work on three "group problems" during weeks 4, 5, and 9 which were associated with letters 3, 4, and 6. Preservice teachers worked on their penpal letters every Friday with the exception being letter #5 which they did on their own time. On a few occasions some preservice teachers finished their letters at home and brought them back to a "drop box" during the weekend. The school students usually worked on reading and writing their letters for 2 class periods. On occasion some of the students were allowed to finish their letters at home and bring them back the next day. Preservice teachers wrote weekly journal entries about their experiences in the course and with the students. They handed in their journals every Monday and received them back on the following Wednesday. There were a total of 9 journal entries. There were no journal entries for weeks 6 and 11. Al l three instructors shared the responsibility for reading and responding to 11 or 12 journals each week, except for the one time preservice teachers responded to each other's entries for week 9. We provided preservice teachers with a series of journal prompts throughout the course (See Appendices C and D). These prompts were meant to help preservice teachers explore and communicate their thinking about mathematics, students' learning, and teaching. At the completion of the first two journal entries we noticed that most preservice teachers were having difficulties reflecting on, or communicating their thinking about, their interactions with the students. Their writing seemed lopsided towards description or interpretation. Many tended to describe rather than analyze their "actions" and their students' actions. Furthermore, their descriptions were often their recollections of the interaction rather than actual excerpts from the letters. At the other extreme, those who were providing analysis of their interactive experiences with the students often neglected to provide references to the actual data they were analyzing. On week 4 (Friday, January -69-27) we introduced a "descriptive/interpretive" tool to help make a clearer distinction between the actual interaction with the students and their interpretations and reflections about the interactions. This was done through an example of my penpal interactions with students from the same class (See Appendix D). The final assignment preservice teachers worked on was a "case study," that is a paper analyzing and synthesizing their learning experiences through the penpal activity. Although the above summary table of the course's syllabus gives the impression that we only spent the last two Friday classes advising the preservice teachers on their final project paper, the reality is that we were providing them with resources, suggestions and advice throughout the course. The information about this assignment was initially provided in the course outline, and more information was provided during class time throughout the course. We allotted a few minutes at the beginning of several classes for discussing and exploring ideas for the investigation projects and the final report. We also held consultation sessions outside of class time and made ourselves available during office hours. We also provided advice through our responses to journal entries dealing with questions and ideas for the major paper. We also provided more documentation (included in Appendix D) to help them organize, analyze, and write this final paper. In addition, several references were provided as illustrations and examples of teachers' narratives of their classroom investigations and experiences. The Participants The Instructors. As I mentioned earlier I collaborated with two other teacher educators in the design and teaching of the methods course. We were: (a) an assistant professor in mathematics education with five years experience teaching elementary mathematics methods courses; (b) a doctoral student with seven years of teaching experience in the school system and two years of experience teaching preservice elementary teachers, and (c) myself, a doctoral student with two years of school teaching experience and two years of experience teaching elementary mathematics methods courses. Our collaborating associate professor was the principal instructor of the course -70-with us, the doctoral students, acting as assistant instructors for the course. We shared all teaching responsibilities except for marking the preservice teachers' work. This latter task rested mainly with our collaborating professor though we all provided our input into this process. Since the main focus of this study is not on any of the instructors they are referred to as a single entity with the pronoun "we" whenever they are referenced in the analysis chapters. This does not include our collaborating mathematics professor since he acted more in the role of observer during our classes and as a reflective partner outside of class. The Penpal Teacher. This teacher had 20 years of teaching experience and was a part-time graduate student in the Master's program at our University. I had been a regular visitor to her mathematics classes since September, 1992. In January of 1994 we had engaged her students and my preservice teacher students in a mathematics letter exchange at her request. The following year she agreed to my request to organize the activity once again for this research study. Her participation in this research study was mainly as a collaborator and advisor to the penpal activity and not as a subject of research. This teacher and I collaboratively designed and coordinated the penpal exchanges to ensure that both her students and my preservice teachers benefitted from the activity. We made joint decisions regarding number of possible letter exchanges, the receive-and-reply rate, possible topics for directed writing, dates for classroom visits, and so on. The Penpal Students. The students in the penpal class were 9 and 10 years of age and were enrolled in Grade 4. Their school was located in an affluent middle-class neighbourhood in the Vancouver School District. Most of the students were Caucasian and fluent (for their school age) in English. Only two students were of Chinese descent and did not speak English at home. The students and their parents were informed of the penpal activity and the research associated with it at the beginning of the school year by their teacher. The parents of all 30 students gave their permission to involve their child in the penpal activity and the research associated with it (See letter to parents enclosed in Appendix A). However, only those students paired with a volunteering preservice teacher -71-are included in this study. Eighteen students participated: 11 girls and 7 boys. The pseudonyms that the preservice teacher penpals assigned to their students are used to ensure their anonymity. The Preservice Teachers. The preservice teachers in our course, as is the case for every preservice teacher admitted to the two-year teacher preparation program, had completed at least 3 years (90 credits) of course work through the Faculties of Arts and Sciences. They had completed at least one university-level mathematics course offered through the Mathematics Department in order to be accepted in the program and in a mathematics methods course. All 34 preservice teachers in our course were informed about, and invited to participate in, the research aspect of the course. Of the 20 preservice teachers who were assigned to the penpal investigation project, 17 volunteered to participate in the research at several levels. These levels included volunteering (a) to provide their written work, (b) to be interviewed, (c) to be video taped during classes, and (d) to be observed during their practicum. The level of participation chosen by each preservice teacher is reported in Appendix A. Of the 17 preservice penpal teachers volunteering for research, only those who volunteered their written work were selected as participants in this study—a total of 13 preservice teachers. These participating preservice teachers include one male and 12 females. They were given pseudonyms to protect their anonymity. III. The Data Analysis Sources of Data Data gathered during the course included 10 videotapes of regular classes (15 hours), 5 videotapes and 5 audiotapes of all penpal writing classes (15 hours), two sets of audiotaped interviews with those who volunteered (9) to be interviewed (18 hours), and all the written work assigned throughout the course. I also kept a journal of my deliberations and observations related to the course, the penpal project, and the research -72-project. In addition, 6 of 9 instructors' meetings were audiotaped (8 hours). All these data sources were available to me for my research study. After reviewing a large portion of the data I made the decision to use only the written records of preservice teachers (penpal letters, journals, case reports). Other data sources, such as excerpts of a class video or my journal, are occasionally used to contextualize or illuminate the analysis of the preservice teachers' written records. Written Records: Penpal Letters, Math Journals, and Case Reports. The main sources of data for the study were: (a) the math penpal letters written by the participating students and preservice teachers, (b) the weekly math journals of the 13 participating preservice teachers, and (c) the final case study of their project investigation submitted at the end of the course. These three components were the main activities associated with preservice teachers' interactive experiences with students in the penpal project. Even though all three of these are in the written form, they each offer different views of the preservice teachers' learning experiences through their interactions with the students. The letters provide records of the actual interactions with the students; the journals capture preservice teachers' plans, reflections, and analysis of their most immediate interaction with students; and the case reports provide reflective data not on any one particular penpal interaction but rather on the whole experience. The decision to focus only on the written records associated with the penpal project was partly motivated by practical reasons. It provided a solution to the problem of narrowing the scope and volume of the data that I needed to single-handedly review, organize, and analyze. My decision was, however, also motivated by my interest in exploring preservice teachers' learning. Preservice teachers' written interactions with students and their journal writing were the prime sources of information for illuminating what and how preservice teachers had learned during our mathematics methods course. These "action" and "reflection-on-action" data provided a wide window into preservice teachers' learning and an opportunity for triangulation. Another reason for using written records as the main source of data for this research study was the fact that I had worked -73-closely with this data while teaching the course. As a teacher, I was very interested in exploring ways of analyzing data that is so readily available in the practice setting in order to inform my future attempts at interpreting and supporting preservice teachers' learning. Looking for Evidence of Learning Learning, as Lampert (1990) says, is an ambiguous term. "It is both the activity of acquiring knowledge and the knowledge that is acquired" (p.59). In this study, learning refers to preservice teachers' evolving understandings about mathematics and its pedagogy constructed through their engagement in an interactive penpal experience. Although learning is considered structurally (Hiebert et al., 1996)—as the residual insights, strategies, and dispositions of active engagement and reflection on activity—the contextual factors that influenced this learning are also examined. In this study I focus on what preservice teachers learned and how they came to learn it. I look for evidence of learning in their written interactions with students as well as in their written reflections on their experiences. Inferring learning, however is much more complicated than this last statement suggests. Inferring learning is not an unproblematic undertaking. How can one make claims about what others learn? It seems like an impossible task, yet we all manage to do it as teachers and researchers. Although there are no sure or agreed-upon guidelines, there are implicit and explicit models and theories of learning that inform the process of interpreting what others learn. One way in which one may infer students' learning is by relying on what students tell us they have learned. After all, no one has more intimate and better access to what and how they know than the students themselves. There are, however, many reasons why relying solely on the students' account of their learning can be problematic. In fact, young students are rarely asked what they think they have learned by educational researchers. -74-One problem in relying solely on students' accounts of their learning is that they tend to have limited experience thinking about, monitoring, and expressing what they are learning. They also have limited opportunities to develop the concepts and tools to talk intelligibly and comprehensively about their learning. Another problem is that the power relations between the teacher or researcher and the students might lead students to say and do what they think the teacher or researcher expects to see and hear (see Davis & Pitkethly, 1990; Hatch, 1990). Therefore, while students' accounts of their learning can provide useful insights into their learning, it is a problematic source of evidence to use on its own. Another way of inferring learning is by contrasting students' performance and discourse before, during, and after engaging in an activity. Changes in the performance and discourse of students are taken as evidence of learning. The best possible scenario is when performance and discourse coincide. However, there are many examples which indicate this is often not the case. Frequently, students who are able to perform a particular task are not able to explain the conceptual underpinnings of their actions. Conversely, students who may be able to talk eloquently about their understanding may not choose, or be able, to use it in particular contexts. Extended observations of students' performance and actions in different contexts and on multiple occasions can help make inferences of students' learning more reliable. In this study, I sought evidence of learning in various ways and through different sources. I looked for evidence of learning in preservice teachers' teaching actions (written interaction with students) and in their reflections on their actions (written discourse in journals and case study). Neither of these was a primary source, rather they are used to complement each other. I looked for evidence of learning in preservice teachers' analysis and accounts of their learning in their journals and their case studies. I also looked for changes in their reflective discourse (in their journals) and in their interaction with students (penpal letters) by contrasting earlier and later work. I also looked for similarities and differences within and between the preservice teachers' data in order to -75-make decisions about what may or may not be included in my final analysis and report of preservice teachers' learning. Analytic Strategy Deciding What