1:. 5.3.4.1.. . at}: .. 49 r Fun » . .2- i... .g , . .. . x ..5. 1 gum.» x: . u ...i .1 :3: J: a . if... a 25...}. .lko...l.fl .L .529 2...: 3 . ..3 t a 1.: . .5 455.... 6...... .JWVR...§ .: .3... . f hammer, . .7 I‘VI’J: . ,x‘ {Ia-«1.9. - 2.3. a! :24... I. ..w. ..£_,..-.......v.~nnfi_ 7». 579. ...L... A x 3...... .fifisfinfl 1..,fi.,.....§.:i In. . .241- 5zi .5 43:5,. . .. $4.99,»: .3 Y (Nb-Y- ; :35: , sh It.(..\ 3|... 5.45 5.1;. VIE... . .2 4 2 OJ. «.3. Sin-5.. .- s 37!: A In! x "90‘. . 5... a 94.: .. ., ti... 1.- .21.}:v {...r: I ' THESIS q 2001p I This is to certify that the dissertation entitled CULTIVATING UNFAMILIAR TERRAINS: A STUDY OF A PRE-SERVICE MATHEMATICS METHODS COURSE DESIGNED FOR ADAPTATION presented by KARA SUZUKA has been accepted towards fulfillment of the requirements for the Doctoral degree in Teacher Education Ilium... Y Ivaajor Professor's Signature ELMW I4 i «1005 Date MSU is an Affinnaflvo Action/Equal Opportunity Institution ._——_——— LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE I 'ALAL’ I I" "‘ ,0 “Mm ,_ LA ‘d UTIL J 1.1.x 2/05 p:/ClRC/DaIeDue.indd-p.1 CULTIVATING UNFAMILIAR TERRAINS: A STUDY OF A PRE-SERVICE ELEMENTARY MATHEMATICS METHODS COURSE DESIGNED FOR ADAPTATION By Kara Suzuka A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Teacher Education 2005 ABSTRACT CULTIVATING UNFAMILIAR TERRAINS: A STUDY OF A PRE-SERVICE ELEMENTARY MATHEMATICS METHODS COURSE DESIGNED FOR ADAPTATION By Kara Suzuka This study investigates one attempt to prepare prospective elementary teachers to teach mathematics differently from the ways in which they were taught. Based on a theoretical approach articulated by Joseph Schwab (1959/1978), an elementary mathematics methods course was designed to provide pre-service teachers with an educational context that embodied the unfamiliar forms of teaching, learning and knowing they were to learn in the class — in Schwab’s terms, a context was designed where the new is presented mainly in its new terms (p.172). The course design was deliberate about the types of work and ways of working encouraged, the tasks and problems used, the tools and material means provided, and the social-intellectual structures supported. These were purposefully targeted to produce a class context where new habits and expectations for mathematical work would be fostered and in which different kinds of knowledge and ways of knowing mathematics would prove valuable. This study looks at what and how changes were initiated in the course: What might it mean to create an educational context for pre-service teachers where “the new is presented mainly in its new terms”? What might be involved in creating such a context? This dissertation also examines the processes by which change occurred over the semester as prospective teachers adapted to — and adapted — the context in which they found themselves. In essence, this study is a test of the theory articulated by Schwab, examining not only whether or not the theory holds in the given case but also to elaborate upon the theory as it is applied to an elementary mathematics methods course. This study contributes to a better understanding of such efforts in mathematics teacher education to create and use educational contexts to communicate new forms of knowing and learning to prospective teachers. Copyright by KARA SUZUKA 2005 To my parents, Reginald and Katherine Suzuka who taught me the value of education and the joy of learning. ACKNOWLEDGEMENTS I’d like to thank a few of the people who have supported me in this effort over the years. I extend my heartfelt gratitude to — - My committee, who has generously given of their time and talents to bring this work to fruition: Deborah Loewenberg Ball, Helen F eatherstone Sharon F eiman-Nemser, Lynn Paine, Brian Delany, and Sandra Wilcox - The Compers of the Round Table, my friends with whom I have shared trials, tribulations, triumphs, and a lot of laughter, gossip, food, and drink: Dirck Roosevelt, Carol Barnes, Sue Poppink, Steve Mattson, and Sarah Lubienski - My friends who have given me a home away from home, incorporating me into their lives: Deborah, Richard, Sarah, Joshua, and Jacob Ball, Helen, Jay, and Miranda F eatherstone, and Sharon, Louis, and Daniel Nemser - My family, who has supported me and continues to support me in all things: Reginald and Katherine Suzuka; Keith, Faith, and Kelly Suzuka; Randall, Corinne, Faith, Ryan, and Aaron Suzuka. And, finally, I’d like to express a special thanks to my dear friend, Mark Rosenberg, whose generosity, kindness, and unfailing friendship never cease to amaze me; to my hero, Helen F eatherstone, who has been my champion and has Provided wisdom, fun and enjoyment in the work, and a safe place to rest, vent, or cry When I’ve sorely needed these things; and to Deborah Ball, whose extraordinary work Started me on this path and whose teaching, support, and friendship have seen me through the long journey... and on to new horizons. vi TABLE OF CONTENTS LIST OF TABLES x CHAPTER 1 Cultivating Unfamiliar Terrains 1 Introduction .............................................................................................................. 1 The Problem: Preparing Teachers to Teach in Unfamiliar Ways ............... 2 Focus of the Study: A Paradoxical Approach to Addressing the Problem ................................................................................................ 11 The Experiment: Cultivating Unfamiliar Terrains ................................................ 14 The Design Experiment ............................................................................. 15 Applying Schwab’s Idea to a Mathematics Methods Course .................... 18 Design Features: Conjectures About and Designs for the “New” ............ 25 Data Collection and Analysis ..................................................................... 51 Overview ................................................................................................................ 6S Analytic Chapters: Chapters 2-4 .............................................................. 65 Concluding Chapters 5-6 ........................................................................... 70 CHAPTER 2 Introduction of the Unfamiliar Terrain 72 Introduction ............................................................................................................ 72 The Primary Analytic Lens: The Thing in Itself and Human Constructions ...................................................................................... 73 Problem Analysis: The Three-Coin Problem ........................................................ 77 Validation ................................................................................................... 79 Creating Mathematical Knowledge ........................................................... 85 Possibilities for Teacher Education ........................................................... 92 Episode Description: The Struggle to Understand ................................................ 96 Set Up of the Problem (Monday) ............................................................... 96 Independent Work on the Problem ............................................................ 97 Whole Group Discussion: Danielle, Iris, and Alisa’s Solutions ............... 98 Whole Group Discussion: Shelley’s Conjecture ..................................... 100 After Class ............................................................................................... 103 Other Explorations (Tuesday — Thursday) ............................................... 104 Whole Group Discussion: Initial Literacy and Math Experiences (Friday) .............................................................................................. 105 Shelly’s Out-Pouring ............................................................................... 110 After Class ............................................................................................... 1 12 Episode Analysis: The Thing in Itself and Human Constructions ...................... 113 Description of Salient Facets ................................................................... 115 vii IIIII’IER 3 huMbhmer I330: P's"! ,‘. i\ L :5 film (HIPIER Dicemmen Intro- PTUI‘, Discussion of Salient Facets .................................................................... 136 New Patterns ............................................................................................ 155 CHAPTER 3 Establishment of the Unfamiliar Terrain 157 Introduction .......................................................................................................... l 57 The Primary Analytic Lens: Constraints and Possibility ........................ 158 Problem Analysis: The Cookie Jar Problem ....................................................... 167 Top-to-Bottom Paths and Bottom-to-Top Paths ...................................... 169 Paths to Answers of 16 and Paths to Answers of 24 ................................ 171 Summary .................................................................................................. 188 Possibilities for Teacher Education ......................................................... 198 Episode Description: Translation of Tentative Understanding into Action ........ 194 Introduction of the Whole Group Discussion .......................................... 195 Kim and Deidre’s Solution for 16 Cookies .............................................. 195 Tess and Megan’s Solution for 24 Cookies ............................................. 198 Teri’s Guess’n’Check Solution for 24 Cookies ....................................... 202 Danielle’s Bid for Iris to Share her Solution ........................................... 204 Kathy’s “Pie Graph” Solution for 24 Cookies ......................................... 205 Shelly’s Distinction Between Cookies EATEN and Cookies LEFT ....... 209 Episode Analysis: Constraints in the Unfamiliar Terrain ................................ 211 Varieties of Constraint ............................................................................. 212 Changing Beliefs vs. Changing Norms .................................................... 222 After Class ............................................................................................... 224 CHAPTER 4 Discernment of the Unfamiliar Terrain 226 Introduction .......................................................................................................... 226 The Primary Analytic Lens: Difference That Makes a Difference ......... 227 Problem Analysis: The 1 3/4 -:— 1/2 Problem ....................................................... 236 The Problem ............................................................................................. 237 Meaning Making ...................................................................................... 238 Error Identification and Analysis ............................................................. 240 Distinctions and Definitions .................................................................... 244 Episode Description: Reflection on Discrepancies ............................................. 248 Preparation for the Whole Group Discussion .......................................... 249 Kathy’s Story for 1 3/4 + 1/2 .................................................................. 250 Discussion of Kathy’s Story .................................................................... 254 Discussion of our Discussion ................................................................... 258 Episode Analysis: Difference That Makes a Difference ..................................... 266 Description of Differences ....................................................................... 268 New Information Flows ........................................................................... 273 viii CHAPTER 5 Epilogue 274 Shaya’s Question (December 6) .......................................................................... 277 CHAPTER 6 Conclusion 231 The Struggle to Understand ................................................................................. 281 The Actions Undertaken Lead to Unexpected Consequences ............................. 282 There’s Reflection on Disparities ........................................................................ 283 New Competencies Are Roused .......................................................................... 283 BIBLIOGRAPHY 285 ix LIST OF TABLES Table 1.1 Types of Work and Ways of Working for Broadening “Mathematics” ...................... 29 Table 1.2 Types of Work and Ways of Working for Teaching Mathematics ............................ 31 Table 1.3 Expectations for Collective Participation .......................................................... 34 Table 1.4 Expectations for Individual Participation ......................................................... 36 Table 1.5 Features of the Mathematics Problems ............................................................ 39 Table 1.6 Features of the Collection of Mathematical Tasks .............................................. 43 Table 1.7 Tools and Material Means for the Class ............................................................ 47 Table 1.8 Tools and Material Means for Individuals ........................................................ 51 Table 1.9 Semester and Data Collection Schedule ........................................................... 54 Table 1.10 , Settings for Investigation and Course Themes Investigated .................................... 61 Table 1.11 Chronological Organization of the Data Analysis Chapters .................................... 63 Table 1.12 Summary of the Analytic Chapters (Chapters 2-4) ............................................. 7O CHAPTER 1 Cultivating Unfamiliar Terrains Introduction Finding effective ways to initiate and sustain fundamental changes in the way mathematics is taught by new generations of teachers has been an important problem in the United States in recent years. In the wake of Significant efforts to articulate ambitious national and local standards for mathematics education,1 the challenges of preparing people to teach mathematics in ways they have not experienced, to use and know mathematics in unfamiliar ways, has gained much attention in the field of teacher education. This study looks closely at one attempt to prepare prospective teachers to teach mathematics differently from the ways in which they were taught. Based on a theoretical approach articulated by Joseph Schwab (1959/ 1978), I designed an elementary mathematics methods course to provide pre-service teachers with an educational context that embodied the unfamiliar forms of teaching, learning and knowing they were to learn in the class. This dissertation examines the approach put forth by Schwab — What does the approach entail? What might it mean to apply the approach to an elementary mathematics teacher education course? It also investigates the teaching and learning processes that were predicted by the theory — and those that resulted. ‘ Perhaps the best-known are the series of “Standards” documents developed by the National Council of Teachers of Mathematics (NCTM). Four documents were released covering different aspects of teaching and learning school mathematics: curriculum and evaluation standards (1989), professional standards (1991), assessments standards (1995), and an integrated document of principles and standards (2000). Tim“, 5', IA“- . -L;’ A 4‘!’ ‘ ‘. rim. .ww' ‘- I-‘IHT‘ ‘ «q HM: 4n“ LA : l u a...“ , . '33,“: 'w I xiglgfi. AA-ur , . . 1 tar-,1? a.) Tiicil§s \ I .4 - 1 ~n‘CJ"vfi.‘ 9'0 LU-ME'Etnig u s L)” (L 'U , l ' . H -. \ .. V :L . . ‘ULII LJI\ . ‘rb, " (Quin _I,' l , . . sites 8:; l 4A.\ 5 Gem} . This study contributes to a better understanding of such efforts in mathematics teacher education to create and use educational contexts to communicate new forms of knowing and learning to prospective teachers. In this introduction I provide a general overview of the problem, discussing several important ways people have viewed and explained the problem. This is followed by a brief description of the particular approach examined in this study for framing and addressing the problem — what I refer to as, “Cultivating unfamiliar terrains”. The Problem: Preparing Teachers to Teach Mathematics in Unfamiliar Ways Despite the enormous changes that have occurred in American society over the last fifty years - changes in culture, technology, and lifestyle; changes at home, work, and places of recreation - mathematics classrooms in the United States have remained much the same (National Commission on Mathematics and Science Teaching for the 21St Century, 2000). Commenting on the videos of US. classrooms that were examined in James Stigler and James Hiebert’s detailed study of 8th grade classrooms in Japan, Germany, and the United States,2 Steve Olson, writes, Watching the US. tapes is like being transported back to your own 8th grade mathematics class. Teachers struggle to keep students’ attention. Kids are as monosyllabic as ever. Sometimes only the nearly ubiquitous presence of overhead projectors marks these classes as different from those of the ‘50s (p.29). 2 The findings of this study were published in Stigler and Hiebert’s ( I 999) book, The Teaching Gap. This video study was part of the Third lntemational Mathematics and Science Study (TIMSS). A follow up to the 1995 video study was conducted four years later; this later study was specifically designed to explore the “distinct patterns of mathematics teaching in different countries” (US. Department of Education, National Center for Education Statistics, 2003, p. 9). The results of the 1999 video study can be found in a publication by the US. Department of Education’s National Center for Education Statistics entitled, Teaching Mathematics in Seven Countries: Results From the TIMSS I999 Video Study. I,\ I . err; In: u . . . M, ~-h. '1"- .L .bJLn ..lu~" -,'.Yj"- fl ‘l 511.3%“ 1 u V ‘ ,. 01.. Vh.‘ , ~ib Lune-l“ - I :anjfl. 1Q; _-u>WL " . 1 I .. ‘33-‘HP’C Li; >§\O‘ld5‘ -' “P" a it Lb; -ihu5\\a- .L' .. ....:?€.L. .\l WI l“ ‘ U - - HM v-~ .. ‘7 15"}.1 En\ : u5.ust\l.. ”V"? "A v LL ..~\"~ i5n\‘b mCIlCCS OI “1):. Tr l.\ Ir LI:“!-)".) 9 ‘. “5.5nL Ln l .‘f‘ V l‘ .34 _ l\ Ah ‘1: 0. {If Among the constants that have persisted over the years, the methods and approaches used to teach mathematics in the United States have proven quite resilient (Fey, 1979; Stigler & Hiebert, 1999). Repeatedly, it has been found that teachers tend to teach as they have been taught (Ball, 1988b, 1990; Cuban, 1984; Lortie, 1975; Wilcox, Lanier, Schram, & Lappan, 1992). Even when apparently significant change has been initiated and observed in teachers’ practices, it has been noted that teachers will often “revert” to the teaching methods used by their own teachers when faced with challenging classroom situations (Hiebert, Morris, & Glass, 2003; Borko, Eisenhart, Brown, Underhill, Jones, & Agard, 1992). Additionally, not only has mathematics teaching been consistent across generations of teachers, but researchers have also found a surprising degree of consistency in how mathematics is taught throughout the United States (Olson, 1999; US. Department of Education, National Center for Education Statistics, 2003).3 This constancy in how mathematics is taught — the persistence of teaching practices over time and across contexts — has been examined and explained in different ways. Three particularly seminal viewpoints have been put forth, each one highlighting a different facet of the problem. One perspective points to important mathematical knowledge, skills, and learning experiences needed by but ofien found wanting among teachers; another position views teaching as a cultural activity involving deeply embedded beliefs and practices; and finally, a third viewpoint considers the ways in which people have understood the nature of knowledge, teaching, and learning. Each of these is briefly described and discussed below. 3 Olson (1999), for example, notes that the Stigler and Hiebert expected to see a great diversity in teaching styles among the American teachers in their video study and were surprised to not find it. “Everywhere, it seemed, American teachers taught using pretty much the same uninspiring methods. Teachers largely drilled their students on low-level procedures.” .' - , Q. I321. 161'- O; 51.: ‘ . '311-‘7‘3 Lkzihflg.\c Lack of Important Mathematical Knowledge, Skills, and Learning Experiences One important perspective on the problem focuses on the needs that surface for teachers as they take on the demands of teaching mathematics. The study of “needs” looks at the disparity between what teachers bring to the work and what is required of them. In the following discussion, I briefly summarize some of the relevant findings that have surfaced from research on what teachers bring to teaching; I then go on to describe two lines of inquiry that have examined, from different angles, the demands placed upon teachers of mathematics and the needs that thereby arise. Studies that shed light upon the mathematical knowledge, skills, and dispositions — along with the beliefs and assumptions — that people bring to teaching have served as an important foundation for other work. In one such study, Deborah Ball (1988a), conducted a series of interviews with prospective elementary and secondary teachers. Among the things uncovered in her work, was a strong, persistent tendency for pre- service teachers to view mathematical knowledge as a static collection of facts, formulas, and procedures - to see problem solving as the correct use of pre-determined steps to arrive at an answer. Ball found her subjects had given little consideration to how mathematical ideas are generated or develop over time. Along with this, she found that the prospective teachers in her study tended to view mathematics teaching as the presentation of a static body of mathematical information and computational procedures; they saw the teacher’s role as that of helping students learn to follow pre-determined steps to find answers to textbook-type questions. In most cases, the prospective teachers had not thought about the tasks of teaching beyond telling and showing students how to do mathematics problems. In one interview, a pre-service teacher asked, “What other 3 ‘9‘ \1 L“ I‘: ‘0‘")? N! 4.. .. m K ' . _l Itchit’lS 1c cxwfic‘“ ‘0 1‘5th task would there be?” Not surprisingly, Ball found her interview subjects unprepared to interpret or respond to students’ work and ideas beyond determining “correctness.” Findings such as these that uncover what people bring to teaching have laid important groundwork for efforts to identify the needs of teachers. One line of research that has been built upon this work has examined the demands that different educational reforms have placed upon teachers and the needs that have surfaced for teachers in their efforts to enact the envisioned changes within their classrooms. Giving careful consideration to the types of mathematical knowledge, skills, and learning experiences teachers were being asked to foster with students and those that teachers typically bring to their work, researchers noted inherent challenges — impediments — to such reforms. For example, in a 1990 study of policy initiatives in the State of California that were designed to bring about ambitious change in instructional practice, David Cohen and Deborah Ball framed a fundamental dilemma: “How can teachers teach a mathematics that they never learned, in ways that they never CXperienced?” (p. 238). As products of the very educational systems targeted for change, teachers often had not had opportunities to develop precisely the types of mathematical knowledge, skills, and experiences for which reformers were calling. Without such opportunities to expand their understanding, teachers drew upon what they had seen, learned and experienced to inform their interpretations of standards and frameworks, their implementation of policy, and their use of available resources. Ofien, this resulted in conserving rather than transforming classroom practices. Another line of inquiry has focused on articulating the special ways in which teachers must know mathematical content for teaching. Ofien labeled “pedagogical .. a'; 78323811. 11' ”Th0 .t‘fi‘. *Lé“ t r241.) vi ~ «4.53.15 . a“; «'s 4‘ ..pe...-.nt l ...L‘ " L \ uq‘u xU\- 'aais- k V 9”,. 0 o "a“ u“‘u\a.-L. w ‘-' 3‘; w' i . "(EH1 [Or {'6‘ if ‘1“ w, . Mddfc’d :‘1 content knowledge” (Shulman, 1986, 1987; Wilson, Shulman, & Richert, 1987), this form of mathematical understanding involves “unpacking” ideas and procedures beyond basic facts and computational steps. For example, in addition to the calculation skills entailed by the procedure for dividing fractions, teachers should understand what it means to divide by a fraction, be able to explain why “invert and multiply” works, and to represent it visually. Unlike the highly generalized, abstracted forms of knowledge that tend to dominate prospective teachers’ experiences and images of mathematics, the mathematics needed for teaching often includes concretized, detailed, and context- dependent forms of knowledge that are not studied or encountered as a normal part of schooling but are, nevertheless, crucial for interpreting and addressing learners’ ways of thinking. Still, this isjust one piece of a much larger puzzle; the picture of what teachers need for teaching mathematics is far from complete. Early efforts to design courses to help teachers extend their mathematical understanding in these ways have often proved disappointing. Researchers found in many cases that although prospective teachers Studied unpacked forms of mathematical knowledge and learned to unpack ideas themselves, the connections to their practice were oftentimes weak (e.g., Borko et al., 1992; Wilcox, Lanier, Schram, & Lappan, 1992; Schram, Wilcox, Lanier, & Lappan, 1988) This work, however, is ongoing. More recently, there have been efforts to develop a deeper and more complex understanding of the mathematical work of teaching mathematics and the specific content-related demands of the job (Ball & Bass, 2001; Lampert, 2001; Ma, 1999). Researchers have examined the mathematics teachers ‘v . mix 3.0 1 “no" ‘. 1 . \I ‘ l"\ '7‘“!- L-- ' .5-; -m 1.)?” . Y \ \ .ob‘fi’” .3.. v "I 7” ;.M w...» ...u ... 1 ,-e - J .Fy .4-..\s ...N Dyva l 3' 4.05131. h r l . . .l‘ \d...a .4 ‘ T:TLI:‘M“‘.I k-wlxllluu\ listen I 19 Hit. . Cluural KI {/1 , o It routinely do in the course of teaching such as, preparing lessons, assessing students’ work, responding to students’ questions, orchestrating class discussions, and following up on students’ ideas. Through this work they have identified forms of mathematical knowledge and modes of knowing mathematics that are quite different from the mathematics people learn as students and bring to their work as teachers — and which are unlike the mathematics needed in other mathematically demanding fields. There is, however, still much to learn about what this means for teacher education. Culturally Embedded Beliefs and Practices A second important perspective has studied pervasive models of teaching that have been found deeply embedded within cultures. In their well-known study of mathematics teaching in Germany, Japan, and United States, James Stigler and James Hiebert (1999) found patterned, shared images of what classrooms should look like and how they should run that permeated throughout, but differed among, the countries they studied — what they have come to refer to as cultural scripts (Stigler & Hiebert, 1999). Cultural scripts have several notable features: ° Scripts are shared among teachers, students, parents, and others in the culture, shaping everyone’s expectations of classrooms and teaching. Stigler and Hiebert note, In fact, one of the reasons classrooms run as smoothly as they do is that students and teachers have the same script in their heads; they know what to expect and what roles to play (p.87). ' Scripts evolve over time, across generations, and fit within a stable web of beliefs about the nature of knowledge and knowing in the subject matter, about learning, and about the roles teachers play in classrooms. ‘ » n‘ f.) '7 gkxl.i) 551“ .y. ‘IWL‘I “7.x . I 5 13'3”“ ‘ ,' 5‘25.dl1‘ M“ I .1\ 111.11 ' , h‘? A-ls J31. \\ ilk P70 1.7. . .;. Luhlelixtx. r it"... I ~- dl‘IWJII it Common 1} 35,3121)“ ' I 1.11, WA Ind-- P" ' ‘- Q If 0 Scripts are distinctive, varying by country. Stigler and Hiebert’s observations about cultural patterns seen in US. classroom lessons reinforce many of the findings noted by researchers, like Ball (1988a) and Ma (1999), who have studied the views, knowledge, and skills of US. teachers through carefully designed tasks and interviews. For example, Stigler and Hiebert note, The typical U.S. lesson is consistent with the belief that school mathematics is a set of procedures. Although teachers might understand that other things must be added to these procedures to get the complete definition of mathematics, many behave as if mathematics is a subject whose use for students, in the end, is as a set of procedures for solving problems (p. 89). Likewise, regarding teaching, Stigler and Hiebert observed — U.S. teachers appear to feel responsible for shaping the task into pieces that are manageable for most students, providing all the information needed to complete the task and assigning plenty of practice. Providing sufficient information means, in many cases, demonstrating how to complete a task just like those assigned for practice (p. 92). Teaching is fundamentally a cultural activity and, as such, is “notoriously difficult” to change (Hiebert, Morris, & Glass, 2003). Some views are so widespread and common that people within the culture fail to see that alternatives might exist. Additionally, such views can be so deeply embedded within the culture, reaching all segments of the population, that significant change to classroom instruction involves not only individual learning but cultural —— societal — change as well. As Deborah Ball (1996) argues in her comments about educational reform efforts in mathematics, Because the mathematics reforms challenge culturally embedded views of mathematics. we will find that realizing the reform visions will require profound and extensive societal and individual learning — and unlearning — not just be teachers, but also by players across the system (p. 501). ‘ vn‘ "L1. ‘1“ -g..\ - rasuzc flurzhcna EKNE aitfilcon: 1 1 MT" ’4)". Lu 1 ’1‘” u MHU \AI " w etha.EL V mcrdil} : “35 “lilt‘l'llf‘ bilurned hamt( Contnucd‘ ‘PilI Epistemological Roots Finally, a third perspective that sheds light on the problem takes a close look at the fundamental views of knowledge, teaching, and learning that lie at the heart of classroom teaching practices, examining their evolution across a long history of human thought. In David K. Cohen’s 1988 essay, Teaching practice: Plus que ca change... , he argues that our modern classrooms practices embody long-lived and deeply rooted ideas about the nature of knowledge, teaching, and learning. For example, he notes that contemporary instructional practices tend to treat knowledge as “objective and stable” — as if “it consists of facts, laws, and procedures that are true, independent of those who learn and entirely authoritative” (p. 39). He traces this view of knowledge back to medieval Europe where educated men worked with texts that were hand-copied, incredibly rare, and often considered sacred. Knowledge, preserved on the written page, was “memorized and analyzed with minute attention and considerable deference” (p. 40) by learned men who endeavored to maintain this knowledge — and pass it on — in its integrity. Cohen goes on to argue that even with the growth of science, knowledge continued to be viewed as “factual, objective, and independent of human distortion” (p.41). In contrast, ideas about knowledge, teaching, and learning that inform a variety of contemporary educational reforms are relatively new on the scene. For example, the idea that disciplinary knowledge is a human creation — refined and revised and, at times, refuted and re-conceptualized within a community of peers — has only gained serious attention in recent years. Scholarly works such as Thomas Kuhn’s (1962) book, The D ."ri )‘ find“: (K v1 p h ' ‘.; Jr,“ hi “'“H I ’1‘|-‘ ‘ilj‘k : 'u u ' ~)G'FK‘V.J ‘ sleL-lfi gush 1' ,' 0‘ MEL. :0? C Ian; «:01; ' Jun: Led-11‘ $1113.11: KNIVES: I were m r 6... nel‘x‘mars. -- It lg 51:35.- \. ‘. T ‘— httunggp installer 1 notes, “I “Mix 3. tr». me. TUEC’E,” » "‘5. Structure of Scientific Revolutions, Karl Popper’s (1962) collected essays, Conjectures and Refutations: The Growth of Scientific Knowledge, and Imre Lakatos’ more recent work (1972), Proofs and Refutations: The Logic of Mathematical Discovery have contributed enormously to re-conceptualizing knowledge as emergent and tentative. Kuhn, for example, traced through compelling historic examples of scientific progress to repeatedly show how paradigms of scientific thought that were dominant at one period of time came to be scrutinized by scientists and were eventually replaced — Kuhn’s idea of the “paradigm shifi” raised new possibilities for thinking about the nature of knowledge. Compared with the “venerable inheritance” of ideas about the nature of knowledge, teaching, and learning that have informed the development of schools and schooling through the centuries, the ideas that impel present-day educational reforms “were born yesterday” (Cohen, 1988, p. 43). Herein lies a fundamental challenge for reformers. The roots of current classroom practices are long and deep: This venerable inheritance is certainly part of the cultural script that has been seen and identified in the United States of America — but it also pre-dates the existence of the US. Additionally, the conceptions of knowledge, teaching, and learning that undergird many of today’s educational reforms are neither widely shared nor well established. As David Cohen notes, “These conceptions. . .. are also a relatively recent, still controversial, and very weakly developed product of modern intellectual culture” (p. 35). These three perspectives for viewing and explaining the resilience of mathematics instruction in the United States offer a variety of insights into the nature of the problem. Together, they reinforce the idea that bringing about change in how mathematics is taught requires extraordinary learning for teachers. It involves learning forms of mathematics 10 ' I I ~fi'lp‘\- “1.“..111‘ “I 4:": ‘I' I" ‘I l. ‘-u\flli .\.r We. list" P01; Mi and modes of doing mathematics which teachers have not had opportunity to learn or experience before. And, perhaps more significantly, it involves developing new understandings about mathematical knowledge and knowing — about teaching and learning mathematics - that depart significantly from what they, and those around them, have always known and believed. Focus of the Study: A Paradoxical Approaph to Addressing the Problem Unlike so many other things that are simply learned by augmenting or refining what has been learned in the past, this type of learning places teachers in an unusual and difficult situation. As Sharon Feiman-Nemser and Janine Remillard (1996) note — While current beliefs and conceptions can serve as barriers to change, they also provide frameworks for interpreting and assessing new and potentially conflicting information. That is the paradoxical role of prior beliefs. Like all learners, teachers can only learn by drawing upon their own beliefs and prior experiences, but their beliefs may not help them learn new views of teaching and learning advocated by teacher educators (pp. 80-81). Such learning involves developing new conceptions of knowledge — while struggling to suspend and call into question one’s own views about what counts as knowledge and what it means to know. It involves developing new modes of knowing and learning when one’s current ways of knowing and learning are unable to grasp the new. The situation is, in many ways, paradoxical. In the follow discussion, I describe an analysis written by Joseph Schwab in 1959 of a comparable paradox encountered by John Dewey in his attempts to communicate his progressive vision of knowing, teaching and learning. 11 Ii. vii. ‘ H“ "JJ5MII‘H‘I‘I b \ D's“: “up. Au¥t ,‘m. “It II‘I 1“” 1' M; I w I l t‘ ‘7. hi: 5“ . ' L ' ‘1‘] ‘, .11 ixmm L.“ iiifilfi,‘ I ‘ ‘w-IA‘A“ n _ s g c. . l »b» . I) ARV" £115.) In 'H. ' """xmt ”1““!4H\itl.) “Ii. inn 1:, 1:". 11:: a: f [I ll‘ icct'iiw . ‘»su.,,_ 93-01 Var; n,‘ “nth d“ W In Schwab’s essay, The “impossible” role of the teacher in progressive education, he describes a fundamental paradox of learning — Dewey seeks to persuade men to teach a mode of learning and knowing which they themselves do not know and which they cannot grasp by their habitual ways of learning. It is the same problem of breaking the apparently unbreakable circle which Plato faces in Meno and Augustine in his treaties, 0n the Teacher (Schwab, 1959/1978, p. 170). A special challenge exists when learning new forms of knowledge and new modes of learning. In such efforts, a paradoxical situation arises where one’s understandings and beliefs about knowledge and one’s ways of knowing and learning are, at once, both instruments and objects of change. Schwab plays out this idea, explaining two different ways a teacher might approach this problem. One approach a teacher might take is to convert the new into familiar terms and present it in ways that are accessible to the learner. This, however, will inevitably fail: Schwab explains — If the new... is entirely converted into the terms of the old, a static and unrecognized misunderstanding is likely to result. The poisoning occurs at the source. The hearers experience no struggle to comprehend. Hence, they do not know that they do not fully know (p. 171) According to Schwab, not only will misunderstanding result as learners come to grasp a poor variant of the new, but they will have no inkling that their understanding is flawed: “they do not know that they do not fully know.” A second possible approach a teacher might pursue is to leave the new in its own terms, despite the fact that the learners do not currently have the background or resources needed for comprehension - 12 “ mat t“ ‘ ”I 3.1.1:; J. L\ ‘5 WW! v ‘1".1‘ ML ..ti. uh. \\..~ .. . . w...€1‘.101‘.> h: H. ‘- w 1' _.j L0hi.dul\._ iiiht'wh ' \ul , 5 wt; : ,, “fih‘ElIiil, I‘li‘ mi... “‘“ “Us. ‘ 1371613 T? Plitcss i, If the new. . . be described in its own new terms, its hearers must struggle hard for understanding by whatever means they have. These means, however, are the old modes of understanding, stemming from the old logic. Inevitably, the new will be altered and distorted in this process of communication, converted into some semblance of the old (p. 171). In this case, although learners will encounter the new firsthand — rather than a simulacrum of the new — it will inevitably be distorted and altered in their minds as they attempt to comprehend it, equipped with only the inadequate conceptions and modes of learning they currently possess. Either way, it seems what is learned is merely a distorted version of the new conceptions of knowledge and the new modes of learning and knowing they were to learn. It seems to be an impossible situation... and yet, such learning does occur. Schwab’s Resolution: Putting the New in Terms of the New This apparent contradiction, as Schwab goes on to show, is not actually a contradiction: The impossible situation is not as impossible as it seems at first glance. Although the approach of converting the new into more accessible forms for easy acquisition is indeed a dead-end — it results in “static and unrecognized misunderstanding” — Schwab explains that the omissions and distortions that occur as learners try to understand the new in its own terms, are not “fatal flaws” but are part of a process leading to new understanding. He writes — If the new is presented mainly in its new terms, a... situation is created, uncomfortable but productive. From the first, hearers must struggle to understand. As they translate their tentative understanding into action, a powerful stimulus to thought and reflection is created. This stimulus acts in two ways. On the one hand, it creates new food for thought. The actions undertaken lead to unexpected consequences, effects on teachers and students, which cry for explanation. There is reflection on the disparities between ends envisaged and the consequences which actually ensue. There is reflection on the means used and the reasons for why the 13 ‘11,". ”k I \ Ntc Pr: “J .. \pVI' trebit. s l ~‘0‘u I: V 11“.. “Lao“; ,. ... ‘ vvh.\ \ .n ‘ Hist“ till... ‘ s: ”9 .3 m6 SALOA- L0 nnLemI 335. mm .i Mi . ‘,‘ lit-t. tlil'Cttf “3}. Sth‘a outcomes were as they were. At the same time new competences for taking thought are roused. The new actions change old habits of thought and observation. Facts formerly ignored or deemed irrelevant take on significance. Energies are mobilized; new empathies are roused. There thus arises a new and fuller understanding of the situation and a better grasp of the ideas which led to it. A revised practice is undertaken. The cycle renews itself (pp. 172-173). Schwab argues that if learners encounter the new in its new terms, a struggle will ensue; and, although it may be uncomfortable, this struggle to understand is both the means by which change will occur and the very change that needs to occur. As people engage in the effort to make sense of the new, they engage in practices of testing ideas, reflecting upon the things tried and undergone. At the same time they begin to develop new actions and form new ways of hearing, seeing, and being in response to the new. Through their struggles, they have the opportunity to see firsthand the emergence of new knowledge and directly experience new modes of knowing and learning in operation. It is in this way, Schwab suggests, that this fundamental paradox of learning is mediated. The Experiment: Cultivating Unfamiliar Terrains This dissertation examines an attempt to mediate this paradox within the context of an elementary mathematics methods course for pre-service teachers. The course was designed to present “new” ways of doing and knowing mathematics — teaching and learning mathematics — in its new terms. Elements of the course such as the tasks and problems used, the tools and material means provided, and the social-intellectual structures of class were purposefully constructed to produce a context where new habits and expectations for mathematical work would be encouraged; different kinds of knowledge and ways of knowing mathematics would prove valuable. This study looks at 14 essence. 11“.; I ‘IfiflL Luff 01 it I; win. u - “I : 1:5 ln 016313“ I x. ' 5M3?“ ‘I’I ' - ltlh‘k‘ Y . {Apt 'Illl‘u' ”'Intv- " , LLHL‘Uu‘ 1 . $1131". :3 what and how changes were initiated in the course: What might it mean to create an educational context for pre-service teachers where “the new is presented mainly in its new terms”? What might be involved in creating such a context? This dissertation also examines the processes by which change occurred over the semester as prospective teachers adapted to -— and adapted — the context in which they found themselves. In essence, this study is a test of the theory articulated by Schwab, examining not only whether or not the theory holds in the given case but also to elaborate upon the theory as it is applied to an elementary mathematics methods course. In this section, I discuss how the study was conducted, beginning with a brief overview of the methodology used. I then go on to describe in more detail three key aspects of the study: The theory underlying - and being scrutinized through —the work; the design and implementation of interventions that embody the theory; and the systematic study of the interventions. The Design Experiment This study uses a method in educational research ofien referred to as a design experiment. Notably, design experiments involve interventions that are based upon and serve to test theory. Additionally, in many instances, design experiments are conducted with an eye for increasing knowledge about the learning environments that influence and result from the interventions. Rather than testing a single innovation in a tightly controlled setting, design experiments are ofien characterized by efforts to implement interventions within naturalistic settings, such as classrooms, where “the purity of research method is compromised by simultaneous changes in multiple inputs” and generalization is made difficult by the “idiosyncratic circumstances of a particular site” 15 (inc. 31.1.13. . ‘ ! attain) it O .. "l ' . build. liiil't I02: the: 5151: C0171 Ip. l “$137.. (Clune, 2000, p. 10). However, despite the challenges and limitations of the method — particularly for studying causes and effects — it offers ways to carefully examine instructional interventions as well as to systematically test theory within and about classroom environments. In their 2002 report, Scientific Research in Education, the National Research Council introduced this genre of educational research in the following way — (One) analytic approach for examining mechanism... begins with theoretical ideas that are tested through the design, implementation, and systematic study of educational tools (curriculum, teaching methods, computer applets) that embody the initial conjectured mechanism. (p. 120). Three features of the method, as described here, are noteworthy. First, as the NRC report underscores, design experiments begin with theory. In this dissertation, the study begins with and sets out to test Schwab’s ideas about the pedagogical approach of presenting the “new” in its new terms and the process of change that ensues from this approach. This theory is discussed in more detail in the next section. Second, theory is tested through the design and implementation of interventions that embody the conjectured mechanism. In this study, several key elements of a mathematics methods course were designed and implemented to set up a context embodying the paradoxical situation described by Schwab — a learning environment where new, unfamiliar mathematics, teaching, and learning were put in their new terms. These designed elements of the course are specified and explained below. Third, the interventions are systematically studied. For this dissertation, data were methodically collected around the designed features of the course as well as on the larger class context in which the designs were embedded and encountered. The data 16 “")..0: T - st “Symon to a; 323.3 51: l 7“” ' 1 Jul tutors xx iil e - '4 Xx: It‘A—cn collection for this dissertation, along with notable selections and omissions that focus the data analysis, are discussed in the section, Data Collection and Analysis. There are a couple of ways in which this study is particularly well suited to the design experiment approach. As mentioned above, the method provides an analytic approach for examining mechanism (National Research Council, 2002) — that is, for investigating the processes by which certain things are brought about or naturally take place. This is important for this study with its dual focus on the processes by which the new comes to be presented mainly in its new terms and the processes by which change occurs with and within this context. Likewise, the careful consideration that design experiments pay to contextual elements and their interactions is vital for this study where multiple facets of class environment were designed as part of the intervention and the context contributing to and resulting from the interventions was focal. As Paul Cobb and his colleagues (2003) note, Design experiments ideally result in greater understanding of a learning ecology — a complex, interacting system involving multiple elements of different types and levels. . .. Design experiments therefore constitute a means of addressing the complexity that is a hallmark of educational settings (p. 9). Rather than being something to avoid, this interplay of elements is an important part of what is examined in this study. I turn next to discuss the theoretical ideas that were at the heart of this design experiment. 17 I .. ~ fifiiciOI§ ll‘. (mung — 1| iii-35'. 3‘3 I: drawn: A L -. ,. IL. prep.“ I . .3: ‘h . . ththSf‘ \\ , u III. the 3i: p—‘ .- r4 described. III idea - Smile Applying Scwh_ab’s Idea to a Mathematics Methods Course The paradox of learning described by Schwab and the resolution he put forth in his (1959/ 1978) essay were part of a discussion on the challenges faced by Progressive educators in their efforts to communicate new ideas about teaching, learning, and knowing — to teach new modes of knowing and coming to know. Although the essay, almost 50 years old, does not specifically address problems of mathematics teacher education in the 21St century, the paradox resonates with modern challenges associated with preparing teachers to teach mathematics in unfamiliar ways. And Schwab’s proposed resolution has similarities with a range of contemporary efforts that provide teachers with opportunities to experience new modes of doing and learning mathematics with the aim of helping them develop important knowledge and skills for teaching. In the following, I discuss how Schwab’s ideas — the paradoxical situation he described, the pedagogical approach he put forth, and the process of change predicted by his ideas — were applied to elementary mathematics teacher preparation. A Paradoxical Situation The problem raised in this dissertation is much like the paradoxical situation that, according to Schwab, was faced by Progressive educator, John Dewey — Dewey seeks to persuade men to teach a mode of learning and knowing which they themselves do not know and which they cannot grasp by their habitual ways of learning. It is the same problem of breaking the apparently unbreakable circle which Plato faces in Meno and Augustine in his treaties, 0n the Teacher (Schwab, 1959/ 1978, p. 170). Similarly, those who seek to prepare teachers to teach a mode of learning and knowing mathematics that is unfamiliar and which teachers cannot grasp using their habitual ways of learning, also face a “problem of breaking (an) apparently unbreakable circle.” 18 haunet wwwml flaunt prisxctitc it com etc; ha . v ‘ ‘I’QI 43?; 4 bh'1_,.bg‘ \h but I‘LL-rte: “ 1 Inner c.) tannr 5 ‘1" . - . him: .15 m; -, In J. ? Unit“ '.I 1112thde 3131 the} Prospective teachers will use their current modes of learning and knowing — and will draw upon their extant understanding — to make sense of the “new” things they encounter and experience in teacher education. However, as they use these personal resources, prospective teachers inevitably miss and distort important facets of the “new” that was to be conveyed. In some cases, unfamiliar ideas and ways of working may not be seriously considered by pre-service teachers because they seem nonsensical or imprudent given their current understanding of mathematical knowledge and learning. For example, in a 1991 study that looked at the beliefs and understandings prospective teachers bring to teacher education, Diane Holt-Reynolds described how prospective teachers in a content area reading course had difficulty considering alternatives to teacher-telling for subjects such as mathematics where the content seems “sequential in nature” and children seem unlikely to “discover” the next steps. The prospective teachers’ views of the nature of mathematical knowledge limited the range of alternative views of teaching and learning that they were able, or willing, to consider. In many other cases, the “new” may simply not be seen by prospective teachers as new; but instead, important distinctions that delineate the new may go unnoticed or become lost in translation. There is a tendency to distort new and different things to fit within a framework of expectations (this psychological phenomenon is discussed in more detail in Chapter 2). Additionally, much can be lost in the flexible ways we use language in everyday life to cover a range of ideas. For example, in her study of the mathematical knowledge and ways of reasoning prospective teachers bring to teacher education, Deborah Ball (1988a) discussed common words used in her interview items that 19 gangsta e I . an ”L‘J;' "I 33.. Ms“ L“ I 3"“. 9 4‘ ,1“ ) May“ a: fi‘ . I .jv-r" ‘ ~\p_ti...-.1§A.. l 5 - C gym 9'“ h:wm\~lL‘ Bil notes. to 1110510 MIGHT nu prospective teachers interpreted and used differently than she had intended. Drawing upon their understanding of what mathematics is and entails, the prospective teachers in Ball’s study heard and meant very different things by familiar words and phrases like, 99 66 9, 66 “explaining, showing,” “figuring something out on one’s own, problem,” ’9 66 “argument, concrete experience,” and “problem-solving.” As one specific example, Ball notes, Explaining something in mathematics means clarifying an idea by unpacking underlying concepts as well as giving reasons that reveal its meaning and the logic. When teacher candidates talked about “explaining”(which they did frequently) they seemed to mean something much weaker, something that much more resembled simple telling — as in giving directions for the steps of a procedure or repeating a definition. Neither did that telling ever seem to include telling about the nature of the mathematics at hand: Its sources or what made it make sense, for example (pp. 181-182). It seems an impossible situation: If prior knowledge and ways of knowing serve to transform the “new” into likenesses of the “old,” how can new forms and modes of knowing be learned? The New is Presented Mainly in its New Terms Despite the apparent contradiction, people somehow do come to see and know in new ways while drawing, as they must, upon their existing knowledge and ways of knowing. In his analysis of the paradox, Schwab shows that although alterations and distortions inevitably take place in the process of communication it is fallacious to assume that misunderstanding is the unavoidable end result. Instead, Schwab puts forth another possibility: “If the new is presented mainly in its new terms, a... situation is created, uncomfortable but productive” (pp. 172-173). In this alternative scenario, rather than being a final outcome, misunderstanding is part of a larger, longer struggle to 20 V I . 4 .. u?" "‘ - 13.1..)qu raters. The , II'. . Loni 1310:;- v Jfixfl' Y “' \ to 5 I but. I - “-10:- Jmn‘Or “bdibdlu‘nh -\u 5‘ . 3.190132. ’ECECTS C3 mania“ M. understand - an early phase in a process of developing fuller understandings and revised practices. The idea of presenting the “new... mainly in its new terms” is a potentially compelling one for mathematics teacher education, resonating in many ways with modern efforts to provide teachers with opportunities to see and experience forms of mathematical work, learning, and teaching unlike those they’ve experienced in their own schooling. For example, in describing a professional development program for practicing teachers called, SummerMath for Teachers, Deborah Schifter and Catherine Fosnot (1993) write, Teachers must be given experiences as students in classrooms that embody the new teaching paradigm. Such classes must provide learning experiences powerful enough to challenge l6-plus years of traditional education. When teachers are challenged at their own level of mathematics competence, are confronted with mathematical concepts and problems they have not encountered before, they both increase their mathematical knowledge and, more importantly, experience a depth of learning that is for many of them unprecedented. Such activities allow teachers, often for the first time, to encounter mathematics as an activity of construction, of exploration, of debate, of a complex interplay of convention and necessity, rather than as a finished body of results to be accepted, accumulated, and reproduced (pp. 16-17). Rather than lecturing on Constructivist principles, assigning readings on teaching and learning mathematics, or providing opportunities to work through activities and lessons designed for children, a significant portion of the SummerMath program was designed to place teachers in a context where the “new” ideas and ways of working that developers wished to convey were in operation — a context where teachers could encounter mathematics in its new terms. Such attempts to present the “new... in its new terms” involve conscious efforts to create and sustain an educational context where new forms of work and ways of 21 statistics A 0*. if“ EEIITA’ Oi elh‘ll‘tti 2:11) :37 rim m- i “(vi " ' " ‘8) Alf 111 11.15 81 (‘2 (‘2 working are supported, unfamiliar roles and interactions are encouraged, and unconventional meanings and expectations are reinforced. It involves a re-engineering of what is “normal” to provide prospective and practicing teachers an opportunity to experience the new firsthand. Over time, I have come to refer to this approach of presenting the “new... in its new terms” as "cultivating unfamiliar terrains." Although this term does not offer a timesaving shortcut for writing or speaking, it does offer two advantages. First, this three-word phrase provides both terminology for an action or activity (i.e. the cultivating or cultivation of an unfamiliar terrain) as well as for a th_ing (i.e. an unfamiliar terrain). This terminological benefit is difficult to gain with the more unwieldy phrase, presenting the new in its new terms. And second, the words of the phrase offer more accuracy than words like “presenting” or “new”, providing a better representation of the ideas targeted in this study — ' "Cultivating" refers to the act of designing and constructing an educational context as well as sustaining and developing it over time. It involves an ongoing commitment of pedagogical thought and effort toward regulating the context for and with prospective teachers. ' "Unfamiliar" highlights the diflerence between the educational context being developed and that to which prospective teachers have been accustomed. Newness or novelty in itself is not what matters; rather, it is the contrast between what pre-service teachers expect to find within the walls of the classroom, within the hours set aside for class, and what they encounter. 22 0 "Tel mr All. :r. -(b I 11‘" 1\ I“, - . 141.7111» A“‘ l! "’h. a'~\1w_“_ ° "Terrain" refers to the educational context being developed but with particular emphasis on the features designed by the teacher educator and those that come to the fore through prospective teachers’ interactions with and within the context: the obstacles and obstructions, the promising paths, the confounding trails, etc. Although innumerable things affect people’s experiences and opportunities to learn within an educational context, the use of the word, “terrain” is used to draw attention to the constructed features — to the things constructed through the intentions of the teacher educator and prospective teachers as well as those things subjectively constructed through pre-service teachers’ perceptions and interpretations. Beyond suggesting a promising approach for resolving the paradox, Schwab’s analysis also includes predictions about the processes by which change takes place within unfamiliar terrains. A SLuation is Created. Uncomfortable but Productive By design and necessity, the educational context proposed by Schwab’s approach would create a situation that is, to some degree, uncomfortable. Unfamiliar terrains are created to facilitate direct engagement with the new — to move people beyond what is familiar, predictable, and easily comprehensible. Schwab notes, “From the first, (there is a) struggle to understand” (1959/1978, p. 173). This struggle can cause discomfort, requiring people to work through conceptual confusion and uncertainties; however, it can also be much more than “uncomfortable.” At times, the struggle can be painful and frightening, introducing confusion and uncertainty around teachers’ identities and sense 23 likewise com. C :2.me (L 1. teachers . .'I Cguii‘ ( of self, undermining confidence in their own abilities and knowledge. For example, commenting on the SummerMath for Teachers program, Schifier and Fosnot (1993) note, Teachers are frequently frustrated, at times angered, by program experiences that create disequilibrium, thereby challenging their professional identities. (“I always thought I was a good teacher because I explained things so clearly. Now I see how I’ve cheated my students out of the opportunity to explain”) (p. 19). Likewise, in describing the experiences of pre-service teachers in a mathematics methods course, Cynthia Nicol (1997) writes, For some prospective teachers breaking free from their familiar experiences of learning mathematics built up over years of schooling was not worth the time, effort, discomfort, and pain. Although they seemed to embrace the idea of teaching for understanding, they at the same time found the possibility of teaching in ways which are responsive to students’ thinking, frightening and unsettling. . .. (Additionally,) the activities we had designed helped to make some prospective teachers feel less confident about teaching mathematics rather than more. We had, as Kendra said, helped to rip the solid floor from underneath her and shoved her into some very unfamiliar territory (pp. 183-184). Although the viewpoints and immediate concerns of prospective and practicing teachers may center on different things, their struggles within unfamiliar terrains can be equally difficult - at times, evoking anger, anxiety, and pain. Nevertheless, despite the fact that it may create an uncomfortable situation, it is also, according to Schwab, a productive situation that provides a powerful stimulus to thought and reflection. He elaborates on this, explaining two things that occur as people translate their tentative understanding into action within an unfamiliar terrain — 1. “Food for thought” is generated ° “The actions undertaken lead to unexpected consequences, effects on teachers and students, which cry for explanation 24 y: «1, 1u\L&‘ . g. '1 Willi. Stitt5 16min. ° There is reflection on the disparities between ends envisaged and the consequences which actually ensue ° There is reflection on the means used and the reasons for why the outcomes were as they were” (p. 173). 2. “New competencies for taking thought are roused ° New actions change old habits of thought and observation ° Facts formerly ignored or deemed irrelevant take on significance 0 Energies are mobilized; new empathies are roused” (p. 173). Schwab sees this as a self-renewing cycle: With new food for thought, and new competencies for taking thought, there arises “a new and fuller understanding of the situation and a better grasp of the ideas which led to it” Q). 173). This will lead to revised practices within the unfamiliar terrain that will lead to new food for thought... and so it continues. A central purpose of this dissertation is to examine these processes, predicted by Schwab’s analysis, to better understand change as it occurs with and within an unfamiliar terrain. I turn next to discuss in more detail, the design of the unfamiliar terrain that was created for this study. Design Features: Conjectures About and Designs for the “New” The course was designed to present prospective teachers a context where unfamiliar forms of mathematical knowledge and ways of knowing would be called for, supported, and valued. In particular, several key elements of the course were carefully thought through and constructed for this purpose. 25 ‘1'7'.I M17”: LI‘ . ~ ,_ crtrsdt than: “W v q ‘ .9. on r 9, ..Lqu‘ ‘ “ . .V?! H” ' 11.0.5.1. \ R‘ ‘.\.I \O..‘er.\~1l u i “CI\ Meb 0.1 J":""\ Kla'g 'i Sikh‘s C0 The most overarching of these designed elements was the types of work and ways of working that were to be encouraged in the class. More than anything else, this element captured the learning goals of the class and characterized the ways in which the class would work towards these goals. In many ways, this element is similar to what some might refer to as “types of discourse” however, it is meant to cover a wider array of actions and activities than “discourse,” which is often limited to written and spoken conversation. Another element of the course consisted of the social-intellectual structures that were designed to support the envisioned work and ways of working. This element overlaps with the idea of “participation norms.” Like participation norms, it includes the social configurations routinely used and the expectations for participating in class but it places an explicit emphasis on the ways of thinking and interacting with others that were reinforced. Additionally, the tasks and problems used in the course and the tools and material means that were provided comprise two final elements that were designed to facilitate the envisioned course work. In their description of a “learning ecology,” Paul Cobb and his colleagues (2003) nicely capture of these elements when they write, Elements of a learning ecology typically include the tasks or problems that students are asked too solve, the kinds of discourse that are encouraged, the norms of participation that are established, the tools and related materials means provided, and the practical means by which classroom teachers can orchestrate relations among these elements (p. 9). 26 “3.15 06.} .1, k ,3 ' Li's il TL...” 9“ Jul“ [A ._ Wm II rm ~- p‘ “-1an ” "'1. 9mm E I ”’ 1.3-w “I: 5k§\ With the exception of the last item on the above list,4 these elements of the “learning ecology” were carefully considered and developed for this design experiment. In the sections below, I discuss the design of each element in turn. Types of Work m Ways of Working The mathematical work prospective teachers were to experience in the class was designed to embody the very types of mathematics they were to learn through the course. By engaging in this mathematical work and these ways of working on mathematics, it was believed the pre-service teachers would begin to develop the mathematical capacities — the knowledge and skills, the practices and dispositions — needed for doing such work. Therefore, this element of the course was especially significant: The types of work and ways of working designed for the course served as aims of the course as well as the primary means for achieving those aims. For discussion purposes, the different facets of work that comprise this element are grouped into two clusters: Those that focus on broadening prospective teachers’ understanding of what mathematics entails, that expand their abilities to engage in more types of mathematical activities; and those that highlight the development of pre-service teachers’ mathematical knowledge and skills for the work of teaching. This is, however, an artificial distinction. The types of mathematical work discussed under the heading, Mathematics for Broadening “Mathematics, ” are vital for the work of teaching; likewise, the types mathematical work labeled, Mathematics for Teaching, are crucial for developing a broader understanding of mathematics from and with others. Nevertheless, " The last element listed by Cobb and his colleagues, “the practical means by which classroom teachers can orchestrate relations among these elements,” is a focus of this study but was not a part of the design. Rather, it was an aspect of the course that was not well understood at the beginning of the study but developed further through the study. 27 s’T‘JwI': A I" l A ' U". .kiov r [SAL snarl I ' | "21".) renul‘sdl I :‘J Uri u Rhona t ILJIR'CIT‘I.‘ ‘ 213326111. lJ-‘I‘Hggi “EN-nu, b x in \ lurch-ml) \ 1 1011px: molt e. and III; separating them in this way is useful for talking about the reasons for their selection and highlighting shared characteristics. flithematics for Broadening “Mathematics”. One recurrent theme that has surfaced from research efforts to uncover teachers’ views of mathematics and mathematical work is a strong tendency for people in the United States to view and treat mathematical knowledge as a static body of rules and procedures to be acquired and retained (Ball, 1988a; Cohen, 1988; Stigler & Hiebert, 1999). Coupled with this has been a tendency for people in the US. to give little or no consideration to those aspects of mathematics dealing with the creation, testing, validation, and ongoing development of mathematical knowledge or the field of mathematics.5 In response to findings such as these, the mathematical work of the class was designed to engage prospective teachers in doing more than simply using mathematical products — rules, procedures, definitions, formulas, etc. — developed by others to complete mathematical exercises. Instead, the teacher education students would also be involved in developing mathematical ideas through exploration, mathematical argument, and the extension and refinement of mathematical ideas within an intellectual community. 5 As a poignant example of this, Ball (1988a) shared an interchange she had with a research subject around a “card sort” item in her elementary teacher interview. The prospective teacher was to respond to the statement, “Some problems in mathematics have no answers,” commenting whether she agreed with the statement, disagreed with it, or was not sure about it. The teacher candidate agreed with it and continued on to say -— Mei Ling: I’m thinking is this the obvious, like just some math problems you are going to find in the textbooks have no answers? Ball: Oh, no, it’s like are there problems in the field that no one can solve? Mei Ling: As far as teaching it? Ball: No, as math as a subject. Like a math problem that no mathematician can solve. Problems like that. Mei Ling: I am sure there are. I don’t know if we’ve found any, but the possibility exists. It’s like saying, “Is there life on other planets?” (laughs). We don’t know, but . . . it could be! (p. 185). 28 I258 801-“ £111,”ng 1"]?th 'i .sflus‘i.iu v rare": ' “15311! I'-.V..vm.. ‘.-§‘ .AA‘. y EAT“. a, H—a.‘ . . These correspond to essential phases in the development of mathematical knowledge. Exploration entails the work involved in generating new mathematical ideas, giving them form and substance, considering their potential for further development. Mathematical argument comprises a variety of activities for presenting, scrutinizing, and evaluating mathematical ideas with others. Extension and refinement involves re-visiting mathematical ideas that have been developed in the past — using them novels ways, drawing upon them to create new knowledge, or improving upon them in some way. Examples of the mathematical activities this work entails are listed in Table 1.1. Table 1.1 Types of work and ways of working for broadening “mathematics” Exploration ' Questioning and wondering 0 Making observations and developing explanations ' Making tentative attempts and forming partial solutions ' Producing conjectures, generalizations, and formulations ° Testing ideas and examining the results ' Generating descriptions and developing working definitions ' Working through confusion and making mistakes Mathematical Agument ' Developing proofs and refutations ' Providing explanations and justifications ' Coming up with examples and counterexamples ' Presenting as well as questioning claims and ideas Extension and Refinement 0 Building and expanding on previous work ° Integrating new information and perspectives ° Revising and reframing earlier work ' Making finer distinctions It was anticipated that much of this mathematical work would not only be different from what prospective teachers saw as “mathematics” but might, in some cases, conflict with their conception of doing mathematics. For example, working through confusion and uncertainty is fundamental to mathematical exploration; however, 29 ~1‘a'“. ‘\fi u. 1.. U)“ " I r“ IYh‘Jfi‘ I" .¥.AJV ”u“. . I ‘ '4 ‘ .‘ .- ut.LI'.‘.r-’ I I or] I y I ' O I “1 Jun 5“ 11.1.1le confusion and uncertainty are frequently considered things to be avoided in US. mathematics lessons (Stigler & Hiebert, 1999). Even familiar sounding items such as developing proofs or providing explanations would likely have very different meanings within the context of this work. Mathematics for Teaching. The design of the mathematical work of the course was also influenced by an important theme that surfaced from research on the mathematics needed for teaching. In recent years, there has been a growing body of research that illumines and begins to articulate the special content-related demands of mathematics teaching (Ball, 2000; Ball & Bass, 2003a, 2003b; Carpenter & Franke, 2001; Conference Board of the Mathematical Sciences, 2001; Hill & Ball, 2004; Hill, Rowan, & Ball, 2005; RAND Mathematics Study Panel, 2003). These findings include both the ways in which teachers need to know mathematical content (this is discussed in the section, Task and Problems) as well as the types of mathematical work teachers need to be able to do. Taking these findings into account, the class was designed to engage prospective teachers in mathematical practices and problems that are endemic to teaching — that are crucial for those who ms; carefully attend to the development of mathematical understanding in the minds of others. In particular, the work of teaching includes a range of activities for accessing others’ thoughts and ways of thinking, making ideas accessible to others, and facilitating the development of better mathematical understanding with individuals as well as with the class as a community. As Deborah Ball (2003) notes, this tYpe of mathematical work is special to teaching — 30 The mathematical problems and challenges of teaching are not the same as those faced by engineers, nurses, physicists, or astronauts. Interpreting someone else’s error, representing ideas in multiple forms, developing alternative explanations, choosing a usable definition — these are all examples of the problems that teachers must solve. These are genuine mathematical problems central to the work of teaching (p. 8). Examples of the kind of work this entails are listed in Table 1.2. Table 1.2 Types of work and ways of working for teaching mathematics Teaching ° Eliciting questions and fiirther explanation from others 0 Explaining, translating, and using (i.e. playing out) other’s ideas ° Reframing, summarizing, and integrating questions and other’s contributions ' Comparing, contrasting, and making distinctions among others’ ideas ' Tailoring an explanation or argument to another’s way of thinking This type of mathematical work was expected to be quite unfamiliar to the teacher education students. Unlike the mathematics that is typically learned and experienced by students in US. classrooms, the mathematical work of teaching includes a range of practices for attending to and working with the emergent understandings and skills of others. This element provides a vision of the “new” that the course aimed to develop for and with prospective teachers; but it provides only the briefest glimpse into M the new would be presented in its new terms. The ways in which this mathematical work was to be supported — through the structuring of time and interactions, the problems and tasks posed, and the tools and materials means provided— is addressed below. 31 wv - r) 10 311 5, gas» 1 9P.) ‘ ‘-" HA ‘0 ’az‘t‘, "Q . .3 bhuNLAEA - Irj'L-pn 11;»; m1. expect. Social-Intellectual Structures The social-intellectual structures for participating within the class were designed to create social contexts and expectations that would support the mathematical work of the course. Through the structuring of time and group configurations, prospective teachers would routinely find themselves in whole group discussions, collaborating in small groups, or working alone in their notebooks, trying to better understand the mathematical issues and problems at hand. Additionally, through the careful shaping of expectations for what it means to “participate,” teacher education students would be called upon to play active roles both in developing mathematical knowledge within the class community as well as in encouraging the growth of new mathematical understanding within individuals, helping other students to learn. Although a variety of group configurations were to be used in the class, it is useful for discussion purposes to group them into two categories: the structures designed for supporting collective participation with two or more people, and those structures for supporting individual participation when students would work alone. The various configurations - along with when they were to be used and for what purposes — are described below. Also, the expectations for participating within these configurations are briefly discussed. Structures for Collective Participation. Participation in class discussions and in various forms of group work is often valued as an important vehicle for learning subject matter content. Such interactions are believed to enrich students’ learning experiences and result in a range of favorable learning outcomes such as better retention of information, higher motivation for learning, improved interpersonal skills, enhanced 32 I: "ng' raw 5 .51 III higher-order thinking skills, and increased self-efficacy.6 However, in the design of this course, “collective participation” was not only desirable as means to enhance student learning, it was also a fundamental part of the content to be learned, engendering key practices and ways of interacting with ideas and people that prospective teachers were to learn through the class. One type of configuration that would be used extensively is the small group. Small group work would typically involve 2 - 4 students and, rather than assigning people to groups with highly specified roles and responsibilities, much of the small group work would be informally arranged, created for particular purposes or situations. Often, the prospective teachers would be allowed to form their own ad hoc groups in response to the demands of the task at hand. One common situation wheresmall groups would be used is in the early stages of working on a mathematics problem: Groups would be formed to confer about the problem — to clarify what is being asked or to share initial ideas about solution approaches. Small groups would also be routinely used to prepare for the whole group discussions. During such times, the pre-service teachers would begin to articulate their ideas to others, explore and raise questions about each other’s solutions, compare approaches, and find effective ways to present their ideas. Another important configuration that would regularly be used is the whole class discussion. One common way these discussions would be used is to introduce a new mathematics problem. In most cases, new problems would be posed and discussed in the whole group before beginning work on them. During these introductions, the class might solve a related-but-simpler problem together or individuals might share their initial 6 See, for example, Johnson, Johnson, and Smith (1998) and Bonwell and Eison (1991). 33 . A.1ua¢ n.5,, thoughts about possible solution approaches. Additionally, on the other end, whole group discussions would ail—waya be used to talk through the work that had been done on the problem. These discussions would provide important occasions for prospective teachers to do the mathematical work focused upon in the course: Pre-service teachers’ ideas and approaches would be presented and explored; mathematical arguments would be made and considered; and earlier work would be extended and refined. Additionally, within the context of the whole class discussions, prospective teachers would be called upon to practice important mathematical skills for teaching — to engage in mathematical practices for accessing others’ ideas, making ideas accessible, and facilitating the growth of others’ mathematical understanding. In all these different group configurations, the expectations for participation would be the same: Prospective teachers would be expected to actively engage in the shared work of developing mathematical knowledge and understanding. This would involve both working with the ideas of others as well as contributing their own work and insights. Examples of such expectations are listed in Table 1.3 Table 1.3 Expectations for Collective Participation Collective Participation ' Actively listening to, using, building upon, and integrating others’ ideas and ways of thinking ' Contributing to the collective work of developing mathematical ideas through questioning, objecting, refining, reframing, proposing, etc. Although “class participation” and “group work,” are not unusual features in undergraduate education — and are nearly ubiquitous in teacher education — the pre-service teachers’ roles within these configurations were expected to be quite unfamiliar. Instead of the roles typically played in such configurations (especially in 34 . .; . maxilla-I» ‘ ‘ *Irserstm -" a If] 1.; 51mm .1 w Int 0,116 3110 he leaned 5:". maintain. indix'iiual this course teach rs III computatio. in‘jj‘lldlltlllj 33111;?“ cm; 0n: immediate}. g1 “up “(mix Oblenaih “f be“ ~ I. mathematics classes) where students simply go over their answers, demonstrate that they “understood” how to do the assignment, and spend time providing extra assistance to slower individuals, participation within the large and small groups of this course would involve more than follow up on already completed work. Instead, much of what was to be learned would take place in these group contexts. Structures for Individual Participation. Individual work is a mainstay of mathematics lessons. Typically, a significant portion of class time is devoted to individual work on mathematics exercises and almost all homework is done alone. In this course as well, there would be a great deal of mathematical work that prospective teachers would do on his/her own. However, in addition to providing learners with computational practice and the instructor an opportunity to assess student learning, individual participation in this course was designed to encourage pre-service teachers to actively engage in a broad array of mathematical work for developing knowledge. One place in the lesson where individual work would commonly be called for is immediately after a new mathematics problem is introduced. Before launching into small group work or a whole class discussion, the prospective teachers would always be given time to work the problem alone — to look for one or more solutions, to record their observations and tentative conjectures, to find ways to explain or prove their ideas. Another place where teacher education students would be asked to work alone is after hearing a number of ideas and solutions from others. They would be given time to consider the various perspectives and write their current thoughts given what they’ve heard —- to record the questions they have, the objections and counterexamples that come to mind, the revisions that might be made to their own earlier ideas. And, of course, in 35 thEWOIk ; of explorizi Des to collectn expected It class work W “011mg CilI “out hid the film In homework assignments they would routinely be asked to work alone, doing similar types of explorations and reflections. Despite the fact this work is done alone, it would still be fundamentally connected to collective work of the class. Even in individual work, prospective teachers would be expected to engage in the issues and ideas raised by the class — to consider and respond to others’ ideas as well as to develop their own positions and contributions in light of the class work. Examples of expectations for individual participation are listed in Table 1.4. Table 1.4 Expectations for Individual Participation Individual Participation - ° Reading/examining others’ ideas (including assigned readings as well as the solutions and ideas of students) ° Writing reflections on and responses to — — Others’ ideas — Work done to date — New issues, problems, and considerations surfacing from work done or to prepare for upcoming work Although teacher education students would have had a great deal of experience working alone on mathematics, the expectations for individual participation in the course would likely be strange to them, deeply connected to the collective mathematical work of the class in ways they would not have likely seen or experienced before. Problems and Tasks The featured mathematics problems of the class were chosen7 and designed to create necessities for doing the types of mathematical work and engaging in the particular ways of working that prospective teachers were to learn through the course. The 7 Deborah Ball authored of many of the problems used in the class. Some problems were adaptations of interview questions developed for her 1988 study on the what prospective teachers bring to teacher education; other mathematics problems developed by Ball were written specifically for use in similar teacher education courses. 36 matter It ere as rather taxis. it higher.“ minem 3."- 1 .- uixbgh 35111335 1 otter. cen problem use these I problems I “Merits. , 11“th s“ mailer“. .. A 1"; J C0111 . 01m ”151' at mathematics problems, by nature of the questions they raise, would serve to initiate tentative explorations, inspire mathematical argument and, in many cases, lead pre- service teachers to revisit things they’ve learned in the past. Additionally, these mathematics problems —— along with the other math-related assignments of the course — were assembled to help prospective teachers begin to develop resources for doing the mathematical work of teaching. Through their work on this collection of mathematical tasks, it was anticipated that pre-service teachers would start to develop the kinds of mathematical knowledge and skills needed for helping others learn mathematics. The special features that were sought after and designed into each of the focal mathematics problems used in class are described briefly below. This is followed by a discussion of the collection of mathematical tasks assembled for the course and the key features of its design. Mathematics Problems. The focus of elementary mathematics methods courses is often centered on the materials used for teaching elementary mathematics — the books, problems, activities, and concrete math manipulatives used with children — and ways to use these materials to help children learn. Frequently in such classes, the mathematics problems that are studied and worked upon are problems designed for elementary students. Although prospective teachers might learn new things about mathematics through such efforts, the work is primarily designed to familiarize them with the mathematics of elementary school and techniques for teaching this content. In this course, however, a large emphasis was placed upon developing prospective teachers’ own mathematical capacities — broadening their understanding of mathematics and abilities to do a wider array of mathematical work; helping them learn mathematical 37 311d 3115“ IIithout I work of meant: .’ 5 an em; problem: chiller; tad-bu: pie-semi explorati $50 nee; .llhllllilt'I dltlerenz ll C1913768: 39.310th Irate-m;1 ”131};ch 11."...1 l knowledge, skills, and dispositions for the work of teaching. Rather than solving and examining mathematics problems designed for children, this type of work required problems that would genuinely be problematic for adult students. If the mathematics problems could be easily solved without any exploratory work or revisions, the solutions and answers verified without argument, the mathematical issues and ideas grasped without the benefit of others’ ideas and ways of thinking, then the core mathematical work of the course would either be circumvented as unnecessary or engaged in as a meaningless exercise. Therefore, in order to support the mathematical work of the course, the mathematics problems would need three fundamental features. First, the mathematics problems would have to present appropriate challenge to adults. An “appropriate challenge” must do more than simply present prospective teachers with problems that are hard-but-possible for them to solve: In addition, these should be problems that pre-service teachers can only truly resolve and understand as they engage in processes of exploration, communal scrutiny, and revision. Second, the mathematics problems would also need to provide multifaceted work that can be approached from various angles. Multifaceted work would not only support a diversity of learners, allowing people with different background knowledge, perspectives, and mathematical strengths and weaknesses to engage in the problem, but it would also d_epeid upon the diverse approaches, interpretations, and understandings that would result. And third, the mathematics problems would need to provide multilayered work, embedding multiple mathematical issues that unfold through continued work on the problem. New questions would surface as individual explorations expanded to collaborative examinations of 38 multiple 31’: 1112131 QbSCI new chiller Impositior $01.16 the p trough V3 Tn pffllde 31' ”I: " beWiles - multiple and varied solutions; new problems would arise from attempts to move from initial observations toward greater generalizations and increasingly elegant formulations; new challenges would be faced in efforts to progress from a tentative idea to an accepted proposition, method, definition. Rather than ending with an individual’s initial attempt to solve the problem, these problems would reveal new issues — and provide new insights — through various stages of work. These features of the mathematics problems and examples of the support they provide are listed in Table 1.5. Table 1.5 Features of the Mathematics Problems Mathematics Problems Mathematics problems designed with the following features: 0 Presents appropriate challenge to adult learners, requiring tentative initial work (conjectures, working definitions, experimenting and testing of ideas, false starts, etc.), justification and argument, refinement and revision. ' Multifaceted, allowing for some variety among answers, conclusions, solution approaches, means of justification/verification, perspectives or interpretations, etc. 0 Multilayered, allowing for unfolding insights through various stages of work on the problem In many ways, the mathematics problems would undoubtedly look commonplace, resembling word problems or other types of application and extension tasks often provided in math textbooks. However, there were other things about the problems that would likely be quite unfamiliar to prospective teachers: the work elicited by the problems; the ways that varied answers and solutions for the problem would be used to move the work forward; the continued unfolding of problems to reveal other issues besides the original question and allow for further insights beyond the answer. 39 ed design of rings prospc comprised or together as a mathematics both useful 2 mahematie: foster nets L Entourage 11 collection 0 teachers 3:, semester. [ and elamrc Collection of Mathematical Tasks. Considerable attention went into the selection and design of every mathematics problem the class would focus upon, however, the things prospective teachers would learn from their work on these individual problems comprised only part of a larger course of study. These problems were designed to work together as a set, allowing prospective teachers to closely study a particular area of mathematics and, in the process, develop an understanding of the content that would be both useful and useable for the work of teaching. Additionally, the assemblage of mathematics problems would be complemented with other math-related assignments to foster new understandings of the content as it relates to learners and learning as well as to encourage the use and development of mathematical skills for teaching. Together, the collection of various mathematical tasks for the class was designed to help prospective teachers acquire important mathematical resources for teaching over the course of the semester. Early work would lay a foundation of knowledge and skills to be built upon and elaborated over time; later in the course, these resources would be drawn upon and used in prospective teachers’ beginning efforts to encourage and support the learning of others. Important features of the collection of mathematical tasks, along with specific design decisions that were made, are discussed below. One fundamental feature involved early and continued opportunities to examine the meanings and root concepts of mathematical ideas and procedures. Through such studies into the conceptual underpinnings of school mathematics, utilizing the types of math problems described in the previous section, it was believed prospective teachers would begin to acquire the type of unpacked, interconnected, multi-perspective understanding of mathematics that is critical to teaching (Ball, 2003; Ma, 1999; National 40 3.353.132 the topII ofnumt mart} ad arithrnet be encat; appropri. 11253115 IO denomin; intolx in; Operation not hat e l IBall. 193 familiar a. prc‘dUCIIII A : 11an II‘IIS I COmEm 35 “1110115 12 liar) I J: b Research Council, 2001). To support this ongoing examination of underlying concepts, the topic area of fractions was chosen as the primary focus of work. Of the various types of numbers studied in the elementary grades, fractions can be especially challenging for many adults in the United States. Although it is often the case that people can solve arithmetic problems involving fractions, their capacity for working with fractions tends to be encapsulated in a collection of rules they have memorized and learned to use in appropriate situations: invert-and-multiply (when doing division with fractions); “of” means to “multiply” (in word problems involving fractions); make common denominators (when adding or subtracting fractions). The list goes on. Operations involving fractions seem to go by a special set of rules, unconnected from "normal" operations with whole numbers. Pre-service and practicing teachers in the US. ofien do not have good intuitions, explanations, or real-world applications for such operations (Ball, 1988a; Ma, 1999). Because of this, the use of fractions can make almost any familiar arithmetic operation seem strange and, as a result, has the potential to productively call into question the fundamental meanings of numbers and operations. A second significant feature of the collection of mathematical tasks would build upon this conceptual foundation, encouraging prospective teachers to examine the math content as it relates to learners and learning. Over time, and through their work on the various tasks, it was expected that pre-service teachers would begin to develop important knowledge and skills vital for “hearing” — and sorting through the implications of —- learners’ ideas about specific mathematical content. Certainly one important site for developing such resources would be the class itself. As pre-service teachers worked through the mathematics problems, sharing and scrutinizing each others’ ideas and 41 damn: confuse : . ” townu studies 1 Videos c respons; assicnrn ‘ - mittens With fraci area 0f m; benefits fr up 'thclr ur learners. i discover} abOUi lira. {fathom solutions, there would be many occasions for uncovering facets of the content that confuse and trip-up — as well as those that stir interest and provide insight. However, in conjunction with this work, teacher education students would also be doing a variety of studies into mathematics teaching and learning that target children’s ideas and ways of thinking about fractions. Such work would include activities like closely examining videos of children working on fractions in a mathematics class or analyzing children’s responses on a carefully designed fractions quiz. There would also be a fairly large assignment involving the analysis and adaptation of a fractions interview for elementary students and the implementation, documentation, and analysis of the interviews. A final notable feature involved the structuring and staging of the work that would allow prospective teachers to draw upon their emergent knowledge and skills and begin to use these resources in early attempts to help others learn mathematics. By grounding the collection of mathematical tasks almost entirely in fractions and operations with fractions it was expected that prospective teachers would make serious gains in this area of mathematical study during the semester — enough so that they would reap clear benefits from this deep and broad understanding. Over time, as pre-service teachers built up their understanding of foundational concepts and mathematical content as it relates to learners, it was anticipated that they would use this knowledge in subsequent tasks, discovering in themselves an increasing ability to hear and appreciate children's thinking about fractions, a growing capacity for analyzing and adapting problems involving fractions, and an expanding set of skills for accessing and addressing others’ ways of understanding fractions. 42 '96}qu if F. support 1: These three features of the collection of mathematical tasks are summarized below in Table 1.6. Table 1.6 Features of the Collection of Mathematical Tasks Collection of Mathematical Tasks As a collection, the mathematics tasks of the course should have the following features: ' Examines the meanings and root concepts of mathematical ideas and procedures that are taken for granted; unpacks the subtleties and complexities of the mathematics content ° Illuminates challenges learners face when studying the content 0 Allows prospective teachers to use mathematical practices that are central to the work of teaching The collection of mathematical tasks for the course was expected to initiate and support learning that would be quite unfamiliar to the prospective teachers. For example, their study of fractions in the course would almost certainly be unprecedented for its depth and breadth; such detailed and extended examinations of an “elementary” topic are rare. Additionally, the mathematical understanding that pre-service teachers would be developing was likely to be quite different from what they would have learned — than they would have had to learn — as students in mathematics classes. As noted by the National Research Council (2001) — Knowing mathematics for teaching... entails more than knowing mathematics for oneself. Teachers certainly need to be able to understand concepts correctly and perform procedures accurately, but they also must be able to understand the conceptual foundations of that knowledge. In the course of their work as teachers they must understand mathematics in ways that allow them to explain and unpack ideas in ways not needed in ordinary adult life (p. 371). 43 that xx oul. designed ' present 1 prospecti doing “0 mathema perspecti A. utilized a the group recording ‘ {he grou Support It “ (liking .. marhem: "1am la; Tools and Material Means A final designed element of the course consisted of the tools and materials means that would be provided. These special instruments for thought and collaboration were designed to enable the production of class records that capture the evolution of ideas and preserve knowledge developed over time as well as to provide active work “spaces ” for prospective teachers to note and present their ideas in the moment. These spaces for doing work and records of past work are essential for the supporting mathematical learning that builds from and upon past efforts; they are necessary for encouraging mathematical understanding that incorporates, accounts for, and addresses the ideas and perspectives of others. Although a variety of materials and resources would be made available and utilized at different points in the class, two types of tools and material means would be used routinely, integrated into the day-to-day work of the class. One type of tool would specifically support the work of the class as a whole, providing an active work space for the group during whole class discussions (i.e. large Whiteboards) as well as a means for recording and preserving the knowledge developed by — and the unresolved questions of — the group (i.e. class generated posters). The other type of tool would more directly support the work of individuals, providing them a personalized and less formal space for working out their thoughts and building a permanent record of their — and others’ — mathematical work (i.e. composition notebooks and pens). These two types of tools and material means are discussed below. Tools and Material Means for the Clas_s_. One of the primary tools of this class would simply consist of the large classroom Whiteboards running along the front of the room. 1 chili” such W1 anemic “omit $6.15 a lesson. This us: tend to Hiebert room. The Whiteboards themselves were unremarkable, similar to Whiteboards and chalkboards that line the walls of classrooms everywhere. However, unlike classes where such writing surfaces are primarily used by teachers as a visual aid to keep students’ attention directed toward the information to be learned at the moment, these Whiteboards would be used by the entire class as a shared writing space for laying out the questions, ideas, and solutions under consideration by the group — for looking across the work of the lesson, considering juxtaposed alternatives, looking for relationships and connections. This use of the Whiteboards would overlap in many ways with how Japanese teachers tend to use chalkboards in mathematics lessons. For example, James Stigler and James Hiebert (1999) note, Japanese teachers lead class discussions, asking questions about the solution methods presented, pointing out important features of students’ methods and presenting methods themselves. Because they seem to believe that learning mathematics means constructing relationships between facts, procedures, and ideas, they try to create a visual record of these different methods as the lesson proceeds. Apparently, it is not as important for students to attend at each moment of the lesson as it is for them to be able to go back and think again about earlier events, and to see connections between the different parts of the lessons (p. 93). In the same spirit, the Whiteboards would be used in this class as a significant material means for looking back and looking across the work of the class. However, unlike the carefully designed layouts of the Japanese chalkboards — unfolding from left to right — resulting in a complete record of the lesson constructed by the teacher (or under the teachers’ watchful direction), the Whiteboards in this class would be used more loosely. Teacher education students would frequently write on the boards themselves, ofien filling up the expansive writing surface before the end of the lesson and requiring that they be intermittently erased. However, careful decisions would be made about which pieces to 45 tar} dill ‘ 3 m0“; 0 tea: not imprint. ephemer. lesson —. idea to t the and r: proxen f0 L‘Sances. unprm en methods I back and Willem: Eitii‘oram Lilith} .\ t and copy: and Biff: preserve — at times, re-constructing portions of the boards in different locations when needed and, at other times, simply highlighting pieces for comparison that may lie on very different parts of the Whiteboards. Rather than keeping a complete and organized record of the lesson on the Whiteboards, individuals would construct such a record in their notebooks (discussed below in the next section). The use of Whiteboards would be complemented by the use of posters displaying important mathematical ideas proposed and developed by the class over time. Unlike the ephemeral content of the Whiteboards that fluidly changes, evolving over the course of lesson — and then is removed before the next lesson —— posters would preserve important ideas to be carried over into subsequent lessons. In some cases, a poster could contain the end results of mathematical work carried out by the class: accepted propositions, proven formulas and algorithms, established definitions, and the like. However, in many instances, the posters would contain the seeds of new mathematical work — for example, unproven conjectures, potentially significant observations, working definitions, and methods that appear to work on a sampling of problems. Posters would allow for looking back and looking across not only a single lesson but over the span of the entire mathematics methods course, allowing for mathematical knowledge to be extended and elaborated - revised and refined — over the semester. Accordingly, these enduring displays of the group’s mathematical work would allow the class to pursue more difficult and complex mathematical problems across multiple lessons, growing from nascent ideas and simple observations to formalized and proven concepts. These tools and material means for the supporting the mathematical work of the class as a whole are summarized below. 46 . u? uni. the t 0) Cl {ECU note will l Adi note 1 ma ~ Ct": Table 1.7 Tools and Material Means for the Class Whiteboards and Posters 0 Whiteboard used by the students and the instructor as a shared presentation space 0 Posters of conjectures, working definitions, methods/algorithms, etc. developed and used by the class Although large writing spaces, like Whiteboards and chalkboards, are found in almost every classroom and posters frequently line school room walls, the ways in which these tools would be used to build mathematical knowledge over time — featuring and focusing the efforts of the group upon the ideas they’ve generated — was expected to be unfamiliar to the pre-service teachers. Tools and Material Means for Individufl An indispensable tool for supporting the mathematical work of individuals in the class would consist of a bound laboratory notebook — or “composition notebook” - that would be filled in by prospective teachers over the semester. This is the same type of notebook that is used by scientists for recording their experiments and tracking their work. The pages are sewn in making the notebooks quite durable. This means, however, it is not possible to remove pages without compromising the binding. Neither can the pages be re-ordered; instead, the notebooks would largely be chronological, each entry picking up where the previous one left off. Along with these notebooks, prospective teachers would also be using pens for their mathematical writing, creating a more “permanent” record than would be produced with the more commonly used, graphite pencil. Although these tools may seem limiting and prohibitive in some ways, many of the constraints found in these materials were deliberately chosen to support the 47 rim item — a ham quilt} . pages. L discard feature reduce I resent alien en lemme pros-m 30 Captu mmfis t mathematical work of the class. The permanence of the pages and the things written on them — along with the book-like form and the fixed ordering of pages — were each important features of this set of tools. The writing spaces students typically used in mathematics classes often have a quality of imperrnanence to them: scratch paper, spiral notebooks with easy-to-remove pages, loose-leaf paper and loose worksheets placed in 3-ring binders. The capacity to discard pages — to winnow out preliminary work from final work — is an important feature. To different degrees and in different ways, such writing spaces routinely serve to reduce the messiness that results when figuring things out, allowing students to neatly present their answers along with the correct steps used. Likewise, the use of pencils is ofien encouraged to enable students to get rid of mistakes, erase unnecessary steps, and remove failed attempts. In this class, however, the composition notebooks in which Prospective teachers would be writing — and the pens they would be using - would serve to Capture and preserve these important aspects of mathematical work. The notebook entries written as in-class and homework assignments would include, - Explorations of new problems, ideas, and alternatives Extensions and elaborations upon work already done — including additional practice with methods and skills, the development of greater generalizations, and the further refinement of ideas Reflections upon current understandings, observations that can be made, and unsettled issues or questions. The composition notebooks would be a staging area to prepare for whole class discussions; a testing ground to try out things; a personal space to clarify one’s own 48 ideas, responses, hunches, and confusion. Instead of supporting the presentation of only conclusions and end results, the notebooks would facilitate the representation of prospective teachers’ efforts from beginning to end. This way of using notebooks for mathematical work often resonates, for many people, with the idea of “journals.” There is a personal feel to the notebooks and they function, as journals often do, to give voice to things that have yet to be articulated, to record experiences (things seen and heard), and to work through those things that are only partially understood. While the notebooks for this class would share these features with journals, there would also be a much more public and formal component to them as well. Although the notebooks would be personal — customized for and by an individual — they would not be private. And though each notebook would be designed to support the work of one person, it would need to be composed and organized so that it could be understood by another person (such as the instructor) —- and useful to the author her/hil'nself, across time. Additionally, along with individuals’ thoughts and efforts, the rIOtebOoks would also be comprised of class notes, recording the thoughts and efforts of the group as a whole. In very real ways, the mathematics notebooks would serve as a primary textbook 0f the course. However, unlike printed books — “perfect printed volumes that exist in tho“Sands of identical copies,” where “the writing is stable, monumental, and controlled exclusively by the author” (Bolter, 1991, p. 11) — these notebook-textbooks would emphasize the constructed, evolving nature of mathematical knowledge, highlighting the role of a community in developing knowledge over time. Additionally, unlike most 49 innit: 4 Oi llltli' V Pi€~$cr~ textbooks, the content of these notebooks would be largely organized by problem8 and ordered chronologically in a way that traces the unfolding of new understanding. The physical constraints of the bound notebooks, coupled with the problem-centric work of the class, would reinforce this. Among other things, this alternative structuring of the content would cluster together the underlying mathematical ideas that inform a problem. It would present alternative solutions for — and different perspectives on — the problem side-by-side. And it would assemble the difficulties and common mistakes learners encounter with the content. For prospective teachers - as students of subject matter, learners, and learning — this organization of the mathematical content is advantageous for drawing connections and seeing important relationships. The ways in which the notebooks and pens would support the mathematical work of individuals in the class is summarized below in Table 1.8. 