to Look For. I began to analyze preservice teachers' data as one of the teachers of the mathematics methods course. I read many of the penpal letters and read and responded to many journal entries throughout the course. During the course, I observed changes in the penpal interactions of our preservice teachers and their students as well as noticed some developments in their reflective journals. In the letters, for example, I had noticed that some preservice teachers' interactive discourse with students had changed from a "question-response-new question" format to more elaborate and longer communications where both students and preservice teachers seemed to be exploring mathematical ideas. In the journals, I had noticed that many of the preservice teachers' reflections had become more elaborate, explicit, and investigative. I made many such observations in my journal. My insights and reflections at the time of teaching, therefore, became very useful and informative to the later analysis of data. An excerpt from my journal serves to illustrate this point. Something I have been intrigued by for quite some time is the "collaboration," or lack thereof, in the penpal group. The structure doesn't seem to allow for much whole group sharing but more table sharing. The times when the discussion was open to everyone, most people were quiet. ... I've been thinking lately that perhaps there's some sort of guilt involved in their lack of sharing. Sometimes I've gotten the sense that they feel it is their fault—that they're doing something wrong—that their penpal hasn't responded the way they "should have." I'm also intrigued by their expectations about the students' written work, and what they wanted as "evidence" of the students' understandings. I'm also intrigued about their expectations of the children's written responses compared to their own responses to the students. Their expectations seem to apply one way in that their responses were perhaps even briefer that those of their students. [Sandra's journal, March 24, 1995] -76-My own experience as a penpal also proved useful in helping me decide what to look for in the data. My penpal exchanges had raised various pedagogical challenges which then turned into further reflection and investigation. I became aware of how certain penpal interactions stood out from others and became recurring episodes and sources of deliberation. Penpal exchanges which had some surprising and unexpected results were particularly salient. But situations in which I became dissatisfied with my actions also provided much fuel for reflection and inquiry. I noticed that my students' mathematical work had helped me raise questions about my purposes and goals and about how I had responded to their mathematical queries at the time. An excerpt from my penpal investigation will serve to illustrate. A major goal I had set for working with my two penpals had to do with providing them with a safe environment for sharing unfinished work and tentative thinking, a place where we could provide honest feedback, and where we could push each other's thinking and understanding further. After a couple of penpal exchanges with "Tommy" regarding a particular problem—"the broken key"7—I became concerned with the affective consequences of continuing to pursue this particular problem with this student. Knowing how fragile students are, in terms of their attitudes and beliefs about mathematics, I was unsure about how much more I could push this particular student to continue thinking about a single problem without losing his willingness to engage in future problems and to share his tentative thinking. Tommy's first reply to the broken key problem read: Dear Sandra: Thank you for your letter. The problem that you give [gave] me is too hard but I can do it on paper: And I like too [to] know in that problem do I have to use a calculator? or I can do it any ways I want? [Tommy, ml28] 7.5 is the answer 6 ]~45 _42 30 30 The + key in the calculator is broken. Find a way of calculating 45-5-6 with the calculator. This coding (ml followed by number 1-6) is used throughout the thesis to refer to the math letters of student penpals and preservice teacher penpals. Other coding references include: M J followed by 1-9 to reference preservice teachers' journals; and CR followed by a page number to indicate a quote from a preservice teacher's case report. -77-By rephrasing and elaborating on the original question, and by suggesting a simpler problem, I managed to keep Tommy's interest. He continued to ask me questions about the problem and shared his attempts to solve it over two letter exchanges. Then in his fourth letter Tommy wrote: The calculator problem I still don't get it, so I asked my dad. He said I can use the x button to divide and I can't understand all about it but I can understand a little bit. I need to multiply 45 and 0.5 three times if we do 45-^ 6. [Tommy, ml 4] Now that Tommy's father was also involved in trying to solve this problem, my goals of honesty and of pushing my students' mathematical thinking were put to the test. I couldn't let Tommy accept a mathematical solution at face value. Moreover, the suggested solution was flawed. But, could I risk damaging the child-parent mathematical relationship by pointing out this mistake to Tommy? Faced with this dilemma, I struggled to accommodate both concerns in my reply. After much deliberation I settled for the following response: I tried your dad's suggestion on the calculator—to multiply 45 and 0.5 three times—and I get 5.625 instead of 7.5 which we know is the answer. Maybe you can check that yourself as well to see if I made a mistake. My friend Jay sent that problem to Tara and I think Beth also worked on this question. Maybe you could consult with them to see what they suggest. I would like to hear what you learn from them. Maybe it would help you to think of 45-^ 6 as, "How many sixes are there in 45"? For example, 25-H5, how many 5's are there in 25? I would start counting 5+5+5+5+5, there are 5 fives in 25, which in a way means 5x5=25. If you can get 45 counters and count how many groups of 6 you can make, I think it would be easier to visualize it. Want to try it? Would you like to try 24^ -3 first? Does this help you to see how you may do it in the calculator? [Sandra to Tommy, ml 4] I was not completely satisfied with my response but it was the best response I could come up with at the time. I could have told Tommy that his method was not a bad one. I could have explained what his dad was trying to do—use the invert and multiply rule (45x1/6)—and where it had gone awry (1/6^1/2x1/2x1/2). However, I thought that Tommy did not -78-understand much of the procedure anyway and the concepts involved in this method were beyond a fourth grade curriculum. I decided instead to show that the method would yield an incorrect answer. This, unfortunately, emphasizes the answer and not the method of solution. I still think that under the circumstances, I made a good decision. I am dissatisfied however that I did not place more emphasis on the solution method which was indeed the point of this problem. Later I realized that 25+5 was not exactly the best example to pick (too many 5's signifying different things). I also wondered about how my penpal would connect the idea of counting groups of 6 with counters to subtracting groups when dividing and using this to solve the defective calculator problem. In his next letter Tommy's reply read: "Now I get the calculator question. I will try 24+3 first, the answer is 8! Now I want to try 42+6 the answer is 7!" I, however, cannot help but wonder what it was that Tommy now understood about the defective calculator question. [Sandra's case study, May 1995] This incident represents one of the major quandaries I faced during my penpal investigation. It stands out as an important part of my investigation for the many questions and issues this episode raised and the many insights I gained about myself, my students, and the tasks I chose to pose. In a similar fashion, preservice teachers faced many "pedagogical puzzles" of their own. Those which they chose to reflect and act upon were, therefore, important to identify as they would reveal a great deal about what preservice teachers had focused and reflected on during their penpal investigations. Identifying such puzzles also made sense from a theoretical view point. According to theories of cognition, learning is said to occur as a result of cognitive conflict, surprise, and contradiction. Cognitive conflict leads to problematization which in turn leads people to reorganize, reconsider, or change their prior ways of thinking about a phenomena (see Hiebert et al., 1996). This is also similar to the way teachers are said to construct knowledge in the context of their practice (see Lampert, 1985; Schon, 1983). With this in mind, I began the analysis of the data by looking for incidents which indicated (either explicitly or implicitly) puzzlement, tensions, and difficulties raised by the interactions with students. -79-Organizing the Data. To aid in the process of identifying "pedagogical puzzles" in the preservice teachers' data, I began to summarize the content of the penpal letters and journals, for each of the 13 preservice teachers, by noting the highlights of their weekly journal entries and letters. These were arranged in a series of columns (one per week) fitted onto two pages (examples are included in Appendix E) in order to have visual access to each of the preservice teachers' data all at once. Looking back at this process, I identify with Wolcott (1990) who says that the irony of conducting qualitative research is that one spends a great deal of energy trying to get data in the field only to spend an equally or greater amount of energy trying to get rid of much of it. My study is a good example of this phenomenon. As part of this process of "getting rid of data" I constructed an in-depth case study of one preservice teacher, Linda, highlighting the pedagogical puzzles she had faced and the sorts of things she was reflecting upon and investigating. I constructed similar case studies for all thirteen preservice teachers, though not in as much detail. In the process, I began to identify commonalities among the pedagogical puzzles these preservice teachers had been wrestling with in their journals and penpal interactions. I could see that these puzzles could be grouped into three broad and interrelated categories which, as it turned out, mirrored their interactions with the students. Preservice teachers' puzzles and subsequent investigations related to issues regarding their: (1) posing of mathematical problems to students, (2) interpretations of the students' mathematical thinking, and (3) responses to the students' work. Analytical Framework. Preservice teachers' pedagogical puzzles which centered on their interactive practices with students (posing, interpreting, responding) became the broad analytical frame for examining everyone's data. This framework was not an "a priori" frame since it was derived from the data itself. It is, nevertheless, my construction of how the preservice teachers' data is interconnected. Different from other frameworks used in the literature to analyze preservice teachers' understandings (e.g., Shulman's model of teachers' knowledge, or the NCTM's categories of knowledge needed for teaching) this posing, interpreting, and responding framework does not separate mathematics from its pedagogy. This framework, however, is simply an analytical and writing tool. It has helped me look through the data with three different but related lenses and attend to specific features in the data. It is important, however, to recognize that as with any model or framework, it can help focus and magnify certain features while it also serves to hide and obscure others. Martin Simon (1994) explained it as follows: The creation of a framework is not an attempt to represent "reality." Indeed we have no direct access to reality. Rather it is an attempt to create a model which is useful and generative in thinking about the phenomena under study. As such, the framework is necessarily simplified (i.e., omits some of the perceived complexity of the situation that it describes) and emphasizes only some of the aspects of knowledge about that situation, (p. 73) Writing Strategy I did not start this study with the intent of including everyone who volunteered to be a participant. Originally I thought I would be constructing case studies of 3 or 4 preservice teachers' learning experiences as is often the analytic strategy for studies of preservice teachers' experiences in their teacher preparation programs. Instead, I have ended up including all 13 and constructing case studies following the three themes of the analytical framework—posing mathematical problems, interpreting students' thinking, and responding to students' work. Therefore I inverted (extended) the analytic strategy. I constructed case studies of the learning themes (posing-interpreting-responding), instead of individual preservice teachers, in order to explore in more depth each of the learning themes. I settled on this framework after having extensively explored the data for all 13 preservice teachers. I had constructed rough case studies for all of the preservice teachers and had begun to see these three areas as common patterns in their reflections and investigations. Learning about all of the preservice teachers in order to make an informed selection led to my getting to know all of them very well. Each of them had something -81-unique to contribute yet they had much in common. I decided then that including everyone who chose to participate in the study would provide a richer and greater pool of examples and experiences from which to draw. This was particularly appealing to me because, as I mentioned earlier, there is little documentation on preservice teachers' learning in the context of their mathematics methods courses. However, I underestimated the challenge that such a large data set would present to writing a coherent and concise analysis that would represent what everyone was thinking, doing, and learning. I do not claim to have achieved this. In fact I do not know whether this is possible within the confines of paper and print. I can only assure the reader that I took great care to include everyone in the analysis of the data though they may not be represented equally either quantitatively or qualitatively in the analytical narrative I have provided. The next three chapters, labeled analysis chapters, are case studies for each of the three elements in the analytical framework. Writing these chapters was not easy. I found no models in the literature to guide me in writing about a large group of preservice teachers who were reflecting on and investigating quite different things. Furthermore, having 13 participants made the process of selecting quotes and examples more challenging. To reference everyone all the time is an impossible task. However, I paid careful attention to the examples I picked and the reasons for selecting them. I tried not to leave anyone out and to select quotes which represented the group or extended the narrative in some way. In terms of the structure of the narrative, I followed Wolcott's (1990) advice, "In the absence of a more compelling alternative, relate the story according to either of two chronologies readily at hand: events as