8 Pre—service teachers will often introduce alternative means for organizing their notebooks that enable them to navigate the content in different ways: Examples include, indices, table of contents, and pages set aside in the front or back for frequently referenced information 50 4;! l. ' lt’l‘lfi.‘ finer only ' muti: dmm Table 1.8 Tools and Material Means for Individuals Notebooks and Pens9 ' The notebooks used as a “space” for working through problems and on ideas, for forming responses and next moves. 0 ' The notebooks constructed by prospective teachers as the primary text of the course. a. A record of the individual’s evolving ideas b. An organized record of the accumulated understandings and investigations, conjectures and arguments, of the class In various ways, the use of the notebooks and pens would likely be unfamiliar and challenging for the teacher education students. The careful recording and preservation of tentative thoughts and rough ideas — of mistakes and false starts — could seem risky and uncomfortable after years of diligently omitting and removing such signs of uncertainty and struggle. Or, on some deep level, it could simply seem distasteful: intellectually “messy.” Additionally, the use and development of the notebook-as-textbook was expected to be new for the prospective teachers, requiring new ways of recording of not only their own work but also that of the class as a whole. Unlike journal-writing or routine note-taking, some thought and creativity would need to be devoted to creating a record that could be understood by another and used weeks, months, and perhaps years down-the-road to support one’s future mathematical work. These designed elements of the course — the types of work and ways of working encouraged, the social-intellectual structures developed, the tasks and problems used, and the tools and material means provided — were conceptualized and constructed to 9 In addition to the functions listed here, the notebooks also served as an additional means of communication between individual students and the instructor. 51 iii" CORE f. Y. KUIAK in '11 Palm 1 Clip" complement one another. Together, these elements were to comprise an educational context where unfamiliar forms mathematical knowledge and ways of knowing mathematics would be called for, supported, and valued. The section that follows takes up the specific ways in which this educational context was documented and discusses how the changes with and within the context were studied. Mollection a_nd Analysis The data collection and analysis were designed to allow the close examination of the unfamiliar terrain and its development over time as well as to enable a careful investigation into the processes by which prospective teachers adapted to the class context — and adapted the context itself — over the course of the semester. In the following discussion, details of the data collection site are described along with the ways in which the class was documented. Special attention is given to data collection efforts focused on capturing the design and development of the class context and also to those focused on capturing students’ reception of and responses to the class context. Finally, important selections and omissions are discussed, explaining the decisions made for narrowing the collection of classroom records that were generated and gathered to a focused subset for fine-grained analysis. The Site The elementary mathematics methods, TE401: Inquiry and Practice: Curriculum, Pedagogy, and Social Life in Classrooms, was the first of two subject-specific teaching methods courses required for elementary education seniors at Michigan State University. The two-course series was intended to provide prospective teachers with an introduction to key issues and practices associated with the teaching of school subjects — in particular, 52 i” _‘/u’ Sill; ll 66 the series concentrated on mathematics, literacy/language arts, science, and social studies. In any single course, two of the four subject matter areas were paired and taught together by two instructors, each a specialist in one of the focal fields of study. For this particular class, I was the instructor for the mathematics portion of the course and my teaching partner, F enice Boyd, was the literacy/language arts instructor. Although, over the years, some pairs of TE401 instructors have structured the course to closely integrate the teaching and learning of the two subject matters, Fenice and I chose to split the semester into three discrete “blocks.” The first block was a short segment that took place during the first week of the semester (August 29 — September 2). This was an unusual week in that the prospective teachers’ school placements had not yet started and we were able to have class sessions at the University, for five consecutive days. During this first block of the class, F enice and I taught together, using the time to introduce the teacher education students to the work we would be doing in mathematics and language arts as well as to take up common themes that would weave through their studies in these two subject matter areas. 10 Fenice and I divided the remaining fourteen weeks of the semester into two seven-week blocks: Fenice taught the “Literacy Block” during the first seven weeks (September 5 — October 21); I taught the “Mathematics Block” during the second (October 25 — December 9). During the Literacy and Mathematics blocks, the class was to meet at the University three times a week — on Tuesday, Thursday, and Friday — for two-hour class sessions. On Monday and ‘0 This is discussed in more detail in Chapter 2. 53 a.“ \ .1.‘ ll. \ u . l V .1 ~th . . mild rd H“ 21 VIM. Mlk WA I rum. G . l ., T b Wednesday, the teacher education students were to spend two hours each day in an assigned classroom at a local elementary school. ” The primary data collection took place during the first week of the semester when F enice and I co-taught the class and the last seven weeks when I was teaching the Mathematics Block. Table 1.9 lays out the three blocks of the semester and highlights the class sessions that were documented for the design experiment. Table 1.912 Semester and Data Collection Schedule Course Introduction _M. Li _w n .1: August/ September 29 30 31 l 2 Literacy Block Seven Weeks (Sept. 5 — Oct. 21) Mathematics Block: October 24____ 25 .29; 27 28 November ..__31_ 1 f 2 3 4 " __Z____ 8 ____9 10 ll ”Ii” 15 j 16 17 18 i..-?—.L- 22 23 ..J‘ Thanksgiving November/December 1-... 28 29 “39“ 1 2 December t, 5 6 7 8 9 The specific data collection activities that took place were focused around two related but arguably distinct sets of premises and research goals. These are discussed separately below. ” Two prospective teachers were assigned to each elementary classroom that participated in this program. The assignments associated with the school placements were typically small tasks that used the classroom as a site for observation and study. On a few rare occasions, the pre-service teachers were asked to make special arrangements to do special activities with the children. 12 There are tables and images in this dissertation, such as this one, that are presented in color. 54 [(‘3 (L) fen ”1:, ‘l 11;.“ Capturing the Design and Development of Class Context One fundamental premise for this study is that people have a tendency to adapt to the contexts in which they are placed. Learning contexts matter. Not just because some contexts are more effective than others in promoting the acquisition and retention of the things we want students to learn but also because students are continually learning contexts. In every moment of every class, they are learning “how things work” within the environment they find themselves, developing habits that effectively achieve their needs and desires in that context. Over time, they also begin to form assumptions and beliefs about themselves and their capacities to act within and on such settings. This way of conceptualizing of the educational environment as a medium that calls forth certain responses from learners - that encourages the development of perspectives, beliefs, and actions that might not otherwise form — was an important idea that John Dewey proposed and on which he built in his writings. For example, in one essay (1916/ 1944) he writes, The particular medium in which an individual exists leads him to see and feel one thing rather than another; it leads him to have certain plans in order that he may act successfully with others; it strengthens some beliefs and weakens others as a condition of winning the approval of others. Thus it gradually produces in him a certain system of behavior, a certain disposition of action. The words "environment," "medium" denote something more than surroundings which encompass an individual. They denote the specific continuity of the surroundings with his own active tendencies (p. 11). In this design experiment, the role of the “environment” — of the unfamiliar terrain — in the process of human adaptation was a significant focus of the work. A major goal of the study was to better understand what it might mean to create an educational context for the preparation of teachers where new forms of mathematics and unfamiliar modes of knowing and doing math are presented in their new terms. What 55 laptt called the \‘i might be involved in the formation and development of such a context? How might such a context function as medium to encourage the development of new capacities for thought and action, different understandings and beliefs, alternatives ways to see and feel? With this goal in mind, the data collection was designed to capture the class context over time, documenting it in a variety of ways. One of the primary data sources for this study consisted of richly annotated video of the class sessions, collected by two colleagues — Angia and Steve. Angia, who was also a doctoral student and a TE401 mathematics instructor for a different section of the course, videotaped the classes. Along with the video record, she kept notes on her thoughts and impressions of the lessons as a highly informed and deeply interested observer. Steve was another doctoral student with interest in teacher education; he brought significant experience with conducting interviews and doing classroom observations for research. During the class sessions, Steve kept detailed fieldnotes on a laptop computer that was connected to the video camera. Using a piece of software called CVideo by Envisionology, Steve was able to “link” his notes to specific times on the videotape. This enhanced our ability to later navigate the video and facilitated its analysis. Steve’s netes focused primarily on capturing the flow of events in the class throughout the two-hour sessions, noting transitions between activities and detailing the things said and done in the large group discussions. He supplemented these observations of the class with more focused notes on small group interactions. On many occasions, Steve would also direct Angia to focus the videotaping on particular small groups located at various locations throughout the room. Four boundary microphones "permanently" mounted on each of the four walls of the classroom were wired into a sound mixer near 56 [fiftl With 1 :31;le notes. records disillbi the video camera. Whiteboards Micro hone 0- ' . O . o I Microphone . c . O Microphone . O ' 3 . l: . . Wdeo Camera/ 000 0.. ... AudioMixer Microphone . chair E: table half-circle tables that small groups used when working together With this arrangement, it was possible for Angia to have some degree of control over the audio and video that was picked up by the camera. Along with creating a video record of the lessons and generating accompanying notes, Steve and Angia made audio recordings of the class sessions, created detailed records of what was written on the Whiteboards, and collected all handouts that were distributed. These efforts to document what occurred during the class sessions were complemented by other efforts to capture the intentions and motivations that informed the lesson designs and that guided my instructional decisions. One way this was done was through journal entries I wrote to prepare for and reflect upon each lesson. In addition, Angia, Steve, and I made audio recordings our own conversations about the class as we planned for the data collection. Steve and Angia would routinely raise questions and 57 In de “T; make observations about the lessons, eliciting my perspectives about the class sessions or about particular students; and, almost invariably, they would press me to articulate the rationales for my past decisions as well as for my future plans. Capturing Students’ Reception of an_d Responses to the Class Context Another fundamental premise of this study is that people adapt the environment in which they find themselves, transforming the context as they learn about it and work within it. One way this occurs in a classroom is through students’ attempts to give meaning to their surroundings and activities. They construct interpretations of the things they encounter and experience using their current understandings, beliefs, and values. People also transform contexts through their actions and reactions — people are, afier all, a part of the context themselves. In a class, students influence the discussions through the things they say and do; they affect the way time is spent; they influence the tone and atmosphere of the moment. As Dewey (1916/1944) notes, “Adaptation, in fine, is quite as much adaptation of the environment to our own activities as of our activities to the environment” (p. 47). A central goal of this design experiment was to better understand the processes by which prospective teachers shape the class context through their perceptions and interpretations — as well as through their actions and words. To some extent, these processes were captured in the documentation of the class sessions, as described above; however, in order to closely examine interpretations and experiences, much more detailed data about the prospective teachers was needed. One important data source for such analyses consisted of the pre-service teachers’ writing. Copies were made of their projects, papers, quizzes, and the composition 58 notebooks they kept for the class. Along with this, we also conducted a variety of structured and unstructured interviews at various points in the semester. For example, prior to the start of the Mathematics Block, during the weeks the teacher education students were studying language arts with Fenice, I arranged to meet with each of them individually for a fifteen-minute "conference" outside of class time. During these interactions, I asked them questions about their experiences with mathematics and teaching; and inquired about their expectations for the course. As another example, Steve and Angia occasionally arranged informal interviews with prospective teachers before and after class, creating opportunities for the teacher education students to discuss their thoughts about the ideas raised during the lesson or their reactions to interactions that took place. Selections and Omissions These efforts to document the design and development of the class context along with the students’ reception of and responses to it yielded a relatively large and very rich set of records of classroom practice. This collection included approximately fifly hours of class video, more than twenty-five hours of conversations recorded with only audio, over a thousand pages of student writing, and almost two hundred pages of typed observation notes. After an initial survey and cataloging of the entire collection, a few crucial decisions were made about the study — two, in particular, significantly narrowed and sharpened the focus of the data analysis. One decision centered attention on the mathematical work of the course; the other concentrated efforts on four particular episodes of the class. These selections, along with their accompanying omissions, are discussed below. 59 it Focus on the mathematical work. Doing and learning mathematics was important in the math methods course however this work was only one piece of a larger curriculum. In addition to helping prospective teachers learn and experience unfamiliar types of mathematical work, the time spent working on math problems and tasks was intended to serve as a shared, “live” instance of classroom practice for study — to be a site for students to use, explore, and reflect upon ideas and practices raised in the class. Three key dimensions — or “themes” — of mathematics teaching and learning were examined across the semester: the formation and functioning of the classroom culture, the role and development of students and meaning making, and the design and use of learning tasks. Approximately two weeks was spent on each theme, studying different theoretical perspectives and pursuing practice-based explorations through reading assignments, class discussions, notebook entries, and investigative tasks. In addition to this firsthand mathematical work, prospective teachers were also provided with a set of K-8 classroom examples that was to serve as yet another site for investigating mathematics teaching and learning. These elementary school examples included written cases of mathematics classroom practices; a large, rich multimedia case of a third grade mathematics class taught by Deborah Balll3 ; and the prospective teachers’ school placements. In Table 1.10, the major themes investigated in the class are laid out along with the practice-based sites —- or “settings” -— in which these themes were explored. '3 This “multimedia case” was an interactive multimedia environment containing a variety of classroom records spanning approximately two months of instruction. The environment was developed with the support of the National Science Foundation (NSF Grant Number TPE-8954724) under the leadership of Deborah Ball and Magdalene Lampert. A detailed description of the multimedia environment, the project that developed it, and the results from this work can be found in Lampert and Ball’s (1998) book, Mathematics, Teaching, and Multimedia: Investigations of Real Practice. 60 Table 1.10 Settings for Investigation and Course Themes Investigated Settings for Course Themes Investigation Investigating Investigating Students & Meaning Classroom Culture . Making Investigating Learning Tasks Firsthand mathematical work K-8 classroom examples The mathematical work of the class that is focused upon in this study makes up a relatively small part of the curriculum. Rather than being a central theme of the course, it served more as a context for and an object of investigation. However, more than any other part of the class, the prospective teachers’ work on mathematics placed them in unfamiliar territory that contrasted in significant ways from what they had experienced in the past — from those things they had come know and expect through their schooling. Although many parts of the TE401 curriculum were “new” to the pre-service teachers the types of mathematics and ways of knowing math they encountered in the course involved a mismatch between old and new -— between familiar and unfamiliar — that gave rise to the type of paradoxical situation described by Schwab and focused upon in this research. Consequently, those portions of the class sessions and the students’ work that focused on mathematics were central to the data analysis. Focus on episodes. Of the twenty-four class sessions that were documented during the first week of class and throughout the Mathematics Block, approximately half of those sessions devoted some portion of the two-hour lesson to doing mathematics. These lesson segments involving mathematical work were given special attention: The 61 videos, observer notes, teacher’s journal, and students’ work associated with these segments were collected and examined. Several “maps” were created from and for this work that detailed the mathematics discussed, the various issues raised, and the events that occurred during each segment. Based upon this preliminary analysis, four episodes from different points in time were selected for detailed study. Together, these episodes portray the unfamiliar terrain as it evolved across the semester. Additionally, each episode highlights a key facet of the process of change theorized by Schwab, corresponding to some aspect of “what happens” when the new is presented mainly in its new terms. Table 1.11 is a color-coded calendar depicting when the episodes occurred — at times, spanning more than one class session. It also indicates thechapters in which these episodes are discussed and the predicted process of change upon which the chapter is focused. 62 . $.- 5 Nail Table 1.11 Chronological Organization of the Data Analysis Chapters* M Tu W Th F Aug/Sept. 22 30 31 l l Literacy Block — Seven \Veeks Mathematics Block 1 Oct: “4 25 :(x 27 3 Nov? 31 _1_ 2 g g i l 7 g 9 10 11 ; 14 15 to 11 Q l 21 g; :3 lli;111|\sgi\'ing; Nov/Dec: ”8 29 30 l _2_ 1 Dec.l 5 g 7 8 9 l Chapter 2 From the first hearers much struggle to understand Chapter 3 As they translate tentative understanding to action, it leads to unexpected consequences Chapter 4 There’s reflection on disparities between ends envisaged and actual consequences Chapter 5 New competencies for taking thought are roused. The new actions change old habits of thought and observation... * NOTE: The underlined dates with bold typeface indicate class sessions involving some amount of mathematical work It is notable that these four episodes were chosen not because they were “typical” but because they were each best suited to make visible the processes of change under examination. In most cases, this meant that normally subtle processes came to the forefront and were more apparent than usual in these lesson segments. At times, such as in the episodes featured in Chapters 2 and 4, the processes of change were problematic and therefore highly visible. In other instances, such as the episodes featured in Chapters 63 innodu desatr. 8 and the 3 and 5, the processes were central and necessary to the work at hand. More details about these episodes and chapters are provided in the Overview section below. Overview This dissertation is organized into six chapters. The first chapter is an introductory chapter that presents the problems and theories under investigation, the design experiment that was developed to explore these ideas within a classroom setting, and the structure of the dissertation itself. The next three chapters (Chapters 2 - 4) are analytic chapters that each focus upon one key aspect of the process of change theorized by Schwab, exploring it through the close examination of a selected class episode. The last two chapters (Chapters 5 and 6) are concluding chapters that round out the investigations discussed in earlier chapters: Chapter 5 is an epilogue, providing a glimpse of the class context at the end of the semester and the prospective teachers’ work within a terrain that has come to be more familiar; Chapter 6 summarizes the findings of the analyses and discusses their implications for teacher education and future research. These remaining chapters are described more fully in the following pages. Analytic Chapters: Chapters 2 - 4 The analyses of the classroom episodes were grounded in several foundational ideas borrowed from an area of systems science called, “Cybernetics.” Many branches of modern systems theory trace their early interdisciplinary roots to the work of cybernetic scholars such as W. Ross Ashby, Norbert Wiener, Warren McCulloch, Heinz von Foerster, Gregory Bateson, and Margaret Mead whose work across disciplines during and immediately after World War II contributed to the growth of the fledgling field. In particular, cybemetics was characterized by its emphasis on examining the functioning of goal-directed systems - both natural systems like individual organisms, communities, and ecologies; as well as artificial systems like thermostats, engines, and artificial intelligence machines — and the ways they achieve and maintain their aimed-for conditions, or “goal states” (Heylighen & Joslyn, 2001). Three conceptual tools, or lenses, drawn from these seminal efforts to study goal-directed systems and how they function proved invaluable for analyzing the unfamiliar terrain, helping to shed light upon the complex classroom ecology as it changed over time. Together, these ideas from cybemetics informed the ways of looking, interpreting, and organizing information that were used throughout the study. However, in addition, each of these lenses also played a special role, serving as the primary lens for one of the analytic chapters. In the following discussion, a synopsis is provided for each of the three analytic chapters describing the classroom episode examined, the theoretical process of change focused upon, and the primary analytic lens used. Chapter 2: Introduction of the Unfamiliar Terrain This chapter focuses on the introduction of the unfamiliar terrain at the very start of the semester when F enice and I co-taught a series of lessons previewing the work of the course and its main themes. On the very first day of class (August 29), I had prospective teachers work together on a mathematics problem that was chosen, in large part, for its potential to engage people in the types of mathematical work — the kinds of problem solving, reasoning, and discussion — that would be central in the class. Equally important, it was chosen for its capacity to create opportunities to learn about the pre- service teachers as they responded to the unfamiliar mathematical work they were asked 65 (4* .r—; his film .lt‘t‘t min the m l5. ol‘ HOiic COnSll' .‘4 . ”lip; 25 _ Tiling r3350!) “ (. to do. The episode examined in this chapter centers attention on this early encounter with the unfamiliar terrain, looking at the work done on that first day of class as well as the prospective teachers’ reflections on the work several days later (September 2). The ways in which the teacher education students received and responded to the unfamiliar terrain during this initial coming-together are carefully analyzed. Additionally, Schwab’s theory that, “From the first, (they) must struggle to understand” (1978, p. 173), is examined in light of what actually happened, “from the first,” in this mathematics methods course. This analysis of the unfamiliar terrain at the beginning of the semester and students’ efforts to give meaning to it was aided by the use of a fundamental perspective of cybemetics that draws a distinction between the thing in itself'4 and individuals' constructions — their perceptions, interpretations, and experiences — of the thing. According to this view, “things” (objects, events, moments of time) as they exist are different from that which we perceive as individual human beings. The thing in itself can only be perceived in part - and our interpretations and experiences of “it” are always based upon this partial picture. In this study, it was presumed that people spoke and acted within a framework — within a construction of the world — that imbued their words and actions with purposefulness and rationality. The analytic challenge was to piece together the meanings individuals created, and the realities they perceived and experienced.15 It is, of course, impossible to directly access people's perceptions and experiences. However, actions and words within a given context can provide glimpses of the constructed or “lived” realities that individuals move within, react to, and act upon. In ‘4 The phrase, “The thing in itself” is a translation of Immanuel Kant’s term, ding an sich. '5 This is similar to a form of pedagogical inquiry Eleanor Duckworth (1987) refers to as “giving children reason” or what Peter Elbow (1986) calls “methodological believing.” 66 11] SUP-p11 this chapter, and throughout the study, people’s actions and words were closely examined as clues about the unfamiliar terrain that was being encountered and to which they were responding. Chapter 3: Establishment of the Unfamiliar Terrain This chapter focuses on the early work of establishing the unfamiliar terrain at the start of the seven-week Mathematics Block. It was important, as we launched our mathematical studies together, to begin communicating expectations, establishing roles, and developing norms and routines for working within the class context. On the third day of the Mathematics Block (October 28), I began with a problem called, “The Cookie Jar Problem.” One of the most notable features of this mathematics problem is the way it tends to constrain the approaches people use to solve the problem and the answers at which they arrive. It sets up, in fairly predictable ways, a productive situation for pursuing the types of discussion, collective problem-solving, and individual reflection that were central to the class. The episode analyzed in this chapter centers upon the Cookie Jar problem and the discussion of answers and solutions that ensued. The processes by which prospective teachers came to develop new ways of talking and acting as doers of mathematics — to form different expectations, use new words and phrases, and experiment with unfamiliar practices — are examined in this episode. In addition, Schwab’s proposition that, “As (learners) translate tentative understanding to action, it leads to unexpected consequences” (1978, p. 173), is scrutinized to better understand what this entails. This examination of how the unfamiliar terrain starts to be established was supported by the use of another important perspective of cybemetics — an explanatory 67 approach Gregory Bateson (1972) refers to as, “Cybernetic explanation.” Cybernetic explanation gives serious attention to alternative outcomes besides those that were observed. It assumes we are always dealing with a set of possibilities and tries to explain why a particular one seems to come up more easily, is observed more frequently, than others (Ashby, R., 1956/ 1964). In this analytic work, constraints are examined as possible explanatory influences that hinder or eliminate certain outcomes in favor or others. A classic example of this viewpoint is the theory of natural selection for explaining evolutionary change (Bateson, 1972). Unlike the many naturalists who asked questions like, "Why do giraffes have long necks?" Charles Darwin asked, "Why do we not see giraffes with short necks?" This led him in a different direction than his contemporaries who developed theories about how particular traits might be "acquired." Instead, Darwin's questions led him to consider what might be preventing other possible giraffe forms from existing (or from being numerous enough to be observed): He sought relationships, processes, conditions that would eliminate alternatives to the long-necked giraffe. In this chapter, and throughout this study, observations were made with careful attention to alternative outcomes and the question of why certain ones were prevalent while others were rare was always present. Additionally constraints were consistently sought after and examined as possible influences affecting what was (and was not) happening within the unfamiliar terrain. Chapter 4: Discernment of the Unfamiliai Terrain This chapter focuses on the subtler work of discerning and beginning to understand the unfamiliar aspects of the unfamiliar terrain. Once roles and routines began to set in — once students started to know what to expect during lessons — it became 68 important for the class to begin to make sense of the underlying ways in which the mathematical work of the methods course differed from the mathematics with which prospective teachers were already familiar. On the fifth day of the Mathematics Block (November 3), I had the class work on a problem called, “The 1% + 1/2 Problem.” This problem is particularly notable for the consistent way it tends to confront people with an area of elementary mathematics that is often not well understood by adults in the United States, involving mathematical ideas on which they have only a tenuous grasp. The episode analyzed in this chapter centers upon the class discussion that ensued from the prospective teachers’ work on the 1% + 1/2 Problem. In particular, it focuses on their struggles to make sense of the mathematical work in which they were engaged, examining the efforts of the students and instructor to discern — and make visible — the fundamental differences between their past experiences and present circumstances. Additionally, Schwab’s hypothesis that, “There’s reflection on disparities between ends envisaged and actual consequences,” is probed in light of this episode, to develop a more detailed understanding of what this “reflection on disparities” entails. This analysis of the processes involved in discerning and giving meaning to the unfamiliar terrain was aided by the fundamental idea that cybernetic systems respond and adapt to perceived differences. This is an idea that Gregory Bateson discusses in several places (1972, 1979, 1991) but which his daughter, Mary Catherine Bateson, describes eloquently in her biography of her parents (1984/1994). In her discussion of critical ideas that attracted and influenced her father, she writes — The biological world, Jung’s creatura, is the world of growth, adaptation, and communication, the world in which events are caused by the perception of differences rather than by direct impact. A plant swivels on its stalk toward the sun because there is more light on one side than on 69 [infill C another — it responds to the diflerence in light, not to the light itself. In the biological world an absence —- a kind of difference between the expected and the actual, which is not itself a thing — can be a cause as well (p.64). In this chapter, and throughout this study, perceived diflerences were looked for and examined as potential causes for the adaptive changes that took place. Rather than looking for particular words or actions, certain math problems or pedagogical moves that brought about change, differences — including perceived “absences” — were examined as possible explanations for how and why adaptations occurred as they did. Table 1.12, below, summarizes the three analytic chapters and their focal episodes, along with the theoretical processes examined in each chapter and the primary lenses used. Table l.12 Summary of the Analytic Chapters (Chapters 2 — 4) Theoretical Process Primary Analytic Lens C I1 a ter 2 I d p f h From the first hearers must The thing in itself & ntro “99°“ 0 t, e struggle to understand human constructions unfamrlrar terram ( ‘haptcr 3 As they translate tentative Establishment of the understanding to action, it Alternative outcomes and . . . leads to unexpected possrble constraints unfamiliar terrain consequences Chapter 4 There’s reflection on . disparities between ends Difference that makes a Discernment of the . . . . . envrsaged and actual difference unfamiliar terrain consequences Concluding Chapters: Chapters 5 — 6 The analytic chapters weave together a narrative thread that describes the unfolding of the TE401 course over time, and an analytic thread that investigates the 70 unfamiliar terrain and processes by which change occurs with and within it. In Chapters 5 and 6, these two threads are brought to a conclusion. Chapter 5 is an epilogue that provides a glimpse of the TE401 class during the final two weeks of the semester when the unfamiliar terrain has come to be more familiar. Chapter 6 gives a summary of the analyses and findings described in the earlier chapters and ends with a discussion of the implications of this work for teacher education and research. 71 plt’it’t sluts in the class t 1cm \l3\ — {Wilt CllCou this el Used 1‘. CHAPTER 2 Introduction of the Unfamiliar Terrain Introduction The educational approach articulated by Joseph Schwab (1959/1978) of presenting the new mainly in its new terms, involves designing and creating an educational context that is different in fundamental ways from what students have known in the past. As students try to make sense within — and make sense of — the unfamiliar class context, misapprehension is not only anticipated, it is necessary for initiating new learning in this approach. Schwab (1959/1978) describes the process in the following way - From the first, (learners) must struggle to understand. As they translate their tentative understanding into action, a powerful stimulus to thought and reflection is created (p. 173). In this chapter, I examine the initial coming together of students with the unfamiliar terrain that had been prepared for them, paying particular attention to this early “struggle to understand.” This analysis looks at the first class session (August 29‘") and the prospective teachers’ reflections on this work, several days later (September 2""). Through a careful study of the things said and done during the lesson and the prospective teachers’ attempts to describe and explain the things that stood out to them from this encounter, the chapter provides an image of how the struggle to understand played out in this elementary mathematics methods course — and sheds some light upon why. The chapter begins with a brief explanation of the theoretical perspective that was used for this analysis. It then goes on analyze the mathematics problem that was selected 72 to introdut mathemat descriptio interstai meaning popular ge: and ' distincti systems being a distone llllfil'prc WSpet to introduce students to the type of mathematical work and ways of working on mathematics that would be central to and focused upon in the class. This is followed by a description of the first lesson and the prospective teachers’ observations and reflections. And finally, the chapter concludes with a detailed analysis of the “struggle” that ensued as the teacher education students drew upon their past experiences and current understandings — their histories, personalities, and range of personal resources - to give meaning to the unfamiliar terrain in which they found themselves. The Primary Analytic Lens: The Thing in Itself and Human Constructions Communication is a fundamental concern of cybernetic study. Contrary to popular aphorisms, in the view of cybemetics, what you see is not necessarily what you get and “seeing” is not essentially “believing.” Instead, cybemetics makes a clear distinction between “things” —- objects, events, relationships, living and non-living systems, etc. — and the information that is communicated about these things. Far from being a perfect and accurate match, the cybernetic viewpoint assumes much is lost or distorted in the process of transmission and reception, the acts of perception and interpretation. In the discussion below, I describe the theoretical foundation of this perspective that served a primary analytic lens of this chapter. Qappcities for (Plrecgnion The world we perceive is inevitably different than the objective reality — what Immanuel Kant refers to as the ding an sich, or the “thing in itself” — that exists independently of the human mind. There is so much more to every object, in every moment, to every event than we are capable of taking in. The unaided human eye can only receive wave forms ranging between 380 and 680 milli-microns; the unaided human 73 hat Il€\'\ time don ear is only capable of detecting sound waves above 20 cycles per second and below 20,000 cycles per second -- just two of our many sensory limitations (Bandler & Grinder, 1975) In addition to these sensorial constraints, there is also much to the act of perception that requires physical agility and skill, developed only through practice and use. A very basic example of this that most of us have acquired without notice through years of using our eyes is the skill of using one's eyes to take in "wholes" - whole words rather than individual letters, a whole cat rather than a series of cat parts. Although “naturally” developed by most sighted people this skill is not automatic for those who have spent entire lives without the ability to see. This skill can be difficult to acquire by newly sighted adults who have largely oriented themselves in the world by the passing of time or sequential cues (chair, then wall, then doorway, then 5 steps, then next doorway...) rather than the visual spatial cues upon which sighted people so heavily depend. Oliver Sacks (1995) describes in An Anthropologist on Mars: Seven Paradoxical Tales an array of neurological cases that illuminate the vast learning required to use one's sensory receptors in ways that inform. For example, in describing the case of Virgil, a man who had been blind since infancy but regained his sight through surgery forty-five years later, Sacks writes — While Virgil could recognize individual letters easily, he could not string them together -- could not read or even see words. I found this puzzling, for he said that they used not only Braille but English in raised or inscribed letters at school -- and that he had learned to read fairly fluently. Indeed, he could still easily read the inscriptions on war memorials and tombstones by touch. But his eyes seemed to fix on particular letters and to be incapable of the easy movement, the scanning, that is needed to read. (p.123) 74 in: 1110' M: “C it: a; This example highlights a way in which “seeing” involves a physical skill -— a fluidity of movement, a practiced ability to focus and refocus them for various kinds of seeing — which is developed as we use our eyes in many contexts, over many years. Rather than being like a light switch flicked “on” or a closed valve being opened, perception involves more than an activation of senses: It involves an array of complex and finely developed skills for properly receiving potential information — and, as shall be discussed below, it involves an array of interpretive skills to properly ascribe meaning to it. Cppgities for Interpretation Aside from the ability of our “sensors” to pick up potential information about our world, there are complex workings of the brain and nervous system that automatically eliminate many signals -- sights, sounds, tastes, smells, and tactile sensations -- which we are fully capable of detecting but would, in their entirety, overwhelm and confuse us (Huxley, 1954). We can pick out a voice in a crowded room and follow a conversation despite the myriad of other sounds that buzz around us. We are able to focus on words on a page regardless of the scene that spans in the periphery. Aldous Huxley (1954) posits that the brain and nervous system work as a "reducing valve," "leaving only that very small and special selection (of sensory input) which is likely to be practically useful." Whatever the objective reality may be, human beings receive a "measly trickle" (Huxley, 1954) of signals; and with this paltry information, we do our best to give it meaning using the language and norms, past experiences and learning, we have available to us. Far from being a passive reception of signals about the world, perceiving and interpreting involve an active process of filtering, filling in gaps, organizing, and interpreting the sensory input we receive. 75 3X1 d0? dot but org 811‘ in de One striking example of this can be seen in Jerome Bruner and Leo Postman's (1949) paper, 0n Perception and Incongruity: A Paradigm. In this classic study on perception, Postman and Bruner showed test subjects playing cards where some of the cards were "normal" but others had anomalous suits such as red spades or black hearts. They found that nearly all tests subjects displayed a reaction to the incongruity they refer to as the "dominance reaction." For some period of the testing, 96 % of the subjects failed to correctly perceive either the color or the shape of the incongruous cards. So, for example, although the test subjects may have correctly reported seeing "black" (color dominance), they incorrectly identified the "heart" form as "spades" or "clubs"; "form dominance" was also possible where the form —- e. g. "hearts" -- was correctly identified but the color (black) was incorrectly identified (as red). The test subjects perceived the anomalous suit as a "normal" suit. In their conclusion, Bruner and Postman state their experiment "is simply a reaffirmation of the general statement that perceptual organization is powerfully determined by expectations built upon past commerce with the environment. When such expectations are violated by the environment, the perceiver's behavior can be described as resistance to the recognition of the unexpected or incongruous." As students, the prospective teachers have spent years in mathematics classes developing ways of attending to, interpreting, and reacting to the stream of “information” that flows across their eyes and ears over the course of a mathematics lesson. Over time, they’ve honed their capacity for picking out the important things, formed habits for dealing with uncertainty and confusion, developed patterns of attention and inattention 76 that have served them well in these contexts. These ways of attending, thinking, and reacting continue to be resources for students, even within the unfamiliar terrain, helping them to filter information, fill in gaps, organize and interpret the sensory input they receive. In the following section, a mathematics problem that was designed to provide glimpses into these ways of perceiving, thinking, and acting that prospective teachers often bring is discussed in detail. Problem Analysis: The Three-Coin Problem For the very first day of class, I chose a mathematics problem especially well suited for learning about students and for introducing several important facets of the unfamiliar terrain. The Three-Coin Problem“ I have pennies, nickels, and dimes in my pocket. If I pull three coins out, what amounts of money could I have? This is a fairly ill specified problem. Students are given little guidance about how to approach the problem: There are no charts for them to fill in, no samples to follow, no patterns to complete. There isn't even any indication about how many solutions they are to find. This lack of specificity leaves much for the students to invent or initiate themselves. They will need to make decisions about how to generate possible solutions, how to verify them, how to keep track of them, and when their work on the problem is done. A wide range of possibilities is opened; students may approach the work in more or ‘6 A version of the Three-Coin Problem appeared in the standards for curriculum and evaluation developed by the National Council of Teachers of Mathematics (1989). I first encountered the Three-Coin Problem in Deborah Ball's third grade mathematics class: Ball had chosen this problem to launch the 1989-90 school year. 77 less challenging ways. For example, some may work empirically, acting out the story and recording the amounts (and/or combinations) they find; others may work more systematically and may develop abstract representations of the coins rather than working with actual coins. Some may focus on merely finding as many different amounts as they can; others may focus on finding them all; and still others may search for methods and formulations that will find all possible amounts for this and similar problems. This lack of specificity allows the teacher to observe students' dispositions and capacities for tackling a problematic situation. However, this problem is not only notable for its potential to uncover some of the "natural" ways students respond to mathematics but also for its potential to engage students in new (unfamiliar) ways of thinking and talking aboutmathematics with some prompting. There are two important aspects of the unfamiliar terrain that are highlighted by the Three-Coin Problem. First, although the Three-Coin Problem is ostensibly about finding various amounts of money (which requires some facility with coin recognition/valuation and addition) it embeds, just below the surface, the interesting problem of validating whether or not you've found all possibilities. Students with widely varying mathematical capabilities can usually develop some form of mathematical validation to prove to themselves and others that they have indeed exhausted every possibility -- but often, they are not inclined to do so. With a little prompting, however, they can be encouraged to seek more rigorous methods, articulate rationales, form convincing arguments. Rather than depending on the teachers' approval or information provided in the textbook, I wanted my students to learn to seek and use mathematical reasoning to validate their answers. Secondly, the Three-Coin Problem also embeds many 78 inter: pot: 1113’. tthK toi he hit i014 interesting patterns that are accessible -- noticeable -- to a broad range of students. Students can use these patterns to create more generalized mathematical knowledge if they are so inclined (or if provided a little nudging). Rather than receiving formulas or algorithms to use and commit to memory, I wanted the prospective teachers to form habits of seeking out patterns and developing conjectures based on their observations -- and, through arguments and refutations, refine these conjectures within a community of peers. In the sections below, I look more closely at the Three-Coin Problem and its potential for encouraging students to seek mathematical validation and create mathematical knowledge within the problem context. Validation For young children who are learning to recognize and associate values with coins, who are still developing proficiency with addition, and are new to working within multiple task constraints, the problem of, "What amounts of money could I have?" is genuinely problematic. Much effort is devoted to simply representing and adding the amounts correctly. It is a challenge to validate that the right number -- and kinds -- of coins have been used. However, as familiarity with the coins and facility with addition increases, this particular question ceases to pose real challenge. It is not unusual for children (and adults) to arrive at ten different amounts for the Three-Coin Problem using fairly random and ill-articulated methods. They may even feel quite confident they've found all possibilities, basing their certainty on their experience: "In all the times I've done it, I've never pulled 13¢ out of my pocket" or, "I've tried finding other amounts and I just keep getting the same ten over and over again." However, while such empirical 79 knowledge may be compelling, it does not stand up as mathematical proof. More is required to prove that the complete set of values has been found than to simply show that the found values are correct. In order to know -- mathematically -- that only ten different amounts of money can be pulled, one must show that all other amounts are impossible. One common way students approach such a proof is by finding all possible coin combinations that can be made in the Three-Coin Problem scenario and then showing these combinations only yield ten different amounts. The main challenge in such an approach is finding all combinations and proving they indeed have all been found. Typically, students use some mixture of systematic patterning and logical reasoning to do this. Another way students sometimes approach the problem uses a process of elimination. These two approaches are discussed below. Generating "All" Possible Combinptions Systematic approaches involve developing some organized way of generating and recording the possible coin combinations. Many adults approach this problem by first determining several categories or types of possible solutions that account for all possibilities. For example, one way to classify all possible solutions is by the number of pennies in the solution. Every possible solution for this problem will fall into one of these categories: Solutions involving 3 pennies, solutions involving exactly 2 pennies, solutions involving exactly 1 penny, or solutions involving 0 pennies. People will then focus their efforts on finding all combinations for each category. This structuring of the possibilities simplifies the work by reducing the realm of possibilities into smaller subsets for consideration. So, for example, one typical organization scheme begins by finding all combinations where the coins pulled are three-of-a-kind: 3-of-a-kind mm = 3¢ nnn= 15¢ ddd = 30¢ Where "p" stands for "1 penny , n stands for "l nickel", and "(1" stands for "1 dime." It is easy to see there are no other possible combinations involving three of the same kind of coin because there are no other types of coins. The possibilities involving three-of-a-kind have been exhausted. '7 Attention can be turned to considering all combinations where two of the three coins are the same (two-of-a-kind). Then, finally, combinations where all three coins are different from one another (one-of-a-kind) can be found.‘8 Systematic approaches also allow for patterns to surface that can aid in generating possible combinations. For example, the search for all combinations involving two-of-a- kinds might begin by finding all combinations involving two pennies -- and specifically, for the example below, all combinations with two pennies and one nickel. 2-of-a-kind ppn= 7 ¢ mm = 7¢ npp = 7¢ The sequence begins with the two pennies together, on the left, with the nickel tacked-on at the end. The nickel then moves to the middle position. Then, finally, it is '7 There is, however, an assumption being made here that all pennies are interchangeable (likewise for nickels and dimes). Therefore, the ordering of the pennies does not matter. If one were to try to count different arrangements of pennies as "different" combinations, there would be many more combinations. '3 Another common approach begins by finding all combinations involving pennies, then all combinations involving nickels, and finally all combinations involving dimes. 81 placed on the far left. All combinations involving two pennies and one nickel have been exhausted because there are no other "spots" to place the nickel. Using this same pattern, one can easily find all combinations involving two pennies and one dime. 2-of-a-kind ppd= 12¢ pdp = 12¢ dpp = 12¢ All combinations involving two pennies have now been found. There are no other types of coins available to combine with the two pennies. This same pattern for finding combinations -- and the same logic for validating that all possibilities have been found -- can be applied to combinations involving two nickels: 2-of-a-kind nnp= 11¢ npn = 11¢ pnn = 11¢ nnd= 20¢ ndn = 20¢ dnn = 20¢ And, finally, it can be used for finding combinations involving two dimes: 2-of-a-kind ddp= 21¢ dpd = 21 ¢ pdd = 21¢ ddn= 25¢ dnd = 25¢ ndd = 25¢ 82 [L14- 1111 liq There are also patterns that surface later, in the results obtained, that allow for further insight. One might notice, for example, in the two-of-a—kind combinations listed above each monetary value is obtained three times. Redundancies in value occur when different orderings of the same coins are used. This small insight would significantly reduce the number of combinations requiring consideration. So, instead of finding all six different orderings of one penny, one nickel, and one dime (i.e. combinations involving one-of-a-kinds), a person could reason that the total value of the coins will be the same regardless of their order. Therefore, only one one-of-a-kind combination would need to be considered. This use of logic that allows one to safely ignore whole groupings of possibilities not only produces "short cuts" but also displays a kind of mathematical power and elegance: It requires mathematical knowledge and a capacity to reason that is noteworthy. This can, perhaps, be seen even more clearly in a second approach students sometimes use to find and validate all possible values for the Three-Coin Problem. A Process of Elimination Students sometimes begin their work on the Three-Coin Problem by attempting to narrow down the realm of possibilities through a process of elimination. Using what they know to be true and some logical reasoning, they try to reduce the set of possible values recIlliring consideration as much as they can. The infinity of possibilities for the Three-Coin Problem can quickly be reduced to something more manageable by considering the maximum and minimum amounts that can be pulled. Since the penny is the smallest denomination and exactly three coins must be pulled out, three cents is the minimum amount that can be pulled. Anything less than three cents need not be considered. Likewise, since the dime is the largest denomination available, thirty cents must be the maximum possible amount. The infinity of numbers 83 de 2111. s'! I‘ be. me; that spans beyond thirty can be dismissed as "impossible." This leaves only twenty-six whole numbers, between three and thirty, that require careful consideration. Although twenty-six is considerably less than infinity it can still be quite a bit of work to go through each of these, testing and proving their "viability" or "impossibility" as solutions. To prove that a particular amount is possible, one need only show that the quantity can be arrived at using only pennies, nickels, and dimes, and exactly three coins. So, for example, to prove that seven cents is a possible amount, it is sufficient to show that seven cents can be made using two pennies and one nickel, a coin combination that satisfies the requirements specified by the problem. However, to prove that a particular amount is impossible is somewhat more difficult. It involves either systematically demonstrating or providing some logical rationale that inarguably shows there is no way to arrive at that amount within the problem constraints. This may mean proving that more (01‘ fewer) than three coins must be used to make the amount. For example, 4¢ can only be made with four pennies (nickels and dimes are too large) and therefore, cannot be grasped by the hand that only pulls out three coins at a time. Or it may mean proving that COins other than pennies, nickels, and dimes are required. For example, 27¢ can be made With three coins only if one of the coins is a quarter (or some other non-existent denomination). ' 9 As it turns out, a little more than 30% (8 out of 26) of the numbers between three and thirty are "possible." '9 Two pennies must be used to make 27¢ (since the other coins will only yield values that are multiple of five, the “7” of 27 must include at least two pennies in its composition). If two pennies are used, this means that the third coin must equal 25¢. @At,£¢,£¢@at,at,aa¢,@ 1C2; 13¢, fl¢,@@ 11¢, 16¢, mggéh, 31¢, man, 25 36¢, 27¢, 28¢,39¢,@ It can be a slow, tedious process to eliminate the 18 "impossible" amounts, reasoning through each case. Indeed, sometimes students skip over necessary proofs, finding verification from their classmates or some other source. However, students can sometimes use reasoning to further eliminate other groups of possibilities. For example, knowing that three pennies produces the lowest possible amount, one could eliminate 4¢, 5¢, and 6¢ as possibilities by noting that the next smallest amount that can be made -- by combining two pennies with a nickel -- is 7¢. Likewise, one might also eliminate 26¢, 27¢, 28¢, and 29¢ as possibilities by noting that the second largest amount that can be made is 25¢. An even more powerful example might be the observation that the nickel and the dime are both multiples of five. Based on this, someone might reason that any combination using nickels or dimes (which is everything except the three-penny combination) can only be one or two cents more than a multiple of five (i.e. two cents more if only one dime/nickel is used; one cent away if two dimes/nickels are used). This means that all numbers ending with three and four (with the exception of 3¢) or eight and nine, are impossible. This eliminates eleven possibilities.20 Logic can be a powerful tool. Creating Mathematical Knowledge In addition to the patterns students use as they generate and eliminate possible amounts for the Three-Coin Problem, there are also patterns to be found when looking at 20 .If eomeone were to use the reasoning described here, fifteen possibilities between 3 and 30 would be elrmmated. This leaves only eleven possibilities to test. Eight of these values can be arrived at within the constraints of the problem and involve a simple proof to show they indeed "work." Only three possibilities - ' 10¢. 17¢, 22¢ -- require some proof to show they are "impossible." 85 ht the Three-Coin Problem in the context of other, similar problems. Although there is nothing in the wording of the Three-Coin Problem to suggest the need -- or even the possibility -- of exploring variations on the problem, the mathematical impulse for seeking out generalizations and formulations can be strong. Indeed, some students feel their work is not "mathematical" unless it involves formulas and abstract representations involving numbers and mathematical symbols. Variations on the Three-Coin Problem are relatively easy to come up with and it is not difficult to transform the Three-Coin Problem into just one example of certain type of problem. Once this shift from the problem to a problem takes place, it is possible to seek out patterns from which to generalize and formulas to express these generalizations. Generalization One common heuristic for beginning a mathematics problem when a good approach is not obvious is to create a simpler version of the problem to work on. So, for example, someone might start by considering a two-coin or one-coin version of the problem to get ideas about how to best approach the Three-Coin Problem. If there are pennies, nickels, and dimes in my pocket, and I pull out ONE coin, how much money could I have? There are three possibilities: Penny-Combinations Nickel-Combinations Dime-Combinations p = M n = 5¢ d = 10¢ Where "p" stands for "1 penny", "n" stands for "1 nickel", and "d" stands for "1 dime." What if TWO coins are pulled out? How much money could I have? 86 In this case, there are six possibilities: Penny-Combinations Nickel-Combinations Dime-Combinations (with no pennies) (with no pennies or nickels) pp=2¢ nn=10¢ dd=20¢ pn = 6¢ nd = 15¢ Pd=11¢ Beginning with simplified versions often allows insight into how one might approach the more complicated problem. One might, for instance, develop a systematic method for working on this type of problem.21 One could even begin to see possible patterns in results obtained. Based on the total number of possibilities arrived at for the one-coin problem (3 possibilities) and the number found for the two-coin problem (6 possibilities), one could speculate the Three-Coin Problem might yield 9 possibilities if it continues to increase by 3. This is, however, not the case. The Three-Coin Problem actually yields 10 possibilities: 2' For example, the 2-coin combinations above were generated by putting a "p" in front of every l-coin combination: pp, pn, pd This was followed by putting an "n" in front of every l-coin combination to the right of the "Penny-Combinations" column: , nn, nd And finally, putting a "d" in front of every l-coin combination to the right of the "Nickel- Combinations" column: dd This same method could be used for generating 3-coin combinations, using the 2-coin combinations to systematically build-up all possibilities. So, all 3-coin combinations involving pennies can be found by simply putting a "p" in front of every 2-coin combination. Likewise, all combinations involving nickels (but no pennies) can be found by putting an "n" in front of every 2-coin combination to the right of the "Penny- Combinations" column. And the process continues on to find all dime combinations (with no pennies or nickels). There are some interesting things to note about the way combinations are generated and organized by this method. For example, the last column of dime-only combinations always yields only 1 possibility: X dimes, where X is the number of coins being pulled out. Additionally, the middle column involving all nickel combinations (with no pennies), always yields X different combinations. And finally, the first column involving all penny combinations always generates exactly the same number of combinations as the X-l version of the coin problem. This organization of the various combinations can quite helpful when considering "Shelly's Conjecture," a conjecture raised by one of my students that is described in the section, “Episode Description: The Struggle to Understand” 87 Penny-Combinations Nickel-Combinations Dime-Combinations (with no pennies) (with no pennies or nickels) ppp=3¢ nnn= 15¢ ddd=30¢ ppn = 7¢ nnd = 20¢ ppd=12¢ ndd= 25¢ pnn = 11¢ pnd = 16¢ pdd = 21¢ Looking across the one-coin, two-coin, and three-coin versions of the problem, it is not uncommon for adult students to recognize a pattern in the total number of possible combinations that result in each: One-Coin: 3 possible combinations Two-Coins: 6 possible combinations Three-Coins: 10 possible combinations Some may notice that the difference in the number of possibilities seems to be increasing by one each time: The difference between one-coin (3 possibilities) and two- coins (6 possibilities) is 3; and the difference between the two-coins (6 possibilities) and three-coins (10 possibilities) is 4. Some may even recognize these numbers -- 3, 6, 10 -- as "triangular numbers"; that is, numbers that can be represented in a regular triangular arrangement with equally spaced points:22 3 6 m o o o o. o. no 000 000 0.0. Based on this, one might predict that a four-coin version of the problem would yield 15 possibilities: 22 This definition of triangular numbers was adapted from a definition found on Swathmore University's Urban Systemic Initiative’s website: "A triangular number is a figurate number: a number that can be represented by a regular geometric arrangement of equally spaced points." 88 This is indeed the case: Penny-Combinations Nickel-Combinations Dime-Combinations (with no pennies) (with no pennies or nickels) pppp = 4¢ nnnn = 20¢ dddd = 40¢ pppn = 8¢ nnnd = 25¢ pppd = 13¢ nndd = 30¢ ppnn = 12¢ nddd = 35¢ ppnd = 17¢ ppdd = 22¢ pnnn = 16¢ pnnd = 21¢ pndd = 26¢ pddd = 31¢ In fact, this pattern plays out when pulling out five coins (21 different combinations), and six coins (28 different combinations).23 This pattern does not specifically addresses the question of "What amounts could I pull out?" but it raises a number of mathematically interesting questions. Will this pattern always work? If so, why? How is the number of coins being pulled out of the pocket related to the number of possible combinations that can be drawn? Is there a fonnula that can predict the number of possible combinations, given the number of coins to be drawn out? Indeed, the discovery of a pattern provides, among other things, starting Points for developing formulas -- and such a formula for determining the possible number of c01n-combinations that could be drawn out for a given number of coins may serve as a Powerful tool when working on this type of problem. ¥ 23 The number of possible combinations consistently follows this pattern however, the number possible values begins to follow more a more complicated pattern starting from a nine-coin version of the problem. The same values begin to be generated by diflerent combination of coins. In the case of nine coins being pulled Ollt fi'0m the pocket, 5 pennies and 4 dimes yields 45¢ as does 9 nickels. 89 Formulation The conversion of words and ideas into mathematical expressions and formulas involves isolating essential information and re-presenting it in new forms that allow for new understanding, generalizations, and possibilities for exploration. For example, in the discussion above, the number of possible amounts at which someone could arrive, were written as single, whole numbers: One-Coin: 3 possible combinations Two-Coins: 6 possible combinations Three-Coins: 10 possible combinations Four-Coins: 15 possible combinations F ive-Coins: 21 possible combinations Six-Coins: 28 possible combinations These whole numbers -— 3, 6, 10, 15, 21, 28 -- are a simple, concise way of representing the possible combinations that can be drawn but they provide little information about how the numbers are related to one another. Although patterns may be found and pointed out -- and may even be immediately recognized by someone with the appropriate background knowledge -- there is nothing in this particular representation of "possible combinations" that makes the patterns easily visible. It is not obvious, for instance, that "the difference in the number of possibilities is increasing by one" as the number of coins-drawn increases by one. The same quantities, however, could also be re-presented as a series of summations that look very similar to one another: 90 One-Coin: 1+2possible combinations Two-Coins: l+2+3possible combinations Three-Coins: l+2+3+4possible combinations F our-Coins: 1+2+3+4+5possible combinations Five-Coins: 1+2+3+4+5+6possible combinations Six-Coins: 1+2+3+4+5+6+7possible combinations In this form, several patterns and relationships become more visible. One thing that stands out is how every one of these "possible combinations" can be represented as a sum of consecutive integers, beginning with "1" and continuing to some other positive integer. It is also notable that the last number of each addition series is always one more than the number of coins being pulled. With these two observations, it is possible to make other predictions. If, for example, seven coins were being pulled out of the pocket each time, then the extension of the pattern would look something like this: Seven-Coins: 1+2+3+4+5+6+7+8 possible combinations The above would predict 36 possible combinations for the seven-coin version of the problem. And indeed, there are 36 possibilities. The general case might be expressed 38-- F or X-Coins: 1+2+3+. . . + (X-l) + X+(X+1) possible combinations (Where X = the number of coins being pulled out from the pocket each time) Although other complexities need to be taken into account in creating a formula that would work for all values of X, this formulation provides a good basis for further exPloration: Are there more concise ways to express this relationship between the number of coins-pulled and the number of possible combinations? Are there ways that would not require adding each of these addends one-by-one?24 Are there cases where this 24 - . . . . . . . . . Another way ‘0 think about the addition of consecutive posmve integers from 1 to A might be to unagme the sum as a collection of dots arranged in a staircase configuration. For example, for A=4, the sum j+2+3+4 might be pictured as: 91 for: 1h formulation would not work? When? Why?25 Rather than marking the end of a mathematical exploration, formulation often marks the beginnings of other paths and possibilities. Possibilities for Teagher Education The Three-Coin Problem is notable for the vast array of possibilities it embeds for mathematical exploration. From young children learning simple addition and coin This "stair-case" however, can also be seen as half of a rectangle: This rectangle has a height of 4 (which equals "A") and a width of 5 (which equals A+1). Therefore, the original "stair-case" has an area of (4x5) x 1/2 -- or, more generally, (A)(A+l)(1/2). This is consistent with the fonnula featured in the famous anecdote of young Carl Friedrich Gauss who was asked, along with his classmates, to find the sum of consecutive integers from 1 to 100. The seven-year-old Gauss represented the summation as a sequence of 50 (i.e. HALF of 100) pairs of addends, all equaling 101 (e.g. 100+], 99+2, 98+3 . - - , 48+53, 49+52, 50+51). He is attributed with creating the formula for consecutive integers from 1 to A: 1+2+3+...+(A)+(A+1)= w 25 . This formula does accurately predict the number of possible values for X=l through X=8 however, after “US, there are instances where two different combinations of pennies, nickels, and dimes yield the same value. For example, when nine coins are being pulled (X=9), there are two ways to obtain 45¢: 5 Pennies, 4 Dimes = 45¢ 9 Nickels = 45¢ The formulation described above would predict l+2+3+4+5+6+7+8+9+10 (or 55) possible amounts when X=9 but there are actually only 54 different possible amounts. Nevertheless, though it fails to cOIISIStently predict the number of possible amounts, the formulation does allow one to accurately predict the number of different combinations of coins. Interestingly, the number of redundant values also seems to increase as a series of triangular numbers, starting with X=9 yielding 1 redundancy -- X=9 1 redundancy X: 10 3 redundancies X=11 6 redundancies X: 12 10 redundancies X: 13 15 redundancies 92 , If; t ‘ tor ori but not: gen. Solu the; 110:1 recognition to adults with sophisticated mathematics backgrounds, anyone can find some way to enter the problem. And virtually anyone, who digs in deeply enough, will be challenged and have opportunities to engage in mathematical discovery and creation. Like a Rorschach inkblot test, the Three-Coin Problem can be many things to many people, depending upon the assumptions, knowledge, dispositions they bring to the work. Students can approach a solution in many different ways -- some more efficient, more elegant, and more powerful than others. It can create an excellent context for instructors to observe students' natural inclinations for approaching a problem situation, organizing their work, using logic, seeking out validation -- and perhaps even going beyond solving the problem, as written, to seeking out mathematical relationships, generalizations, formulations. However, in addition to observing what students do "naturally," instructors can also easily adapt the Three-Coin Problem to offer individuals greater challenge, taking them beyond what they are initially inclined to do. Simple questions, building upon the original problem, can move students on to tasks that are more mathematically demanding but still mathematically connected to the original problem. Rather than having "fast" Students begin different work, explorations of the Three-Coin Problem can be extended in various ways, opening up new avenues for exploration with each new insight raising new questions,“ For example, if a student believes he is "done" because he has found ten different amounts and can't find anymore, the instructor can ask, "Are you sure you have ¥ 26 Deborah Ball and Hyman Bass have done an analysis of this same problem, used with third graders, and no“: a progression from an empirical exploration of the problem, to pattem-use/discovery, to generalization, and to abstraction. Although much of the children's work focuses on finding possible so1utlons (both randomly and systematically) and is limited to creating generalizations of only methods, they argue that there is a strong parallel between the children's explorations and search for pattern and the work that adult mathematicians do (Personal communication, 1997). 93 CC 111i lea 10( them all? How do you know?" This simple call for justification can subtly shift the Three-Coin Problem from simply finding appropriate amounts and validating their correctness, to the slightly more complicated task of validating the completeness of the solution set found. This line of questioning often spurs students to re-examine their work and strive for a better articulation of their ideas. Similarly, the question, "Can you prove you have them all?" also asks for justification but the call for "proof" can be more demanding because of the need to convince other people. This question can encourage students to come up with ordered presentations of their ideas, representations that make their ideas clearer, statements that are irrefutable. Although students may not be inclined to come up with rationales and proofs on their own, this kind of questioning can engage them in this crucial, if not "natural," aspect of mathematical work. Or, if a student believes she is "done" because she can prove she's found every possible amount, an instructor can ask, "What if I pulled out 4 coins -- instead of 3 -- from my pocket filled with pennies, nickels, and dimes?" This can be extended still more by asking, "What if I pulled out 7 coins? What if I pulled out X coins?" With this line of questioning, the Three-Coin Problem shifts from being the problem to simply being one example of a certain type of problem. At minimum, this can engage students in using and extending methods and approaches they’ve developed in their earlier work. It can also, however, lead students to develop other forms of generalizations and create formulations as they look across this family of problems.27 While this type of mathematical work may not 27 The "family of problems" alluded to here, involves maintaining three types of coins -- pennies, nickels, and dimes -- but varies the number of coins being pulled out. This way of extending the problem surfaces Interesting patterns, providing a good degree of challenge for most adults as they try to make generalizations and create formulations from their observations. Another common way people extend their exploration of the Three-Coin Problem is by varying the number of coin types. Although this opens vast possibilities for pattem-finding, generalization, and 94 come naturally to students who have only been given formulas and algorithms created by others, it is an important facet of doing mathematics. In particular, it was an important facet of the mathematics I envisioned for this class. There are many other possibilities for the Three-Coin Problem but this gives a sense of the expandability of the task. The flexibility it afforded was particularly important to me for the first class session. I did not yet know my students; but, still, I wanted a problem they could all work on and would all be challenged by. Also, I wanted a problem that would allow me to learn a bit about my students and the habits, assumptions, and understandings they brought to mathematics. And, finally, I wanted a problem I could use to begin to engage students in important aspects of the mathematical work that would be central in the class -- proof, justification, explanation, generalization, representation, etc. -- taking them beyond what was familiar and natural, into the unfamiliar terrain I was creating for them. In the Episode Description section that follows, I describe the work we did on the Three-Coin Problem during that first day of class. I also go on to discuss students' thoughts and reflections about that first class session, as they surfaced in conversations later in the week. As I had expected -- and hoped -- the Three-Coin Problem allowed me to learn a great deal about my students; it also provided a context for introducing important aspects of the unfamiliar terrain. However, students' interpretations and reactions were far more varied and complicated than I had anticipated. This variety and formulation, it introduces a great deal of complexity to the work. For example, when using one type of coin, the number of possible combinations that can be pulled when X coins are being drawn (assuming the order of the coins does not matter) is simply I. When using two types of coins, the number of combinations can be represented as X+l. When using three types of coins, the number of combinations can be represented as 1+2+3+...+X+(X+1 ). When using four types: 1 + (1+2) + (1+2+3)+...+(1+2+3+...+X)+(1+2+3+...+X+(X+l)). The representations continue to grow in a fractal pattern as the coin types increase. The patterns quickly grow more difficult to identify and represent. 95 complexity shows up in the narrative below, and is examined more thoroughly in the last section, Episode Analysis: The Thing in Itself and Human Constructions. Episode Description: The Struggle to Understand28 Setup of the Problem (Mondav) I put the Three-Coin Problem up on the overhead projector and asked one of the students read it aloud. A few people had already started working on the problem but I wanted to make sure we had a shared understanding of the question before breaking up into individual and small group work. "I have pennies, nickels, and dimes in my pocket," I declared as I fished around in my pocket for some coins. Grasping three coins, I held out my closed hand before them and asked, "If I pulled three coins out, how much money could I have?" "Three cents," suggested Megan. "Sixteen cents," volunteered Kathy "Eleven cents," added Iris. We tested each of these possible amounts to verify that they could indeed be made using only pennies, nickels, and dimes and using exactly three coins. "If I kept doing this, what are the various amounts of money I could pull out?" I posed this question not expecting an answer: This was the problem they were to work on. In the silence that followed, I asked them if they had any questions before we started working. 28 The quotes used in this section are derived from several sets of detailed observation notes taken during the class sessions and are not verbatim quotations. At the time of these first few class sessions, we had not yet introduced the rationales and purposes of documenting the class and did not want to begin recording until we had a chance to do so. 96 One student raised her hand. "You never said how many pennies, nickels, and dimes you had." "Why would that matter?" I asked. "The chance would be different depending on the number of each kind of coin," she explained. I had not intended this to be a probability problem. Although I could (and, perhaps, should) have told them this, I wanted to observe their attempts to figure this out on their own. I believed, if they just tried to start the problem, they'd find they could work toward a solution without getting into probability or the concrete quantities it required. I encouraged them to write down what they were thinking -- about the missing information and why they needed it -- and try to work on the problem with the assumption that they had "a lot " of each type of coin. Eager to get them started on the problem, I instructed them to go ahead and begin working and added, "Feel free to confer with the people seated nearby; I will be coming around (to where you're seated) as you work." Independent Work on the Problem As I walked past the small group sitting in the back comer of the room, one Student asked me, "I was wondering whether or not the pennies, nickels, and dimes are r & placed afier being drawn." Again, this would have been an important consideration for a .nobability problem but not for the problem I intended. There were several others, like this student, who continued to be stuck, uncertain about how to begin without all the information they wanted. However, with the help of people seated nearby or with subtle 97 prompts from me -- e.g. "Given what you do know, what are some of possible amounts that might be drawn?" -- everyone was able get started. For many students, this was an easy problem; they quickly arrived at ten different amounts. As I came across these individuals and groups, I'd ask them, "Are you sure you have them all?" "How do you know?" "Would you be able to prove to someone else that you've found all possibilities?" Some students, who found their solutions in somewhat random or ill-organized ways, responded by trying to develop more systematic and articulated solutions. Other students began to strive for greater generalization in their solutions: developing a method for finding all possible amounts or searching for some pattern in the combinations they generated. A few students went on -- at my prompting or their own impulse -- to consider similar problems such as, "What if I pulled out four coins instead of three?" Or, "What if -- in addition to pennies, nickels, and dimes -- there were also quarters?" After about twenty minutes, I called the students back together to discuss their various solutions as a whole class. Whole Group Discussion: Danielle. Iris, and Alisa's Solutions I launched the whole group discussion by calling on Danielle, Iris, and Alisa to share the solution they had been working on together. They started off by explaining, "We approached the problem in a couple of ways because we weren't sure whether the question was asking, 'How many combinations are Possible? or 'What are the various amounts of money you could pull out?” The first approach they shared involved a vaguely remembered formula. "It went, something like, this --" 98 The number of possible combinations = (The number of pennies) x (the number of nickels) x (the number of dimes)29 They believed this formula would help them address the first question regarding the number of possible combinations, however, they were unable to go very far because they didn't know the number of pennies, nickels, and dimes. At this point they began to consider the other question regarding the various amounts of money that could be pulled out. "We tried to graph (sic) out the possible combinations with their total amounts," explained Danielle, as she pointed to the chart they had created. I99 0 11 15c v.1 J . s k a a d 3 (1 1') 3 Their chart was labeled with "p," "n," "d," and "$" (for "pennies," "nickels," dimes," and "total amount of money") and was partially filled in. They seemed to be working systematically, finding all combinations involving three of the same kind of coin. (Perhaps to be followed by finding all combinations involving two of the same kind of coin; and finally finding combinations involving one of each type of coin.) However, their work was cut short when I called the class together for discussion. 29 This formula does not generate the correct number of possible combinations for the Three-Coin Problem. Indeed, if it did work, then the more pennies, nickels or dimes you had, the more possible combinations you would be able to make. But there are only ten possible combinations regardless of how many coins you have (as long as you have at least three of each type of coin). A formula such as the one Danielle, Iris, and Alisa shared would be more appropriate for a scenario such as: The school council needed one representative from the 6th graders, one from the 7th graders, and one from the 8th graders. How many different combinations of students are possible for the council? Here, the number of possible combinations = (the number of 6th graders) x (the number of 7th graders) x (the number of 8th graders) In this problem, sixth graders are not interchangeable -- each sixth grader will generate a different set of combinations. The same thing goes for the 7th and 8th graders. This is different than the Three-Coin Problem where pennies are interchangeable. It does not matter which penny is pulled. (The same goes for all the nickels and all the dimes.) Also, it is not possible for two sixth graders to serve on the council - only one representative from each grade-level will be allowed. This is different from the Three-Coin Problem where two or three pennies could be drawn. 99 It was a very straightforward approach and similar to the way many other students had worked on the problem. I was sure we could quickly finish filling in their chart together. I asked if others had worked on their solutions in a similar way. More than half the class had. Before focusing our energies on completing the chart, I asked for a quick show of hands of those who also tried to work with a formula, as Danielle, Iris, and Alisa had. Again, a large number of students raised their hands. I had expected this. I'd seen many other people attempt to use a variety of vaguely remembered formulas. While I wanted to place an emphasis on reasoning and thinking through problems rather than merely plugging numbers into mysterious ill-remembered formulas, I wanted them to see that this tendency to grope for "The formula" that would somehow produce "the right answer" was widespread. I intended to return to completing the chart but Shelly had her hand in the air and appeared to have something she urgently wanted to say. Whole Group Discussion: Shellv's Conjecture Shelly said she had also been trying to find a formula. "But," she explained, "I thought it was something like n plus one factorial where n is the number of coins you are pulling out." She went on to elaborate that for three coins it would be: (3+1) + 3 + 2 +1 = 10 possible amounts you could pull out She also predicted that if one were drawing out five coins, the number of possible amounts would be: (5+1) + 5+ 4 +3 +2 +1 possible amounts 100 And for ten coins, it would be: (10+l)+10+9+8+7+6+5+4+3+2+l possibleamounts Although this is not called a "factorial"30 this series does indeed work for finding the number of possible combinations you could pull if you have three types of coins. I didn't say this to the students but I did comment that Shelly's formula was different from the formula Danielle, Iris, and Alisa had proposed. I knew that earlier, when students were working individually and in small groups, several people had begun to explore the case of drawing four coins rather than three. So I proposed, "Shall we try out Shelly's idea for pulling out four coins? Does the formula seem to work?" One group quickly found and reported, "It works.” “There are 15 possibilities," they added. Lindsey, however, found that "it did not work for four coins: there are 26 possibilities." But, unlike the others, Lindsey had introduced a fourth type of coin -- a quarter -- to the mix.31 A few people were starting to explore the case of drawing out five coins. I was pleased that some students seemed to find Shelly's idea interesting and were off and running with it but I worried that things were moving too fast. People in the class were doing very different kinds of explorations but were simply reporting "it works" or "it doesn't work" without being clear about what they were trying out. Also, some 3° The term, "Factorial," symbolized with an "1," is used in connection with multiplication rather than addition. It is the product of consecutive positive integers from 1 to a given number. So, for example, "four factorial" (or 4!) is the product of the consecutive positive integers from 1 to 4: lx2x3x4=24. Formulas involving factorials sometimes come up in permutation scenarios like the "Handshake Problem": You are at a meeting with seven other people. You shake hands with every person in the room. How many different ways could you proceed through the group, shaking hands (once) each of the seven people? There are 7! different permutations of handshake orderings. 3' There are actually 35 possibilities if there are four types of coins being used -- pennies, nickels, dimes, and quarters -- and four coins are being pulled out. 101 students seemed lost or uninterested. I wanted to pull us together again and try to make sure that everyone at least understood what Shelly was proposing. I asked Shelly if I could write down her idea. "We could call it 'Shelly's Conjecture' to signify that it is just a tentative idea, something we can examine together." Shelly agreed and added, "It (the formula) was based on only one trial. . . . It is important," she emphasized, "when doing experiments, to not base things on only one trial." Next to the title, "Shelly's Conjecture," I wrote, "(1 trial)." As a class, we generated a few more examples and I had several people state the conjecture in their own words. Finally, we developed the case for n coins being drawn out. All the while, I kept notes on the whiteboard of the various comments and ideas that were shared: Shelly's Coeniture (1 trial) 3 kinds of coins drawing out n coins (n+1) + n + (n-l) + (n-2) + . . . + (n-n) = # of possible combinations e.g. (8+1)+8+7+6+5+4+3+2+1 for pennies, nickels, dimes -- pull out 4 (4+1)+4+3+2+ 1 =15 . ot 26 with 4 kinds of coins There were only a few more minutes left to this first class session and I didn't think we'd be able to prove (or disprove) Shelly's conjecture in the remaining time. I was, however, curious about what it would take to convince the students, one way or the other, about the validity of Shelly's Conjecture. "How might we come to know if this conjecture were true or not? How might we convince everyone in the class about whether this conjecture worked or did not work in all cases?" I asked the class. 102 Teri suggested, "We could try it with lots and lots of numbers to see if it works" and added that perhaps "we could use a computer to help us." Someone else commented, "We could know if you (the teacher) would tell us!"32 Tasha said, "We could know if it said it in a book and perhaps if we tried out a few examples to test it." Conspicuously missing from this list was anything that might be considered a mathematical justification or proof. I had hoped to return to the Three-Coin Problem at this point but the students were deeply curious about the purpose of our work on this problem and were also divisively at odds about what the problem was really asking. A few students, for example, argued that the problem was "really" only asking for the various amounts that could be made; they claimed to be unable to see the relevance or importance of discussing formulas and combinations. It wasn't long, however, before we ran out of class time. After Class Lindsey and JoAnn came up to me after class to show a formula they had been working on for determining the possible amounts that could be pulled for x types of coins, drawing out y (number of) coins. They were very excited about what they had found so far but they were still not finished. Lindsey explained they were working towards a formula by writing out all the possible amounts for various coin configurations 32 Shaya built on this, describing her experience in her MTH 201/202 class (Mathematical Investigations for Prospective Elementary Teachers) where the teacher gave her the formula for a similar problem and it worked. She also mentioned, "I don't think it (Shelly's Conjecture)'s right. I think she's getting it mixed up with a different formula 'cause we used a formula with a similar problem and this wasn't it. 1 have it in my book." 103 and looking for a pattern in the number of possibilities. I encouraged them to keep at it. It seemed that they had made a nice start into a complex problem. Although we weren't able to get as far on the Three-Coin Problem as I had hoped, I was pleased about what we were able to do. Shelly's Conjecture seemed a terrific example of someone looking for pattern, trying out an idea, and developing a tentative formula that we could explore together over time. I wrote up the conjecture, along with some of the examples and observations students made, on a big sheet of paper. I hoped we would be able to return to it at some point in the fiiture. Or, at the very least, that it would become a part of our collective, intellectual history: An intriguing idea, still ripe for further investigation. Other Explorations (Tuesday — Thursdax) Over the following three days, we turned our attention away from the Three-Coin Problem and spent our class time focusing on the other major components of the course. On Tuesday, we watched a video of a third grade mathematics class and discussed the teaching and learning captured on the video. On Wednesday and Thursday, Fenice led the class in explorations of a language arts activity and analyzing videos from two different language arts classrooms. For the last day of the week (Friday) Fenice and I wanted to have the students reflect upon and discuss their experiences as math and literacy students in this class. So, on Thursday night, we asked them to write a journal entry on the following topic: How has it felt to be a math/literacy student? Write your thoughts/reflections. Try to describe specific events/facets as much as possible. 104 I asked the class to think specifically about how it felt to be a math student in class on Monday (when we did the Three-Coin Problem) and how it felt to be a literacy student on Wednesday (when we did a literacy activity called, "Ordeal By Cheques" which focused on reading and interpretation”). Whole Group Discussion: Initial Literacy and Math Experiences (Friday) Most of our class time on Friday was spent discussing people's experiences of and thoughts about the activities we did earlier in the week. Some students commented on similarities they found in the literacy and mathematics work they did. Kathy, for example, commented, "Both activities encouraged critical thinking and we're not used to that." "What do you mean by 'critical thinking'?" I asked her. Kathy responded, "You know, questioning things -- like when you (Kara) asked, 'Are you sure?” She continued on saying, "We're used to the teacher telling us rather thinking ourselves about why.... I was annoyed with having no answers (given) but many different ideas came up. We're just used to getting everything given to us in school." Beth built upon this, adding, "It made us look at (how) there's more than one answer... you pretty much thought there was only your way until you heard other people." 33 Ordeal by Cheques is a "story" that was published in Reader's Digest consisting entirely of entries into a checkbook registry. From the dates, names, and signatures, the "reader" can piece together multiple interpretations -- guesses -- about the lives of the people who have written checks using this bank account. A large amount paid to a jeweler followed by large sums to florists, caterers, musicians might suggest a wedding. Years later, monthly rent checks to a nearby apartment building and personal checks to a woman who is definitely not his daughter or wife may imply an extra-marital affair. The check registry provides merely a sketch of a story. No further elaboration or detail accompanies the publication. It is simply whatever the reader makes of it. 105 For Kathy and Beth, there were aspects of both the mathematics and literacy work that stood in contrast to what they were "used to." They found themselves thinking in ways and about things they might not have thought on their own. These sorts of comparisons were interesting -- and encouraging -- to both me and Fenice: We had, after all, designed the opening week with the hope of creating learning opportunities which would stand in contrast to prospective teachers' past experiences in these very ways. However, the contrasts students drew between the mathematics work and the literacy work were what I found most remarkable. Alisa, for example, commented, "Imagination (was) required for the literacy activity. For the math one we just used what we knew. It was based on what we've had -- on what we learned in other math classes." Others commented on the "creativity" and "imagination" involved in the literacy activity but Alisa was the first person to contrast this with the mathematics work where they simply "used what (they) knew." I asked, "Did anyone feel creative doing the math problem?" Someone replied, "I think Shelly was, with the formula." "Thank you," Shelly said. "It was just a curiosity thing. Trying various things, working it backwards. I noticed there was a pattern of four, three, two, one. Like I said, it was only based on one situation and was not meant to cover everything." I agreed with whoever made the comment: Shelly had indeed been creative. Her curiosity, her observation of pattern, her inclination to experiment and take risks was notable. In many ways, she stood out among her classmates. 106 Hillary followed Shelly's comment with another, somewhat different contrast between the mathematics and literacy work: "I don't do math well. . .. I lose patience and get confused when it is not clear-cut. Like when someone asked about the number of pennies, nickels and dimes and Kara said, 'a lot'. . .. I felt like if it doesn't matter how many there are, and it doesn't matter if you can put the coins back in your pocket, then it doesn't matter what I get for a solution. I was fi'ustrated. I didn't feel that way with the literacy because I like to make things up." I asked her, "You didn't feel like you could make things up with math?" "Yeah," she replied. I then turned to the class and asked if others felt similarly. "Yes," said Shaya emphatically. "Math seems very concrete," said one student. "It seems there is always an answer so you're always looking for an answer but with literacy, it seemed there was room for imagination because when you create a story there's not always guidelines and you don't have to come up with a set ending for the story. I enjoyed literacy better. I don't like math." Several other students commented along similar lines. In some cases, students felt unable to "make things up" for interpreting or setting up the problem. Without "clear-cut" numbers and quantities, confusion and frustration set in. For others, there was no "room for imagination," no place for creation and invention, with the knowledge that "an answer" was out there somewhere. This was, however, not the case with the literacy activity. 107 It was an interesting discussion but towards the end it turned, as it had on Monday, toward now-familiar topics: "What was the purpose of the mathematics activity on Monday (i.e. the Three-Coin Problem)?" and "What was the Three-Coin Problem really asking?”4 But unlike Monday's discussion that was filled with questions and speculations about the purposes of the activity and attempts to clarify the "real" question, this conversation was riddled with expressions of annoyance and frustration. "You need to understand," explained Hillary toward the end of class, "that people would be annoyed because we've been working so long on it. . .. I've never worked so long on a problem. It's understandable that people are annoyed." "Yeah!" exclaimed several students. "We've been going over that (indicating the mathematics work) for three days," said Liz. Shaya added, "The day after we got the math problem we spent the whole two hours of the next class talking about that math problem again." "(And) today we're talking about it again -- we're irritated about that." We had only spent about an hour on Monday working through the problem ourselves. On Tuesday, we spent most of the class period watching a short (10 minute) videotape of Deborah Ball's third graders working on the same problem and discussing the teaching and learning captured in the video. And on this day (Friday) we had spent 3‘ Although I had tried to articulate my own responses to these questions at the end of class on Monday and again on Tuesday, it remained pressing and puzzling to many students. Included among the various things I said regarding the "purpose" of the mathematics work we did together on Monday, were the following: 0 To provide them with an opportunity to do math "differently" -- where there was not just one right answer or approach or thing to explore; where they were expected to think about how they knew an answer was (or was not) valid; where they might be pushed to go a little further on a mathematics problem than they might normally would go on their own. 0 To provide an opportunity for me to learn about them from their work on the problem. ' To create a context where they could begin to talk and think about mathematics in new ways. 108 much of the class session discussing their experiences with the mathematics and literacy activities. Although we did no further work on the problem beyond what we did on Monday, in the memories and experiences of some, we had spent three days on the problem. But perhaps, even the single hour we had spent working on the Three-Coin Problem seemed excessive and taxing to some. Tammie, for example, explained, "When you feel you have the right answer, you ignore what's going on. I find I'm doing this. (Like) when we talked about the formula (i.e. Shelly's formula) -- 'cause it will confuse me if I think about more things. It has to do with different development levels: If you hear other ideas, you may come to doubt yourself." Indeed, if finding the right answer is the ultimate goal there is little reason to continue listening to others or participating in a class discussion about the problem once the answer is obtained. However, in the context of this class -- within this unfamiliar terrain -- the ultimate goal was less about the arriving at the right answer as it was about creating new knowledge and developing new understandings within a community of critical peers. Arriving at a "right answer" in this context does not mark the end of the work but, instead, often marks the beginning of a much larger process of testing, proving, revising, and generating new lines of inquiry. When Tammie started to "ignore what's going on" and protect herself from "confusion" and "doubts," she had effectively cut herself off from this much larger process. Tammie continued in her explanation of how people respond to long discussions of a math problem: "We've all been in classes where there are few students who don't get 109 it and you sit there and the instructor goes over the same subject six times. And you get bored and stop listening." Shelly's Out-Pouring The discussion continued for a few more minutes. Sometimes it touched upon the subject of literacy, other times mathematics; sometimes students raised things they found frustrating, other times they pointed to things they found interesting and challenging. I found it to be an extremely helpful conversation -- students seemed to be speaking honestly, revealing their understandings and experiences -- however, we were getting close to the end of our class time. Just as Fenice moved into wrapping up the class session, Shelly tentatively raised her hand and, upon catching Fenice's eye, burst through with a stream of words: "I don't know if this is real or not but the way we re-hashed and re-hashed that math problem -- I feel a sense of animosity. I wish I never had said anything about that problem. It drives me nuts. I knew it wasn't going to answer the question -- I can't stand it... it's devouring me. I don't think laying on one person all the time is helpful.... I thought by Wednesday it was over. So many people can't understand why I brought up that formula -- I don't understand any more. I didn't realize it would cause so much discussion." She melted into tears and began sobbing. The class was silent, the atmosphere was heavy and deadly serious. Awkwardly, Liz said, in a soft voice, that Shelly's idea had helped her out a lot. It was a strained moment and something needed to be said. In that instant, I realized what an awkward phase we were all in: We were in the very early stages of establishing new expectations and norms and, to do this, I was acting 110 in a way that operated under new expectations and norms. I assumed the students' experiences in the class and my explanations would help them adjust to the new context. But the things being said and done were being interpreted within students’ own, very different frameworks. I attempted to articulate this to the students, explaining how disorienting it must be in this early phase of the course with "everyone trying to figure out what is going in this class." Continuing, I told them, "The norms of what you have grown up with in your mathematics classes, in many ways, are very different from the ones I'd like you to consider as students in this class." I went on to elaborate on this "difference" a bit: Among other things, I mentioned that I hoped "hearing how other people solved things will become as interesting as -- perhaps more so than -- the way you solved it." Lindsey, concerned about Shelly, said she thought Shelly wasn't receiving much respect. "I don't mean to sound rude but... rather than being praised for throwing out an idea, she's being put down for it. Maybe it would have been nice to ask if we can put it (Shelly's conjecture) up to play with it." Someone interjected, correcting Lindsey, saying that I had indeed asked Shelly if I could put up her idea for us to explore together. I was simply stunned. I saw my actions -- having the class stop to think about Shelly's idea -- as a high form of praise. I didn't see myself -- or the other students -- as "picking on" Shelly, but I could certainly see how it might be interpreted that way. Shelly seemed to get more upset with this exchange. I don't think she meant to whine and blame the class for picking on her. I suspected that Lindsey's comment, intended to defend Shelly, simply made Shelly feel even more like she wasn't being understood. 111 I brought the class session to a close by trying my best to articulate my interpretation and intentions. "My thinking, about labeling it 'Shelly's Conjecture'," I explained, "(Was that) this was a cool idea. There was some mystery around it -- we didn't know what was going on with it. 'Conjecture' was meant to imply a tentative idea. In fact, based on 'one trial.’ And her (Shelly's) name on it, (was) to attribute where that really cool idea came from -- (It was) the interpretation (of these actions) that caused a lot of tension. . .. I wish it had been different." After Class Shelly stayed after class for almost half an hour talking with Fenice and me. It became clear, from our conversation, that Shelly saw it as her fault that we spent as much time on the problems as we did. She thought she was responSible for making the class think about the formula and people were blaming her. Every time someone commented that they didn't see why a formula would help or how trying to find the number of possible combinations was relevant to the question being asked, Shelly heard it as an accusation. Every time someone voiced frustration with working on the problem as long as we did, Shelly heard it as a censure against her. Every time someone complained about the instructors dragging on the conversation because some people -- slower, less able people -- weren't getting it, Shelly thought they were talking about her. It was a personal hell for her to sit through these discussions. Although F enice and I had intended to put our teaching practice and students' experiences on the table for discussion, for Shelly, these conversations were really about her and her idea. It was a tribunal of peers and her instructors. 