they occured, or events as you learned about and recorded them" (p. 28). I wrote the analysis chapters following the chronology of the course (in the background) which helped to highlight the development of preservice teachers' learning (in the foreground) with contrasts between what preservice teachers said and did initially and what they did and said differently later on. The subheadings for each section are an organizational tool rather than a claim about the sequence of -82-preservice teachers' learning. This, for me, was a necessary choice in order to write about and sift through what preservice teachers were learning. Although this may suggest that preservice teachers came to similar understandings at the same time, I have tried to provide a sense of the messiness and individualistic way in which preservice teachers learn. The analysis chapters are reconstructed narratives of preservice teachers' learning. They are written from the teacher-researcher's perspective. However, they rely considerably on preservice teachers' written words. Writing the analysis chapter in a narrative, story-like, format serves multiple purposes. On the one hand it has allowed me to interconnect the multiple sources of data which otherwise would remain separate. This, in turn, facilitated the analysis of the data through the writing process. Furthermore, it also serves to invite the reader to engage in the analysis of the data, which in turn provides opportunities for dialogue and scrutiny of the account as I have told it. -83-C h a p t e r 4 Analysis of Preservice Teachers' Learning Part I: Posing Mathematical Problems The selection and construction of worthwhile mathematical tasks is highlighted in the NCTM Professional Standards (1991) as one of the most important pedagogical decisions a teacher needs to make. The mathematical activities students are regularly engaged in are known to affect their perceptions and learning of mathematics. The reform vision for school mathematics, therefore, has called for teachers to choose tasks "that are likely to promote the development of students' understandings of concepts and procedures in a way that also fosters their ability to solve problems and to reason and communicate mathematically" (p. 25). Teachers' decisions must not be made lightly, the Standards say, but rather they must take into consideration the mathematical content, the students' interests and prior experiences, and the ways in which students learn mathematics. Helping preservice teachers to begin to ask such worthwhile mathematical questions, however, is bound to be a challenge for teacher educators. Learning to pose worthwhile questions is likely not an easy matter for novice teachers. Throughout their schooling, most students grow used to solving and answering the teacher's or the textbook's questions. Therefore, when the tables are then turned and preservice teachers are in a position to construct mathematical problems for their own pupils, it is not surprising that their questions resemble those they have been asked before in their school days. Breaking away from traditional patterns and forms of questioning is one of the challenges preservice teachers face when learning to teach mathematics for understanding. Math penpal letters afforded an opportunity for preservice teachers to select and pose mathematical problems to their students. Most of the mathematics problems preservice teachers used in their letters were problems we had distributed and, or, had worked out during class. The problems which were most commonly used (though they were not -84-necessarily posed at the same time) provided a starting place for examining preservice teachers' reasons and goals for posing and adapting them in their letters. The few original problems preservice teachers constructed on their own, that is problems that were different from those shared during class, are also closely examined. These, as well as the adapted versions, reveal a great deal about the knowledge and ideas preservice teachers brought with them to our methods course and sometimes how these interacted with the new ideas presented in our class. I. Selecting Tasks "Remember that we need to select tasks that will help us learn about students' mathematical thinking and that would entice students to share their thinking with us," I remember saying to our preservice teachers as we were handing them their students' first letters. This was only our fourth class, but our three previous sessions had been filled with activities designed to encourage our preservice teachers to question some of their preconceived ideas and to suggest a different way of teaching and learning mathematics. Today, it was their turn to experiment and try out their developing ideas. They were to come prepared with tentative problems to pose to their students. To that end, we had spent a portion of our last class examining various math problems and discussing their "worthwhileness." And so, we all began the penpal activity with great anticipation and excitement. Choosing Problems Students Can Solve Easily The problems preservice teachers initially chose for their student penpals were problems which were (or seemed) straightforward to them. Six of the 13 preservice teachers in this study chose to send "the horse problem," a problem we had discussed during our third class. Although this is not a straightforward problem, it does look like a simple addition-subtraction problem at first glance. Some of these preservice teachers also attempted to make this problem easier for their students to solve through the -85-adaptations they made to it (see pp. 103-104). Other preservice teachers openly admitted to picking the problem which seemed easiest while others chose instead to construct their own math problem. These "original" questions were mostly exercise drills such as multiplication tables (Mitch), and routine translation word problems (Rosa), or problems similar to those their Grade 4 students had sent to them (Lesley). Some, for example, asked their students to figure out their age given their birthdate and year of birth. Others asked students to solve number operations exercises dealing with subtraction, multiplication, and division of whole numbers (and occasionally fractions). These were single answer arithmetical types of problems, such as: Pavel Bure scored eight goals in four games. In the first game he scored 1 goal. In the second game he scored 3 goals. In the third game he scored 1 goal. How many goals did Pavel score in the 4th game? [Rosa, m i l 9 ] Preservice teachers' early pattern of problem selection suggests they valued problems which were relatively uncomplicated and easy to solve. This is also apparent in some of the preservice teachers' journal comments indicating surprise and distress at their students' unexpected incorrect answers to problems which they thought students "should not have any problem solving." Marcia, for instance, said, "I was surprised to see that John didn't get the answer to my math problem. I think it was deceptively easy" [ M J 3 1 0 ] . Linda, in turn, indicated a sense of guilt for posing a problem her student was not able to solve correctly. She wrote, "I was confident enough to try her on slightly harder problems in the next letter [since she had done previous problems right], but given what she then answered, I see that I was wrong" [MJ4]. Susan's comments (see below) also indicate disappointment at her student reaching an incorrect answer to her carefully chosen problem. 9 m i l stands for math letter #1 (of 6) written by penpal. 1 0 M J is the code used to reference preservice teachers' journals. MJ3 means that it is an excerpt from the third (of 9) journal entry. -86-She tried so hard to get the answer but she just could not do it. As I read over her answer, I realized how difficult it was for her to answer the problem because of one reason only! She could not see that a square is a rectangle. ... If she had known that a square is a rectangle, she would have got the right answer. This brings me to a very important point. It is extremely important that students have grasped the appropriate concepts before providing them with problems to solve about that concept. One simple misunderstanding can throw students in the wrong direction. My pen pal went crazy with trying to figure out the problem, she had solutions everywhere, however, had she known that a square is a rectangle, she would have quite easily come up with the right answer. [Susan, MJ4] Susan's comment also speaks of the expectations and views preservice teachers might hold about the role of problems and of posing them to students. For Susan, problems are not meant to present too much difficulty to students and should only deal with concepts and content which are familiar to them. This provides an explanation for why preservice teachers were selecting problems which could be easily solved by their students. Another explanation is provided by Miriam who, for example, acknowledged, "Most of the time, I try to give Beth questions that I know are not too difficult." Encouraging and promoting her student's success was an important goal for Miriam. She said, "I do not believe in giving questions that the students will not have success with." She questioned, "What is the point of that if the questions are too difficult and they are not going to figure it out anyways." "My purpose is for students to enjoy math," [CR: p. 511] said Miriam to justify her thinking. Marcia also noticed the importance she had placed on her student solving her problems correctly when later examining her letter exchanges. She wrote: It is interesting and insightful to reread the questions I asked and to look for the motivations behind asking them, because my motives were not always clear to me at the immediate point in time. Upon reflection, I can see that at times, my questions were open-ended but that at other times, my questions were not only closed, but that I expected a certain answer from John. [Marcia, CR: p. 3] C R refers to preservice teachers' case report and it is followed by the page number in the document. -87-Preservice teachers were not always clear about their reasons for their selection of math problems, as Marcia indicates, even though in our journal prompts we had suggested that they reflect and write about this. Among such journal prompts, the following questions were suggested: "How did you decide on what questions/tasks to send?"; "Can they be considered 'worthwhile tasks' as defined in Ball's article?"; "How do you expect your penpal to respond to your math activities?" Even when specifically asked about their task selection, most preservice teachers did not seem able to justify (or articulate the reasons for) their choices. It is, however, possible to infer some of the reasons our preservice teachers had for selecting such unproblematic problems for their students. So far, preservice teachers' initial selection of problems (first three letters) and their expectations for these problems (revealed in their journal reflections) have given us clues into their initial views that good problems were meant to be quickly and easily solved by students. Consider Mitch who, for example, sent in his first letter some multiplication-table exercises: l x l , 2x2, ... 12x12. "A good test," Mitch said in his journal, "to see where she is in math right now, since I remember Grade 3 and Grade 4 composed partially of doing and re-doing times tables." Mitch's first choice for a math question to his penpal is interesting, considering that in his math autobiography Mitch talked about times tables with disdain. Believe it or not I have no recollections whatsoever from elementary school regarding this subject except for grade four's hated times tables. 'Two times two equals four, three times three equals nine, four times four equals sixteen,' and on it went, ad nauseam. Why did we always stop at 'twelve times twelve' anyways? [MA p.l] Unaware of this apparent contradiction, Mitch continued to write in his journal: "I have asked her not to use a calculator and to time herself and send me the results in the next letter," adding that "I expect she will do okay but I am interested in seeing how long she takes" [MJ2]. We tried, with little success, to help Mitch better articulate the point of, and to make a defensible argument for, asking these questions and carefully examine the value of asking such questions. We asked in response to his entry: "What will this tell you about her math abilities?" But, in his next entry, Mitch made no comment about our inquiry. When he noted that his penpal had written a very short letter back without including a problem for him to solve, we tried probing his thinking again about the point of his asking the multiplication-table questions: "Do you think that your problem influenced the amount of dialogue? Is there a way to generate discussion around times tables?" [MJ3]. Again, we received no response to our queries. This particular episode provides some important clues into preservice teachers' problem selection strategies. First, it suggests preservice teachers would choose problems which are very familiar to them. For Mitch his recollection of elementary school mathematics, although distasteful, was a resource he trusted to help him select an "appropriate" problem for his student. Furthermore, the problem's attainability (solution accessible to students) seemed to have been a very important feature of the selection process. This is apparent in that Mitch's goal was not to assess for accuracy, since this was a problem he expected the student would be able to solve correctly, but to assess the student's speed. This, Mitch said, would provide him with information about the student's mathematical ability which would help him figure out "what kinds of questions I can ask her next week" [MJ3]. Therefore, aside from promoting students' success in and enjoyment of mathematics (as Miriam suggested), assessing ("testing out") the students' facility with mathematics seems to have been another reason to provide students with problems they could solve. Determining the "level the student is at" was also a theme that ran through many preservice teachers'journals. Preservice teachers often wrote: "I am still struggling to find out where Paul is at when it comes to mathematics" (Rosa); "I'm not sure what her level of ability is in math" (Miriam); "I still don't know what level Lynn is at as far as math skills are concerned" (Lesley). Similar to Mitch, many thought that having this information would help them select appropriate problems for their students. Interestingly, -89-preservice teachers seemed to think that students' correct work on a single mathematical problem would provide them with all the information they sought. The idea of selecting problems which students could correctly solve was also justified in terms of allowing them to gain information about the students' mathematical ability. Marcia, for instance, said: "Although this question is a bit easy, I thought it would be good because John would find it easier to explain how he had solved it" [MJ2]. This comment suggests that students may find it easier to explain their answers to problems which they can solve correctly. This also implies that students' thinking related to wrong answers may not be as clear or informative. This provides further insights into preservice teachers' reasoning for posing