112 One thing that struck me about my conversation with Shelly was how invisible my role, as teacher, seemed to be. She took everything upon herself and viewed the evolution of the discussion as happening of its own course. This reminded me of the way people often completely miss the words actions of teachers when viewing classroom teaching that looks quite different from that with which they are familiar. For example, when watching videos of Deborah Ball’s mathematics classrooms for the first time it is often the case that the only thing viewers notice is a conversation that is apparently being generated and directed by the students. It is not unusual for viewers to fail to take in how the teacher gets some things to be in focus and other things to be ignored. They miss how (s)he introduces some things -- inserting ideas, questions, language into the conversation; and uses others things, from among the students' many utterances, reshaping them for pedagogical purposes. The role the teacher plays in guiding the discussion is often subtle and easily overlooked. Not only was Shelly laying a different interpretation on the class events she was, quite possibly, perceiving a far different set of information about the events. Episode Analysis: The Thing in Itself and Human Constructions Shelly's out-pouring was unexpected and caught me by surprise. Although I was aware -- and anticipating -- that certain aspects of our work would be potentially confusing and uncomfortable for some students, Shelly's pained expression of her experience allowed me to see new possibilities I had not considered for how students might be perceiving and experiencing the unfamiliar terrain. If Shelly failed to see my heavy involvement in the unfolding class discussion, what did others see (and not see)? If Shelly interpreted the discussion that was focused on her "formula" as "laying (in) on one 113 person" rather than attention directed at a generative idea, how had other students interpreted our work? If Shelly experienced blame and animosity directed at her, what did other students experience? While there was no way I could know what students were actually perceiving, thinking, or feeling during our exploration of the Three-Coin Problem, I had some sense for the things that struck them as noteworthy from the journal entries they had written describing their thoughts and reflections on Monday's mathematics work.35 The task posed to them at the of Thursday's class was quite open-ended -- How has it felt to be a math/literacy student? Write your thoughts/reflections. Try to describe specific events/facets as much as possible. Unlike surveys or structured interviews which might provide an overall sense of what every student thought about particular aspects of the class session, these journal entries only show what some students thought about some aspects (i.e. those things individuals chose to write about). Although this makes it impossible to make meaningful claims about the number of people who shared a particular opinion or idea, the journal entries do provide a glimpse of the various things -- among of an infinity of possibilities - - that stood out in some way to the prospective teachers who worked on the Three-Coin Problem.36 3’ Although I am not including the spoken comments students made during our class discussions in this analysis, they fit quite well with their written comments. A good sampling of what students said can be seen in the narrative above. 36 Only twenty-nine of the students' journal were included in this analysis. Two of the thirty-three students missed most of the first week of class and were not able to comment upon the mathematics or literacy activities. Two other students declined to be included in this study; therefore, their work was not included in this analysis. 114 Description of Salient Facets The prospective teachers' comments varied greatly. However, despite the range of things discussed, their journal entries seemed to focus around several main themes. I refer to these dominant topical strands as "salient facets." They are those facets of the class session(s) that were particularly noteworthy (joumal-worthy) to my students. As much as possible, I tried to focus only on those sections pertaining to Monday's mathematics work but, in some cases, students made generalized comments that applied to both the mathematics work (the Three-Coin Problem) and the Literacy activity (Ordeal by Cheques). In the sections below, I describe each of the five salient facets that emerged from a close analysis of the students' journals. Salient Faccet: The Nature/Wordingof the Mathematics Problem The mathematics problem itself was a focus of discussion in several journal entries. Twenty-two words were strung together, displayed on an overhead projector, and seen by every student in the class: I have pennies, nickels, and dimes in my pocket. IfI pull three coins out, what amounts of money could I have? Upon reading these words, several students immediately identified this as a probability problem that was, for some reason, missing crucial information. Indeed, if this were a probability problem, asking about the chance or likelihood of pulling out a specified amount of money, it would be impossible to solve without other information. How many coins are there in the pocket? How many of them are pennies? How many nickels? How many dimes? Are the coins replaced after they are pulled or are they simply removed? As Tasha wrote in her journal, 115 The problem is worded so vaguely. If the problem would have been worded specifically in what it was asking and how many coins of each were present (it) would not have annoyed me. Perhaps a minor revision to the wording of the problem might have helped: I have pennies, nickels, and dimes in my pocket. If I pull three coins out, what are the different amounts of money could I have? Or, perhaps, an addendum, such as, "Find as many possible amounts as you can,"37 would have made things clearer for students such as Tasha who wished, "the problem would have been worded more specifically in what it was asking." Indeed, I found myself repeatedly saying these sorts of things to Tasha and others as I walked among them, trying to help those who were stuck. However, it is possible that despite such elaborations on the wording of the problem, the inclination to "see" a probability problem or feel a strong need for specific quantities of coins would have persisted. Kerrie, for example, wrote -- I felt a frustration on the math lesson because of... how much info the problem contained or the lack of. . .. The problem seemed like one of the problems we had to do in MTH 20138 but missing a few details, but I kept digging and digging my memory to remember the formula. Many students, like Kenie, drew a connection between the Three-Coin Problem and the "problems (they) had to do in MTH 201 but missing a few details." Undoubtedly, there were surface similarities but slight differences in the wording of combinatorial problems matter a great deal for what formulas -- or even if formulas -- can be used to 37 I wanted to hold off asking students about all possibilities until they had a chance to find as many possibilities as they could by whatever methods they chose. I hoped to observe their solution approaches as well as see what, if any, inclinations they had for seeking out "all possibilities" and for proving to themselves (and others) that they indeed had found "all." 38 MTH 201 and MTH 202 are two courses offered by the Mathematics Department at Michigan State University, specifically created for elementary education students. These courses must be taken prior to, or concurrently with, TB 401 unless substituted with higher-level mathematics courses. Typically, only students with minors in mathematics are exempted from MTH 201/202. 116 answer a question. There seemed to be, for several students, an excessively large category for "problems like these" that could be solved with "the formula." However, not everyone perceived the Three-Coin problem as a vague probability problem (or a MTH 201/202 problem) missing critical information. And not everyone experienced annoyance and frustration when they read the Three-Coin Problem off the overhead projector. Shelly, for instance, felt anxiety when she first read the problem. She explained -- While reading the math problem for the first time, I was feeling somewhat anxious and thinking, "I hope I can remember how to do this." For approximately 30 years I haven't had math classes other than college algebra which I took this summer. There were also others, like JoAnn, who dove right into the problem as soon as it was displayed, eager to work on it -- I was very excited about the challenge we were given. As soon as the problem was put on the overhead I began working on it. Some even found the Three-Coin Problem very easy at first. Angela, for example, remarked -- One the first day when we did the math problem I at first felt that the problem was really easy and I did it. Although students' experiences varied greatly, the nature and wording of the mathematics problem stood out, to many students, as a salient facet of Monday's class session. Salient Facet: The Role of Other Students in Shaping the Nature/Experience of the Work Another facet of our work together, noted in prospective teachers' journal entries, was the role of other students in shaping the work they did and influencing their 117 experience of the work. There were two main interactions students had with one another during Monday's class session. One involved an opportunity to work with a small number of people on the Three-Coin Problem, shortly after I introduced it. The other was a whole group discussion, talking about students' solution approaches to the Three-Coin Problem. The opportunity to work with a few other students on the Three-Coin Problem provided some with a sense of relief and bolstered their confidence. For students, like Shelly, who initially felt anxious about their capacity to solve the problem, there seemed to be some security in the possibility of conferring with others. In her notebook, Shelly WI'OIC -- When Kara told the class we could work in groups, I sensed relief and confidence since each of us could rely on each other to check the correctness of our thinking and approach to the problem. After making a sequential list to acquire all possibilities, confirming with Erica that my list matched her table, I felt confident I had all possibilities. Even for students like Kim, who seemed quite confident from the start, the opportunity to work with other students seemed to lend a feeling of satisfaction and assurance: The Three-Coin Problem appeared at first to me to be very simple. I figured out there were only 10 ways to get the number of amounts that was asked for. Everyone in my group did the same thing and we felt very satisfied and confident that we had the correct answer. There were also prospective teachers who described how other students helped them to make progress on the problem when they were stuck or believed they had gone as far as they could. Hillary, for instance, wrote about her own initial confusion and how the other students, seated around her, helped her begin to make sense of the problem -- ...At first I didn't understand the problem, I automatically assumed there was "no solution" because many story problems had no solution. When 118 everyone around me began figuring out problems like 1,5,10 = 16¢ and making columns of numbers I began to do the same thing. As I worked on it for a while, I was able to come up with 10 amounts. The problem made sense. Similarly, Teri reported how she worked alone until she believed she had exhausted all possibilities and then began to work with others. She found, not only that more monetary amounts could be made within the problem constraints but also there existed other questions -- other mathematical ideas -- to be explored: At first I started to work as an individual. I didn't show my answers or ask for anyone else's. After I exhausted all of my options, I looked at Deidre's paper and discovered that I had missed a couple. . .. Shelly's comment on a formula set Deidre and I racing trying 2 coins and 4 coins to test "her theory." It works. Why, we don't know. Although many commented on the confidence they gained by working with other students and several, like Hillary and Teri above, described how others helped them go further on the problem, there were also other remarks in the journals about the doubts, silencing, and frustration that resulted from listening to others' comments in the whole group discussion. Kim, for example, who had commented (above) about how "the Three-Coin Problem appeared at first... to be very simple," and how her group felt "very satisfied and confident that (they) had the correct answer," continued on to report how this changed during the class discussion -- When the class discussion started and people started talking about the number of combinations and such, I began to wonder if our group had done the right thing. Pretty soon I began to get frustrated with the whole problem and discussion. It seemed pointless and at a standstill. Indeed, students had approached the problem in such different ways, focusing on different aspects of the problem, that an "other's" work could easily have seemed utterly 119 disconnected -- irrelevant -- to the task at hand. If a student focused exclusively on finding various amounts that could be drawn without deeply considering the question of how they would prove to someone else they had all possibilities -- or if they considered their empirical results and group consensus as adequate forms of validation -- another students' interest in all possible combinations might seem puzzling or excessive. If students directed all their efforts toward reasoning through various combinations and proving that ten and only ten amounts were possible, they might find other students' interest in finding or remembering a formula a waste of time and energy. This sense of "wrongness" concerning other others' interpretations and efforts to solve the problem was voiced by a number of students. More often than not, despite their belief that something was amiss -- or perhaps because of it -- these students felt silenced. Liz, for example, wrote -- The first day I didn't want to say anything when everyone started talking about formulas and there as got to be an answer. I don't believe that... I was so afraid to say something on Monday because math is not my greatest subject. I don't really like talking about things I don't know about because it makes you sound stupid. So I kept quiet and listened and took notes. Liz described being "afraid" to saying something. Her fear grew, it seemed, from the risk of sounding "stupid" in an area she didn't feel able to speak to ("because math is not my greatest subject"). Although Liz was the only one to talk about being "afraid," others also wrote about being inhibited by their uncertainty. For instance, writing about her experience during the mathematics discussion, Kerrie commented —- I was getting frustrated when I knew or I thought I knew that someone else had a flaw in their explanation, but I didn't know the "real" answer myself, so I couldn't say anything. 120 But it was not only in uncertainty that students kept their silence. JoAnn, who had done an enormous amount of work on the problem in the allotted time (and beyond the allotted time),39 also spoke of her reluctance to give voice to her objections and ideas: I was... somewhat restless when other students were making conjectures I know were wrong. I wanted to keep saying -- No that's not right but I kept my mouth shut instead. Silence seemed to be a common response to the dissension that bubbled up as students listened to one another's ideas. Despite a desire to speak up, there were students who dared not for one reason or another. For Liz, there seemed to be a danger of appearing "stupid" before her peers; for Kerrie, there seemed to be some risk involved in being wrong, in misjudging what the "real answer" might be. And, although JoAnn did not seem particularly concerned about the correctness of her ideas or the appearance of ineptitude, she may have sensed other forms of risk.40 There are many ways to be misperceived, characterized by others in undesirable ways: A know-it-all, a brainiac, a teacher's pet. There are many ways one's "self" can be put in jeopardy. Aside from being "wrong," there are other potential dangers in putting forth one's own thoughts and ideas. 39 As mentioned in the description of the salient facet, The Nature/Wording of the Mathematics Problem, JoAnn was "excited" about the mathematics problem from the very beginning; in fact, she began working on the problem even before we had a chance to read it over together. This enthusiasm continued even into the beginning of the whole group discussion. JoAnn wrote, After everyone stopped working on the problem and began discussing it, I continued to try to work the same problem using 4 coins. I didn't pay great attention to the conversation because I was so into trying to solve the problem. 4° It's not at all clear what kept JoAnn from speaking despite the things she kept wanting to say; many things are possible. She could, for example, have been interested in how 1 as the teacher would handle students' incorrect ideas and hence refrained from voicing her objections; or she may have wanted to give other students a chance to talk before ending the discussion with her own solution and was patiently (or "restlessly") waiting for the right moment. But the possibility of risk -- perceived danger -- is also among the viable possibilities. 121 Shelly, for example, voiced frustration about the way other students had taken her idea -- Shelly's Conjecture -- and raised objections that were grounded in inappropriate applications of her idea: During the discussion of my "assumption formula," I became somewhat frustrated and angry because some of the students were making statements about it for which I had never intended the formula to answer or to be used. Some of their statements included such things as: "It doesn't tell anything about the amounts which the problem is asking for." "It doesn't work for 5." When I introduced my formula, I clearly stated its intent... how it had quickly (unscientifically) been derived (just drawn from one sample), that it might not be correct, etc. Obviously the "attackers" were not listening. There is always the chance, when sharing an idea with others, that they may interpret it, use it, transform it in ways that were not originally imagined or intended. Others could also poke holes in it, rip it apart, revise it without permission or regard for the context fiom which the idea grew. It could, as Shelly points out, feel like an "attack" — oppositional, dangerous, hurtful. Regardless of whether or not anyone intended to attack Shelly's idea -- and despite my own intentions of exploring and developing Shelly's idea because it had the potential to be highly generative, a wonderful example of mathematical curiosity and imagination at work -- Shelly perceived and experienced an adversarial context. And irrespective of whether or not any of the fears and concerns students harbored would have been realized if they voiced their contrary ideas, it is clear that some perceived risk and experienced enough potential danger to silence them. The prospective teachers saw and experienced the role of other students in very different ways. In fact, some students, like Kim and Shelly, had very different experiences at different points in the lesson. While Shelly gained confidence from others 122 during the individual/small group time, she found herself becoming "frustrated and angry" during the whole group discussion when others seemed to interpreting and using "Shelly's Conjecture" in inappropriate ways. Likewise, Kim reported feeling "very satisfied and confident" from the work she did with others in her group but began to lose this confidence during the class discussion when people "started talking about the number of combinations and such" -- and she eventually grew frustrated.“ Nevertheless, despite the variety of viewpoints, the role of others in shaping the nature and experience of the mathematics work was an important facet of Monday's class appearing in many journal entries. Salient Facet: The Effect of the Instructor's Probes While students worked alone or in small groups on the Three-Coin Problem, I kept an eye out for people who seemed to be "done." I was curious to see where their work left off and eager to see where else they could go with a little nudging. I came across many who believed they were finished and asked them questions like, Are you sure have them all? How do you know? Would you be able to prove to someone else you've found all possibilities? The way I questioned them -- pressing them to explain and prove their ideas -- was another important facet of the class session for students. Erica's comments provide a good example of how several prospective teachers interpreted and responded to my probing questions: When I found 10 answers, I figured I was done. But when Kara asked me if I was sure, suddenly I wasn't. I thought I better find a formula to help me make sure I had all possibilities... 4| . . . . . , , . . . This shift in how students experienced the role of others Will be discussed in more detail in the section, “Discussion of Salient Facets.” 123 For Erica, my questions seemed to shake her confidence about the completeness of her solution set, causing her to seek out a different approach to the problem. Some students, like Erica, commented on the uncertainty they experienced when pressed about their solutions. There were, however, others who reported a sense of confidence when I questioned them, stemming from the work they had done with other students. Shelly, for example, reported -- When Kara asked, "Are you sure you have all of the answers?" I felt confident in replying, "Yes," since both Erica and I had the same number although working somewhat independently.42 Whether experiencing a surge of assuredness or doubt, my questions seemed to serve as test of confidence. Many students who commented on their exchanges with me spoke of how their "confidence level" changed because of -- or remained the same despite -- my questioning. Along with interpreting my questions as a test of their confidence, there were some -- like Erica (above) -- who also seemed to see some connection between my probes and the need to find a formula: "I thought I better find a formula to help me make sure I had all possibilities..." Although there are many ways to prove whether or not all possibilities have been found for the Three-Coin Problem, several students immediately responded to my queries by searching out a formula. When Shelly heard my questions, her suspicions about the existence and usefulness of such a formula were confirmed: Kara's question also served to reinforce my belief there is a formula to determine the exact number of possible answers for this and similar 42 Shelly's confidence grew from the verification of her results from a "somewhat" independent source. This is similar to how experimental scientists might obtain verification from an independent laboratory replicating their empirical trials. Although it is understandable how this might lend a sense of confidence, it IS not considered a valid form of mathematical proof. 124 problems... This formula could... be used to check that we had found the correct number of answers. These students believed the right formula would provide the verification they needed. With the formula, they would lmow how many amounts can be made and would therefore know whether or not their ten solutions formed a comprehensive list. The search for formulas was-a very prominent facet of Monday's class, appearing as a topic in many journals; this will be described below in the next salient facet. Although I did not get around to asking probing questions of all the students -- not everyone had gotten through finding, what they believed to be, all the amounts they could find -- among those I questioned further, quite a few of them found this interaction with me to be a significant aspect of their work. Salient Facet: Th_e Search for Formulas_and Combinations Searching for formulas and combinations comprised a large part of students' efforts while they worked alone and in small groups on the Three-Coin Problem. Many wrote about their own attempts to find or remember a formula. Some, like Shelly and Erica (above), believed a formula would help them to verifi/ whether or not they had found all possible amounts. Indeed, for at least one student, the lack of a formula seemed to make it difficult, if not impossible, to be sure of "the answer." In her journal, Iris explained -- When I decided to become an elementary education student, I chose to minor in math because it has always come fairly easily to me. However the majority of my classes consisted of what I call "plug and chug" problems. These types of problems require a formula and then you plug the numbers in and solve the equation... I was... unsure of the answer (to the Three-Coin Problem) because I was unable to recall a formula relating to my "plug and chug" problems. I am 125 used to answering questions by substituting a formula into a set of equations. I could not recall a formula that would apply to this problem though. It is not clear how or why a formula would have helped Iris -- perhaps, like Shelly and Erica, Iris was interested in a formula that would help her know how many possibilities to expect. Or, perhaps, a formula would simply have made the whole problem-solving task easier, by allowing Iris to "plug" in numbers and "chug" out an answer for which she would feel confident. Several other students explicitly discussed how a formula would simplify their work. For instance, Alisa wrote in her journal -- For me, math has always been easiest when there was a set formula to plug in. "Plug and chug" as the saying goes. So my first thought was to find a formula. If there were a formula to recall and use -- like the quadratic formula or the Pythagorean theorem -- Alisa would have been able to find the answer without thinking about pennies, nickels, and dimes. She would have merely needed to plug in numbers and solve the equation without reasoning through various configurations of coins. It would have made the work easier. There were also students whose search for a formula was spurred on by their memories of MTH 201. In her notebook, Elaine explained -- I remembered that I had used (a formula) in MTH 201 so I spent time trying to figure it out. Guided by vague memories of a formula used in MTH 201 for "this kind of problem," many students tried to recall or reconstruct the formula. A few prospective teachers remembered the formula utilized "factorials" in some way. Simply knowing (or believing) that a formula existed, locked away somewhere in their memories, seemed a 126 compelling motivation for some.43 Perhaps, like others, these students believed and hoped that a formula would simplify their work or provide validation about the number of possible amounts. Or, perhaps, the search for and use of a formula seemed the type of work expected in a classroom setting when focusing on mathematics. Erica, for instance, speculated -- ...I don't know why I became so hung up on getting4a formula. Maybe I have been "brainwashed" by years of math classes. Undoubtedly, students' expectations for what kind of work should be done when doing mathematics -- expectations developed through their experiences in other mathematics classes -- played some role in how they approached the problem. Perhaps, as Erica speculated, something in her (and others') past experiences created a strong impulse for finding and using formulas for solving problems like these. And finally, for at least one student, the strong suspicion that a formula existed, coupled with strong curiosity, seemed to provide yet another reason for seeking out a formula. In her journal, Shelly explained, I knew there was a definite number of answers and most likely there was an existing formula which could be used to help us determine the number of answers to be sought. . .. Afier making a sequential list to acquire all possibilities, confirming with Erica that my list matched her table, I felt confident I had all possibilities. I began working the formula backwards because I was very curious to see ifI could find some pattern. The perceptions and interpretations that led students to seek out a formula as they worked independently on the problem were quite varied but, nevertheless, were a ‘3 For some, "knowing" that a formula existed and being unable to recall or find it was the source of frustration. Tasha, for example, reported -- While I was figuring out the pennies, dimes, and nickels problem I became very frustrated because I know there is a formula to simplify the problem. 4‘ Prior to this quote, Erica wrote about her own efforts to find a formula and her frustrations with "having forgot everything from MTH 201." 127 significant part of their descriptions and reflections of Monday's class. The search for formulas and combinations was also an important part of the whole group discussion and was mentioned in several journal entries. Unlike students' descriptions and commentary of the independent work time, which focused primarily on their own efforts to find or recall a formula, students remarks on the whole group discussion focused almost exclusively on others’ attempts to find and use formulas and the frustrations that resulted. For instance, Megan reported, At first I started getting really aggravated when we spent most of the class looking for the different # of combos instead of # of amounts -- which is what the question was asking for. Also as a math student, I always try to avoid using formulas. I hate them. I think if you can solve a problem without one then do it. Too many people get caught in the idea that there must be a formula. Although other students did not voice similar sentiments concerning formulas, Megan’s annoyance with spending class time examining combinations and formulas was shared by several others. There were those who felt the problem was really about amounts and found others' tenacious search for a formula bothersome. In her journal, Angela wrote -- (The work) changed and it wasn't about math anymore. It was about something else like the people were giving answers for getting some formula and not the real answer for the question and I started to get annoyed because I didn't think we had to find a formula if we got it right. This feeling of annoyance was repeated in several journals. There seemed some genuine concern about the "real answer" which formulas and "different # of combos" did not seem to address in any useful way. And there also seemed to be some resistance to formulas -- or perhaps to the emphasis that other students placed on formulas. 128 Angela also went on to describe another concern regarding the search her classmates were engaged in during the whole group discussion -- And I thought that it (Shelly's Conjecture) was a made up after the fact formula in that we came up with the answer 10 so then we just added some numbers together until we got 10 then said it was the formula. There is no proof and no explanation for that formula. Angela was the only person to pen a protest about the validity of the formulas we had been considering. This was a valid concern: A few students had merely tested Shelly's Conjecture and found it worked with three variations of the problem -- i.e. pulling out two coins, three coins, and four coins (using only pennies, nickels, and dimes) -- we had yet to prove when and why this "formula" would work. This uncertainty, it seemed, contributed to Angela's annoyance about the time and energy focused on Shelly's Conjecture. It is possible that Angela was concerned we had not gone far enough with Shelly’s Conjecture, abandoning our efforts before proof and explanation had been developed. But it is very likely that the very endeavor of exploring a formula that was “made up and after the fact” was not seen as a worthwhile pursuit -- that, at some point, “(The work) changed and it wasn't about math anymore” and we had gotten off-track from finding “the real answer for the question.” Whether caught up in or annoyed by the search for formulas (and combinations), this was an important facet of Monday's work that many students commented upon in their journals. Salient Facet: No Solution and/or Answer Given by the Teacher Monday's class session ended without any firm resolution about the Three-Coin Problem. We never did return to completing the table of pennies, nickels, dimes, and total 129 values that Danielle's group had started and, although we had a nice statement of Shelly's Conjecture -- along with several examples of the conjecture -- we still did not know when Shelly's Conjecture could be used nor did we have any proof or explanation for why this observed pattern might (or might not) be valid. There seemed to be a strong sense that most people had arrived at ten different amounts for the Three-Coin Problem but we had not come up with a complete and compelling proof that these were all the possibilities -- and, notably, I did not confirm or deny "ten" as the maximum number of valid amounts. Interestingly, although there are numerous forms of closure that would have been nice to have for the Three-Coin Problem, students' commentaries focused on the instructor not providing an answer, solution, or confirmation. Notably missing -- with the exception of Angela's concerns about Shelly's Conjecture being a "made up after the fact formula" with "no proof and no explanation" (see above) -- are any comments about the kind of "closure" a mathematical proof or some form of mathematical rationale might provide. This ending to the class session, without any solution or answer provided by me as the instructor, was a salient facet for many students. As Iris noted in her journal, this aspect of the class seemed to stand out for its departure from what many considered to be "normal" for mathematics classes: Monday's problem did differ, however, from my other math classes. The teacher always provided the correct answer. But it seems this particular salient facet was especially prominent due to the similar way the literacy activity came to an end without F enice providing any answer or definitive interpretation for the Ordeal by Cheques story. In the case of the Three-Coin Problem, there were certainly a fixed number of possible amounts and a variety of ways to prove only ten possible amounts could be made but I made a decision to not tell 130 students whether or not they were correct or to provide them with a proof. I believed they could and should find validation for their work based on mathematical reasoning and not depend on me for justification. In the case of the Ordeal by Cheques activity, however, there was really no answer for Fenice to give or withhold. The author of the Ordeal by Cheques "story," which consisted entirely of checkbook registries for the Exeter family over decades and generations, had provided no other supporting text to elaborate or provide an authorial perspective on the story. It appeared the author intended the record of checks to be interpreted in diverse ways, to be given meaning through readers' attempts to fill in holes and piece together plausible meaning. This selection and design of activities, requiring prospective teachers to use reasoning and argument to construct sensible solutions with the support and challenge of their peers, was not accidental: For me and F enice, this was an important aspect of the work we wanted to introduce to students during the first week. And, indeed, not only did many students comment on their interpretations of and reactions to this facet of the class, but in most cases their comments encompassed both the mathematics and literacy activities. More than with any other salient facet it was very difficult, if not impossible, to distill students' reflections on the mathematics work from their reflections on the language arts work. Additionally, more so than other salient facets, this lack of instructor-provided answers seemed to evoke comments from the prospective teachers about possible rationales or potential benefits that might help to justify this aspect of the class sessions -- comments that would help explain why the instructors would choose to withhold answers or select problems without answers. For example, Kathy commented -- I also felt a little bit annoyed when I wasn't given a concrete answer. I think these (activities) would work even better if there were answers given 131 in the end -- But then again, many of the ideas don't have concrete answers, and it is good for the students to come up with their own answers. Despite being "annoyed" and initially believing things would "work even better if there were answers given in the end," Kathy concluded there were, perhaps, potential benefits for student learning in not providing answers. This attempt to see beyond feelings of frustration, anger, and disappointment when answers were not given was quite common. And students arrived at many different conclusions. Some, like Kathy, arrived at a new or alternative perspective about how the withholding of "concrete answers" might be beneficial for students. Others searched for hidden lessons -- a potential moral to the event -- that might benefit them as prospective teachers. Lynn, for instance, reported -- I was annoyed when you said you didn't have the answer. After a while, I thought about it and rationalized maybe you gave us these problems to see how some students may experience the same problems as us and want more explanation from the teacher. So as a teacher to be aware of what we are teaching our students. Maybe you want us to understand what we are learning so our students understand. Although I may be totally off and not really understand why you gave us these questions. Lynn seemed to be striving, in a self-conscious way, for a connection between their experiences as prospective teachers in the class and some possible insight into teaching they were supposed to be getting. This facet of the course was difficult for her to understand and she speaks of her deliberate attempts to give meaning to it: "I thought about it and rationalized. . .." She seemed aware of her own struggles to make sense of why answers were not available, and the potential inadequacies of her efforts -- "Although I may be totally off and not really understand. . . ." 132 There were also students who were self-consciously aware of their reactions -- their troubled feelings with having been denied an answer as well as their continued desires for an answer -- and arrived at some insight about themselves as learners and their past experiences as students. In her journal, Kim explained -- I was almost sure that I had the right answer but I would have liked confirmation. . .. I just realized that I have never really done anything like that in college. Everything here seems to be so structured and full of answers. We are basically taught that there is always an answer & we are always in search of it. That is probably why it was so hard to be told there was no answer. Teri, likewise, wrote in her journal -- I was disappointed that the teacher did not tell us the answer. . .. It is a new way to approach learning. It enhances critical thinking which is much different then (sic) we are used to. . Students like Kim and Teri seemed to sense a mismatch between what they've known as students in other classes and this facet of the class -- and in Kim's entry she hypothesizes that their longings were, in part, a product of their past experiences. Other students made similar observations about their experiences and noticed things about themselves as learners however, rather than finding a silver-lining to their clouded struggles in this first week or some new understanding about the source of their difficulties, these prospective teachers simply found frustrating circumstances that hindered their work. Iris, who was quoted above regarding how Monday's class differed from her other math classes because "the teacher always provided the correct answer" went on to explain — This (i.e. being provided with the correct answer) was beneficial because it often helped me to see the answer and then I could work backwards to 133 the beginning of the problem. I was unable to perform this technique with Monday's problem. Kara never offered an exact answer so I consequently was unable to confirm it. After many struggles I gave up. I was too frustrated and did not have the patience to finish the problem. I sat back and waited for the answer, however, it never came. Iris noticed that without access to "the correct answer" an important technique for her own learning was rendered inaccessible to her. With an inability to "work backwards to the beginning of the problem," Iris encountered insurmountable frustration and hit an impasse. From this experience, she surmised it was "beneficial" to be given the answer. Shaya, likewise, made note of her own efforts to cope with this facet of the class but, rather than expounding on the benefits of being provided with correct answers, she commented on the detriment of being left without a clear answer -- We spent the whole class period... discussing this particular problem and Kara never gave us a solution or showed us how to do the problem. . .. I found F enice's assignment fun however she never gave us a solution as well. The engagement of my brain on a problem then have no solution explained to me. I see how frustrating this is so I don't plan to us this strategy in my class. Although most students found some benefit to, what seemed to be a perplexing aspect of our class, there were several like Iris and Shaya who simply found the experience frustrating and undesirable. In addition to all the students described above who at one point or another -- in one way or another -- experienced some anger, frustration, or disappointment concerning the lack of teacher-provided answers (or confirmation), there were also a few prospective teachers who claimed to not share these feelings with their classmates. In their journals, they explicitly noted their observations and musings about the reactions of others and contrasted this with their own perspectives and experiences. For example, Alisa reported — 134 I did notice the frustrations of several of our classmates which were sort of surprising. I know they believed that Kara had the answer and they wanted to know it. Sure, I was curious but I didn't read so deeply into the point. Danielle was another student who noted the apparent feelings of annoyance surrounding her but, rather than simply noting the divergence of her experience from the vocal majority, she went on to articulate a goal for herself -- I was never "annoyed" w/ Kara because she didn't tell us the "correct" answer. This is one thing I want to keep away from: assuming that the teacher has the one & only correct answer. For students like Alisa and Danielle, the discrepancy between their own reactions and those of their classmates was noteworthy and allowed for some discussion about their own thoughts and viewpoints. This was particularly true for Heidi, who wrote a fairly detailed journal entry centered on this contrast— During the math problem I recognized the strategy that Kara was using since I understood that she was trying to get us to think more in depth and be able to explain how we got the answer. I didn't really feel that frustrated about her not giving the answer. With my background knowledge about how there was no one right way to go about getting an answer, I, as a learner, felt comfortable with Kara's strategy. However, I think ifI hadn't been able to disregard my past experiences in school where there was only one right answer (which the teacher had), I would have been annoyed. Annoyed because everything I had learned before was that the teacher had the right answer, and now Kara, my teacher, wouldn't tell us the answer. Therefore, I think this strategy would be more difficult to use with older children, who have already been brainwashed into thinking there was only one right way to solve this problem. First they would have to learn that they should throw out this old notion before they can patiently and understandingly accept this new approach to learning math. I think that as an early childhood educator, I will have an opportunity to start children out with this way of thinking, so that they will grow up thinking how outrageous it is to think there is only one path to each solution and only one solution. Heidi continued on to discuss the Ordeal by Cheques activity, finding many similarities between the mathematics and language arts lessons. 135 Although students had many different ways of interpreting and experiencing the way the class ended with no solutions or answers given by the instructors, more than half the class commented on this salient facet. It was an aspect of our work together during the first week of class that stood out to many. Discussion of Salient Facets Only a handful of the students' comments did not fit within one of these five salient facets. Two students alluded to a connection they saw between previous education courses and the "many new ideas about teaching and learning math and literacy" they had been "exposed to" during this first week (Elaine). For example, Elaine briefly mentioned her experiences in TE301, an introductory teacher education course entitled, "Learners and Leaming in Context." She wrote -— In 301 I was starting to learn some of these theories of teaching but now I'm excited about going more in depth. Additionally, several students wrote about their feelings regarding mathematics and briefly described what it was like be a mathematics student in other contexts. For example, Danielle noted -- I must say that I wasn't too pleased that we would be studying math this term. It seems to me that every term I'm telling my parents that this is the last math class that I'll ever have to take at MSU. Math frustrates me! I never truly learned the basics about math during my own elementary school days. For example, I was taught the multiplication table my memorizing it. However, other than a few comments of this sort, students' descriptions and reflections centered on the five facets of the class discussed in the previous section. Although it is beyond the scope of the data and this analysis to account for the various contextual and social constraints that may have contributed to these foci, the fact that 136 these five themes emerged seemed significant given the fairly open-ended nature of the assignment. Even though no two students expressed the same set of experiences, these themes surfaced among the group and were repeatedly touched upon by many individuals. Among infinite possibilities there seemed to be a kind of patterning in what students attended to and the ways they experienced these things. Seeing patterns among students’ diverse responses to the writing assignment -- beyond the identification of these general themes -- is a challenge. Each response was unique in the set of themes it addressed (or did not address), the events and moments it focused upon, as well as the experiences it described. In the following discussion, I share one way of organizing and representing the diverse array of students' interpretations and reactions against the backdrop of the shifting class context. Although this representation strips away the details and examples discussed in the previous section, Description of Salient Facets, it provides a narrative schematic that allows multiple events to viewed simultaneously. This simplification makes it possible to see patterns and relationships over time, across salient facets, among varied experiences, that are less visible otherwise. The “Occurrence” Diaggm45 Many of the prospective teachers' comments were devoted to describing shifts in their thoughts and feelings as the class session progressed. In their narratives, students discussed the emotional states they entered as new occurrences took place ("I was getting frustrated when. . ."; "When Kara asked, 'Are you sure you have all of the answers?’ I 4’ The term “Occurrence Diagram” and the basic idea of this representational form is an adaptation of the early work of Anatol Holt, a mathematician and programming systems designer. Although this analysis borrows from Holt, it is the simplest, most introductory form of his work. 137 felt. .."; “While reading the math problem for the first time, I was feeling. . .”). They also reflected upon the thought processes and intellectual efforts they engaged in at various points in the lesson ("So my first thought was to find a formula. . ." "After making a sequential list...” "At first I didn't understand the problem. . . "). Five "occurrences" were noted repeatedly in students’ journal entries, marking the start of time intervals when the prospective teachers became engaged in certain kinds of work or began experiencing a notable emotive state. In the figure below, the five occurrences are represented with five vertical lines and are labeled accordingly. The time intervals that followed are indicated with arrows. 138 3325:. scene. .3338... 9.6.5 0.2.! «one... fican. etc! 95.3 a .8885 __asflaause... 88:85... _ E039... TAIIAIII TI ‘ The Instructor’s Probes occurrence is an event that only a portion of the class encountered. Only 139 It is not surprising these particular occurrences were significant in students' narratives. In many ways, the nature of our work had indeed shifted with each of these occurrences, surfacing new challenges and requiring different kinds of intellectual skills and dispositions. In the language of Anatol Holt,46 each "occurrence" initiated a new “interval of condition-holding.” Important contextual shifts had taken place, creating students who appeared to be completely done with the problem were further questioned about their solutions. Therefore, this particular occurrence is represented with a dotted line to indicate its limited reach through the class. ‘6 This analysis was inspired, in large part by the ideas of Anatol Holt ("Tolly") as conveyed in Mary Catherine Bateson's book (1972/1991), Our Own Metaphor: A Personal Account ofa C onference on the Eflects of Conscious Purpose on Human Adaptation. In particular, Tolly's method for representing the back'n'forth swinging motion of a pendulum sparked my imagination and greatly influenced my analysis and efforts at representation in this Chapter. Bateson writes the following account of Tolly's instruction on this representational approach: "Let's imagine a pendulum swinging back and forth." Tolly walked around for chalk and then he drew this picture. F F 0<—-o+—e4——e<—— "This means that for some interval of time the pendulum swings to the right, shown by the arrow labeled R. Here's an occurrence, shown by a point, and then the pendulum swings to the left for some other interval, shown by the arrow labeled L. The occurrence is the end of the swing. You can think of the same picture as representing a billiard ball rolling back and forth on a frictionless table between two reflecting boundaries. Lefi, right, left, right, and the occurrences are the bounces." Horst did a double-take. "You mean the point indicates the moment it changes from right to left?" Tolly nodded gleefully. "Yeah. That's right. Unconventional." Once Horst had called my attention to it, I realized that this was indeed unconventional. The minute I stopped thinking that the arrow indicated the direction of the pendulum (which it did not, because the diagram of a light changing from red to green to red would have looked exactly the same), I realized that Tolly was doing the strange thing of using an arrow to represent something stable (an "interval of condition-holding" he called it) and a point to represent change, the occurrence that initiates new conditions. This was the exact opposite of the convention Barry had used in his diagram, where arrows had represented the transition from say, organic to non-organic nitrogen compounds, or Fred, who had used arrows to represent causation (Bateson, M., 1972/ 1991, p. 166). 140 intervals of time in which new conditions were in operation and, accordingly, students' emotions and efforts at problem solving entered new states. One could imagine these occurrences -- introducing the problem, starting the independent work time, probing students about their solutions, initiating the whole group discussion, dismissing the class —— as "triggers" or “firings,” setting new processes into motion and putting new conditions into play. This way of representing the flow of events is subtly different from commonly used representations such as flow diagrams and tree diagrams that are often used to depict processes. In such diagrams, an emphasis is placed on identifying the major steps or components of the processes and representing their sequencing. A flow diagram of the first day of class might look something like this: Discussion of the Whole group problem and what it Independent work discussuon of the .s asking on the problem problem Individual/small group discussions with the teacher about particular solutions The arrows, here, represent transitions and are meant to simply indicate movement and direction from one step in the process to the next. In decision trees, arrows are used in slightly more complicated ways to depict movement and direction along multiple possible pathways. For example -- 141 (Independent \ work on the problem Discussion of the problem and what it is asking believe he/she is “done”? Teacher Assistance: Work w/ student to generate. test and record examples, etc. If shtdt... / Work on examples It progressi Continue 3- in Problem Whole group discussaon of the problem lfcha How do you k no Workl Ioen justification you have found all solutions: ? How would you prove it to someone? Work on other versions of If not challenged... problem Teacher Probe: What it you pulled out 4 coins instead of 3? What about 5 coins? The arrows of the Occurrence Diagram, however, indicate more than movement and direction along process pathways. Rather than depicting the transitions between significant components of the process, the arrows represent those important intervals of time where certain states and contextual conditions were in operation. In the earlier diagram depicting the five occurrences mentioned in students’ reflections, only the dimension of time had so far been depicted. This picture, however, becomes much more complicated with the addition of the salient facets students perceived and the various interpretations and reactions they described. 142 "K I'\ clan dhrnhcal i i r _ the'rd' Amman mum murmur!" instructor probes whole group discussion laundr IWWM' mirrored-um 1 ‘ ' . anew-w mmwrmm mum-“mama Madam.“ individual/small group work launch problem Introduction " This cluster was raised in students' comments for both the Individual/Small Group Work time interval as well as the Instructor’s Probes interval. Its placement within the independent work time interval but near to the instructor probes occurrence line is meant to indicate its applicability to both intervals. 