problems which students could unproblematically solve. Realizing Need for Better Questions We began our 3rd week-5th class (the class following the first letter writing session) by encouraging our preservice teachers to reflect on the kinds of questions they had posed in their letters. We challenged them to not only pose questions to find out what and how much the students knew about math, but also to promote students' learning. We started by saying, "We noticed that the children posed straight translation problems, problems that quickly translate into a mathematical sentence, and in response, many of you posed similar types of problems." We continued saying, "Two things you need to think about: Why did you respond in that way—and there are lots of good reasons why you would— but then, what, as a teacher, are other possibilities that you should have thought about?" After a brief pause, we offered the following explanation: ...because not only do I want to know who the students are, but another role of the teacher is of course to continue to bring children forward, to continue to challenge them, to continue to have some learning going on. So you don't want to just keep them where they are. [Instructor, Class #4] Students' response letters also challenged the preservice teachers' thinking about the kinds of problems they might pose and the kinds of responses students might give to their -90-questions. One student, for example, told her preservice teacher penpal that, "The question that you give [gave] me was easy. I think the one I gave you is harder than the one that you gave me" [Lesley's penpal, ml2]. Another student said, "Your question was to [too] hard but I'll keep working on it. I'll give the answer next letter" [Mitch's penpal ml3]. Another one stated, "I like word problems because they are very challenging" [Sally's penpal ml4]. Such feedback from students, therefore, helped encourage preservice teachers to attend to, and consider changing, the types of problems they were asking their students to solve. Consider Rosa who, for example, had asked her penpal the following: "1) 6x2= 2) Can you think of two other pairs of numbers that when multiplied together equal the answer you got in #1?" In return, her penpal responded: "1) 6x2=12" and "2) another pair is 2x6." The answer 2x6 was not what Rosa had in mind and so she restated her question and specified that the pairs should be two numbers other than 2 and 6. Similarly, Sally 2 had asked her penpals, "Do you know other ways to write j ?" When one of her student 2 penpals responded: "Another way to write j is 2/3," Sally restated her question to make clear what she really meant: "Do you know any other fraction that means the same thing 2 as j ?" From this experience Rosa and Sally became more aware and concerned with saying what they meant. "This showed me the importance of language in math and how students don't know what teachers are thinking. I will be more precise in the future!" read Sally's journal entry that day [MJ3]. The reality of facing students' work—quite often unexpected work, and at times no work at all—served to focus preservice teachers' attention onto their problem posing practices. Most of them realized that their interactions with students would not be as straightforward as they had originally imagined. Susan, for instance, began to question the expectations she had for her problems and the feasibility of her idealized problem posing strategy. I think that I expected my pen pal to easily solve the problems that I sent and hoped that gradually I would start making them harder instead of -91-giving her really difficult problems on the first day (which would turn her off). However, my plan backfired, and I now am finding myself creating easier problems. [Susan, MJ4] Some others, like Lesley and Terry, began to reconsider their strategy of posing easy-to-solve questions. Lesley, for example, said: "I want to pose questions they will be successful at as well as challenging" [MJ4]. Terry also became concerned with finding challenging problems for her student when after her first letter exchange she realized her student was ready to solve mathematical problems which were more challenging than the ones she seemed ready to pose. After sending a problem which turned out to be more difficult than she had intended, Terry worried about the frustration her student penpal would suffer. To her surprise, her penpal had not only solved the problem, she had also enjoyed the challenge. In her journal Terry wrote: I was quite surprised by her last letter, she found the more difficult answer a lot more fun. Myself and a group of friends worked on it and I regretted giving it to her because it seemed too difficult. A couple of my peers actually gave up. I feared I had given her a too difficult of a problem and she would become frustrated. But I was wrong, she enjoyed the challenge. I will try next time to find challenging questions. [Terry, MJ3] After this experience Terry said: "I am more aware of the questions I ask." Terry also vowed to be more careful not to "put my own difficulty with math on other people" and to not "take it for granted they will find [math problems] as frustrating as I did" [MJ3]. Terry also began to worry whether she would be able to keep her student interested. "I try to use challenging questions," Terry said while admitting: "Sandy seems to like ones that take a bit of trying and are not obvious. This is overwhelming because I am the opposite" [MJ4]. For most preservice teachers, however, it was their dissatisfaction with students' brief responses to their initial problems which helped turn their attention towards examining their problem posing practices and finding better questions to pose to their students. Preservice teachers often wrote about their concerns with students' communication. -92-Miriam, for instance, wrote: "My penpal reveals very little about herself and her math abilities. In her letters, she answers the numerous questions that I ask her about her math very briefly" [MJ5]. Carly also alluded to this problem when she wrote: "I'm having trouble with asking questions that will help me explore what and how the students are thinking. I think I need to ask them more questions, but do not know what kind to ask" [MJ3]. Similarly, Linda said: "I'm trying to probe and question a little better" [MJ4]. Many preservice teachers' investigations, therefore, began to turn towards making their problems and their problem posing practices more inviting for students to share their work and communicate their thinking. Trying New Problem Posing Strategies I am still having difficulty getting Parker and Lynn (Lynn more so) to elaborate on their solution processes in their writing. I tried to demonstrate how I would like them to respond to the questions I pose in answering their questions. For instance I write out each step of solving their problems, draw diagrams, etc., in the hopes that they would follow my example. I've repeated in both letters that I would like them to show their work also. Is there anything else I could do in the letter writing process— at least until I meet them and explain in person? [Lesley, MJ4] The students' sparse mathematical communication became a serious concern for our preservice teachers and a source of much deliberation in their journals. Many had assumed that their students would be willing and able to make their mathematical thinking explicit. When this did not happen, preservice teachers began to request students to "show your work" and "explain your thinking" along with their math problems. This, as it turned out, was often not enough to help most students elaborate on their solution processes. When the students' work did not turn out to be as elaborate and as revealing as they had hoped, preservice teachers became increasingly unsettled and disappointed. Some were quick to blame the lack of face-to-face interaction with their students as the source of their difficulties while others began to investigate the nature of this phenomenon and to devise theories and strategies to deal with it. - 9 3 -Some preservice teachers, like Sally, realized that students may "censor what they write for fear of errors" and therefore they began to develop some strategies to deal with such contingency. They began to develop a pedagogical discourse which could be conducive and inviting to students' communication. Sally, for example, made sure to state in her letters: "Write everything you think down, even the mistakes," and "even if you make mistakes, that's OK" [mil]. Others, as Terry did, tried to give their students some reasons for sharing their thinking, such as: "because I believe it is important to understand how you feel and work through a problem, could you please explain and show me the steps it took" [mil]. Other preservice teachers, such as Thea and Lesley, began to experiment with alternative modes for posing their problems in order to elicit more elaborate responses from their students. Thea for example said: "I tried to ask the student more questions this time to elicit a longer response" [MJ3]. Similarly, Lesley wrote: "To get the students to respond more fully I tried different things this time: i.e., providing graph paper, suggesting ways of solving, suggesting Lynn and Parker work together, tried giving them more encouragement" [MJ5]. There were others who, similar to Lesley, also began to send a common problem to their students (before we suggested this idea) so that students could work together and help each other. Rosa, for instance said: "Perhaps by suggesting that he works with a classmate for a specific problem, he will have the opportunity to reflect on this experience and then respond to my questions" [MJ5]. For Nilsa, however, it was upon reflection that she recognized the potential of group work for helping her student "discover the answer," and as a way to "raise student's interest in responding to our probing questions" [CR: p. 4, 10]. In her second penpal letter, Nilsa along with her group (Marcia, Carly, and Susan) decided to send the same problem to all of their students (6 in total). It was much later, however, that Nilsa realized that she had not revealed to her student the names of the classmates who also had the same problem. In retrospect, Nilsa questioned and regretted this move: -94-I'm not quite sure why I did not give her the names of her peers that she was to work with in letter #2. I think maybe it was because I have been conditioned that math is an individual event and that by giving her the group question in the first place that I might have been giving away too much information towards the solution already. [Nilsa, CR: p. 10] Many preservice teachers also began to model for their students how to communicate their mathematical work and thinking. They made sure to respond to the students' letters and to the problems students were sending the way they expected them to respond to their questions. Such modeling, Terry hoped, "would provide a model as well as [show] respect for [the student's] questions." Such modeling also gave preservice teachers, like Rosa and Linda, insights into the difficulties associated with making one's understanding explicit as they tried to show and explain their work on the problems their students had posed to them. Recognizing this difficulty for themselves—"I didn't do a good job of justifying my answers" [Linda, MJ3], "I often don't know myself why I do things the way I do when it comes to math" [Rosa, MJ9]—helped these preservice teachers begin to develop a deeper appreciation for their students' attempts at explaining their work and thinking. There were a few others, however, who despite having such difficulties themselves (or even neglecting to respond to their students' problems) continued to expect and demand full responses to all their questions, missing, in the process, the opportunity to discover, as Terry did, the challenges and benefits of communicating mathematically—"I was surprised at exactly how helpful this was to me as a teacher. Not only do you have to think about the processes that you already take for granted, it allows you to see it through the eyes of a learner" [CR: p. 3]. Or, to consider, as Thea did, an alternative explanation or point of view that could suggest an unforeseen course of action: [My penpal] often just gives me the answer when I plead with her to show me her work. I think she may have done that last time because I gave her too many questions to work with. I think it would be better to get a good question and try to develop it in a way that the student can show all of -95-their ideas about it, rather than asking all these questions and getting all the answers. [Thea, MJ4] Posing New Kinds of Questions Preservice teachers' concerns with obtaining more elaborate responses from their students, therefore, began to encourage them to explore and experiment with alternative forms of posing their questions to students. Many preservice teachers began to consider, and look for, problems which were quite different from the ones they had posed before. Lesley began looking for problems which "students are interested in" solving. She noticed that her students liked games and puzzle-like problems and began "trying to devise a word problem that was somewhat like a puzzle" [MJ4]. Most preservice teachers also seemed more receptive to trying out the less traditional types of problems we were working on in class. There were also a few others who began to pose more exploratory or open-ended types of problems, that is, problems which had more than one correct answer and/or had multiple solutions. The goal was more than seeing whether the student would reach the right answer. Lesley, for example, said: "I thought this question would push her to explore different ways of solving and recording solutions" [MJ6]. These developing practices indicate that preservice teachers were beginning to broaden their selection process to include problems which elicited and promoted students' mathematical communication. These new expectations for their problems in turn encouraged preservice teachers to venture into posing different kinds of questions. This stimulated a few preservice teachers' to initiate their own investigations into alternative questions to pose to their students. Sally and Thea, for example, asked their students to write a story problem to match a given number sentence. Similarly, Marcia also incorporated changes to her problem posing practices which had not been discussed or encouraged during our classes. "I guess I'm trying to figure out how to get kids ideas out into the spoken word, and how to help them develop their ideas," wrote Marcia in her journal [MJ5]. -96-Marcia, like most preservice teachers, had also been struggling to obtain more information from her student about how he was solving the problems she was sending. In her journal Marcia began to develop a sequence of questions she could pose along with her problems to try to get at students' thinking. In her 5th letter, Marcia tried out her idea and sent a sequence of probing types of questions along with the "Kanga and Pooh Bear" problem12—a problem we had thoroughly explored during class (7th week-14th class) and is enclosed in Appendix C. Their letter exchange read as follows: Marcia: I'm going to send you a problem that might be a bit difficult, but don't worry. If you could, I'd like you to answer the math questions I send along with the problem. Kanga the Kangaroo is standing beside a tree stump and would like to get at Pooh Bear, but she can only hop half the remaining distance each time. Where is Kanga after one hop? Two hops? Seven hops? Does Kanga ever get to where Pooh Bear is standing? Could you please answer these questions when you do the problem and then send this sheet back to me when you write again? 1. What did you think when you first read the problem? 2. Did you know how to start, or did you have to think about it for a while? 3. Did you use paper and pencil to do the work or did you do most of it in your head? 4. What do you think you learned from doing the problem? [ml5] Marcia's Ppal (John): I looked at the problem you gave me. You would never get to Pooh [Bear]. I have how I worked it out on another page [Student's work shows the drawing of a line K B marked at the half point of the progressively smaller segments (7) as K approaches B]. On the problem you gave me, how many times were you suppost [supposed] to make it half? [Answers to attached (see above) questions reads]: 1. I looked hard; 2. I had to think about it for a while; 3.1 worked on pencil and papper [paper]; 4. how to do half with really small numbers. [ml6] This problem is based on Zeno's classical paradoxes—"The Dichotomy" and "Achilles and the Tortoise." -97-Marcia: On the Kanga and Pooh Bear problem, you can keep dividing it in half forever. If you are using a calculator you would still never really reach Pooh Bear. It is strange to think about isn't it? [ml6] Marcia was very pleased with the results of this experiment. "I think I had just discovered a technique to get at some of John's thinking," announced Marcia in her journal. "I sent him a list of questions... and it worked!" she said while adding: "It is too bad that I only discovered this at the end of the term, because I could have probed his thinking further." Marcia then began to explore how this idea might work in an actual classroom: "A tentative idea might be to make up a checklist of some sort which the kids could quickly mark as they worked through a problem."[MJ7]. Later in her case report Marcia returned to reflect on the idea of posing such kinds of questions to students. This is an excerpt of what she wrote: I found that asking this series of specific questions helped me to get a more complete picture of John as he worked through the problem. ... Questions such as those that I asked John can help me to determine if the problem is too easy, too hard, or at an appropriate level—a consideration with which I'd been struggling throughout the Penpal project. Were I to use such a strategy again, I might consider including another question which asked specifically about the student's thinking when they were completely involved in the problem. For example, "When you were doing the problem, did you ever feel confused about the math, or was it always clear to you?" Such a question may not only help me to understand the situation, but also might help the student to be more aware of his or her own thought processes while working. As I looked over these questions, I realized that during the Penpal project, I had progressed in my questioning techniques. In this last set of questions, I had included a series of questions related to one problem, rather than questions about a variety of unrelated things. In addition, the last question on the question sheet "What do you think you learned from doing the problem" was an open question that was helpful to me as a teacher, but also helpful to John in asking him to think about his own learning. [Marcia, CR: pp. 12-13] - 9 8 -Similarly, there were other preservice teachers who also began to pose probing types of questions. The introduction of various problems related to area and perimeter (4th week-7th class) seem to have encouraged some preservice teachers, like Rosa, to begin to investigate for themselves the mathematical concepts involved in such problems. Prior to the class where we were to discuss these problems, we gave our preservice teachers a journal prompt—"Area of Interest"—which instead of listing some problems for preservice teachers to solve and consider posing to their students, had a few open or exploratory-type questions (see Appendix C). This particular format seemed to have encouraged some preservice teachers to engage in the investigation of these concepts. Rosa, for example, wrote: I must admit that I went to our text to get a more complete understanding of what area and perimeter are. I tried doing some of the activities suggested in the book to apply what I knew about these concepts. I then went back to the handout to see if this information could help me in answering the questions: e.g.: "is it possible to have two figures that have the same perimeter but different areas" and "what about two figures that have the same area but different perimeters?" I found myself doodling on scrap paper to try to answer these questions. I drew a square and a long, skinny rectangle, both of which had a perimeter of 16; thus 2 shapes can be different yet have the same perimeter. I then drew a grid to answer the second question and found that it too was possible. [Rosa, MJ3] Their own explorations into the concepts of area and perimeter seemed to have made these preservice teachers curious about students' thinking and understandings about the explored concepts. This work, in addition, seemed to give these preservice teachers the confidence to design their own problems and questions for their students (as Marcia did with the Kanga problem). Sally, for instance, constructed a task (based on a perimeter-and-area activity suggested in our course's textbook) to try out with her students. Sally's problem had a similar open-ended design to the journal prompt we had provided. Her area-perimeter problem read as follows: -99-We also have to bring an area/perimeter question for [our students]. Here is mine: If you have 24 square tiles of equal size, how many ways can you arrange them to make different size rectangles? Do you think they will all have the same perimeter? Why or why not? Do you think they will all have the same size area? Why or why not? Now find the area and perimeter for each rectangle and record: Length Width Perimeter Area Rglel Rgle2 etc.... Do you notice anything about the area? If so, what? Do you notice anything about the perimeter, if so what? I wonder if they will realize that the area will stay constant at 24 units^? I wonder, too, if they will expect the perimeter to be the same? I wonder if they will make 2 rectangles of equal dimensions (e.g., 4x6 and 6x4; 3x8 and 8x3, ...). I am curious to see the results! [Sally, MJ4] Similarly, Thea also became curious about her student's understanding of these concepts. She said: "I'm really interested to hear what the students are learning about area and perimeter, so I can't wait for my next letter," after sending her student some questions about these concepts. Particularly, she said, since "I know that this can be tricky because I was trying to figure out how to do it myself [MJ4]. Here are some of the questions Thea posed and the responses she received from her student. Thea: Our teacher told us that you were going to learn about area and perimeter this week. How is it going? Are you having trouble? Pretend that I have no clue what area and perimeter means. Can you explain what they mean in your own words? What is area? What is perimeter? Try to explain these words to me. [ml3] Thea's Ppal (June): I think that area means space inside the shape and perimeter means distance around the shape. For example: [ml4] -100-Perimeter 28 Area 42 28 1 2 3 4 5 6 7 8 27 26 25 8 9 10 11 1 2 13 14 9 15 16 17 18 19 20 21 10 24 22 23 24 25 26 27 28 1 1 23 22 29 30 3 1 32 33 34 35 12 21 3 6 37 38 39 40 41 42 13 20 19 18 17 16 15 14 Thea: Thank you for answering all the questions I asked in such detail. The way that you explained area and perimeter to me with all of the small squares and numbers really helped me a lot! ... I'm glad that area and perimeter aren't too hard for you either. ... How would you find the area of this triangle? [ml4] 4 cm Can you find the area of this shape? 6 cm 8 cm Thea's Ppal (June): For the area and perimter quest., the triangle you have to make it into a rectangle, for example: J. The area is six: [ml5] 1/2 + 1/2 = 1 whole 1/2 + 1/2 = 1 whole whole whole This and the preceding examples and illustrations make it clear that our preservice teachers were exploring and experimenting with alternative ways of posing questions to students and had become more receptive to posing unfamiliar and more open-ended types of problems to their students. Various reasons have been suggested as to how and why these changes occurred. Students' unexpected feedback and the introduction to new problems during our classes are factors associated with these preservice teachers' reflections on, and changes to, their problem posing practices. Several preservice teachers, for instance, also expressed a view similar to Marcia who wrote: -101 -As the penpal project progressed, I began to pose problems that were more open, and focused less on getting a single correct answer than those I had posed earlier. I believe that it was the influence of my math class and the types of problems we were studying as student teachers that led me to try some new approaches. [Marcia, CR: p.9] For Thea, it was the student's brief responses to her questions which led her to the continual search for ways to encourage and invite students to communicate their thinking. In her case report she discussed different strategies for eliciting students' mathematical communication. The importance of posing good problems and the teacher's questioning skills are highlighted. A perfect example of a question that can be used for these purposes is one that came from my MAED 320 class at UBC. The question is called "Investigating Foot Area," and it goes as follows: ... [See Appendix C]. I have seen this question work well on three different occasions; once in inciting a lively discussion among my fellow UBC student teachers, the second time this question resulted in another interesting discussion among grade five students, and thirdly when I sent the question to my Math penpal, June, and received a fairly lengthy explanation in return from a student who is usually short on words. ... There are many more problems out there that can be useful to initiate true discussion and create valuable learning experiences in the classroom. I recently read an article about a teacher who is doing the kind of thing that I am talking about with the simplest of questions.. The teacher I read about would transform himself into a "kimp"—"(one that is) so stupid that (they) are not aware of (their own) stupidity" (Berkman, 1994)—right in the middle of Math class and "use seemingly stupid questions to goad students into thinking" (Berkman, 1994). ... one really needs the commitment and the time to build up an atmosphere in the classroom where the students are willing to talk in this way. It will also take time for the teacher to come up with the appropriate language and the right questions to initiate these kinds of discussions. Considering all the benefits for students as well as for the teacher, I am sure that most would agree that it would be well worth the time and effort to make student talk work for you in the classroom. [Thea, CR: pp. 7-10] - 102-II. Making Adaptations During their penpal interactions preservice teachers also made decisions related to the reformulation or modification of some mathematical questions. In their decision making, preservice teachers had to contend with the constraining features of the penpal activity. This meant they would pose their questions and receive their students' responses in a written format. It also meant that there was no opportunity for intervention while the student worked out the problems they had sent. These, undoubtedly, played a part in some of the decisions our preservice teachers made. Modifications ranged from simple to substantial changes to the look, wording, and content of the problems. Generally these were made for the purpose of making the problem more appealing, easier, and/or clearer. Adaptations to the original problems were often the result of collaborative discussions with peers prior to the letter writing activity. At times, they were made in anticipation of the potential difficulties students may encounter with the material covered by the question. At other times they were made in response to their students' difficulties with the originally posed question. Simplifying Problems Similar to their initial problem selection practices preservice teachers' adaptations to problems were initially meant to make their problems simpler and easier for students to solve. Linda's and Megan's adaptations to "the horse problem13," the first problem they sent to their student penpals, serves to illustrate. This problem reads: A man bought a horse for $50. He sold it for $60. Then he bought the horse for $70. He sold it again for $80. What is the financial outcome of these transactions? From: Marilyn Burns (1987). A Collection of Math Lessons: From Grades 3 through 6. Math Solutions Publications: Sausalito, C A . I have used this problem extensively since first introduced to it by a fellow math methods instructor several years ago. I have used it with school students from Grades 4 through 7 as well as with preservice (elementary and secondary) and practicing teachers. -103-This was a problem we had discussed during our third class, prior to the first letter writing class. This deceptively simple-looking problem never fails to cause uncertainty and confusion as to how the problem might be interpreted and solved. Usually students generate alternative ways of solving it and come up with reasonable-sounding explanations for right and wrong answers (Makes $30, $20, $10, breaks even, and loses $10, $30 are common answers to this problem). The situation seems clear enough and the numbers are as simple as they can be, "yet deciding precisely what to do isn't clear to everyone" (Burns, 1990; p. 12). The fact that the problem is open to multiple interpretations and that it raises a plethora of solution methods and answers makes this particular problem a good choice for promoting mathematical communication and encouraging students to justify and explain their answers to mathematical problems. Linda and Megan, who sat at the same table during regular classes and during the letter reading/writing sessions (along with Miriam and Terry), chose to send the horse problem to all of their student penpals (they each had two). While they kept the scenario of the problem intact, they changed the problem's question from: "What is the outcome of this transaction?," to: "If he had $100 when he started, how much money does he have in the end?" It is interesting to note that all six preservice teachers who chose this problem for their student penpals also changed the original problem's question. These were their new questions: "How much money is he left with?" (Thea); "How much money does he have in the end, or did he lose money?" (Sally); "Did the man lose or make money? How much?" (Miriam); "Does the man make, lose, or break even? If he makes or loses money, how much?" (Terry). While it is true that the original question is perhaps too broad and can lead to a non-mathematical