143 In this diagram, two more "dimensions"47 have been added to the dimension of time. First, five colored "planes" have been added to the diagram, each one representing a different salient facet of the class discussed in students' journals. One thing to notice is each of the salient facet planes intersects with at least one but no more than three of the occurrence lines. In students' comments, different salient facets came to the fore at different times. So, for example, the nature and wording of the mathematics problem seemed particularly salient to students not long after the problem was introduced. Students were off and running "as soon as the problem was put on the overhead" (J oAnn) or encountered confiision "because at first (they) did not understand the problem" (Hillary). However, there were no other intervals of time for which this particular facet of class was a primary focus. On the other hand, the search for formulas and combinations was an important topic in three different intervals: It was focal for students following the start of the individual/small group work, afier my further questioning of individuals and groups, and after the whole group discussion was launched. The final dimension that has been added to the diagram is the varied thoughts and reactions students had toward a given salient facet. Each type of response is represented as a "cluster" of interpretations and reactions, indicated with half-spheres (A) and ° These clusters were actually described in the section, “Salient Facet: The Effect of the Instructor's Probes,” and could be placed on the Instructor's Probes salient facet plane. However, their positions on these other salient facet planes are also very apt: They are examples of commentary that encompass more than one salient facet. d This salient facet, No Solution and/or Answer Given by the Teacher, does not have a right-hand side border. Unlike other salient facets, this one refers to an issue that surfaced sometime after the class ended. There is, however, no specific occurrence to mark the end of the class dismissal time interval. The absence of the right border is intended to represent this lack of clarity. ‘7 The word, "Dimension," is used here to refer to the each of different kinds of information contained on the diagram. The Occurrence Diagram portrays three "dimensions" of analysis: time, salient facets, and clusters of students’ interpretation and reactions. While the diagram was drawn three-dimensionally, the x, y, z coordinates of the salient facets have no meaning; only the relative placement of the planes along the time axis matters. Likewise, the particular placement of the clusters has no specific, quantifiable meaning: Only their position, relative to the salient facet planes and the time intervals, matters. 144 scattered across the various salient facet planes. These clusters are each labeled with one type of response. So, for example, afier the dismissal of the class (an “occurrence”), there were several different ways students responded to the absence of teacher-provided answers or solutions (a “salient facet”). Some were frustrated about not receiving an answer but saw benefits for their own learning of mathematics or a potential lesson for them as prospective teachers. Others were frustrated but gained some insight into their past experiences as students or into him/herself as a mathematics doer and learner. There were also some students who were simply annoyed and found the experience undesirable. And, in contrast to these, there were several students who noted the frustrations of others but claimed to not share them: Instead, they reflected upon their own thoughts and feelings regarding this salient facet. Reading the Occurrence Diagram: A Few Observations This way of reorganizing and representing students' comments does two rather important things. First, it groups students' comments thematically, placing certain comments upon the same (colored) plane thereby associating them with a particular salient facet. This allows for a salient facet to be viewed across time and makes it possible to compare how students' experiences of a salient facet may or may not have differed at different intervals. Second, it groups the comments chronologically within a specific time interval. This allows each time-interval to be examined separately, highlighting the range of experiences students reported for that interval. Examples of the kinds of observations that can be made from these three ways of examining the Occurrence Diagram are pointed out below. 145 Different experiences at different times. A perusal of the clusters, A, arranged together on a single salient facet plane gives a sense for the variety of ways students interpreted and experienced this particular aspect of the class. More often than not, such an examination reveals a mixture of frustration and pleasure, of insight and confusion. This range of possibility is quite interesting in itself, however, there are a few instances where a salient facet spans more than one time interval and notable differences can be seen in students' comments depending on the time interval with which the remarks are associated. For example, the role others students played in the class was experienced differently after the start of the independent work time than it was experienced after the launch of the whole group discussion. 146 . earlier work no discusion individual/small group work launch whole group instructor discussion launch probes During the interval when the prospective teachers were working independently on the Three-Coin Problem, they spoke appreciatively of the role others students played in alleviating and overcoming their uncertainties. Some gained confidence and felt a sense of relief from the opportunity to confer with their classmates. Some received help from other students, allowing them to go further with the problem than they were able to on their own. However, this was not the case when viewing the role of others for the whole group discussion interval. Students commented, instead, on the doubts that began to form as they listened to their classmates. Some spoke of the silence they kept rather than voicing their dissent to others. And, one student (Shelly) commented on the frustration 147 she experienced as other students used her idea (Shelly's Conjecture) inappropriately. Unlike the supportive role others played earlier it seemed, in this later time interval, other students created a context of greater uncertainty and risk. Diverse experiences of Daniella time intervals. Another look at the clusters (1"), focusing this time on those gathered within a particular time interval, provides a sense for how the class -- comprised of many individuals «described their thoughts and reactions within a given segment of time. Rather than looking at change or difference over time, this focuses attention on the diversity and range of experiences during an interval of condition-holding. It paints a portrait of the multiple things going on concurrently. This way of viewing an interval of time is especially interesting when more than one salient facet was discussed. For example, for the interval following the launch of the whole group discussion, there are several clusters of comments regarding the role of other students in shaping the nature/experience of the work as well as several about the search for formulas and combinations. 148 a Frustrated by Inappropriate use of her ' The Role of Other Students a Developed doubts r . earlier work a Kept silent d ing discussion Irritated by others persi search for formulas Annoyed by the lack Kl proof or explanatio or formula The Search for Formulas] Combinations class dismissal whole group discussion launch 149 As pointed out earlier, there seemed to be a more threatening quality to the role others played during the whole group discussion. The students who commented on this salient facet for this time interval described the doubts, silencing, and frustration they experienced. In addition to this, prospective teachers also expressed frustration and annoyance with other students' efforts to pursue formulas and combinations. Some were frustrated about the apparent irrelevance of formulas: It seemed, to them, formulas did not address the "real" question. Some were irritated by other students' persistence. And one student (Angela) expressed annoyance with the lack of proof or explanation for the formula (Shelly's Conjecture) that was being considered. Although it is not possible to draw causal links within this data, there seems to be complementarities between the irritation and frustration some students felt regarding the search for formulas/combinations and the increasing doubts, silencing, and frustrations that some connected with the role of others. Clipped words, voices edged with impatience, questions about why someone would want to approach the problem the way they did could all contribute to a sense of risk and uncertainty. Likewise, the unvoiced arguments along with the unarticulated connections and explanations could contribute to the suspicion that good reasons and valid connections do not exist -- or for others, reinforce their sense of being alone in their divergent views and in the assumption that the thoughts and feelings being voiced are generally shared by everyone. The Occurrence Diagram gives a bird’s-eye view of the interpretations and reactions students expressed. This vantage point allows one to look for differences and relationships across time and among varied experiences, highlighting where important contextual shifts seemed to take place for the students and laying out the mixture of responses that characterized the intervals of time. Other significant differences and relationships are evident, however, in the students’ writing that are not seen -- or seen only in part -- on the Occurrence Diagram. Bevond the Diagram The unfamiliar terrain is designed to be different, in important ways, from other learning contexts to which students have become accustomed. It is the contrast between what students bring to the new class context and the kinds of responses the context calls 150 forth that give the terrain its “unfamiliar” features. For a variety of reasons, these types of differences between students and the context were not captured well on the Occurrence Diagram. Sometimes comments addressing such difi‘erences were simply not included on the diagram because they lacked the necessary time-referents to be included on the Occurrence Diagram. This happened most frequently when students discussed the mathematics and literacy activities together, addressing the shared features of the Three- Coin Problem and the Ordeal By Cheques activity that stood in contrast to schoolwork they had done in the past. In other cases, however, these types of differences were just not addressed directly in the things students wrote. For example, the prospective teachers did not comment upon the differences between their underlying assumptions or their fundamental ways of knowing mathematics and those being called for by the class context. Nevertheless, signs or indicators could be seen in students’ comments that point to the existence of such differences. In the discussion below, I describe some of the new things students saw in the class context and noted in their journals that differed from what they thought of as “normal “ or common for school. I also go on to talk about some of the indicators of ill- fittedness that can be seen in their entries — signs pointing to differences between the things that came naturally to students and the things they encountered in class. The notablyl “new.” Some prospective teachers noted “new” things they encountered in Monday’s mathematics class that stood in contrast to more typical kinds of problems or work they had done in the past. For example, Elaine wrote -- Math was always boring for me since I was just adding numbers such as 26+32. These new problems made me think about what I was trying to do rather than simply adding 2 numbers. 151 Several students noted similar observations about the “new problems” they were given and the kind of thinking they were encouraged to do that seemed different from other mathematics classes. Along with this, a few prospective teachers also commented on the serious attention paid to multiple valid solutions as something new and notable about our class context. Heidi, for example, wrote a long journal entry commenting on how “older children. . .have already been brainwashed into thinking there was only one right way to solve this problem” and her hopes to provide her own students with something different — (Older children) would have to learn that they should throw out this old notion before they can patiently and understandingly accept this new approach to learning math. I think that as an early childhood educator, I will have an opportunity to start children out with this way of thinking, so that they will grow up thinking how outrageous it is to think there is only one path to each solution and only one solution. The notably new things students perceived and commented directly upon were largely limited to things they felt were beneficial or enjoyable, contrasting with things that were less so in past experience. This does not mean, however, that other sorts of differences -- less pleasing differences -- did not exist. Indeed, it may have seemed risky or inappropriate to write about facets of the class that seemed detrimental or objectionable. Other ways of thinking about differences in past and present learning experiences may not even have occurred to some. It is also possible that many “differences” were not recognized in any definable way that could be articulated but were only experienced as feelings of discord or being ill at ease. Indicators of ill-fittedness. Mismatches between what students bring and what is required by the class context are to be expected in a class context that is truly unfamiliar 152 to students. As Schwab (1959/1978) notes, "From the first, hearers must struggle to understand. As they translate their tentative understanding into action. . .. the actions undertaken lead to unexpected consequences, effects on teachers and students, which cry for explanation" (p.172). These types of mismatches -- i.e. that surface when students' efforts and expectations fail to result in the things they believe would (or should) follow - - tend to be experienced as feelings of confusion, uncertainty, frustration, annoyance, and perhaps even a sense of meaningless or futility. Similar to ill-fitted shoes that rub and irritate, leaving red marks or painful lesions where adaptation is required but has yet to happen, places-of-rub can be seen in prospective teachers’ comments where something was not quite right in the fit between students and their new environment. Five different types of indicators are mentioned below. 1. Frustration A very common reaction to the puzzling and confusing aspects of class students experienced was frustration. (For example, “I felt a frustration on the math lesson because of the lack of information,” (Kerrie); “My thoughts about being a math student the other day in class could be described as frustrated and annoyed. . .. I felt confused because at first I did not understand the problem” (Hillary).)48 2. Attempts to ascribe meaning or purpose Another way prospective teachers frequently responded to their uncertainty was to seek out some larger, hidden purpose that would explain the puzzling facets of the class context. Liz, for example, wrote -- 48 This particular indicator, “frustration,” is one of the few that shows up on the Occurrence Diagram. Various clusters of frustration can be seen on four of the five salient facet planes and in three of the five time intervals of the Occurrence Diagram. 153 The math portion of the first day shocked me. I thought it had nothing to do with math. I was looking for something else. I thought it was an assignment on understanding. The first day was confusing and... I was curious and anxious to find out the meaning behind this activity. 3. Cautionary Tales Sometimes students went beyond wondering about the purpose and meaning of the class to developing a rationale in the form of a cautionary tale, a lesson to be learned from their experiences within this unfamiliar context. Marcia, for example, arrived at the following — As I look back though I realized how important good, clear "communication" is with our students. Often times students feel as fi'ustrated as we did and are too shy to speak up. This is when problems arise. 4. Positive Spins Similarly, prospective teachers’ attempts to give meaning to an unusual facet of the class often took the form of a "positive spin," a silver lining to the cloud of uncertainty that hung over them. For example, commenting on both the mathematics and literacy activities, Lindsey wrote -- Although it was rather frustrating, I liked the fact that there were no CORRECT answers in each exercise. This placed emphasis upon process rather than result. 5. Pointlessness There was also a student whose uncertainty and frustration seemed to leave her not only questioning what the purpose may have been but wondering if there was any purpose at all for these unfamiliar facets of the class. She wrote — I get very frustrated when I can’t figure out problems or if I think I have the solution and then I am still not told if I am right or wrong. Then I begin to question what’s the point of doing the problem if you are not going to be told the solution. 154 New Patterns In the analysis above, I focused only on those things students chose to write about but for everything that was written, noticed, attended to, there were myriad other things left out. Much of what students undoubtedly perceived was quite run—of-the-mill and not particularly noteworthy: There was a whiteboard. The teacher wrote on it. Students raised their hands when they wanted to speak. These, and many other things, simply fell into familiar patterns of "school," "college class, mathematics work," which students have developed over years of experience in such settings. When departures from these patterns occurred -- or when our work together created other differences -- such things were sometimes noticed and noted. But whether patterned or divergent, noted or omitted, these many things students were aware of comprised only a smattering of the possibilities. A vast array of things to perceive, to attend to, simply fell into the backdrop, filtered out in the process of human perception and experience. There was, however, much in the things students filtered out (or wanted to filter out) that were intended as "message" rather than "noise." For example, the issues of mathematical reasoning, justification, and argument underlying the questions I posed about how they knew they had all possible amounts were an important aspect of the content I wanted them to consider. This was also true of my question near the end of class about what it would take to convince them of the validity of Shelly’s Conjecture, as well as my decision to end class without providing them answers or any indication regarding the validity of their work. These pedagogical “moves” contained explicit and implicit messages about mathematical knowing that may have simply been filtered out as nonsensical or lost as other things clamored for attention. Indeed, it seemed in general 155 that the moves and decisions F enice and I made as their teachers were largely lost or not considered noteworthy. Such pedagogical facets of our class, however, were intended to be a focal part of what was to be attended to, analyzed, and discussed. Mary Catherine Bateson (1994) points out, Even as we compete to receive attention or struggle to know where to give it, it remains the elusive prerequisite of all thought and learning, always selective and always based on some implicit theory of relevance, of connection. Patterns of attention and inattention cluster in every setting and are packaged and pummeled into new forms in school and in the work place (p.101). "Patterns of attention and inattention, packaged and pummeled" through students' experiences in other contexts, were indeed evidenced in the students' written observations and reflections. However, it was my hope that the class -- and the individuals who comprise the class -- would develop new patterns in the weeks and months ahead. A variety of new patterns would indeed be embedded in the unfamiliar terrain: Class would always end without the instructor providing an answer, they would always be asked for proof and explanations, they would never be given a primary problem that simply involved "plugging and chugging" a number through a formula -- but what was less clear was how the students would adapt to these new patterns. What new patterns would they form? 156 CHAPTER 3 Establishment of the Unfamiliar Terrain Introduction Although the educational approach of presenting the new mainly in its new terms can involve confusion, uncertainty, and frustration as students struggle to make sense of the unfamiliar context in which they find themselves, this struggle to understand is crucial for initiating new learning. Schwab (1959/1978) describes the process that is begun as students start to respond to the new context and attempt to act within the unfamiliar terrain — As they translate their tentative understanding into action a powerful stimulus to thought and reflection is created... The actions undertaken lead to unexpected consequences, effects on teachers and students, which cry for explanation (p. 173). This chapter examines the early efforts of the prospective teachers and the instructor to shape and give meaning to various features of the class context. Such an examination involves, in part, looking closely at the pre-service teachers’ efforts to act upon their developing understanding of the class and its requirements — to “translate their tentative understanding into action” (Schwab, 1959/1978, p. 173). However, it also involves carefully studying the ways in which the unfamiliar terrain is cultivated through the instructors’ words and actions. This analysis focuses upon the beginning of the mathematics block (October 25th), seven weeks afier the events described in Chapter 2, when I took over the instruction and began working with the class to develop the routines, expectations, roles, shared language and meanings specific to the mathematics portion of the course. Through a careful study of things said and done during the first 157 CHAPTER 4 Discernment of the Unfamiliar Terrain Introduction The educational approach of presenting the new mainly in its new terms involves more than simply placing students within a strange context to which they must adapt. It must go beyond simply creating a sense of being off-balance or ill at ease that will be reduced through the development of new behaviors. Rather, it is crucial that a process of action and reflection is initiated; a process that will ultimately lead to new understanding and modes of knowing. Schwab notes (1959/1978) that as the unfamiliar terrain is encountered and actions undertaken lead to unexpected consequences... There is reflection on the disparities between ends envisaged and the consequences that actually ensue. There is reflection on the means used and the reasons for why the outcomes were as they were (p. 173). In this chapter, I look closely at this process of “reflection on the disparities,” paying particular attention to what is involved in initiating —— and productively learning from — such reflective processes. This analysis focuses upon the fifth lesson of the Mathematics Block (November 3), approximately one week afier the events discussed in Chapter 3. During this lesson, the class first began to express and examine their understandings of the mathematical work and class context. Although students had begun to adapt in noticeable ways to the unfamiliar terrain — they had, for example, become much more careful in their explanations and had begun to raise alternative approaches and representations (alongside arithmetic and algebraic manipulations) without my prompting — what students perceived and understood about our work together 226 lesson of the mathematics block, this chapter provides an image of how the unfamiliar mathematical work and context of the class began to be cultivated. It sheds light upon the importance of constraints in cultivating unfamiliar terrains, illuminating the crucial role they play in limiting the vast array of possibilities (including the possibility of working in old and familiar ways), reducing uncertainty and confusion, and opening new possibilities for working and learning. The chapter begins with a brief overview of the theoretical perspective that was drawn upon for this analysis. It then goes on to analyze the mathematics problem that was used to begin shaping the mathematical work of the course and forming important features of the unfamiliar terrain. This is then followed by a description of the lesson outlining the set up and initial work on the problem, and moving on to a close examination of the class discussion of the prospective teachers’ solutions. Finally, the chapter provides a detailed analysis of the constraints seen in the lesson and how these functioned to reduce variety and narrow the range of alternatives available to the prospective teachers as they began to “translate their tentative understanding into action” (Schwab, 1959/1978, p. 173) within the unfamiliar class context. The Primary An_alytic Lens: Constraints and Possibility An important characteristic of the cybernetic viewpoint is a fundamental interest in considering a broad array of possibilities, beyond the processes and outcomes actually observed in any given system. This interest is coupled with a tendency to question not only why things were as they were but, more importantly, why things did not work out differently —— why other possibilities did not occur. In his seminal book, An Introduction to Cybernetics, Ross Ashby (1956/1964) gives the example of how embryonic 158 development is viewed differently in cybemetics from more traditional biologic perspectives. From a classic viewpoint, a biologist might view an ovum develop into a rabbit and ask, “Why does it do this? — Why does it just not stay an ovum?” This type of questioning led to a line of study that has uncovered important insights into how energy is supplied for such transformations (p. 3). He contrasts this with the cybernetic perspective, writing — Quite different, though equally valid, is the point of view of cybemetics. It takes for granted that the ovum has abundant free energy, and that it is so delicately poised metabolically as to be, in a sense, explosive. Growth of some form there will be; cybemetics asks “why should the changes be to the rabbit-form, and not to a dog-form, a fish-form, or even to a teratoma-form? Cybernetics envisages a set of possibilities much wider than the actual, and then asks why the particular case should conform to its usual particular restriction (p. 3). Due to this interest in why some possibilities do not occur — or occur only rarely — the idea of constraints that reduce the likelihood of some occurrences plays a central role in the cybernetic perspective. Anywhere there is predictability — laws of nature, patterned behaviors, significant correlations, etc. — it is possible to gain insight into these relationships and outcomes through an examination of the constraints that are present and in operation. This analytic perspective that looks for and studies constraints and the ways in which they function was important in this study — particularly in this chapter focused upon efforts to shape and give meaning to important facets of the unfamiliar terrain. In myriad ways, this work of establishing and learning a new class context involves a variety of constraints - constraints not unlike the development of routines, expectations, shared understandings that takes place at the start of all classes. 159 In particular, this chapter focuses upon the constraints found within the materials and questions the teacher provides, the knowledge, skills, and dispositions students bring, and the pedagogical moves a teacher makes and the ways in which these combine to constrain students’ mathematical work. These are discussed in more detail below. Constraints in the Materials and the Question Constraints are always present in the mathematics materials use by students — the manipulatives, tools, resources — as well as in the questions and problems that are posed. The materials and assignment can be tightly constrained, narrowing the range of possible solution paths to a very limited set: For example, perhaps only one or two solutions might result or be considered acceptable. For example, if students are asked to use fraction bars49 to find fractions equivalent to 1/2, identifying which fraction bars have the same shading as the 1/2 fraction bar is all that is required. 1:]: ICE [EDIDII Find fractions equivalent to 1/2 ‘9 Fraction bars are a manipulative designed for teaching fractions. They consist of equal-sized rectangular strips (of paper/cardboard/plastic) marked off to represent halves, thirds, fourths, fifihs, sixths, tenths and twelfihs. Because of their design, it is quite easy to compare the areas of different fraction bars. When fraction bars arejuxtaposed, the relative-sizes of the shaded and un-shaded areas become quite visible For example, when comparing the fraction bars -- it is easy to see that the two-thirds and four-sixth fraction bars are equivalent, and the shaded region of the three~fourths fraction bar is slightly bigger than the shaded region of the other two. 160 These materials constrain many facets of fractions that one might consider in determining whether two fractions are equivalent, making them unimportant. For instance, the size of the whole, which is a crucial aspect of working with fractions, is not an issue when working with fraction bars because all the fraction bars are the same size. Also, problems associated with representing fractional quantities — such as ensuring the "parts" are the same size and attending to the number of parts that make up the whole — also become unimportant since the fraction bars are pre-made representations, rendered with factory-made precision. Many of the difficulties children with which typically struggle are obscured and rendered inconsequential. Despite the strengths and weaknesses, the understandings and misunderstandings, students bring to their work with fractions, they are likely to arrive at the same set of fraction bars in response to the problem. Constraints embedded in the task and materials narrow the realm of possible responses to a very small set.50 The combination of mathematical materials and questions can also be loosely constrained providing few limitations on the solution approaches and generating a wide range of responses. For example, students could be asked to explore equivalence using pattern blocks;5 I this task opens up many possibilities for students. They could, for 50 This example appears in Ball, D. (I992). Magical Hopes: Manipulatives and the Reform of Math Education. American Educator I6 (2), I4-18; 46—47. In describing children's work on this task, Ball notes that, "It is very hard to go wrong with these materials. Students' answers will likely be what we want: e.g., 4/8, 2/4, and so on." She argues that these materials are, "relatively rigid... (forcing) you to get the right answers." 5‘ Patterns blocks are a manipulative commonly used for teaching about patterns, symmetry, linear and area measurement, and fractions. There are six different pattem-block shapes, each shape is associated with one 9-0 m 161 example, look for relationships between the blocks, attending to the way the sides and angles of various shapes fit together. Or they might attempt to build certain shapes using other shapes: the yellow hexagon O can be made with two red trapezoids 9, or three blue rhombi 9, or six green triangles .. And perhaps students might go on to consider the relationships among the areas of these shapes -- e.g., 1 red trapezoid = 1/2 a yellow hexagon; 3 green triangles = 1 red trapezoid. Or they may create pictures with the pattern blocks: 0 (house), 0% (duck). They might even smell and taste the pattern blocks, try to break or cut them into smaller pieces, see how they react to heat, test their buoyancy in the toilet, or see how many green triangles can be stuffed into Sam's mouth. There are, after all, innumerable ways to "explore" using pattern blocks to an unfettered mind. And of course an infinite spectrum of possibility lies between "tightly" and "loosely" constrained; questions and materials can be constrained in many ways to varying degrees. For our first mathematics work together in the "mathematics block" of the course I chose a problem called the Cookie Jar Problem: The shapes have some interesting features worth noting. First, all the sides of all the shapes are the same length -- with the exception of the base of the red trapezoid which is twice the length of the other sides. Additionally, four of the six shapes —- the yellow hexagon, the red trapezoid, the blue rhombus, and the green triangle -- have interior angles that are a multiples of 60 degrees. This combination of features allows many of the shapes to be constructed from other shapes. It also allows the shapes to "fit together" in many different ways. 162 There was a cookie jar on the table. First, Chris came along. Chris was very hungry because he hadn't had any lunch so he ate half the cookies in the jar. Lynn came along next. She had missed dessert so she ate one-third of what was left. At that point, Kim came along and ate three-fourths of the remaining cookies. Then Elaine came along and took one cookie to munch on. If one cookie was left at that point, how many cookies were there in the jar to begin with? The Cookie Jar Problem was one of the more constrained mathematics problems used in this class. This is not to say the problem was simple or unproductive. It is also not meant to imply there was only one way to correctly solve the problem. On the contrary, it was a rich problem with quite a few viable solution paths. There were, however, various facets of the problem that worked together to effectively restrict the ways people progressed through the problem and the answers at which they arrived. Although at the time I only had the vaguest sense for how or why the problem worked as it did, I knew it tended to produce the interesting effect of landing people who are facile with arithmetical manipulation but who think little about what the manipulations mean, at an incorrect answer. It also had the strange effect of landing people who are less good with manipulating abstractions but are inclined think carefully about the concrete situation (often people who have not done well in math classes), at the correct answer. ’2 This problem was originally developed by Deborah Ball to be used with her elementary mathematics methods classes. Only the names of the characters in the story have been changed to match the names of students in my class. 163 There were, of course, exceptions to this. But the results were quite consistent across time and groups of students. The presence of trends, patterns, predictability indicates the presence of constraints reducing the full range of infinite, random possibility into observed recurrences and tendencies”. In the next section, Problem Analysis: The Cookie Jar Problem, I examine some of the different paths students have taken as they worked through this problem (as well as some hypothesized paths where no travelers have yet been spotted). By attending to where people do and do not go, we can gain a better sense for various constraints in operation — constraints that bring certain mathematical issues to the forefront and make others inconsequential. By analyzing the solution paths taken and avoided, we can make inferences about where conceptual leaps are required, where fragile understandings tend to trip people, where the going is smooth and little challenge exists. Some paths require less knowledge and mathematical savvy; others (often the ’ shortest, most elegant paths) require more. Constraints Students Bringllmpose In addition to the constraints embedded within the materials and questions, there are also constraints that reside within the expectations and beliefs, the habits and understandings, students bring with them. Again, this is not to say students' beliefs and 5’ Ross Ashby (I956/ 1964) made a similar claim in his discussion of constraints. He explained, That something is "predictable" implies that there exists a constraint. If an aircraft, for instance, were able to move, second by second, from any point in the sky to any other point, then the best anti-aircraft prediction would be helpless and useless. The latter can give useful information only because an aircraft cannot so move, but must move subject to several constraints. There is that due to continuity-- an aircraft cannot suddenly jump, either in position or speed or direction. There is the constraint due to the aircraft's individuality of design, which makes this aircrafl behave like an A-lO and that one behave like a Z-20. There is the constraint due to the pilot's individuality; and so on. An aircraft's future position is thus always somewhat constrained, and it is to just this extent that a predictor can be useful (p.132). 164 understandings are simplistic or limited. They do, however, serve to limit the realm of possibilities for interpreting the task and approaching a solution. For example, when working on the Three-Coin Problem, a number of students interpreted the task as primarily being an assignment to correctly recall and properly use an unstated formula. Previous experiences with "this kind of problem" in their MTH 201 and 202 courses narrowed the efforts of many students' to finding some formula they had learned in their earlier studies of probability. Additionally, many of them undoubtedly had a long history of mathematics work that required the accurate recall and use of formulas. This interpretation of the Three-Coin Problem constrained their problem-solving efforts to "testing out" various formulas constructed from vague recollections and information provided in the Three-Coin Problem. Through trial and error, students slowly eliminated formulas that did not work. Another, different, interpretation of the Three-Coin Problem framed it as an exercise in finding and developing a systematic method. This perspective tended to focus students' efforts on creating clear, patterned ways of producing and recording all combinations. The Three-Coin Problem was also interpreted, literally, as finding various amounts that could be pulled. This understanding of the problem directed students' energies toward making sure each and every amount they considered was indeed possible given the problem parameters, and carefully checking to ensure there were no duplicate amounts. A single mathematics task can be many things to many different people. Much depends upon how the problem is perceived and interpreted by the learner. A host of assumptions and values students bring about mathematics and mathematics classes also constrain their interpretations of their work and the solution 165 paths that are available. These include beliefs about what counts as mathematical work. For example, Pattern Block experiments involving taste and smell, heat and water, or green triangles in Sam's mouth, are likely to be considered inappropriate ways to "explore" with Patterns Blocks in many math classes. Past experience has also developed tacit sensibilities about the features of better and worse solutions, beliefs about the criteria for justification and validity, a sense of when work on a mathematics problem is "completed," knowledge about the nature of mathematical tasks and the form responses should take. Such values and assumptions can prevent students from settling on the first representation they come up with —- as it did with Lucy, the third grader whose first drawing of 4/2 depicted two circles which neither showed "four" nor "halves" (Chapter 1). Or it can constrain further exploration of a problem — as it did with students who settled on "ten" (different combinations) or "3¢, 7¢, 11¢, 12¢, 15¢, 16¢, 20¢, 21¢, and 25¢" as their answer for the Three-Coin Problem -- with the belief that the problem is solved and there is nothing more of interest or value to pursue. In the section, Problem Analysis: The Cookie Jar Problem, I begin to discuss some of the constraints prospective teachers ofien bring to the Cookie Jar Problem which interact with the constraints within the problem itself. Constraints Created Pedagogically Finally, a third important source of constraints which influence the solution paths students traverse, can be found in the pedagogical moves and decisions teachers make as the lesson unfolds. With words and actions, placed at various points in time, teachers reduce the vast possibilities for what students do and don't do, encounter and circumvent, focus on and ignore, as they work through the mathematics task. 166 One way that teachers influence students' solution paths is through the use of questions and probes. For example, when students believed they were "done" working on the Three Coin Problem I asked them questions like, "Are you sure you have them all?" "How do you know?" "How could you prove it to someone else?" For many students, these probes served to rule-out the option of settling on answers arrived at by vague, unarticulated methods. With this option eliminated, most continued to work a bit more: Some went on to develop systematic methods to prove no other combinations were possible; others sought out formulas that would confirm their findings. Teachers also influence the solution paths students traverse with the comments they make, the instructions they give, the points they reemphasize, the ideas and questions they highlight. In the many moves and decisions teachers make, they constrict the realm of possibilities for how students can interpret the work and move towards a solution. In the section, Episode Description: Translation of Tentative Understanding Into Action, I describe various moves and decisions I made as I orchestrated the class discussion of the Cookie Jar Problem. I then go on in the section, Episode Analysis: Constraints in the Unfamiliar Terrain, to unpack various ways in which my words and actions as the instructor constrained the mathematical work we did together. Problem Analysis: The Cookie Jar Problem There are several features of the Cookie Jar Problem that must be confronted and accounted for in every solution path. One important feature is the goal -- or end point -- that must be reached in order for students to "complete" their work on the problem: Problem-solvers must figure out how many cookies there were in the jar before people began eating from it. All solution paths will eventually lead toward this unknown 167 quantity. Another feature -- or perhaps, a set of features -- can be found in the array of fractions that must be dealt with to arrive at an answer. Students must work with halves, thirds, and fourths; they must also grapple with both unit fractions (1/2, 1/3, and possibly 1/4) and non-unit fractions (3/4 and possibly 2/3 ). This variation embeds challenges that would not otherwise exist if the set of fractions were more patterned. In the Cookie Jar Problem, each "eating" is different from previous ones and, in many solution paths, must be treated a little differently.54 This would not be the case if 1/2 of the cookies Were eaten every time or if a unit fraction (1/2 ,1/3 ,1/4 , etc.) were consumed by every person -- this would make it much easier to fall into a correct pattern, a repeated set of steps, to account for the amount of cookies consumed by each person. There are other ways this 5“ As a contrasting alternative, consider the Three Hungry Monsters problem (Nicol, C., 1997) Three tired and hungry monsters went to sleep with a bag of cookies. One monster woke up, ate l/3 of the cookies, then went back to sleep. Later a second monster woke up and ate l/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? In this problem, one-third is used at each step. The idea that each monster eats one-third of the cookies but a difierent number of cookies is eaten each time may be somewhat misleading (particularly for children), however, beyond this particular facet of the problem it is, in many ways, an easier problem to solve than the Cookie Jar Problem. The exclusive use of a unit fraction avoids some of the complexity that is introduced with non-unit fractions. Also, the repeated use of one-third allows for patterned ways of working on the problem. Another interesting problem that is patterned differently than the Three Hungry Monsters problem, is the Mangoes problem: One night the King couldn't sleep, so he went down into the Royal kitchen, where he found a bowl full of mangoes. Being hungry, he took 1/6 of the mangoes. Later that same night, the Queen was hungry and couldn't sleep. She, too, found the mangoes and took US of what the King had left. Still later, the first Prince awoke, went to the kitchen and ate l/4 of the remaining mangoes. Even later, his brother, the second Prince, ate 1/3 of what was then left. Finally, the third Prince ate 1/2 of what was lefi, leaving only three mangoes for the servants. How many mangoes were originally in the bowl? (Stonewater, 1., 1994) Here, everyone eats the same quantity of mangos (i.e. three) although the fraction eaten changes incrementally at each "eating." A pattern can be arrived at readily and used to solve the problem (and make predictions). 168 collection of fractions provides varying degrees of challenge and differing forms of influence that are discussed more below but, generally speaking, these fractions offer a nice set of obstacles for adults. These features are written into the problem itself and constrain all solution paths. There are, however, an assortment of features that only come into stark relief as they are encountered by problem-solvers on a particular solution path, serving as obstructions or footholds. In this section, I discuss some of these features. Top-to-Bottom Paths and Bottom-to-Tg) Paths Solution paths for the Cookie Jar Problem begin at either the "top" of the problem with a cookie jar containing an unknown quantity of cookies, or at the "bottom" of the problem with a single cookie remaining in the jar. And, because of the way quantities "eaten" are expressed relative to previous amounts eaten (e.g. 1/3 of what was left fiom before, 3/4 of the remaining cookies) one must work in seriatim, progressing through the problem from top-to-bottom, or from bottom-to-top. This is a tightly constrained aspect of the Cookie Jar Problem and all solution paths run in one of these two ways. I Solutions to the Cookie Jar Problem I /\ I Top-to—Bottom Solutions I I Bottom-to-Top Solutions 169 Top-to-bottom routes begin with, and include in each step of the solution, the unknown quantity of cookies that originally filled the jar: "There was a cookie jar on the table. . ." In one way or another, the unknown must be represented and accounted for with each "eating." So, for example, one might start by designating "x" to represent the original amount of cookies in the jar: Let x = the amount of cookies in the jar at the start. The problem-solver would then proceed through the problem using x: 1/2 x = the amount Chris ate. And so on. One challenge of top-to-bottom solutions lies in this necessity to work with an unknown at each step. There is a quality of abstractness in working with such unknowns -- where "x" is not a specific quantity of cookies that can be counted -- that people sometimes find uncomfortable and difficult. On the other hand, there are others who have developed robust habits for setting up representations involving unknown quantities and solving for the unknowns; top-to-bottom solution paths are likely to seem natural and familiar to such people. Bottom-to-top routes, on the other hand, allow for a specific number of cookies to be considered at each step of the solution. These solution paths begin with a specific number of cookies: "One cookie was left. . . " And the information provided in the problem allows one to proceed upward, solving for the specific amount of cookies in the jar before the arrival of each cookie-eater. For example, knowing that there was one cookie left in the jar in the end and that "Elaine came along and took one cookie to munch on" just prior, one can deduce that there were two cookies in the jar before Elaine's arrival. In so far as it is easier to work with specific numbers rather than an unknown, bottom-to-top routes provide some alleviation to problem-solvers. Bottom-to- 170 top paths offer good alternatives to those who find working with unknowns awkward or challenging.55 Paths to Answers of 16 and Paths to Answers of 24 The two most common answers people arrive at when solving the Cookie Jar Problem are sixteen and twenty-four. Although there are exceptions -- calculation errors made along the way, inaccuracies when switching to decimals and rounding up or down, etc. -- most students come to either sixteen or twenty-four at the end of their solution path. There is an important distinction that must be made in solving the Cookie Jar Problem between the cookies that are eaten and the cookies that are left. The fractional quantities given in the problem -- one half, one-third, and three-fourths -- refer to quantities that have been eaten. However, the amount eaten is expressed relative to what was left in the cookie jar from the prior "eating." There is a subtle shift here that must be taken into account: The amount eaten must be converted into an amount remaining after 5’ It is, of course, possible to embark on a bottom-to-top solution path that, while providing the opportunity to calculate a specific amount at each step, does not actually involve doing these calculations. For example, working from the bottom-to-top one might do the following -- Let x = the amount left in the jar before Elaine eats one cookie so, x - l = I Let y = the amount lefi in the jar before Kim eats 3/4 of the remaining cookies so, x = (1/4) y and, ((1/4) y)- 1 =1 Let 2 = the amount left in the jar before Lynn eats 1/3 of the remaining cookies so, y = (2/3) 2 and, ((1/4) (2/3) 2) - l = 1 Let q = the amount left in the jar before Chris eats 1/2 of the cookies in the jar so, 2 = (1/2) q and, ((l/4) (2/3) (1/2) q) - l = 1 (2/24) q = 2 q = 24 cookies Here, it would have been possible to solve for x (i.e. x = 2), y (i.e. y =8), and z (i.e. z = l2) along the way and work with these specific numbers and shorter equations than those found above. 171 each eating. A failure to do so will result in nonsensical calculations of "cookies eaten" from a quantity of already-consumed cookies (i.e. rather than from cookies that were left behind.) The answer of sixteen is found by failing to make such a distinction. Oflen, people simply do not realize there is a distinction to be made; instead, they merely work with the numbers they are given, concentrating on trying to select and perform the "correct" mathematical operations without questioning what those numbers and operations mean in the concrete context of the story. Sometimes, people do realize there is an important distinction to be made but have trouble representing or carrying out that distinction throughout their entire solution. For example, one possible solution might start off like this: There was a cookie jar on the table. Let x = the amount of cookies originally in the jar First, Chris came along. Chris was very hungry because he hadn't had any lunch so he ate half the cookies in the jar. 1/2 x = the amount in the jar after Chris ate The amount that Chris ate as well the amount that was left after Chris ate can both be represented as 1/2 x. "One half," more than any other fraction, obscures the distinction between the amount of cookies eaten and the amount of cookies left; they are, after all, the same quantity. Here, 1/2 x is labeled as "the amount in the jar after Chris ate" (i.e. the amount left); it seems a distinction is (possibly) being made. In these initial steps, it doesn't matter if the amount remaining is or is not calculated from the information given. In subsequent steps, however, it does matter. 172 Lynn came along next. She had missed dessert so she ate one-third of what was left. 1/3 (1/2 x) = the amount in the jar after Lynn ate 1/3 (1/2 x) is actually not the amount left in the cookie jar after Lynn ate. Rather, it is the amount that Lynn ate. An additional calculation would be required to convert the amount eaten into the amount remaining. If Lynn had eaten 1/3 (1/2 x), then she must have left 2/3 (1/2 x) in the jar for the next person to eat. However, without seeing that a distinction must be made and accounted for, this solution path would continue on with compounding errors of this sort to arrive at an answer of sixteen: At that point. Kim came along and ate three-fourths of the remaining cookies. 3/4 (1/3)(1/2 x) = the amount in the jar after Kim ate Then Elaine came along and took one cookie to munch on. (3/4 (l/3)(1/2 x)) - l = the amount in the jar after Elaine ate If one cookie was left at that point, how many cookies were there in the jar to begin with? Since, the amount in the jar after Elaine ate = 1, then (3/4 (1/3)(1/2 x)) - 1 = 1 Solving for x, we get: (3/4 (1/3) (1/2 x)) = 2 1/3 (1/2 x) = 8/3 1/2 x = 8 x = 16 cookies originally in the jar On the other hand, those who arrive at twenty-four cookies as their answer have usually been successful at distinguishing between what was eaten and what was left throughout their entire solution. So, for example, one such solution path might start off like the one above however rather than mistaking 1/3 (1/2 x) for the amount left in the jar 173 after Lynn had eaten, 1/3 (1/2 x) would correctly be identified as the amount Lynn had eaten and, subsequently, 2/3 (1/2 x) = the amount left in the jar after Lynn ate. Continuing on, in like manner, this solution path would eventually arrive at an answer of twenty-four: At that point, Kim came along and ate three-fourths of the remaining cookies. 3/4 (2/3) (1/2 x) = the amount Kim ate and 1/4 (2/3) (1/2 x) = the amount left in the jar after Kim ate Then Elaine came along and took one cookie to munch on. 1/4 (2/3) (1/2 x) - 1 = the amount left in the jar after Elaine ate lfone cookie was left at that point. how many cookies were there in the jar to begin with? Since, the amount left in the jar afier Elaine ate = 1, then (1/4 (2/3) (1/2 x)) - 1 =1 Solving for x, we get: 1/4 (2/3) (1/2 x) = 2 2/3 (1/2 x) = 8 l/2 x = 12 x = 24 cookies originally in the jar This crucial distinction between the cookies left and the cookies eaten plays out consistently across solutions paths; it separates the problem-solvers, taking those who successfully make the distinction to an answer of twenty-four and taking those who do not to an answer of sixteen. 