answer (e.g.; the statement "the man has no horse" is an outcome of the transaction), Linda's and Megan's new question, in particular, does not invite students to speculate and consider possible solutions to this problem. Suggesting that the man had $100 to start with, not only prevents students from pondering about such things, it limits the ways in which students could interpret the problem. In addition, by asking "how -104-much money does the man have in the end?," instead of allowing the student to figure out whether and how much money the man made or lost, these preservice teachers have limited the work the students must do to arrive to the correct answer. Linda's and Megan's adaptations to the horse problem provide further evidence of our preservice teachers' tendencies (discussed earlier) to pose problems which students could solve with relative ease. Linda's and Megan's journal entries (prior to the horse problem incident) provide further insights into their attempts to make math easier for their students. These particular journal entries were made after our second class when we hosted an interactive teaching session with four guest students. In this session, we (the instructors) investigated the students' understandings of fractions and worked on the "Three Hungry Monsters14" problem. Megan's and Linda's observations (see below) reveal their attempts to simplify the questions and problems we had asked to the students. They began to think of ways they could change the questions to make it easier for students to answer them correctly: Megan wrote: When the students were asked to represent one fourth of the marbles to be white out of eight, Marion was the only one to get the answer correct. The other three students coloured four of the circles in. I believe if the question was rephrased to "represent one quarter of the eight marbles as white" all of the students would have answered the question correctly. [Megan, MJ1] From Jane M . Watson (1988). Three Hungry Men and strategies for problem solving. For the Learning of Mathematics, 8(3), 20-27. The problem reads: Three tired and hungry monsters went to sleep with a bag of cookies. One monster woke up and ate 1/3 of the cookies, then went back to sleep. Later a second monster woke up and ate 1/3 of the remaining cookies, then went back to sleep. Finally, the third monster woke up and ate 1/3 of the remaining cookies. When she was finished there were 8 cookies left. How many cookies were in the bag originally? -105-Linda wrote: If the problem had been broken up into 3 parts, I wonder if it would have been easier? i.e.; a) A red monster wakes up, and eats 1/3 of a bag of cookies. There are 8 cookies left. How many were there in the beginning? b) The blue monster ate 1/3 of the cookies, before the red monster. Use your answer in part a) How many cookies did the blue monster eat? How many were in the bag before? etc. This would have made the kids work backwards, but it might be more convoluted than the original. Maybe I will have to find someone to test it out. Another thought—what if they were given pie drawings of the problem? Would that have made it easier? This also could have forced them to work backwards. [Linda, MJ1] Linda's and Megan's adaptations to the horse problem, and to the fractions questions, reveal that while they were considering and anticipating the potential difficulties students may have with these problems, they felt they must try to prevent such difficulties from arising. Interestingly, while the idea of acting out the horse problem with some play money was a possible teacher intervention strategy considered during our class discussions in order to try to help students rethink their interpretation of the problem, Linda and Megan used it to try to bypass and avoid their students' errors. They seemed to be trying to spare students from becoming confused and getting a wrong answer. Yet, when discussing her reasoning for choosing the horse problem, Linda reported having much bigger and greater goals than to simply ensure students reached the answer to her problems. She reported choosing this problem because "it has more possible routes to the solutions (& more solutions!)," and had hoped that "it would provide me with a lot of information on how the children 'do' math" [Linda, MJ2]. Removing Ambiguity and Providing Clues Similar to Linda and Megan, our preservice teachers' initial adaptations to problems can be characterized as removing ambiguities and potential difficulties for students. Such adaptations tended to restrict and narrow the work of the students and seldom to widen the mathematical scope of the original question. The "Checkerboard" problem (See Appendix C), for example, which could be used to generate a pattern to predict the -106-number of squares in a variable-size square, became, instead, a problem of finding (counting) the number of squares on a specific-size square. Other examples are found in the adaptations many preservice teachers made to the "Squares are In" and the "Staking your Claim" problems. These problems were meant to provide a context for exploring patterns in the perimeter and area of growing squares and rectangles and for investigating their relationships. These problems read: Squares are In. Here is a square covered with a piece of pink paper and outlined with green ribbon. How much more paper would be needed to cover another square that is twice as long and twice as wide and how much more ribbon would be needed to outline it? Explain. How much more paper and ribbon would be needed for a square that is three times the size of the original square? Four times? Explain. Staking Your Claim. Suppose you were a homesteader in the Old West and you were about to stake out your homestead. The rules say that you can stake any piece of land in the shape of a rectangle as long as the perimeter of your rectangle is exactly 120 m. What dimensions would you choose? Explain. Preservice teachers who chose the first problem tended to limit the problems' potential in a variety of ways. Some, for example, limited the problem to finding the perimeter of the squares, leaving out questions regarding the area of the squares. Some also felt it necessary to clarify what "twice the size" meant or to illustrate it with drawings. Others asked students to find the area and perimeter of a square only twice the size of the first one, avoiding the part of the problem that dealt with bigger squares and the exploration of patterns. One preservice teacher chose instead to ask the student "to draw" the second square rather than figure out the relationship between the first and the second, third, and so on, squares. Some of those who chose the second problem made what can be also considered to be very explicit and leading adaptations. Megan, for example, made sure to tell her students: "Remember farmers want as much land as possible." Many others also felt it necessary to - 107-clarify or avoid the words "dimension" and "perimeter" in their wording of the problem. Some resorted to less obvious but equally leading adaptations such as highlighting important wording in the text (holding or underlining)15 to warn their students of potential difficulties or emphasize information they should keep in mind. Some others, such as Susan and Nilsa, figured that reminding students that "a square is also a rectangle" was an important clue to help their students to solve more quickly the "Staking your Claim" problem. These adaptations further illustrate preservice teachers' concerns for students' success and enjoyment of mathematics. They highlight preservice teachers' attempts to make their problems less problematic and more attainable to their students. These adaptations, however, were not frivolously made. Preservice teachers spent a lot of thought and work in figuring out and anticipating what would be helpful to students without giving away the answers. Even though many adaptations tended to be of the leading and imposing type (as those reported above), some adaptations were also quite subtle and potentially useful for students. Some preservice teachers, for example, gave their students visual clues (such as drawing the squares on grid paper) or sent manipulatives or models to help their students represent the problem more concretely or visually. Others provided a table for students to write down their measurements and later try to see relationships and patterns in the data. For the "staking your claim" problem, Linda, for example, suggested the following to one of her students. You might find that some of the following will help you: grid paper, string, a geoboard, drawing pictures, cutting out shapes. Let me know what you use, why you chose it, and if you have a special strategy that's your own idea, please tell me! [Linda, ml 3] Not unlike strategies teachers are known to use with their voice such as a change in intonation of speech during classroom discourse. -108-We can sense in Linda's hint the tension between trying to help the student work on the problem without becoming too prescriptive and imposing. Linda's solution to this pedagogical tension was to both provide very specific ideas about tools the student could use to solve the problem while also leaving the door open for her student to choose her own strategy and tools for solving the problem. In their reflective journals preservice teachers' deliberations about the changes they were making to problems reveal they were also struggling to become less prescriptive and allow students to make their own sense of the problems. Even though preservice teachers' adaptations to problems did not seem to be radically changing, their reflections and deliberations, on the other hand, about their reasons and goals for making such modifications seemed to be shifting. Preservice teachers, therefore, began to write about their intentions to, not necessarily steer students towards the answer, but rather to make their problems understandable to the students and provide enough but not too much help to students. Susan, for example, wrote: Going through the squares are in problems I realized how difficult a lot of the problems were for Grade 4 and 5 students. I did not think that she [student penpal] would be able to answer any of the problems correctly. It was a challenge for me to evaluate the problems and rewrite them in such a way that my penpal would understand the problem enough to be able to set her thinking skill to work and attempt to solve it. [Susan, MJ4] Preservice teachers' intentions therefore had taken a slightly different turn. Rather than attempting to make the problem easier for the student to solve, preservice teachers were concerned with ensuring the students had sufficient information to work on the problem and attempt to solve it. Carly, for example, also said she had made adaptations so that "students will understand and not be baffled by how the problem is written" [MJ5]. And, though preservice teachers felt their questions needed to be clear "so that children can understand exactly what we're asking for," as Nilsa said, they also wanted their questions to be a challenge for students. - 109-Students' feedback and workings on their adapted problems also helped preservice teachers to both focus on and question the changes they were making to their questions. For Mitch, for example, examining his student's workings on his version of the "Squares are In" problem served to reveal some of the "technical errors" he had inadvertently made. I used inches instead of cms. (sacrilege!) and since Kelly's ruler had no inches on it, it became a point of confusion. As well, the square I drew and colored so nicely was in Kelly's view not a square. How was I to know she would measure it exactly? [Mitch, MJ5] For Terry, reading her student's work and feedback helped her realize that her attempt at clarifying her chosen problem had instead caused more confusion. She had chosen the "Twice as much for twice the price" problem which originally read: "When the circumference of a pizza pan doubles, should the price double? Explain your reasoning." Terry preambled this question with: "Imagine having a pizza that is twice as big as another, should this larger pizza be twice the price? In other words ... (original text)." For Terry's penpal doubling the circumference of a pizza and making it twice as big meant two different things. This she said in her reply: Dear Terry: If the pizza's size doubles then I would double the price because I would get the same amount of money as I would if I sold two small pizzas. When I read this [pointing to the "In other words ..." part of the question] I got confused because this is not the same as the one above. When the circumference doubles the size is more than doubled. I looked at John's [Marcia's penpal] sheet to help me along. Then I understood. If I were selling the pizza I would do more than double the price because it is more than twice as much pizza. [Terry's penpal (Sandy), ml4] "Oops," wrote Terry in her journal realizing her "mistake" while also admitting: "I feel very bad that I was so confusing." After examining Marcia's version of the problem, Terry felt that "Marcia had given the same problem to John explained in a 100% better - no-way." Yet, at the same time, Terry also felt that "although John's question is more clear, it isn't very challenging" [see both questions in Appendix F]. This helped Terry see a fine line between making a problem clearer and making it trivial. Terry framed the issue as follows: "Somehow I need to make the questions simple to understand but still challenging to solve" [CR: p.3]. Reconsidering How to Make Problems Interesting Selecting problems students would be interested in solving was always an important consideration for our preservice teachers. Initially most of our preservice teachers, however, seemed to think that students would be interested in solving problems which they could solve easily. This was implied by their selection and adaptation to problems, and it was explicitly expressed by preservice teachers, such as Marcia and Terry (reported earlier). The idea, it seemed, was that for their students to really enjoy and therefore learn mathematics, they needed to make it unproblematic, unambiguous, easy, and clear. For Sally, however, who throughout the penpal activity had been working very hard to make her questions very clear and unambiguous to her students (and demanding the same from her students' problems), the realization that ambiguity "could be looked upon as a valuable tool for further exploring mathematics" came afterwards when writing up her case report [CR: p. 9]. Sally wrote: It was not until I was given a copy of an article entitled Beyond Problem Solving: Problem Posing, that I became aware of the opportunities that exist for furthering students' conceptions and understandings in the classroom by using such ambiguous questions. The goal should be not just to "catch" the ambiguity as if is merely an error but to use it productively. ... Ambiguity leaves room for curiosity, imagination, and generating one's own ideas. Ambiguity is also inevitable. Where it arises, we should make it useful rather than a failure (Moses, Bjork, and Goldberg, 1990). [Sally, CR: pp. 9-10] - i n -Similarly, for those preservice teachers who themselves had begun to investigate some of the mathematical concepts and ideas in order to understand them (not simply for the purpose of solving math problems), their notion as to what would make mathematics interesting for students also began to change. The idea that students would enjoy and learn mathematics better if problems were made unproblematic and were removed of any ambiguity began to lose face for these preservice teachers. Instead, they began to realize that becoming confused did not necessarily lead to frustration, disengagement, and loss of interest in the activity, but that it rather created interest, curiosity, and the need to understand and construct meaning. Some examples of this can be found in the area-and-perimeter investigations which sprung out of the "Area of Interest" journal prompt (reported earlier). Another, perhaps more salient, example is also found in Thea's self-initiated investigation into the comparison of fractions with unlike denominators. During the penpal visit, Thea shared with her student a method for comparing fractions. She devised a visual method (as opposed to symbolic manipulation) to compare two fractions with unlike denominators. Thea's method relied on the drawings of two wholes that were the exact same size. The wholes would then be cut into a different number of pieces (to reflect their respective denominators) and later shaded to match each of the fractions' numerators. She then visually determined which was the larger fraction by comparing the sizes of the shaded regions. After receiving her student's next letter and examining her work comparing fractions (student had neglected to make two wholes of equal size), Thea began to have second thoughts about her method. She wondered in her journal: "what if the two wholes are not the same size? Do you just take it for granted that they are?," then added: "If you did my method and drew your wholes different sizes, you might end up picking the wrong fraction [as her student had done]" [MJ8]. Investigating this further, Thea made an important mathematical discovery: "when you use common denominators to determine which fraction is bigger you are ensuring that each 'whole' is the same size! What a break through!" [MJ8]. This exhilarating experience had a profound effect on Thea who - 112-in turn began to consider that exploring mathematical ideas could be in itself an interesting activity for students. The thing is that in taking this long road to figure it out I am learning and understanding a lot more. These kind(s) of things are what makes math interesting to me. Everything you do in math does make sense, you just have to figure out how it makes sense. I think that it is good that I am figuring out things like this because it seems to me that this is the way that you hope that we will teach math. [Thea, MJ8] For many preservice teachers, our class work on the "Foot Measuring" and "Packaging" problems seemed to have a strong impact on their ideas and attempts to make mathematics (in particular their math problems) more interesting for students. We introduced these problems during the 5th week-9th class of the course. These problems followed the area-and-perimeter investigations ("Squares are In" set) and they were meant to help revisit and expand these concepts (to surface area and volume) and to also introduce our preservice teachers to the idea of connecting mathematics with other school subjects. We structured this particular class as a simulated math lesson with half of the preservice teachers playing the role of school students and the other half the role of observers. We spent the first half of the class working on a foot measuring problem16 using a well-known children's book—"Jim and the Beanstalk17"—as a context for the mathematical investigation. The "Foot measuring" problem was set within the context of the story in the book which tells about a boy who befriends a lonely old giant. To help the giant get a wig, glasses and teeth Jim had to find ways to take the giant's measurements in order to have these items fit his size. To get the giant's shoe size (this is an extension of the story) Jim took a string and measured the perimeter of the giant's foot and took the string to the shoe store. Jim told the shoemaker that he needed a pair of shoes whose soles From Marilyn Burns (1990). About Teaching Mathematics: A K-8 Resource. Math Solutions Publications: Sausalito, C A . and also in Marilyn Burns (1987). A Collection of Math Lessons From Grades 3 Through 6. Math Solutions: Sausalito, C A . Raymond Briggs (1970). Jim and the Beanstalk. Puffin Books: Trento, Italy. -113-were the size of the rectangle he had made with the string. The mathematical question we asked the preservice teachers to discuss and explore was "Is Jim's method a good one?" We also asked them to give a convincing arguments as to how and why this method would or would not work. For the second half of the class we turned to another problem which related and extended the concepts of area and perimeter to the concepts of surface area and volume. This problem was set in the context of the environmental issue of waste management. We showed our preservice teachers various boxes and while holding two of the boxes up, we said: "Do you ever wonder why they make boxes which hold the same amount of stuff yet one is much bigger than the other?" (paraphrase). With this preamble we launched our planned investigation to find a box which would hold the most but used the least amount of cardboard. Using 12 interlocking cubes as models we asked the preservice teachers to find the dimensions of a box which would produce the least amount of cardboard waste. This class raised a lot of interest for our preservice teachers who wrote extensive journal entries exploring and extending on the idea of using interesting and relevant contexts for mathematical problems. "This was the most helpful class yet" wrote Nilsa afterwards, also adding: "I understand the 'why' or the theories behind teaching maths in this way but it was invaluable to experience and observe how it is to be done and showing/giving us ideas on how to implement it!" [MJ5]. Similarly, for Marcia, this class activity had not only "showed us that [children's] literature can be used in conjunction with math," it had also helped her construct an image for how she might go about teaching this new kind of mathematics. She wrote: "I could see that kids could be taught to think about math, not necessarily as numbers and correct answers, but math as ideas" [MJ5]. This experience also triggered the critique of the previously posed problems, particularly the ones students had worked on last. While reviewing and analyzing the last letter they had received from their students, many preservice teachers, for instance, wrote -114-in their journals about how interesting or non-interesting their chosen problems were and how they could be set in more appealing contexts for students. Carly, for example, wrote: I do not think that there was much of a hook for this problem ["Staking your claim"] seeing as it may be difficult for her to imagine staking out a homestead. Something that may be more interesting could be: Try mapping out a room (new bedroom) with a certain perimeter, making a swimming pool, buying a trampoline, etc. I think that it would be better to relate the questions to real life and to things students can imagine. [Carly, MJ5] Some preservice teachers were also inspired by the previously mentioned class to experiment and try out their ideas for making problems more interesting in their next letters. The "staking your claim" problem, for example, was given several new contexts, such as: fencing a dog pen (Linda), and constructing a poster (Sally). Even the more reluctant preservice teachers (e.g., Rosa and Mitch) were inspired to venture into posing less clear-cut types of problems in their next letters. Preservice teachers' adaptations to problems therefore began to broaden, rather than to narrow, the mathematical scope of their problems (see Terry's and Lesley's problems in the Appendix F). These daring adaptations, however, were not always mathematically sound or viable. Terry's version of the "Packaging" problem (see Appendix F), for example, manages to capture and connect both the contextual and mathematical aspects of the problem in a way that makes her problem interesting contextually and mathematically. For other preservice teachers, however, the context of the problem rather than the mathematics took center stage. For the foot measuring problem, for example, some preservice teachers neglected (either forgot or did not see the significance of the information) to include a crucial element of the problem, that is, that to order the shoes the person had reshaped the string into a rectangle to find the area of the foot. Instead, these preservice teachers said that the person had given the shoemaker the string with the perimeter of the foot. This changed the nature of the problem in a significant way and was reflected in the kinds of -115-responses the students provided in return (See Appendix F). Preservice teachers became aware of this discrepancy only when we, the instructors, had pointed it out to them. In Summary: From Problems to Solve to Situations to Investigate Preservice teachers' initial selection and adaptations of problems revealed their preference for unproblematic problems—problems which could be easy and quick to solve. Their problem posing practices were characterized by their attempts to remove potential difficulties and avoid students' errors. The main reasons for posing such easy-to-solve problems, as it turned out, were: (a) to promote the students' success and enjoyment of mathematics, (b) to test out the students' facility with mathematics, and (c) to make it possible for students to explain their work. Preservice teachers' assumptions, however, that students' would enjoy working on easy problems and that they would be turned off by challenging questions were soon challenged by the feedback and work the students were providing in their letters. Students' workings also served to challenge preservice teachers' ideas about the kinds of problems students would be ready and willing to solve and the kinds of responses they would be able and willing to provide. Students' brief and unexpected responses to problems, in particular, encouraged preservice teachers to refocus their expectations for their problems to not only be solved correctly, but to also invite mathematical communication and to provide access into students' mathematical thinking. Preservice teachers therefore were beginning to see problems not only as tools for promoting students' success in and enjoyment of mathematics, but also as means to investigate students' mathematical thinking. The students' lack of communication prompted preservice teachers to explore alternative ways of posing their problems in order to encourage students to share their thinking. Features of the problems which were previously taken for granted (i.e., wording, content, context, etc.) began to be taken into consideration when selecting and adapting problems. Preservice teachers also explored alternative ways of posing their -116-problems in order to elicit more elaborate responses from students. Among the strategies considered and explored were: establishing a trusting relationship with the students, posing group problems and encouraging students to work with other students, modeling the sort of written communication they expected, and trying out different formats (i.e., depth vs. breadth) for their questions. In the process, preservice teachers were gaining a deeper appreciation for the challenges involved in eliciting mathematical communication and in posing better problems to their students. Feedback from their students coupled with the introduction of problems of a more investigative nature during our regular classes began to encourage preservice teachers to try posing less straightforward types of problems to their students. Preservice teachers, therefore, began to be more receptive to the idea of posing the more exploratory types of problems we were introducing in class. They also began to question the adaptations they had made to problems which had served to constrict and to ease the work of the students. They began, instead, to incorporate less imposing and less leading types of changes to their problems and began instead to engage students in the investigation of mathematical ideas. These changing practices signaled (and were consistent with) an important shift in preservice teachers' views of their role as problem posers. Rather than attempting to make their problems easier for students to solve, preservice teachers were trying, instead, to make their problems understandable for students to engage in the problem solving process. Therefore, instead of anticipating potential difficulties in order to prevent their students' errors, preservice teachers had begun to anticipate what could be helpful for their students to understand and engage in their problems. Preservice teachers' own mathematical investigations played an important role in helping them gain confidence and ideas to explore alternative ways of posing mathematics problems to students. -117-Chapter 5 Analysis of Preservice Teachers' Learning Part II: Interpreting Students' Mathematical Thinking Images of students learning mathematics with understanding have far reaching implications for how teachers are to help students learn. Rather than the repetitive and silent activities associated with traditional mathematics classes, the reform vision considers group and classroom discussions to be central to students' learning. Promoting and sustaining such discussions are at the heart of teaching students for understanding. Teachers, then, are faced with the task of orchestrating substantive and productive mathematical discussions among students. This is no small feat for it involves, among other things, the capacity to hear and interpret what students say and do, and to respond in ways that extend, support, and challenge their mathematical thinking. The importance of teachers listening to, and understanding, students' thinking has been widely promoted and supported by the mathematics education community as it is believed to enhance students' learning. This "listening" orientation towards teaching mathematics, many believe, promotes a learning environment conducive to and respectful of students' own sense making and intellectual autonomy (Davis, 1996; Kamii, 1989). When teachers do not listen to, or do not understand, their students' thinking they tend to dismiss it by imposing their own formalized mathematical constructions onto their students (Cobb, 1988; Maher & Davis, 1990). Therefore, listening to and understanding students' mathematical thinking is an important goal in preservice teacher education. Preservice teachers' unfamiliarity listening to students' thinking makes it particularly difficult to envision themselves teaching in such a way. Their past experiences as students of mathematics, their knowledge and beliefs about what mathematics is and how it should be taught serve as lenses and filters that help interpret classroom events and give meaning to students' words and actions. As Suzanne Wilson (1990) relates: when preservice teachers watch videotaped "classroom conversations brimming with false - 118-starts, tangential discussions, [and] seemingly endless arguments" they often conclude that such discussions have gone on for too long and that the teacher should explain and