174 I Solutions to the Cookie Jar Problem I /\ I Top-to-Bottom Solutions I I Bottom—to-Top Solutions I Answer of 16 ' Answer of 24 . Answer of 16 . , Answer of 24 Paths Usin Pictures and Paths Usin Arithmetic/Al ebraic E uations Many solution paths involve the use of drawings56 to represent and calculate the cookies being eaten and left behind. Pictures are used, in these solutions, to reenact the steps of the story. So, for example, one solution path involving pictures might begin with a drawing of a "whole" cookie jar containing some unknown quantity of cookies: There was a cookie jar on the table. cookie jar With each fractional "eating" a corresponding quantity would be "removed" -- perhaps by redrawing the cookie jar with the appropriate amount missing or by shading in the quantity that has been consumed. First. Chris came along. Chris was very hungry because he hadn't had any lunch so he ate half the cookies in the jar. 56 Objects are sometimes used instead of drawings. For example, rather than drawing lines and shading pieces to represent the removal of cookies from the cookiejar, a sheet of paper might be cut with scissors to show the same thing. Although I only refer to "pictures" in the following text, the argument could 175 The amount eaten (the shaded region) can then be ignored and all attention focused only on dividing and removing some quantity from the un-shaded, remaining cookies. The un-shaded region becomes the new "whole" from which the next person eats. The drawing itself provides strong visual cues -- supports -- for distinguishing between and accounting for the cookies eaten and remaining. Lynn came along next. She had missed dessert so she ate one-third of what was left. At that point, Kim came along and ate three-fourths of the remaining cookies. Then Elaine came along and took one cookie to munch on. There was one left at that point. . . one left Elaine Drawings such as these can help to constrain students' attention in helpful ways, diminishing the confusion that surrounds the eating and leaving of cookies. The drawing contains information about the relative amounts that were eaten (e.g. Lynn ate less than Kim, Chris ate more than Lynn and Kim put together, etc.) and visual guides concerning the relationship between the quantity of cookies eaten and the amount available for eating (e.g. two cookies is one-fourth of the amount Kim had available when she came upon the cookie jar). All that remains on this particular solution path is to calculate the various amounts Kim, Lynn, and Chris ate. (However, as we shall see in the section Episode Description: Translation of Tentative Understanding Into Action, the remaining stretch on this solution path can be quite non-trivial for people.) There are also solution paths involving only equations. These equation-only solutions lack the constraints, described above, that accompany the use of drawings. As a result, there are more possibilities for going astray -- for missing an important consideration or bypassing a crucial distinction in the problem context. This makes these equation-only paths more challenging in some ways. Equations have the power and drawback of transforming a story about cookies and consumptive actions to a much more abstract representation devoid of specific objects and human action. There is power in the abstractions to move beyond the particular case; also, abstractions often allow for 177 patterns and new insights to surface. However abstractions can also strip away many of the details of the situation, making important distinctions and subtleties easy to overlook. Nevertheless, despite the constraints drawings provide, students can still lose their way en route to a correct answer of twenty-four. None of these constraints are so binding or so vital that they guarantee success. Rather, they merely serve to make some things easier to see than others. 178 < _lmcozaom nobouéobom; VM\ 7:328 Eofioméfifl _ < E0595 ..2. 2x80 wfi B mcozaom _ 179 Picture Paths Using Sets of Objects and Paths Using a Single Whole Among the various solution paths involving the use of drawings, a number of them utilize drawings of a single "whole," like the example above. Almost without exception, single wholes are used in top-to-bottom routes. With a single whole, it doesn't matter how many cookies make up the whole -- it is a simple thing to divide the single object into fractional pieces. The single whole, like "x," serves as an abstraction for the unknown quantity: Any amount could be contained within the circle (or rectangle, or other cookie-j ar-shaped whole). This fits well with top-to-bottom routes that begin with, and include in each step of the solution, the unknown quantity of cookies that originally filled the jar. There are also solution paths that use sets of objects to represent "wholes," rather than single objects. These solutions are almost exclusively bottom-to—top routes. Using a set of objects for the cookie jar problem is, in many ways, more "true" to the story context and allows for more concrete consideration of the problem. There are, for example, a number of "wholes" in this story. There are whole individual cookies, there is a whole cookie jar (containing the original set of cookies), and there are various wholes across each step of the problem as some fraction (of some whole) is eaten. Sets of objects allow for all these different wholes to be accounted for pictorially. The objects can be grouped and regrouped in multiple and varied ways. So, for example, such a solution path may begin in the following way: (There was) one cookie...left. llefl: O 180 Here, the single circle represents a single, whole cookie. In the next step, the single cookie becomes one, in a set of two. Elaine came along and took one cookie to munch on Elaine's 1 left Although the single circle continues to represent a single whole cookie, the set has doubled in size and the single circle is now one-half of the set. The solution path continues on this way with the set-size changing and the circles representing a different proportion of the set. At that point. Kim came along and ate three-fourths of the remaining cookies. 3/4 that Kristin ate 1/4 left '00 GO DO1 '06 Here, two circles represent one-fourth of the cookies in the large set (i.e. the amount left after Kim ate). Three groups-of-two-circles, represent the three -fourths Kim ate.57 ’7 The remaining stretch of this solution path might look something like this: Lynn came along next. She had missed dessert so she ate one-third of what was left. 1/3 that Lisa ate 2/3 left @000 '00” 00001 Here, the eight cookies that were in the jar prior to Kim's arrival are regrouped into two equal groups of four to represent the two -thirds that were left after Lynn's departure. Since Lynn ate one -third of what was in the jar when she arrived, she must have eaten only four cookies. And, as the picture shows, there must have been twelve cookies in the jar before Lynn dipped in. Chris came along. Chris was very hungry because he hadn't had any lunch so he ate half the cookies in thejar 181 Bottom-to-top solutions allow for a specific number of cookies to be considered at each step of the solution; using sets of objects, lends further support to this. These drawings of sets provide visual cues that aid in calculating the quantity of cookies eaten and left; they also require the number of cookies to be calculated at each step of the solution. Without this computation, it is difficult -- if not impossible -- to proceed to the next drawing. 1/2 that Cory ate 'OOOOOOOOOOOO' 00000000009394 1/2 left The twelve cookies left in the jar after Chris ate (and before Lynn came along) are redrawn here in white to represent one half -- the half that Chris had left behind. Therefore, as can be seen in the picture, there must have been twenty-four cookies in the jar to begin with. 182 E mcozaom nobouéofiom _ _ mcouaom Eofiombuaohl— _ 828...”. .5 a_xoou 9: S 20:28 _ 183 Equation Paths Using a Series of Equations and Paths Using Single Equations Among the various solution paths involving only equations, many utilize a succession of short equations. A separate equation is set up for each step of the solution and where ever possible, whenever possible, specific values and reduced expressions are substituted for unknowns. This allows the problem-solver to simply use the results of the previous step into the current one and ignore the rest of their previous work. In contrast to this, there are also equation-only solution paths involving single, "long, " equations that embed information about every step of the solution. As the problem—solver progresses through the solution path, more and more is accounted for in the single equation: The equation grows. There are a number of challenges involved in representing the entire Cookie Jar Problem in one equation that are otherwise effectively constrained in solution paths that utilize a series of short equations. For example, consider the following: Solution path involving a series of short Solution path involving a single long equations equation There was a cookie jar on the table Chris came along. Chris was very hungry because he hadn't had any lunch so he ate half the cookies in the jar Let x = the amount of cookies originally in Let x = the amount of cookies originally in the jar the jar Let C = the amount left after Chris ate The amount left after Chris ate = C=x-°l/2x x-1/2x C = 1/2 x The two solution paths are quite similar to start with but this changes with the next step. The solution path involving a series of short equations continues on to introduce and solve for new unknowns with each step; the solution path involving a 184 single equation continues on with its single unknown, x, to encompass more of the Cookie Jar Problem with every step. Lynn came along next. She had missed dessert so she ate one-third of what was left Let L = the amount left after Lisa ate The amount left after Lisa ate = L=C-1/3C (x-l/2x)-l/3(x-1/2x) L = 2/3 C L = 2/3 (1/2 x) L=2/6xorL=l/3x Although the two solution paths perform essentially the same operation -- i.e. subtracting one-third of the amount left by Chris, from the total left by Chris -- the equations on the left are "simpler." Not only does this case the demands for solving any particular calculation encountered on this path, it also allows for methodological patterns to surface more easily: The step above can serve as a template for setting up and solving the next step. Although this is possible to do with the longer, more complicated equation, it tends to be less easy to see a pattern within the lengthening string of numbers and symbols. Kim came along and ate three—fourths ofthe remaining cookies Let K = the amount left after Kim ate The amount left after Kim ate = K=L-3/4L [(x-l/2x)-1/3(x-l/2x)]— K =1/4 L l/4 [(x -1/2 x) - 1/3 (x -1/2 x)] K=1/4(1/3 x) K: l/12x All that remains on these solution paths is to account for the last two cookies -- the cookie Elaine ate and the cookie left in the jar at the end -- and solve for x. Most adult students find the remaining calculations on the left quite trivial and easily arrive at an 185 answer of twenty-four. The remaining calculations on the right, however, present many with a tedious and time-consuming problem that some find quite daunting.58 58 Although it is certainly possible (and likely) for students to simplify the single, long equation in different ways, at different points, the challenges of working with an equation that attends to and embeds information about all the acts of cookie-consumption remain (but, perhaps to a lesser degree). 186 fl mcozaom monorEofiom _ _ mcozaom Eofiomérqoh _ < E285 ..2 9x08 9: B mconaoml— 187 Summqry There are other kinds of solution paths, not represented here. For example, some students use a "guess'n'check" approach where they choose a number -- twelve, sixteen, twenty-four, etc. -- and "plug it in" to the Cookie Jar story. If there is one cookie remaining at the end, as described in the story, they conclude they've found their answer; if there isn't, they eliminate that number and try another.59 There are also a myriad of possible solution paths that run between the trails I've laid out here: there are picture paths that join with equation paths, paths involving sets of objects that merge with single objects, paths that weave between a single equation and a sequence of equations. In many ways, the variety of possible solution paths is less like a neatly branching tree diagram than a wildly flowing stream with rivulets of water flowing along infinitely many plausible courses. And, while it may be impossible to predict the path of single water molecules -- or individual students -- much can be said about how and where they are likely to go and not go. Features of the landscape make some course ways more "main stream" than others. 59 Although this solution involves a "top-to-bottom" path, it works with specific numbers (arrived at through some kind of "guessing") rather than unknown quantities, such as x or (circle). In this way, it is different from other top-to-bottom routes. 188 189 For example, the wording of the story constrains "all" solutions to working, in seriatim, from top-to-bottom or from bottom-to-top. Like the high banks of a stream, this feature makes it difficult -- perhaps impossible -- under normal circumstances to not eventually progress in one of these two ways. Similarly, the necessity for distinguishing between cookies eaten and cookies remaining creates an unavoidable obstruction. It is like a steep or high mountain that forces most routes to circle around to the left or the right: this feature separates the solutions which lead to sixteen (for those who fail to make the distinction) from the routes that lead to twenty-four (for those who successfully make the distinction). There are also many challenges along the way that pose other obstacles. Sometimes these can be so daunting they compel people to abandon their present course and go in search of alternatives: certain calculations can be especially difficult to do, some equations can be hard to set up, particular steps can be tough to reason through. Indeed, some routes are rarely traveled all the way to a final solution: The grayed-out paths in the tree diagrams above indicate such roads-less-traveled.60 50 However, even within the non-grayed branches of the tree diagram, there are some paths that are less- traveled. For example, one very elegant top-to-bottom path, leading to 24, and utilizing a single equation goes something like this -- Let x = the amount of cookies originally in the jar Chris came along. Chris was very hungry because he hadn't had any lunch so he ate half the cookies in thejar The amount left after Chris ate = l/2 x Lynn came along next. She had missed dessert so she ate one-third of what was left. The amount left after Lynn ate = 2/3 (1/2 x) Kim came along and ate three-fourths ofthe remaining cookies. The amount left after Kim ate = 1/4 (2/3 (1/2 x) Elaine came along and took one cookie to munch on. The amount left after Elaine ate = l/4 (2/3 (1/2 x) - l lfone cookie was left at that point, how many cookies were there in the jar to begin with? 190 Features of the problem give texture to the landscape. However, whether these features are encountered as mountains or molehills depends a great deal on what students bring. Working with unknowns, as is required of top-to-bottom solutions, may seem "natural" to some but overwhelming to others. Using drawings may seem awkward and inappropriate (not really "mathematical," or perhaps "childish") to certain people but comfortable and "normal " to others. In a similar manner, conventions for representing fractions, well-developed heuristics for solving story problems, doubts about one's own ability to "do" certain kinds of mathematics -- and many other things of this sort -- contribute to the choices problem-solvers make as they wend their way toward an answer. Students' understandings and skills, their habits and assumptions, make some paths viable and inviting while rendering others inappropriate or intimidating. Possibilities for Teacher Education There are a number of ways the Cookie Jar Problem seemed to be suited for our early mathematics work together. First, it is a problem likely to provide genuine challenge around important mathematics for adult-students. Although the prospective teachers may have learned to set up equations and solve for x during their schooling, the 1/4 (2/3 (1/2 x) -1 =1 By solving for x, we arrive at: 1/4 (2/3 (1/2 x) = 2 2/24 x = 2 x = 24 cookies This solution path requires the problem-solver immediately apprehend things about the problem situation that are not always apparent to prospective teachers. First, a distinction between cookies eaten and cookies left needs to be made. Second, the values given in the problem must be transformed from cookies eaten to cookies left. This can be done by subtracting the fraction of cookies eaten from whole amount of cookies available for eating. Although this "boils down to" what is done in the example, "Solution path involving a single long equation," it is more elegant in that it results in a simpler -- simpler to write, simpler to solve -- equation. As is often the case with elegant solution paths, the mathematical knowledge and connections required can make it a less-traveled path. 191 Cookie Jar Problem demands extra careful attention to the representations --pictorial, algebraic, arithmetic —- people develop as well as their connection to the story context. The meaning of the operations and fractions, as they relate to the story, matter. The Cookie Jar Problem also has the potential to focus attention on issues that are significant in the study of fractions. For example, the issue of parts (eaten and remaining) and wholes (varied and shifting) play a crucial role in this problem. And, almost without exception, the various solution paths available to students involve calculations and leaps of logical reasoning that adults find somewhat challenging. If they were to have firsthand experience with creating mathematical knowledge within an intellectual community, the prospective teachers would need opportunities to work together around problems they found truly problematic.61 Finding an answer to the Cookie Jar Problem is only one facet of this rich problem, however. The large diversity of ways people can -- and do -- solve this problem presents another challenge well-suited for teacher education: There is the invaluable opportunity to develop one's ability to access and understand other ways of thinking quite 6' Deborah Schifter and Catherine F osnot make a more extensive argument for this in their book, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform . They write: When teachers are challenged at their own level of mathematics competence, are confronted with mathematical concepts and problems they have not encountered before, they both increase their mathematical knowledge and, more importantly, experience a depth of learning that is for many of them unprecedented. Such activities allow teachers, often for the first time, to encounter mathematics as an activity of construction, of exploration, of debate, of a complex interplay of convention and necessity, rather than as a finished body of results to be accepted, accumulated, and reproduced. Experiences like these are by far the most persuasive impetus for a changed mathematics pedagogy and serve as vital touchstones of what is possible when the uncertainties and frustrations of change threaten to overwhelm teachers' best intentions. Teachers are often surprised to discover that the topics of the elementary school mathematics curriculum provide rich opportunities for challenging mathematical exploration. By digging more deeply into what, on the surface, seems like familiar territory, they develop a broader sense of the conceptual issues their students confront. (Schifter, D., & Fosnot, C., 1993, p.16-17) 192 different from one's own. Regardless of whether one is a teacher trying to understand a child's way of thinking or whether one is a learner of mathematics scrutinizing the idea of a peer or trying to make sense of a "new" mathematical idea, the capacity and disposition to ask generative questions and follow along while someone explains a "different" line of reasoning is enormously important. And, more than "following along," there is much to be gained -- as both teacher and learner -- by being able to use and value a variety of ideas and ways of reasoning about any given mathematical situation. Finally, and perhaps most importantly, the Cookie Jar Problem sets up a situation where students are likely to encounter unexpected results in their efforts to solve the problem. As mentioned earlier, it is often the case that people who are quite good at working with abstractions and equations -- but who do not have good ways of making sense of them -- end up at an incorrect answer of sixteen. On the other hand, those who are more comfortable using drawings and objects to reenact the story (often those who are less facile with manipulating numbers and equations), typically arrive at a correct answer of twenty-four. The Cookie Jar Problem is constrained in such a way that many "good" math students who have successfully solved story problems by ingrained heuristics and memorized keywords (for example, the word "of" in a mathematics story problem means to multiply) are often surprised to find themselves in the wrong. And, likewise, many "poor" math students who have had trouble getting down the rules for manipulating numbers and struggled with the steps for setting up equations, are surprised to find themselves among those who have successfully found the right answer. This kind of dissonance that arises "when routines and normal ways of working fail to achieve expected results" is an important part of establishing an unfamiliar terrain. It is, once 193 again, the "uncomfortable but productive" situation described by Schwab (1959/1978) describes where, The actions undertaken lead to unexpected consequences, effects on teachers and students, which cry for explanation. There is reflection on the disparities between ends envisaged and the consequences which actually ensue. There is reflection on the means used and the reasons why their outcomes were as they were (p. 173). In the following section, I describe some of the work we did on the Cookie Jar Problem in the TE401 class. In addition to the constraints embedded within the problem and the constraints students brought with them, there were also many constraints introduced as the lesson unfolded through my words and actions as the teacher. These can be seen in my description of the lesson, below ("Where did the class actually go?"). Although the constraints themselves are not analyzed or discussed until the section, Episode Analysis: Constraints in the Unfamiliar Terrain, they are marked with colored dots in the unfolding narrative. There are four different kinds of pedagogical constraints I focus on: constraints by specification (0), constraints by contrast (0), constraints by patterning (0), and a subtle form of constraint that I refer to as "punctuating the discourse" (0). Following the narrative, each of these is discussed in turn. Episode Description: Translation of Tentative Understanding into Action This was our first session devoted to doing mathematics since the start of the "math block" and I was eager to get started. I knew we would need as much class time as I could possibly carve out to work on the problem. So, after a few brief announcements, I handed out copies of the Cookie Jar Problem. Tasha read it aloud to the class and we briefly discussed what the problem was asking: How many cookies were in the jar to 194 begin with? The question seemed quite straightforward. No one had any questions so I asked them get to started. They could confer or work alone but they needed to be able to explain their solution to others. Introduction of the Whole Group Discussion After half an hour of working independently, I called them together. Kara: What I'd like you to do is try to explain your solution. If you want to come up as a group or whatever, that's fine. But try to explain. . . what your answer is, and how you thought about that. And try to work it through. Help us understand the way you went about solving it. 0 It was important that we begin, right from the start, to establish norms of talking to one another, listening carefully to each other, and striving to make ideas understandable. I didn't want this to be a class session where students merely presented their solutions to demonstrate that they could do the problems or to show others how to do the problems. I wanted us to learn to work together creating new ideas and understandings of mathematics. I took a quick survey of the various answers people had arrived at. More than half the students had gotten twenty-four as their answer and quite a few had found sixteen. In order to explore some of the complexities of this problem, I thought it important to see at least one solution that involved sixteen cookies as the answer. Several were willing to share their solutions for sixteen. I asked Kim and Deidre to start us off. Kim and Deidre's Solution for 16 Cookies Kim began by writing out, on the whiteboard, all the various amounts people in the story had eaten. 19S Chris: 1/2 Lynn: 1/3 Kim: 3/4 Elaine: l 1 left Then, pointing to the last line, she started explaining there was one cookie left at the end. "So," she reasoned, "There must have been two cookies in the jar when I (the "Kim" in the story) ate three-fourths." With Deidre's help, she continued on to describe, in a patterned way, the remaining calculations they did: What would you multiply three-fourths by to get two?62 2 + 3/4 = 2 x 4/3 =8/3 (Therefore, there were 8/3 cookies in the jar before Kim ate.) What would you multiply one-third by to get eight-thirds? 8/3 +1/3 =8/3 x 3 = 8 (Therefore, there were 8 cookies in the jar before Lynn ate.) What would you multiply one-half by to get eight? 8+l/2=8x2= 16 (Therefore, there were 16 cookies in the jar before Lynn ate.) "And that," she explained, "Is howl figured it out!" Kim immediately went on to demonstrate to us how they "put it (the sixteen) back in" to double-check it: There were 16 cookies Chris ate 1/2 of them, so there were 8 left Lynn then ate 1/3 of them, so there were 8/3 left I had 3/4 of the, so there were 2 left and then Elaine ate 1 and that leaves 1 Kim's explanation was fast and filled with many numeric manipulations which, I suspected, flew by most people. She made a rather convincing case for sixteen -- her set 62 Although they didn't write or say it, I suspect their question, "What would you multiply three-fourths by to get two?" might have been represented in their minds as Y x 3/4 = 2, where Y is the unknown they are looking for. So, if Y x 3/4 = 2, then 2 + 3/4 = Y (which is what they had written) 196 up of the equations seemed "soun ", utilizing all the numbers given in the story; all her calculations seemed correct; her answer even checked out when put back into the story. If one weren’t paying close attention to what the numbers and numeric manipulations meant in the story scenario, it would have seemed that sixteen was indeed the answer. This would (I hoped) be puzzling to those who arrived at twenty-four cookies as the answer. I wanted us to spend some time trying to understand what Kim and Deidre had done, so I opened up the discussion to the rest of the class: Kara: Could people please ask Kim some questions that you might have about her solution? 0 Kim: Okay. Kara: Why don't- you go ahead and just call on whoever you want. Kim: Megan? Rather than working to get clearer about Kim and Deidre's solution, however, Megan began to suggest a different, "simpler" way to do the problem: "If you just looked at that and figured there's two left and we know that Kim ate three-fourths. . ." I was determined to focus on Kim and Deidre's solution for a while. It seemed important that we make a serious effort to understand others' ways of thinking and the mathematical issues entailed in various solutions. I stopped Megan mid-sentence. Kara: So are you- you're proposing another way to solve it? Megan: (Yes) Kara: Why don't we hold that for a second. 0 What I'd like people to... do is to try and understand what it is that Kim and her group did and um» 0 I bet there's a lot of questions out there about like, "Where did you get that?" or "Why did you decide to divide?" or things like that. So why don't people go ahead and-- 9 Let's first try to figure out what it is that they did-- why that makes sense as something to do-- and then we can try to explore some alternative ways of working on it. I think that would be really useful; but let's try to first get inside of this one. o 197 Kim went on to call on Heidi who asked whether they were focusing on pieces eaten or on what was left. It was an excellent question. Those who arrived at sixteen as an answer typically did so by failing to make this very distinction. In response to Heidi's question, Kim walked us through her "double-check" again, starting with sixteen cookies and ending with one cookie remaining in the jar. Heidi continued to press them, " Eight- thirds would be how many she ate, not how many are left." But it seemed that Kim and Deidre were not hearing Heidi and it also began to feel as if the class discussion was deteriorating into many small side conversations. I wanted to get the whole class to think about what Kim and Deidre had done and about this distinction of cookies eaten vs. cookies left in the jar. I stepped in addressed the whole class: Kara: Let's try to walk through this step by step. . . . (to Kim and Deidre) maybe use pictures or something to try to walk us through step by step?. .. Take us through who's eating what? 00 Deidre: Wait, what are you asking? Kara: Could you walk us through each step of your thing and try to maybe use a picture or something to show us a little bit clearer about who's eating what... 0 Deidre asked that I go on to someone else while she worked it out. Tess and Megan immediately volunteered. I had put Megan on hold earlier when she wanted to show Kim a "simpler" way to solve the problem and she was eager to show what they did. Tessfi and Megan's Solution for 24 Cookies While Tess and Megan prepared to present their solution, I turned my attention to the other students, many of whom also had twenty-four as their answer. 198 Kara: ...Could the other people with twenty-four. .. watch really carefully and try to figure out how the way they (Tess and Megan) did it might be different than the way you did it because I'm... interested in the different approaches you might have used to arrive at the same answer. Could you pay really close attention and see if you can draw some distinctions between what they did and what you did? 0 I wanted to encourage everyone to be a part of this conversation, regardless of whether or not they had the same answer. Learning to access and follow others' reasoning, noticing subtleties of various solutions, and accounting for differences seemed at least as important as arriving at the "right" answer. Tess started explaining their solution, which began much like Kim and Deidre's: There was one cookie left at the end, and Elaine ate one cookie, so there were two cookies. 1 cookie left 1 cookie to Elaine these 2 0001063 = 1/4 She continued on, "We know that (the two cookies) equals one-fourth of what is remaining so the other three-fourths. . ." "Wait," interrupted Megan, "Tell them why. .." Tess started again, "Oh, because Kim had three-fourths. .. we know that this (the two cookies) is going to be the one-fourth that is left after she had the three-fourths. Does that make sense? We can draw pictures too." Megan and Tess seemed to be moving beyond merely "presenting" their solution. They appeared to be sensitive to the fact that they were trying to make their solution understandable, accessible, to other people. I didn't have to ask Tess "why" -- Megan prompted her. I didn't have to ask if other people were following them --Tess asked them herself. They even offered to draw pictures. 199 Megan continued on with their explanation: "If Kim ate three-fourths and we know there were two left, then the two would have to be the one-fourth that was left at that point. So the two, equals one-fourth. Three-fourths would equal—" "Six," said Tess, completing the sentence. They were making an important and complicated point. I wasn't sure everyone was following their reasoning so I pressed them on this. Kara : Could you prove that to us? That if two equals one-fourths, three-fourths equals six? 9 "We started off with a cookie jar, Okay?" Tess began, drawing a cylindrical cookie jar on the whiteboard. "And you have your one that's left. And then you have the one that Elaine had... And this (the two cookies) is one-fourth because you know that she had three-fourths. That, um, Kim had three-fourths." Tess then drew equal four segments in the cookie jar and labeled three of them as the "3/4 that Kim ate" and the remaining one as the "1/4 left after Kim." A V 2 cookies left W 3/4 that Kim ate after Kim ate V \V 1/4 left after Kim She continued, "You know that one-fourth is going to equal two. Okay? The next one-fourth's going to be two. The next one-fourth. The next one-fourth. They're all gOing to be two." 200 Megan jumped in saying, "So it's eight all together." Tess echoed this, "So it's eight all together. She (Kim) had six... Okay?" As she completed her explanation, I once again prompted the class -- Kara : Any questions about that? 0 Rachel and Beth both seemed to want clarification, some help in understanding what was being presented, but Megan and Tess were eager to finish what they had started. Megan seemed confident that if they could just finish it up, she had a different way she could explain it that would make a lot more sense. So Megan quickly walked through the rest of their solution: "Before Kim, there were eight. And so, Lynn ate one-third. .. So we know before Lynn ate her one-third there would have been twelve. 80,. .. if we had twelve left, then we had Chris eat half, there would have been twenty-four to start with." I was quite sure that few, if any, could follow that hasty explanation. But Megan went back and, like Kim had done earlier, she plugged twenty-four into the story and demonstrated that it indeed worked out: There were 24 cookies Chris ate 1/2 of that, and there were 12 cookies left Lynn ate 1/3 of that (which is 4 cookies) and there were 8 cookies left Kim ate 3/4 of that (which is 6 cookies) and there were 2 cookies left Elaine ate 1 cookie, and there was 1 cookie left It was a compelling argument for twenty-four as the answer but I was still not satisfied with this as an explanation for how they knew there were twenty-four cookies in the jar to start with. Kara : I think when you're working it backwards it-- I kind of follow how it might check out but I'm not quite clear about how you got that (the twenty-four). 00 201 Megan asked if there were other questions and many hands went up in the air. This was rather uncharacteristic for so many people to want to comment on a solution so I asked, "Do people have questions for them? Or what --" And all the hands went down. I probed further, "Okay, so you have another way to show it?" And all the hand went back up. Among the many hands that were raised was Teri's. She had been raising her hand for quite a while and I was pleased she wanted to share something. She had commented to me more than once about how weak she is in mathematics and how she struggles with the subject. She said she had something to show that was related to what Tess and Megan had shared and would make what they were saying clearer. Teri's Guess'n'Check Solution for 24 Cookies Teri went up to the front of the room and explained to the class how she thought about those numbers "by the bottom (that) you divide by" (i.e. the denominators of ll; IQ, and 3/1) and came up with twenty-four. "'Cause like four goes into twenty-four, three goes into twenty-four, and two goes into twenty-four." A couple of students asked her why she didn't try twelve and Teri explained that she had tried sixteen early on and realized that "sixteen wasn't big enough." "So," she reasoned, "I needed a bigger one than sixteen." She continued on, showing the class how she drew twenty-four blocks to represent the cookies in the jar. After she drew out her blocks, she paused and said, "There are twenty-four cookies. Each block is one. Did everybody get that?" She glanced around the class and, not seeing any questions, she went on, "And then Chris ate half (she crossed out 12 of the blocks). So then we have twelve left." 202 use "And then Lynn ate a third of what's left... a third of twelve is four. So these are plans "And then Kim ate three-fourth, right? Am I doing this right? Okay, so three- gone (she crossed out 4)" fourths of eight. There is six so «well, (she somewhat haphazardly crossed out some squares). Do you guys understand that? Six?" Tammy jumped in and spoke directly to Teri about how she might revise her "No," said Chris immediately. picture. Working together, Teri and Tammy developed a revised representation: "Does that make sense, Chris?" Teri asked. With Chris' confusion cleared up, she proceeded, " And then there's two blocks left. And Elaine ate one. And then there's one left in the cookie jar. . ." flu When she was done drawing, Teri turned to face the class. I asked them if anyone had any comments or questions for her. Heidi commented, "I liked the way you did that." Teri looked enormously pleased. I was pleased as well. Not only had Teri done a careful job of explaining her reasoning, I was pleased with the way the class discussion was 203 going, in general. The students were talking directly to one another: Raising questions, voicing their confusion, helping Teri with her solution (i.e. rather than presenting her with different ones). And Teri seemed mindful of her audience, trying to make sure that people understood her at each step. I was, however, concerned that Teri's solution was a guess'n'check type of solution. While it "worked" for this problem, I wanted us to find other ways of reasoning through the story that dealt with and solved for the unknown quantity (rather than fishing around for one that worked.) I raised this with the class. Kara : How many people. . .. guessed around a bit, landed on twenty-four, then worked it through and found out that it worked? That's kind of what I was thinking that Teri did when she got twenty-four. . .. Did other people do something like that too? (Lindsey nods her head)... That's one really logical way of approaching a problem. . .. It just seems like somehow you're lucky to get twenty-four, to some degree. So, that's another issue. 0 I asked if other people had comments or questions for Teri. Danielle said, "Kinda." Danielle's Bid for Iris toLhme Her Solution Upon getting my attention, Danielle launched in saying that Iris had done the "same thing" that Teri did "but she worked from the front to the back." She continued on, with a big smile on her face, "And I don't understand math at all and she made me understand it so I think she should go up there." Iris blushed, rolled her eyes, and said, "Oh my God. . ." I wasn't clear about what was going on but it seemed as if Danielle was trying to get Iris up to the board despite Iris' reluctance to do so -- like someone volunteering a shy 204 friend to be the magician's assistant from the audience. I tried to find out more about what Iris had done to solve the problem. Kara: So, Iris, you did something similar but instead of (pause) Danielle: Instead of starting from one cookie and working up she um, she worked from like, from the top of the equation to the bottom. Kara: Did you start with twenty-four? Iris: No. . .. I started with like the fractions and kind of made like a mathema— mathematical equation, kind of? Kara: Okay. I think several other people did that too, right? Other people used some algebra to try to come up with some kind of formula? 0 It seemed she had used a series of algebraic equations (rather than using pictures, as Teri had done). And Iris had worked with an unknown (rather than starting with an concrete quantity -- arrived at through some informed guessing -- as Teri had done). There actually seemed to be little resemblance between Iris' solution and Teri's, despite Danielle's claim that they were basically "the same." Although I was interested in exploring algebraic solutions to the problem -- many students had approached the problem using algebraic equations —- I was also mindful of the time. I wanted to hold off on the algebra until the next class period when we could give it more serious consideration. I was about to say this when Beth jumped in with a question about Teri's solution. The discussion moved on without further comment from me. chv's "Pie Graph" Solution for 24 Cooki_es_ We were getting close to the end of class but I thought we could do one more quick solution. Kathy had one she had wanted to share from a while back that utilized a picture and a solution path that ran from top-to-bottom. In these ways, it was similar to Terri’s but, unlike Terri’s solution, Kathy’s began with an unknown quantity. I asked her 205 if she wanted to talk about hers. "Yeah," She replied and made her way to the whiteboard, carrying her notebook. "What I did was I started with, like, just a pie graph and drew out the different fractions." Drawing a big circle on the whiteboard, she began her explanation: "'Kay, I made just a big circle for the cookie jar." cookie jar "And then it says, 'First Chris came along. Chris was hungry so he ate half of the cookie jar.‘ So I split this (the circle) in half." She drew a line, dividing the circle in two. Then she shaded and labeled the picture. "Here's what Chris ate." "'Kay, 'Then Lynn came along and she ate one-third of what was left.’ So here's what was left," Kathy said as she pointed to the un-shaded portion that remained. "And she (Lynn) ate one third of (it)." Kathy proceeded to divide the half-circle into thirds then Shaded and labeled the one "third" Lynn ate. 206 "'Kay, 'Kim came along and she ate three-fourths of the remaining.’ So here's the remaining cookies." Kathy explained as she pointed out the un-shaded -— uneaten -- regions. "And Kim ate three-fourths of this. So we'll split this up into four pieces because she ate three-fourths." Kathy drew lines to split the remaining sections into fourths then she shaded and labeled three of the "fourths." V "Now it says, 'Then Elaine came along and took one cookie... and then there's one cookie left over." So, in this remaining," she said as she pointed to the small, un- shaded wedge, "There's actually two in there." She drew a line, cutting the last piece into one left 7 Elaine two thin slivers and labeled them. "So right now," she continued, "I'm gonna work my way backwards." Kathy then began to calculate the amount of cookies represented by the various shadings and labeling in her drawing. Two cookies is one "fourth" of what was left after Lynn ate. one left Elaine So three "fourths" (the amount that Kim ate) must be six cookies. That means there were eight cookies left after Lynn ate. 207 Eight cookies is two "thirds" of what was left after Chris ate. So one "third" (the amount Lynn ate) must be... Kathy struggled with this part of her explanation. She had a sense that Lynn ate four cookies and that this somehow represented one-third of the remaining cookies but she wasn't clear about why this was the case. She said she tried a number of various calculations but eventually ended up with " just thinking, 'Okay, maybe each part was four.'" She attempted to explain why four might be reasonable but gave up saying, "I didn't explain that part real well. I guess for that part,. .. that was just a guess. I just tried that number. And then I just came up with that." She continued on. So one "thir " (the amount Lynn ate) most be four cookies. That means there were twelve cookies left after Chris ate. Twelve cookies is one half of what was in the jar to begin with. That means there were twenty-four cookies originally in the jar. As Kathy finished up her explanation, she turned and asked the class, "Does everybody follow that?" Kathy's approach was a nice addition to our discussion and she had explained her reasoning quite well. Although she struggled a bit at one point she was sharing a tentative idea, something she was not quite clear about herself. I was pleased about this. I wanted people to be able to share their uncertainties and their ideas-in-progress. Megan began to comment, "I think that was a really good way to kind of explain what we were trying to do. But," she continued, "The thing I don't understand is why would you have to guess when you're saying you didn't know why you picked four?" 208 Megan explained that if you know that two thirds equals eight, then one third must equal four. Two-thirds is simply one-third plus one-third. It seemed obvious to Megan. "Simple." I appreciated Megan's clear explanation but I also worried that Megan's comments about how "simple" it was, would make some people feel stupid if it was not simple to them. I felt I needed to address this. Kara : I just want to make one comment about that... the reason I think that it was hard for Kathy is because it is hard. I think Megan had an insight... (It's) an insight to be able to look at that and to see. .. what she's seeing about, "Oh, eight is two- thirds, and that's really two fours". .. That's not something that's simple or easy. I think you'll find that will happen a lot with (your) students- - they'll (have) some cool insight: They'll be able to look at something and see something there that other people can't see. One thing you might want to be careful about is to (realize) that insights are special. And that insights are really a kind of a gift, in some ways, that you get. . . And you should treat it that way. It's not that there's something about it that's inherently easy. It's because somebody had a really cool idea. So, (if) it wasn't simple, that makes sense. But also that (Megan) had an insight, that's pretty cool. I just wanted to make sure that nobody felt like, "I'm so stupid!" "Oh, that is easy. . .." You were working with something that's really complicated. 0 Shelly's Distinction Between Cookies EATEN and Cook_i£s LEFT A number of students commented, saying that they were really beginning to understand the problem now. This seemed like a good place to end our discussion but there were still a few people with their hands raised. I called on Shelly. She had a lot of things to say but, amidst it all, she commented that she realized she had been taking fractions of what was eaten rather than of what was left and that is 209 why She came up with the wrong answer. This was a very important idea but I feared that it was lost in her deluge of words and ideas. Also, time was running short and there were still several more people who had comments. I spoke directly to Shelly: Kara : That's really important. So hold on to that. Okay? We'll come back to that. 0 I took the few remaining comments and then returned to Shelly. Kara : Shelly, do you think you can really quickly explain. . .. that comment you made about what you were working with because that seems like an important thing. . .. As I walked around a lot of people were doing what Shelly was doing and she's noticing something interesting about the way she was worked the problem. 0 Would you mind talking about that for just a little while? Shelly seemed uncomfortable being put in the spotlight but she went ahead and explained the distinction she had originally failed to make between the cookies that were eaten and the cookies that were left. A combination of Shelly's discomfort and the other students' apparent disinterest in Shelly's observation (or perhaps it was their interest our imminent dismissal) led me to feel like I needed to justify why this was important. Kara : The next person (in the story problem) isn't eating what was EATEN. The fraction that (the next person) is eating is what '5 still LEFT in the cookie jar. . . (Shelly) wasn't really taking that into account. . .. That's an important thing to take into account. So I just wanted to raise that because I think a lot of people were getting stuck on that very same idea. 0 And, with that said, we ended class. 210 Episode Analysis: Constraints in the Unfamiliar Terrain The Cookie Jar Problem embeds a useful array of constraints that can challenge and surprise students in productive ways however there is nothing inherent in the problem itself that compels people to View mathematics (or their relationship to mathematics) differently. There's nothing special about working through one of the solution paths that fundamentally alters students' existing patterns of perceiving, interpreting, and responding to mathematics. Indeed, the routines and habits -- the assumptions and beliefs -- prospective teachers bring to their work are powerful constraints, that operate quite smoothly with almost any mathematical problem. If someone came to class believing the only thing that matters in doing mathematical work is finding the right answer, they could certainly finish the Cookie Jar Problem with this belief intact. If they entered the class discussion with the habit of tuning out their classmates or ignoring solutions that seemed too complicated, too childish, or too different, there is nothing in the Cookie Jar Problem that would change this habit. The option to work, think, and talk as they always have is certainly present. The selection of "good" tasks which provide opportunities for exploring mathematics in new ways is a crucial part of cultivating unfamiliar terrains; but equally important are the pedagogical decisions teachers make which constrain what is routine and "normal" for students. With their words and actions, teachers reduce the realm of possibilities and help students develop shared expectations for how to work, think, and talk about mathematics in this class. I examine some of the various ways my words and actions constrained the class discussion of the Cookie Jar Problem in the section below. 211 ill-4t— Varieties of Constraint My actions as the teacher in this early mathematics discussion involved, what felt like, a great deal of constraining. Beginning class sessions often do. Students use their past experiences -- and the assumptions, habits, and understandings they've developed over time -- to make sense of the work immediately before them. But if they are to develop new ways of thinking and talking, learn different expectations and beliefs, they will need to venture beyond the habitual and familiar. Often, it is not enough to simply provide opportunities to change -- old patterns sometimes need to be constricted. During this first mathematics discussion of the "math block" of the course, four different kinds of pedagogical constraints surfaced in my own attempts shape the expectations and behaviors of the students as they engaged in the class discussion of the Cookie Jar Problem. 0 Constraints By Specification One important way teachers communicate their expectations and intentions for the class is through (attempts at) clear explication: telling students straightforwardly what is being required -- requested -- of them. But even with the best efforts at unambiguous description and explanation, there always exist multiple possibilities for interpretation. The risk of miscommunication is ever-present -- and, indeed, it is heightened in unfamiliar terrains where much of what is "normal" is marked for change. Several examples of my attempts to provide constraints by specification and students' ill-fitted- but-reasonable responses can be seen in the discussion surrounding Kim and Deidre's solution for sixteen cookies. 212 I began the discussion by trying to clearly state howl wanted people to present their solutions: Kara: What I'd like you to do is try to explain your solution. If you want to come up as a group or whatever, that's fine. But try to explain... what your answer is, and how you thought about that. And try to work it through. Help us understand the way you went about solving it. However, despite my intentions for students to work carefully through each step of their reasoning with close attention to the audience -— perhaps with questions and answers along the way as the solution unfolded -- Kim's explanation proceeded quickly with numbers flying and flipping everywhere. In what seemed like one deep breath, she explained her solution and double-checked it before our eyes. It could have been that Kim ignored or missed my instructions but it could also be that she was indeed doing her best to follow them. The things she had written on the whiteboard, the questions she asked herself in order to set up the equations (e. g. "What would you multiply three- fourths by to get two?"), her efforts to show that sixteen indeed "checks out" could have all arisen from her best sense of what it means to explain her thinking and help others understand what she did.63 Soon after Kim finished her explanation, I again tried to explicate what I wanted students to do now that Kim's solution was before us. Kara: Could people please ask Kim some questions that you might have about her solution? 63 This way of thinking about what it means to “explain” something in mathematics fits well with how prospective teachers used the word, “explaining,” in the interviews conducted by Deborah Ball in her 1988 study of what prospective teachers bring to teacher education. Ball writes — When the teacher candidates talked about “explaining” (which they did frequently) they seemed to mean something much weaker, something that much more closely resembled simple telling — as in giving directions for the steps of a procedure or repeating a definition. Neither did that telling ever seem to include telling about the nature of the mathematics at hand: its sources or what made it make sense, for example. 213 Rather than evaluating what Kim had shown or questioning Kim myself, I hoped to engage the class in discussing Kim's solution, having them raise their own questions and challenges. In my mind, this was the beginning of our examination of Kim's ideas, not the conclusion of it. Megan, however, responded to my request by proposing a different, "simpler" way to solve the problem, one that seemed likely to move collective our focus away from Kim's solution. Although Megan could have been deliberately ignoring or rebelling against my request, it is also possible that she heard my instructions as a petition for people to "help" Kim, or perhaps Megan was simply doing what students often do when "discussing" mathematical solution: Show how to solve the problem, with little (or no) effort directed toward getting inside of others' ways of thinking about the problem. Such efforts at specifying what students are to do -- articulating directly and straightforwardly, "Do this " -- can serve as a form of constraint, narrowing the realm of appropriate alternatives for thought and action. However, as the examples above show, attempts to be clear and straightforward do not necessarily result in clear, straightforward communication. It is, of course, possible to go into more detail and to choose different -- perhaps better -- words and phrases but, nevertheless, what is said is only a part of what is heard. At times, this is sufficient for the general idea to be conveyed (particularly when what is being said fits with what students already expect or have some familiarity with) but, at other times, other forms of constraint are needed. 