clarify. Furthermore, Ball (1990a) notes: When watching children struggle with a difficult problem, many of my students infer that the children are frustrated—and, therefore, uncomfortable and unhappy. When they see children disagree and argue about a solution to a problem, they think that the children are confused, that the teacher should step in and explain. When they see a child revise a solution in front of the class because of something that another child pointed out about his approach, many of them assume that he is embarrassed about having been wrong, (p. 13) It is these ways of seeing and interpreting that teacher educators hope to "shake," "uproot," and "inform" (Holt-Reynolds, 1994; Wilson, 1990). They hope, as Ball (1990a) notes, to help preservice teachers "reconsider these conclusions and understand their sources" and to help them "[develop] alternative frames of references" (p. 13) for making inferences and drawing conclusions from such classroom interactions. In this study, preservice teachers interacted with students mostly through written letters. This medium afforded them more time to decipher their students' work and to carefully think of an appropriate way to respond. This, however, did not make the task of interpreting and inferring meaning from students' work any simpler or less problematic. Knowing what to look for and what to do with students' mathematical utterances is not a trivial task, even for experienced teachers. This may be an even greater challenge for preservice teachers who have grown accustomed to letting their mathematical procedures and symbolism do the explaining for them. Furthermore, listening to students explain and try to make sense of an idea, aloud or on paper, can be a daunting expectation for preservice teachers used to classrooms where the dominant form of mathematical discourse has been the teacher's and textbook's polished expositions. Their past experiences both enable and limit what preservice teachers look for and are able to see in their students' work. Penpal interactions, therefore, provided preservice - 119-teachers not only with needed experience communicating mathematically, they also served as a context for investigating and interpreting students' thinking. An examination of some of preservice teachers' interpretations of their students' mathematical work will serve to highlight preservice teachers' interpretive frames (what they attended to, ignored, and possibly remained unaware of) and to trace some of their developing ideas about students' mathematical abilities and understandings. I. Interpreting Students' Work Initial Expectations Learning to understand students' ways of thinking was not among the things our preservice teachers said they hoped to learn in our methods course. It was not until after our second class, when they observed us doing an interactive math activity with a group of four Grade 5 students we had invited into our class, that many of our preservice teachers began to consider that listening to students' thinking could be a strategy for teaching. "I liked hearing the 'why' of choosing answers," observed Nilsa in her journal afterwards. "There was a sort of logic in [their] answers that I never would have predicted, nor would I have understood if the students hadn't explained," also noted Sally in her journal. Such "prob[ing] into the child's mind" had some appeal to our preservice teachers and many agreed with Susan's rationale that "to understand and make sense of what the child is thinking" was helpful because it would allow her "to correct flawed logic" or, as Sally said, "show why [an answer] isn't correct." While the idea of eliciting and understanding students' thinking was beginning to appeal to many of our preservice teachers, no one explicitly investigated or explored further what this approach may mean or entail. It was in our fifth class that our preservice teachers would begin to foresee the demands this approach placed on the teacher and to question their ability to enact it. A showing of a video clip of a 2nd grade teacher who had asked her students to solve 27-18 mentally, and then proceeded to have the students - 120-explain and defend their answers (regardless of their correctness), raised a commotion in our class. The video, Sally later wrote, "blew me away" and was seen by many as a mixture of being "helpful and intimidating," or "exciting but scary." Many expressed sentiments similar to Sally's: "I would love to have my students explain, challenge, argue, justify, the way they did in the video," but admitted to having "no idea how I would go about trying to get my students to think about subtraction the way this teacher did." Others, like Terry, wondered and hoped that "I'll be able to understand what they are saying to me when describing how they reached a particular answer." The idea of respecting and listening to students' thinking, many of our preservice teachers thought, was intriguing and inspiring. However, they also realized, as Marcia said, that teaching in this way "is not as simple as teaching a formula anymore," and that it would require them to "learn far more math than we had realized." We have to be able to understand a variety of methods of getting to an answer and we have to be able to see where kids are going wrong in all of these methods. Given that many of us had bad experiences, it is not surprising that this sometimes appears to be an overwhelming task. [Marcia, MJ3] Experience and practice, some of our preservice teachers thought, would provide them the skills to be able to teach in such a way. They welcomed the opportunity of working with students during our methods course so that they could learn and get familiar with "children's methods," and "children's views and opinions about mathematics." Having read and responded to their students' first letter, our preservice teachers wondered and looked forward to their students' responses. Many, like Linda, were "curious to know their answers and how they justified them" [MJ2]. Not many, however, seemed to expect or anticipate having much difficulty interpreting their students' written work. Only Megan and Sally wrote in their journals about their concerns that their students may not share their thinking with them either because they may "fear making errors" (Sally), they may "not find it fun" (Megan), or due to a "language barrier—can students express what - 121 -they are thinking on paper?" (Megan). It was, therefore, with great anticipation that preservice teachers awaited for their students' responses. Focusing on General Features of Students' Work Despite indications that our preservice teachers were inspired by the idea of listening to their students' mathematical thinking, once faced with their students' workings, our preservice teachers focused instead on the general features of their students' work. In particular, our preservice teachers tended to focus on the correctness of their students' answers. "Yes! They got it," "This kid screwed up," "Wow, was she ever off were some of the comments our preservice teachers made aloud while reading their students' response letters. Most preservice teachers' initial comments (in their journals and in their letters) referring to their students' work also tended to focus on the overall correctness of the students' answer. They tended to accept students' right answers as evidence of understanding and to assume students' wrong answers as signs of students' confusion and carelessness. This is apparent in their initial responses to their student penpals' work. In their letters preservice teachers congratulated students for getting the right answers and corrected those with incorrect answers (this is explored further in the next chapter). These tendencies were also apparent in their initial journal entries where preservice teachers wrote little about the meaning of students' work. They noted, for example, whether students were successful answering their questions and made no specific inferences about students' mathematical understanding. Here is an example of what Lesley wrote about her student's work after the following penpal exchange: 2 Lesley: Draw a picture to explain 3 x —. [ml3] Lesley's Ppal (Lynn): The teacher helped me figher [figure] it out this one. [ml4] - 122-In her journal Lesley later commented: I'm hoping she just got confused as to where to put the numerator and the denominator because her diagram is accurate. But the written form should be j according to her diagram. She may have just been careless. [Lesley, MJ4] In their journal entries preservice teachers' comments about the students' work focused on describing surface features—particularly correctness—of the students' work. This concern for, and focus on, such general features tended to draw preservice teachers' attention away from the details and meaning of the students' work. Our preservice teachers' initial difficulties seeing more than the correctness of their students' answers is illustrated by Nilsa's journal observation (below) about her student's mistake adding large numbers. Nilsa had missed the first letter writing class and I had replied to her student's letter instead. Having copies of both her student's letter and my reply to it Nilsa commented (both orally and in her journal) on the differences between what she and I had noticed. It was interesting that in her first letter she said "math in this class is prity isey [pretty easy]." Yet in her two letters it is very apparent that the math is not coming easy for her. She gave these problems [as an example of the mathematics she can do]. letter #1 71539642 At first I thought "wow was she ever off!" perhaps she had put the wrong sign down and had subtracted the numbers rather than adding them. Sandra caught this and pointed it out to her. [Nilsa, MJ3] Nilsa's inattention to the details in the students' work was not an isolated case nor confined to students' wrong answers. Preservice teachers whose students had reached the correct answers also seemed unaware of, and inattentive to, what they could learn from their students' work. Some simply commented that their student: "did a good job at explaining how she got the answer" (Linda); was "successful" answering their question + 16475265 55064377 Obviously this answer is wrong. But I did not even think [until I read Sandra's reply] that - 123 -(Mitch); or "seems advanced in math and writing compared to others' letters" (Terry). Others, like Thea and Carly tended to focus on what they felt was missing in the students' work: "[students'] answers are short and contain little information, "or that "their comments were fairly brief and didn't cover why they thought what they did." Likewise, Megan noted that her students had reached the correct answer and without delving into their work's meaning she focused on the kind of information she needed in order to • understand her students' thinking. Mary and Jake both got the horse problem question correct, but Mary just gave me her answer and not how she figured it out. Jake did show me his work, but it would have been more helpful to me if they gave me some of the rough work. [Megan, MJ3] While Thea, Carly, and Megan were beginning to expect more than just the final correct solution from their students—they expected more explicit and elaborate articulations of their students' thinking—they did not investigate or explore what their students' solutions may reveal. This could be because, as Megan pointed out, their students had "solved the problem the same way [they] did" or perhaps because, like Nilsa, they were not able to recognize meaning in their students' work when explained in unexpected or unfamiliar ways. Marcia and Sally are good examples of the latter. Consider Marcia whose only reference to her students' incorrect work read: "I'm surprised John didn't get the answer to my math problem" [MJ3]. Similarly, Sally admitted to being baffled by her student's work. "I'm not sure why she did this, and to be honest, I was so pressed for time in reading and responding to 2 letters, that I didn't think to ask her," wrote Sally in her journal afterwards, making no further written attempts to make sense of her student's thinking [MJ3]. Looking for Clues into Students' Understandings Preservice teachers' comments about their students' work reveal that they had very few tools and strategies for interpreting students' mathematical work. Preservice teachers seemed, initially, more willing to focus on the correctness of their students' answers - 1 2 4 -rather than on trying to figure out what and how the student may be thinking or understanding. They were paying little attention to the details in their students' workings, which limited their insight into their students' thinking. Students' unfamiliar and inexplicit work, however, began to encourage our preservice teachers to focus on more than the correctness of their students' work and to look for other clues into students' mathematical understanding. The challenges involved in making sense out of student's mathematical work came as a surprise to many preservice teachers who seemed to be expecting the interactions with their students to be a smooth and effortless transaction. Miriam, for instance wrote in her journal: "If she does not get the problems, I will probably lower the level of difficulty. Or if she does understand the problem, I will probably have to make up some more difficult questions" [MJ2]. This comment suggests that Miriam had conceived the task of interpreting students' work as a matter of deciding whether the student had gotten the answer to the problems she had posed. For our preservice teachers, like Susan, the complexity and the inherent fallibility involved in interpreting their students' work came as a slow and late realization for which they were found unprepared. My penpal does not hesitate to ask others for help, it appears. She asked her mother to help her on the first one and her teacher to help her in the following letter. However I am wondering how much external help she receives from these other sources. If she receives quite a lot of help, I am not able to evaluate her accurately in terms of her performance in math, Thus, I am unable to find out what kinds of math problems she can and cannot do by herself. As a result my expectations of her are distorted. If I ask her not to ask anyone for help, she may not be able to accomplish the task by herself. It seems like it is a double-edged sword. I am not sure what to do. [Susan, MJ4] These experiences were emotionally and intellectually challenging for our preservice teachers but provided them with insights into the difficulties involved in interpreting and understanding students' mathematical thinking. Many of our preservice teachers began to realize that deciding what and how much a student understands is not as simple as - 125-categorizing their work into right or wrong. Sally, for example, said: "They may get the right answer but it doesn't mean that they understood the question" [MJ4]. For others, these difficulties were encouraging them to see that our interpretations of students' thinking are enhanced and limited by how and what the students choose to share (and their ability to communicate their thinking) as well as our willingness and ability to notice and make meaning out of their work. Carly, for instance, said: "I find it difficult to understand how other people think through math. ... Usually peop