0 Constraints Bv Contrast A second type of constraint involves setting up contrasts, narrowing the realm of possibilities by juxtaposing what is not wanted with what is wanted. One example of 214 constraint by contrast can be seen in my response to Megan's attempt to introduce a new solution before we had gotten the chance to discuss Kim's: Kara: Why don't we hold that for a second? What I'd like people to... do is to try and understand what it is that Kim and her group did and um-- I bet there's a lot of questions out there about like, "Where did you get that?" or "Why did you decide to divide?" or things like that so why don't people go ahead and-- let's first try to figure out what it is that they did- - why that makes sense as something to do-- and then we can try to explore some alternative ways of working on it. I think that would be really useful; but let's try to first get inside of this one. Although Megan's "simpler" way to do it might have been appropriate and welcomed in other contexts, I was determined to have them think about Kim's solution and how/why she arrived at an answer of sixteen. By stopping Megan's description of her solution, mid-sentence, and redirecting the class, I had effectively ruled-out the possibility of sharing new solutions at that point in time. Two important things were addressed with this constraint. First, it pointed to the idea that something different might be called for ("understanding what... Kim and her group did") than what they were inclined to do (share "alternative ways of working" on the problem). It seemed, for many students, their first reaction to the task of "discussing" another student's solution is to propose their own, different solution. And, second, this constraint ruled out -- at least for the momen -- the possibility of sharing different solutions. Moving away from simply sharing one’s own solutions to engaging in someone else's solution continued to be a challenge, a point of tension, throughout most of the class session. After Tess and Megan's shared their solution, a large group of students again raised their hands to share a different solution rather than comment on or question what Tess and Megan had shown. This time, however, they (and I) seemed to be aware 215 that a distinction could be made. After Teri shared her solution, Danielle tried to draw a connection to between Teri's solution and what her friend, Iris had done, in order to get Iris an opportunity to share her solution. Although I failed to see the connection (and Iris failed to get her opportunity), Danielle seemed aware that they were expected to discuss and build upon T eri's solution at that time. Contrasts between what students are inclined to do and what is desired can be a powerful constraint in an unfamiliar terrain. Although this distinction doesn't necessarily provide greater clarity about what students are to do, it can serve the purpose of identifying students' natural inclination as being merely one alternative. This is important if students are to develop new ways of responding, new expectations, for learning and doing mathematics in the class context. a Constraints By Patterning The establishment of patterns in things said and done is another form constraints can take. Patterns can provide strong limitations that allow students to anticipate or make good guesses about what should come next. Certainly one way patterns are used to shape classroom discourse is by creating routines -- sequences -- that play out repeatedly. For example, after a student shares a solution, it is always followed by the opportunity (and expectation) for others to ask questions about what was shown. By repeatedly using prompts like, Kara: Any questions about that? A pattern of pausing after presentations for questions and clarifications began to form. 216 Another way patterns can be used to constrict responses is by offering examples -- models -- of the kinds of questions and comments students might appropriately raise. A brief example of this can be seen (in yellow) above, in my comment to Megan after I interrupted her: "I bet there's a lot of questions out there about like, 'Where did you get that?‘ or 'Why did you decide to divide?’ or things like that." Beyond telling the students what I wanted them to do (i.e. "understand what it is that Kim and her group did") and what not to do (i.e. introduce new solutions), this constraint by patterning provided examples of the kinds of questions people might ask of Kim. Sometimes, the models are offered in less overt ways: not marked as examples in any way but simply put forth as a question or comment in the flow of conversation. For example, during Tess and Megan's description of their solution, I asked the two women, Kara: Could you prove that to us? That if two equals one-fourths, three-fourths equals six? In subtle ways, these kinds of insertions into the class discussion limit the scope of acceptable questions and comments to ones that are somehow like these. They provide students with things to pattern their own responses after. The constraints described above -- constraints by specification, contrast, and patterning -- were aimed at shaping students' words and actions.64 However, later in the class session, after students had begun to anticipate the requests and questions I'd raise, a different kind of constraint began to surface in my interactions with students. This fourth 6" This pedagogical work of developing shared expectations within the class and achieving some clarity in the roles played by the teacher and students, is studied and discussed in Louis Smith and William Geoffrey’s (1968) book, “The Complexities of an Urban Classroom: An Analysis Toward a General Theory of Teaching.” In this work, they use the phrase, “Grooving the children” (p. 68) to describe these types of interactions between the teacher and students that move the class toward a shared sensibility for “what exists” (p 71). and the way things operate within the class. 217 kind of pedagogical constraint deals less with what students say and do and more with how they perceive and interpret what has been said and done. 0 Punctuating the Discourse"5 The words and actions generated during a class discussion, flow by in an undifferentiated stream that students interpret in whatever way seems natural to them: Certain things are stressed, other things ignored, and meaning is created in their minds. But teachers can attempt to influence these interpretations by the ways in which they (re)frame the exchanges that occur. With words and actions, teachers can punctuate the stream of events, stressing some of the students' utterances while downplaying others; emphasizing some questions and issues that arise and bypass others. Much like the symbols that transform long, undifferentiated strings of words into sentences and clauses conveying certain meanings, teachers' words and actions can likewise transform a student's undifferentiated stream of words into a single important point or key question or they can distill three central ideas from half an hour of student discussion. I refer to these pedagogical efforts to bring certain things to attention, to break up the flow of ideas and events, to imbue certain things with more importance than others, as punctuating the discourse. 65 An important question in Gregory Bateson's psychology work (1972) focused on how people (and animals) "acquire a habit of punctuating or apperceiving the infinitely complex stream of events... so that this stream appears to be made up of one type of short sequences rather than another" (p.163). According to Bateson, fatalistic views, optimistic tendencies, instrumentalist perceptions, are patterned ways of perceiving and interpreting events as they arise. In the "infinitely complex stream" of sights, sounds, feelings that flow past in the happenings of life, people have habitual ways of emphasizing certain things, downplaying other things, and giving it all meaning -- they have habitual ways of punctuating the stream of events . Although I use the language of "punctuating" in a slightly different way to describe how a teacher might highlight, stress, obscure, segment events for students in order to help them develop new perspectives and ways of thinking, I am indebted to Bateson for this way of talking and thinking about actively giving meaning to events. 218 Punctuating constraints can take the form of simple comments that flag the importance of something a student says or does. For example, near the end of class, Shelly had made a statement about her own failure to distinguish between cookies eaten and cookies left. This was a key issue of the Cookie Jar Problem Shelly was raising. However, this one critical statement was likely to get lost in Shelly's characteristic deluge of words and in the rushed atmosphere of the end of class. I responded by requesting that Shelly explain this particular comment a bit more, claiming (and providing a rationale why) she was making a very important point. Kara: Shelly, do you think you can really quickly explain. . . . that comment you made about what you were working with because that seems like an important thing. . . . As I walked around a lot of people were doing what Shelly was doing and she's noticing something interesting about the way she was worked the problem. Would you mind talking about that for just a little while? By constraining students' attention -- and Shelly's comments -- to one important point embedded in the more verbose and multifaceted comment Shelly made earlier, there was (hopefully) a greater likelihood that students would come to see something of significance there. A more subtle form of punctuating constraints involves allowing the conversation to flow by a students' comment or idea without emphasizing it or focusing on it in any deep way. One of the clearer examples of this can be seen in my exchange with Danielle and Iris. For reasons that were not clear to me, Danielle seemed to be pressing for Iris to go up to the board and share her algebraic solution. Although I did want to spend some time discussing algebraic solutions, I didn't want to introduce them at that time: It was too close to the end of class and there were still a number of unresolved issues I wanted 219 us to pursue before dismissal. I asked Iris a few questions to find out a bit more about her approach and then simply stated, Kara: Okay. I think several other people did that too, right? other people used some algebra to try to come up with some kind of formula? After that comment, I allowed the conversation to continue on. Iris didn't go up to the board at that moment as Danielle had hoped. Rather, we returned to discussing Teri's solution (as we were doing prior to Danielle's attempt to involve Iris) and eventually moved on to see another solution and hear comments that addressed particular mathematical issues as I had hoped. Finally, punctuating constraints can also take the form of alternative interpretations suggested by the teacher. By superimposing other plausible ways to compose meaning, teachers can attempt to influence students' interpretations of things said and done. One example of this can be seen in my response to Megan after she provided a "simple" way to think about something Kathy had been struggling with. I worried that Megan's framing of the mathematics as "simple" might make some of those who had struggled -- and have spent a lifetime struggling with mathematics -- feel even more inadequate than they already felt. Following Megan's comment, I inserted another way to think about the mathematics and Megan's contribution: Kara: I just want to make one comment about that... the reason I think that it was hard for Kathy is because it is hard. I think Megan had an insight... (It's) an insight to be able to look at that and to see... what she's seeing about, "Oh, eight is two- thirds, and that's really two fours"... That's not something that's simple or easy. . .. I think you'll find that will happen a lot with (your) students- - they'll (have) some cool insight: They'll be able to look at something and see something there that other people can't see. One thing you might want to be careful about is to 220 (realize) that insights are special. And that insights are really a kind of a gift, in some ways, that you get. . . And you should treat it that way. It's not that there's something about it that's inherently easy. It's because somebody had a really cool idea. So, (if) it wasn't simple, that makes sense. But also that (Megan) had an insight, that's pretty cool. I just wanted to make sure that nobody felt like, "I'm so stupid!" "Oh, that is easy. ..." You were working with something that's really complicated. These are subtle constraints -- constraining attention, focusing the topics of conversation, creating space for alternative viewpoints -- but they are significant because they go beyond addressing students' behavior to addressing how students see and think about the mathematical work. 81mm: There are a variety of pedagogical constraints that teachers can and do use to limit how students work on and talk about mathematics problems. These kinds of constraints are particularly important when establishing an unfamiliar terrain ; without providing limitations on students' deeply ingrained patterns, they are less likely to develop new ones. The list of pedagogical constraints described above is by no means exhaustive but it does give some sense for the variety that exists. In addition to varying in form, pedagogical constraints can also vary in their capacity to limit the realm of alternatives —- they can vary in "strength." For example, many of the constraints by specification 1 used early on in the class discussion proved to be weak constraints, allowing the students to continue to "explain" and "discuss" mathematics solutions in old, familiar ways. Weak constraints are not necessarily bad. Indeed, weak constraints are often highly desirable because they allow students to make 221 choices and decisions. And even if (when) mistakes arise, they can be used to set up stronger constraints (e.g. constraints by contrast) that make contrasts between the "more desirable" and the "less desirable" easier to see. It is, however, 11861111 to have some sense for how strongly or weakly a pedagogical move will constrain students' options. If constraints are too strong too often, students are robbed the opportunity to exercise decision-making and the chance to experience the results of their decisions. Overly constrained contexts provide few opportunities for students to exercise their capacity to reason through options, learn from mistakes, take risks, act in uncertainty. On the other hand, if constraints are too weak too often, many students will fail to discover and develop new ways of perceiving, interpreting, and experiencing that lie beyond what their current assumptions and habits will allow. Changing Beliefs Vs. Changing Norms Students adjust in response to the pedagogical constraints they encounter. Some of these adjustments can be seen within the progression of solutions presented over our forty-five minute discussion. From our beginnings with Kim's non-stop whirlwind tour of her solution, to the next solution shared by Tess and Megan, subtle changes can be seen in the expectations for, and understandings of, how mathematics gets discussed in this class. For example, during Tess' explanation, Megan interrupted Tess, mid-sentence, just as she was about to move on from one step in the solution to the next: "Wait," said Megan, "Tell them why . . . " Megan seemed to be anticipating the inevitable, oncoming press for reasons to explain or justify each step of their solution. 222 In response to Megan's prompting, Tess backed up and began to explain the step she had hurried over: "Oh, because Kim had three-fourths. . . we know that this (the two cookies) is going to be the one-fourth that is left after she had the three- fourths." Tess then turned to the class and asked, "Does that make sense? We can draw pictures too." Like Megan, Tess seemed to be anticipating questions and requests unasked but likely to come. Indeed, if Tess had not checked with the class before moving on, I would have. And, although Tess and Megan did not use pictures at this point in their explanation, they did use them soon after in response to my probing: Could you prove that to us? The use of pictures and periodic "checks" with the class to see whether or not they were following along seemed to grow increasingly more "natural" as the class discussion proceeded. The pace of sharing solutions slowed as presenters (i.e. Teri, followed by Kathy) shaped their explanations to questions asked and unasked. In these, and other ways, students began to anticipate with greater accuracy the kinds of actions and responses that were expected of them in this class. And, over the duration of this first mathematics lesson in the math block, their behaviors as "presenters" and members of the "audience" began to adjust appropriately. However, as Smith and Geoffrey (1968) have pointed out in their study of classroom structure development, an important distinction can be made between the development of beliefs and norms within a classroom. In their terminology, “A belief is a generalized perception of what exists, while a norm is a generalized or group expectation 223 of what ought to exist” (p.71, emphasis added). Applying this to Geoffrey’s classroom, the authors continue on to elaborate -- As Geoffrey made the class rules clear, he was dealing with belief systems; as he tried to build an emotional commitment on the part of the children to these beliefs, he was engaged in the more complex task of shaping normative structure (p.71). Although there were indications that the prospective teachers were developing a well-fitted sense "what exists" -- the constraints, the inevitable questions and requests, the expectations I had for them -- these were not necessarily signs that students were developing any "emotional commitment" to the kind of work we were doing together. The changes in their expectations and behaviors did not guarantee that an accompanying shift in their ways of perceiving or experiencing the unfamiliar terrain had taken (or would take) place. Indeed, instead of everything smoothly falling into place -- new values and commitments forming to match the new expectations and behaviors -- there was some evidence that a gap or tension was forming. After Class As usual, there were a few people milling about, waiting to ask me a question or tell me something before leaving. Danielle came up to me after everyone else had left. She wanted to let me know that "all of them sitting over by (her)" felt "blown off" because I didn't have Iris go up to the board. As a result, she explained, they were "checked out for the rest of class." I tried to explain how large and important the work with algebra seemed and how I was hoping to pick it up the next day when we had more time. Danielle, however, seemed less interested in my rationales or assurances than in simply pointing out a fact to me: I blew them off; they checked out. 224 This was an unexpected way to end, what I thought was, a rather good class session. Over the hour we worked on the problem together, it seemed students were getting better at listening to one another, asking questions of each other, explaining their thinking to an audience. I felt we were beginning to make some real headway with developing shared expectations and norms. We were, I thought, starting to work together as a nascent intellectual community. Danielle's comment seemed to point to some incongruity, to something that didn't fit with my interpretations and impressions of the day. 225 CHAPTER 4 Discernment of the Unfamiliar Terrain Introduction The educational approach of presenting the new mainly in its new terms involves more than simply placing students within a strange context to which they must adapt. It must go beyond simply creating a sense of being off-balance or ill at case that will be reduced through the development of new behaviors. Rather, it is crucial that a process of action and reflection is initiated; a process that will ultimately lead to new understanding and modes of knowing. Schwab notes (1959/1978) that as the unfamiliar terrain is encountered and actions undertaken lead to unexpected consequences... There is reflection on the disparities between ends envisaged and the consequences that actually ensue. There is reflection on the means used and the reasons for why the outcomes were as they were (p. 173). In this chapter, I look closely at this process of “reflection on the disparities,” paying particular attention to what is involved in initiating — and productively learning from — such reflective processes. This analysis focuses upon the fifth lesson of the Mathematics Block (November 3), approximately one week after the events discussed in Chapter 3. During this lesson, the class first began to express and examine their understandings of the mathematical work and class context. Although students had begun to adapt in noticeable ways to the unfamiliar terrain - they had, for example, become much more careful in their explanations and had begun to raise alternative approaches and representations (alongside arithmetic and algebraic manipulations) without my prompting — what students perceived and understood about our work together 226 remained, until then, largely unvoiced and unexamined. Through a careful study of the things said and done during this significant lesson, this chapter provides an image of the processes leading up to our initial efforts to reflect on the disparities and sheds light upon some of the challenges involved in such efforts. This chapter begins with a brief explanation of the theoretical perspective used for this analysis. It then goes on to analyze the mathematics problem that was selected for this lesson. Originally, it was chosen to encourage prospective teachers to notice discrepancies — inconsistencies and fragilities — in their own understanding of a basic piece of elementary mathematics (division). However, the problem also became a catalyst for noticing disparities in the ways people perceived and understood the mathematical work of the class. This analysis of the math problem is followed by a description of the lesson, detailing the class discussion of the problem and the conversation that arose about the class itself and the type of work we were doing. Finally, the chapter concludes with a detailed analysis of the processes by which underlying disparities became perceptible and examinable — the processes that allowed for reflection on the disparities to be initiated and pursued. The Primm Analytic Lens: Difference That MakLes a Difference The detection of difference is a crucial element in the functioning of cybernetic systems. The thermostat regulates a heater in response to the difference between a goal temperature (or a temperature range) and the ambient temperature; the pituitary gland regulates the production of thyroid stimulating hormone (TSH) in response to difference between expected levels of thyrotropin-releasing hormone (TRH) and current levels. Even in many unstable systems that are in a state of accelerating change - a group of 227 countries locked in a nuclear arms race, a human being caught in a drug addiction, a microphone and speaker system linked in audio feedback — the detection of difference, between a current state and some other state that involves “more” or “greater” levels, can drive change towards an extreme. In particular, the perception of difference is essential in the growth and adaptation of living systems. Mary Catherine Bateson (1984/1994) provides a nice example this in her description of the way a plant will swivel on its stalk as it responds to differences in the amount of light available: It twists because more light shines on one side — not simply because of the light itself. For humans, the perception of difference is not only important for physical growth and adaptation, it also plays a significant role in human behavior and learning: We respond to myriad differences we detect within our bodies, in our surroundings, in our relationships, in our own understanding of things — sometimes making unconscious adjustments; at times making deliberate decisions to take certain courses of action (including the decision to not act). News of difference informs us, influencing our thoughts and actions. In fact, early cybemetics scholar, Gregory Bateson, proposed that, “Information can be defined as difference that makes a difference” (Bateson, G., & Bateson, M., 1987, p.17). In the discussion below, I briefly describe of a few key ideas about the perception of difference and its role in growth, adaptation, and communication that were used for the analysis found in this chapter. The Role of Difference in Experience and Learning The difference between what was intended and what actually results, between the conditions at time, t], and those at t;, between what has been known and what is being 228 suggested are a few types of dijferences that often play an important role in our capacity to experience and learn from experience. A classic example of a perceived difference making a difference in human thought and action is told by Donella Meadows (1999), an environmental scientist and systems theorist, known for her work on global systems and issues of sustainability. She describes a housing sub-division where all the houses were identical except for the location of a meter that displayed information about energy consumption in the home: The indicator would go around faster or slower as more or less electricity was used by the occupants. In some homes, the meter was installed in the r basement; in other homes, it was installed in the main front hall. It was reported that 30 percent less electricity was consumed in those homes where the meter was displayed in the high-traffic area of the hall. Difference-Reducing Feedback. The scenario above exemplifies an important type of feedback loop,"6 known as a negative feedback loop. As the residents of the homes with hallway-installed meters pass by the meter, they have frequent opportunities to see the levels of electricity they are consuming. Over time, they develop a sense for what are “usual” or “desirable” levels and can adjust their use of electricity accordingly when there are significant departures from this. This type of feedback loop allows for the regulation of systems, making it possible to adjust for fluctuations that occur. It serves to reduce discrepancies between “actual” and “intended,” making constancy possible despite changes that may be occurring in and around the system. Negative feedback 66 “Feedback” is information about a process (or set of processes) that is fed back to the system and used to influence future iterations of the same — or a similar — process. This flow of information is often depicted and referred to as a (feedback) loop (Heylighen, 1992/2000). 229 loops provide self-correction and are crucial to the development of stable, sustainable systems that can endure a variety of change and impacts over time. At the same time, however, this capacity for maintaining stability and reducing differences also creates resilience, making deep or significant change difficult. There is, for example, a negative feedback loop involved in the acts of perception and interpretation. In their book on language and therapy, Richard Bandler and John Grinder (1978) described the powerful self-reinforcing cycle — the negative feedback loop — that can occur: A person’s generalizations or expectations filter out and distort his experience to make it consistent with those expectations. As he has no experiences which challenge his generalizations, his expectations are confirmed and the cycle continues (p.16). As mentioned in Chapter 2, the brain and nervous system serve as a “reducing value,” functioning to limit perception and attention to those things that are most “likely to be practically useful” (Huxley, 1954). While this reduction makes thought and action possible ~allowing for quick apprehension and inferential leaps — it can also work against seeing new things or understanding in new ways, inhibiting the perception of difference that is necessary for initiating and sustaining significant change. Difference-Amplifling Feedback. A second type of feedback loop, known as positive feedback, functions to amplifi/ differences occurring within and around systems. Unlike negative feedback loops that create stability and allow for self-correction, positive feedback loops create instability, accelerating change to unsustainable extremes. Donella Meadows (1999) notes— 230 Positive feedback loops are sources of growth, explosion, erosion, and collapse in systems. A system with an unchecked positive loop ultimately will destroy itself. That’s why there are so few of them (p. 11). Very often, examples of positive feedback loops are destructive with costly consequences — vicious cycles that not only repeat but worsen with each iteration. The person suffering from drug addiction not only continues to use drugs but comes to need greater dosages to produce the desired effect. The country involved in an arms race not only needs to continue to invest resources into arms development and production but requires increasingly greater investments as other countries respond by building up their own arsenals. Additionally, positive feedback loops are often associated with miscommunication and unintended (or reverse) effects. For example, a person with undiagnosed Type I diabetes might attempt to relieve their symptomatic thirst by drinking more sugar-filled soft drinks, thereby increasing their sugar levels; this, in turn, intensifies the body’s efforts to flush out excess sugar, resulting in a greater need to drink and pass liquid. Despite the person’s efforts to relieve their thirst, their efforts create a physical response that only increases dehydration. If this positive feedback loop continues without intervention, the body will collapse. Positive feedback loops entail gains that increase with each iteration. Although there is often the potential for destructive consequences associated with them, positive feedback loops also have the potential to drive changes that might not otherwise take place. There is, for example, the possibility for paradigm shifts as anomalies, contradictions, and inadequacies of “normal science” accumulate and new observations, questions, and theories build up (Kuhn, 1962). 231 In this chapter, the perception of difference — along with the feedback loops that reduce and amplify differences — served a primary lens for the analysis. However, perceiving difference is only the beginning. The impact of a difference also depends upon one’s “readiness” to receive such news of difference and act upon it. This is particularly true in the work of teaching where the ability to access and address thinking that is quite different from one’s own is an inherent challenge of the job. This capacity to perceive, receive, and use news of difference in teaching is given closer attention below. The Role of Diflerence in Teaching Mathem_atics More than most walks of life, the profession of teaching exposes its practitioners to mathematical ideas and ways of reasoning that are different from the ideas and methods they may have learned as students and from those they use in their day-to-day lives as adults. As they prepare lessons, choose and adapt materials, consider standards and learning goals, weigh the pitfalls and possibilities of different representational contexts, teachers routinely encounter alternatives to their own understanding and ways of thinking. Teachers also encounter a wide range of mathematical novelty and oddity in the ideas, representations, and procedures their students generate and use. The "stuff" teachers must work with is sometimes muddled and ill-articulated, sometimes incredibly similar to convention with small-but-significant differences, sometimes utterly different than what has been seen or thought about before. With their current understandings, teachers must assess and address “new” mathematics with an eye toward students' present and future learning. Deborah Ball (1993) discussed some of the challenges involved in following, assessing, and addressing students' mathematical ideas in her article, With an Eye on the 232 Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics. Drawing upon her own experiences as a third grade mathematics teacher, Ball noted -- Third graders tread frequently on "mathematically sacred ground" (Hawkins, 1974). They also tread on mathematically uncharted ground. Surely, "respecting children's thinking" in mathematics does not mean ignoring nonstandard insights or unconventional ideas; neither must it mean correcting them. But hearing those ideas is challenging. For one thing, teachers are responsible for helping children acquire standard tools and concepts -- ideas of mathematical heritage. However, the unusual and novel may consequently be out of earshot. For another, making sense of children's ideas is not so easy. Children use their own words and their own frames of reference in ways that are not necessarily congruent with the teacher's ways of thinking. ...Often my problem is to figure out where they are in their thinking and understanding. Then I must help to build bridges between what they already know and what there is to learn. Sometimes my problem is that it is very difficult to figure out what some students know or believe -- either because they cannot put into words what they are thinking or because I cannot track what they are saying. And sometimes... students present ideas that are very different from standard mathematics. The ability to hear what children are saying transcends disposition, aural acuity, and knowledge although it also depends on all of these. And even when you think you have heard, deciding what to do is often a trek over uncharted and uncertain ground. Consider the following example. One scenario that commonly surfaces for teachers involves them in the mathematical analysis of unexpected procedures or algorithms used by students. A teacher might encounter something like this -- 233 A student (says). “To divide fractions, I just divide the tops and the bottoms... Ihave 1% + iwhichlcan write as: = I g 1 4 ' 2 and then I just divide the tops and bottoms: _ 7 + 1 = 7 ‘ 4 + 2 = 2 which I can rewrite as 3 l/2, which is the right answer.” (Ball 2002) 67 This procedure is similar to the rule many teachers may have learned for multiplying two fractions: When multiplying two fractions, multiply the numerator of the multiplicand with the numerator of the multiplier — this will give the numerator of the product; then multiply the denominators of the multiplicand and the multiplier — this will give the denominator of the product. i.e. g x In the scenario above, however, the student has invoked this procedure for dividing two fractions. Additionally, the procedure is markedly different from the invert-and-multiply rule commonly learned in schools and used by adults for dividing two fractions. For many teachers, this procedure would fall through the cracks of their prior learning and the mathematical issues or possibilities surrounding this procedure might not be at all apparent. If no significant difference is detected -- for example, perhaps only the answer receives serious attention and the method is not scrutinized -- then both the student and (”Used with special permission from the, Learning Mathematics for Teaching Project (LMT). Not for reproduction or use without written consent of the LMT (for contact information, see hgpd/wwwsii.soe.umich.edu[l. 234 teacher will proceed as they had been, not suspecting there might be more to the students’ work than met the eye. If, on the other hand, the teacher does detect notable differences, new possibilities are opened: There are alternatives, options, and decisions-to-be-made. It is certainly possible the teacher might not know what to make of these differences between the student’s idea and her own -- much less what to do about them -- and, rather than allow the student to use an incorrect or flawed method, the teacher might simply insist that the student use standard rules and algorithms. But even within that pedagogical decision, there is a sense (even for a brief moment) that there might be more to understand than what is'already understood. These encounters with different ideas and ways of reasoning are vital in the work of teaching for their potential to help teachers expand and deepen their own mathematical understanding, correct flawed ideas and ways of reasoning, refine muddled and informal meanings. However, to take advantage of this potential — rather than to simply fall back on what is already known and has been done in the past — prior learning must be used in somewhat different ways to uncover things unknown and not-yet-understood. Rather than aiding immediate apprehension and bringing quick closure, prior learning must serve to initiate and guide the pursuit of new understanding. For many prospective teachers, this is a different way of learning and using mathematical knowledge — dissimilar from the work on “plug and chug” problems with which they are accustomed or the familiar effort to remember and follow the steps of a procedure. Some mathematical problems, more than others, provide a good context for using prior learning to uncover gaps, limits, errors of understanding and vagueness in meaning — and for initiating and guiding attempts to learn. As the second week of the mathematics 235 block drew to a close, I chose a mathematics problem known for its potential to do these very things. Problem Analysis: The 1 3/4 + 1/2 Problem The problem that was selected for this lesson is known as the 1 3/4 + 1/2 Problem: 1 3/4 + 1/2 Problem Write a story problem for 1 3/4 + 1/2. Include your answer. Be prepared to explain what your answer means within the context of your story. This problem was originally created for studies on prospective teachers' knowledge and ways of reasoning about mathematical pedagogy, conducted by the National Center for Research on Teacher Education/Learning and reported in Deborah Ball's (1988) study, Knowledge and Reasoning in Mathematical Pedagogy: Examining What Prospective Teachers Bring to Teaching Education.68 Ball also did further work with a very similar problem‘”, using it in a teacher education course. In 1990 article entitled, Breaking With Experience in Learning to Teach Mathematics, Ball discusses her findings of what and how prospective teachers can learn from working on this problem in the context of 6" In Ball's study (1988), the task of writing a story problem for 1 3/4 + 1/2 was one of a series of interview questions posed to prospective teachers. The original task was worded as follows: 1 3/4 + “‘2 Something that many mathematics teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story problems to show the application of some particular piece of content, or examples or models that make clear what something means. Sometimes this is pretty challenging. ° Could you think of a good situation or story or model for 1 3/4 + 1/2? (i.e. something real for which 1 3/4 + 1/2 is the appropriate mathematical formulation? ' How does that fit with 1 3/4 -=- 1/2 0 Would this be a good way to help students learn about fractions? (p. l 12) 69 In this article, Ball discusses prospective teachers’ efforts to write a story problem for 2 1/4 + 1/2. 236 teacher education. In the problem analysis below, I use and expand upon Ball’s work to discuss the curriculum potential of the 1 3/4 -:— 1/2 Problem. The Problem The 1 3/4 + 1/2 Problem allows a number of different discrepancies to surface within the solutions (the stories) people develop. This is a rather special characteristic of this problem: Although it is not particularly unusual for a mathematics problem, like the Cookie Jar Problem, to have multiple valid solutions or more than one likely answer that will surface discrepancies when looking across different students’ responses, it is less common for a problem to surface discrepancies — surprises, puzzles, apparent contradictions — within the single solutions produced by many individual students. A part of what makes the 1 3/4 + 1/2 Problem consistently challenging for American pre-service teachers is the fairly consistent way division by fractions is treated in the school curriculum: It tends to be taught and learned primarily (and, in many cases, exclusively) as an algorithm that transforms division into multiplication. Rather than studying division by fractions as a variant of division, meaningfully connected to division with whole numbers, students are frequently given the “invert and multiply” rule and simply instructed in its application and use. Deborah Ball (1988) notes -- A typical sixth grade textbook page introducing division of fractions says simply, “Dividing by a fraction is the same as multiplying by its reciprocal.” No or little attention is given to the meaning of division with fractions and no connections are made between division with fractions and division with whole numbers. Each is treated as a special case (p. 87) People usually begin their work on the 1 3/4 + 1/2 Problem by using this rule to calculate the answer, 3 1/2. However, obtaining the quotient is usually not a problem for most pre- service teachers — and it is only a small piece the assigned problem. 237 MeanMaking The task of writing a story problem requires students to ground a particular mathematical expression within a sensible context, ascribing meaning to the given quantities, to the operation, and to the unknown being sought. Although the form of the mathematical expression may be very familiar, similar to many problems solved in the past, the effort to assign meaning to its parts and closely attend to those meanings may not be. Indeed, the task of writing a story problem for 1 3/4 + 1/2 has repeatedly proven to be a challenge for prospective (and many practicing) teachers in the United States (Ball, D., 1988; Ball, D., & Wilson, 8., 1990; Ma, L., 1999). Some prospective teachers, sticking closely to the invert-and-multiply algorithm, create a scenario for 1 3/4 _x_2 instead of working with the “+ 1/2.” For example — Dave went to the store to buy hamburger for the barbeque party. He grabbed a 1 3/4 pound package and asked the person behind the counter, “How many burgers can this make?” The butcher replied, “It depends on how big you make them but on average maybe somewhere between 5-7 burgers.” Dave thought he needed about twice (two times) that amount. How many pounds of hamburger would that be? 1 3/4 pounds of hamburger x 2 = 3 1/2 pounds In some cases, prospective teachers perceive no significant difference between the two mathematical expressions: 1 3/4 + 1/2 is 1 3/4 x 2. More than a rule to help solve division problems with fractional divisors, they see the inverted-and-multiplied form of the expression as having the same meaning as the original expression — similar to the way “cat” is gato is katze is chat is 3'5. The invert-and-multiply rule makes it possible for people to get by without ever having to work directly with a fractional divisor (they instead work with a fractional multiplier) or give meaning to an expression that contains one. 238 This raises another important aspect of .the 1 3/4 + 1/2 Problem worth highlighting: It is not just that the problem -- with its requirement of creating a story context -- focuses attention on meaning but it is also significant that the chosen mathematical expression involves a fractional divisor. This presents a particularly difficult challenge for many Americans whose experiences with division story problems primarily involve only whole number divisors. Deborah Ball (1988) found that some prospective teachers, using what they knew about division with whole numbers, encountered a nonsensical situation when considering a fractional divisor. One prospective teacher she interviewed commented -- I guess when I think of division problems, I think of dividing it between people and it’s hard to have half a person! (laughs) or between bins or between discrete things, as opposed to between a fraction of a thing” (Ball, 1988,p.l25) Another stated -- You don’t think in fractions, you think more in whole numbers” (Ball, 1988,p.12l) One way people respond to the 1 3/4 + 1/2 Problem is to simply not produce any story: They find themselves unable to create a sensible, real-world context, not knowing where to begin or how to proceed. Some, believing it is not possible to produce a reasonable story context for 1 3/4 + 1/2, settle instead for something they believe to be equivalent and create a story context for 1 3/4 x 2. This discrepancy between what is known and what there is to know surfaces in the gaps and at the edges of understanding. It indicates places where current understandings and meanings can be broadened and expanded. In this case, not only is there the potential 239 to develop to a fuller understanding of division but there is also the potential for developing a more complex understanding of what it means to “understand” something in mathematics. Using a similar problem — i.e. Write a story problem for 2 1/4 + 1/2 — with prospective teachers, Deborah Ball (1990) notes the following types of comments made in the written reflections of her teacher education students -- I realize now that I didn’t really understand many of the manipulations that I could produce the correct response for. Working with the [fraction problem] was a real eye opener for me. While I could quickly come up with a correct answer to the problem, I had no idea how to write a story for it. Finally, through discussion with others and my own thinking out loud, I realized what the problem was asking me to do. After 16 plus years of school I understood division of fractions for the first time (Ball, 1990, p.14). Error Identification and Analysis Another important feature of the design of the problem is its use of “1/2” as the divisor. More so than any other fractional divisor, 1/2 poses significant and unique challenges to American prospective teachers. The prospective teachers tended to confuse dividing IN half with dividing BY one-half and they did not seem to be aware of the difference... The teacher candidates’ error may have resulted from a common but problematic confounding of everyday with mathematical language (Ball, 1988,p.121) Many people develop story problems portraying division by two rather than division by one-half, unaware of the mismatch between their story and the mathematical expression. A typical example might look something like this: There were 1 3/4 pies leftover from the party. Donna and Drew, the hostesses, decided to divide the remaining pies in half so they could each take some home for their families. How much pie did each woman get? This story appropriately uses 1 3/4 as the dividend -- it also invokes “division” and uses the word “half” -- however, it is a scenario for dividing by two rather than 240 dividing by one-half: The 1 3/4 pics are being divided equally between the two women.70 Although prospective teachers may struggle to give meaning to division by fractions, they often bring an understanding of division by whole numbers that allows them to recognize stories, like the one above, as depictions of 1 3/4 + 2. If this connection to 1 3/4 + 2 is pointed out, many find it surprising and puzzling: Does the story represent 1 3/4 + 1/2 or 1 3/4 + 2? Is it possible that it represents both? Sometimes another type of discrepancy surfaces in students’ attempts to make a pictorial representation of a story like the one above. For example, There were 1 3/4 pies leftover from the party. 0‘ ~ Donna and Drew, the hostesses, decided to divide the remaining pics in half so they could each take some home for their families. How much pie did each woman get? They each get one-half of the whole pie, a one-quarter piece, and half of a quarter piece (i.e. a one-eighth piece of pie): 1/2 +1/8 +1/4 = 4/8 +1/8 + 2/8 = 7/8 70 Similarly, students often respond to the 1 3/4 + W Problem by inadvertently creating a story problem for 1 3/4 + 1/2. An example might look something like this: There were 1 and 3/4 pizzas left when John finally arrived. He knew Dave was also running late so he only ate half of what was there, leaving the rest for Dave. How much pizza did John eat? Although this story involves splitting up — sharing -- a given quantity of pizza (1 3/4) the scenario represents multiplication by a fraction (1/2), or possibly, division by an unstated whole number (i.e. 2). 241 7/8, however, is contrary to the answer calculated with the invert-and-multiply method (3 1/2). Sometimes this unexpected mismatch between the two answers, arrived at by two different methods, catches the eye and causes people to raise questions and doubts. Why are there two different answers? Another special feature of the problem is that the fractions used - 1 3/4 (or 7/4) and l/2 have the following special relationship: a/b + c/d = (a/b + d/c) (b) One thing to notice about this relationship is that the quotient for the left-hand side of the equation, a/b -:- c/d, is a multiple of what would result from dividing the same dividend (a/b) by the reciprocal of the divisor (d/c) (see the right-hand side of the equation). So, in the case of the 1 3/4 + 1/2 Problem, we can say that the quotient of 1 3/4 + 1/2 is a multiple of 1 3/4 + 2. A second thing to notice is that this “multiple” is not just any multiple -- specifically, the quotient for the left-side, a/b + c/d, is exactly “b” times the embedded quotient (a/b + d/c) on the right-side. For the 1 3/4 + 1/2 Problem, this means that the quotient of 1 3/4 + 1/2 is exactly four times the result of 1 3/4 + 2. This special relationship becomes important as people try to make and interpret pictorial representations of what they believe to be 1 3/4 + 1/2 -- pictures that actually represent 1 3/4 -'- 2. Consider, for example, the following representation. 242 This is a typical representation created when people attempt to depict a story context for 1 3/4 + 1/2 that is actually a story for dividing in half or dividing by two. As mentioned earlier, this representation can reveal an unexpected quotient of 7/8. However, because of the special relationship among the numbers, it is also possible (and fairly common) to see an answer of 3 1/2 in this drawing: There are three and a half quarters (or “pieces”) shown. Liping Ma (1999) describes this response in her interview with Ms. Francine. After creating a drawing for her incorrect story problem, Ms. Francine talks through her representation -- Would we get three and one half, did I do that right? [She is looking at what she wrote and mumbling to herself] Let us see one, two, three, yes, that is right, one two, three. They would each get three quarters and then one half of the other quarter (Ma, 1999, p.68). The relationship between the fractions chosen for the dividend and the divisor is a special feature of this problem that can add another layer of mathematical challenge to the work. Why does a story context for 1 3/4 + 2 appear to result in a valid illustration for 1 3/4 + 1/2 = 3 1/2? Was a mistake made? Is this a special case? Is this a feature of problems involving division by fractions? Other pairs of the fractions, sharing the same relationship,71 would also yield a similar scenario where an apparently valid picture would result from an incorrect story context. For example, 2 4/9 -=- 1/3 = 7 1/3 is another equation that is also open to the possibility of being interpreted as 2 4/9 divided into thirds (rather than dividing by 1/3) and being represented with a picture that shows both 22/27 (the answer to 2 4/9 + 3) as 7 . . . l The same relatronshrp can also be rewritten as: b = (d/c)2 Although this obscures the relationships among the quotients and the fractions used, it does simplify things for finding other possible numbers that would create the same scenario. 243 well as the misleading result of 7 1/3 ninths. However, this potentially confounding result is not seen with most pairs of fractions. For example, consider -- 13/4+ 1/3 = 51/4 If the operation was misconstrued as 1 3/4 divided into thirds (or divided among 3), the resulting picture might like something like -- Here, each portion (or “third”) consists of two “quarter” pieces and a third of a quarter piece (i.e. the answer to 1 3/4 + 3).72 This is clearly different than the 5 1/4 that should result from 1 3/4 + U3. 5 1/4 quarter pieces is not visible in this drawing in the same way that 3 1/2 quarter pieces were visible when depicting 1 3/4 + 2 instead of 1 3/4 + 1/2. The discrepancy between expectations and result -- between different answers and different representations -- raises questions about what might account for the mismatches. It indicates places where possible errors, faulty assumptions, or flawed ways of reasoning may exist. Distinctions and Definitions Finally, one of the most fundamental features of the 1 3/4 + 1/2 Problem worth highlighting is that the problem is centrally about the meaning of “division.” This in itself poses challenges for American pre-service teachers who often have only an informal, intuitive understanding of this arithmetic operation. Such understanding is 72 1 3/4 + 3 actually equals 7/ 12. However, 7/ 12, can also be represented as: 3/12 (or “one quarter”) + 3/12 (“one quarter”) + I/12 (or “one third of one quarter”) 244 developed tacitly through exposure with multiple examples over time. Although this informal understanding may prove adequate for passing tests and progressing through mathematics classes, it is also tends to be quite vague in nature and unnecessarily limited in scope and usefulness. John Dewey notes -- Vague meanings are too gelatinous to offer matter for analysis, and too pulpy to afford support to other beliefs. They evade testing and responsibility. Vagueness disguises the unconscious mixing together of different meanings, and facilitates the substitution of one meaning for another, and covers up failure to have any precise meaning at all. It is the aboriginal logical sin -- the source from which flow most bad intellectual consequences (Dewey, 1910/1990, p. 130). Much of what we “understand” in and about our world is understood in this vague, informal, example-based way. The ambiguity largely goes unnoticed until it makes a difference in some way -- perhaps giving rise to surprise, confusion, or mistake. The 1 3/4 + 1/2 Problem is a particularly good problem for bringing muddled meanings to the foreground. Confounding a “Division Problem” With a “Fraction Problem” In addition to confounding dividing by half with dividing in half in the ways described above, this confusion is sometimes further compounded when prospective teachers mix up “dividing” (an operation) with representing a “fraction” (a quantity). An example of this might start with drawing the dividend, 1 3/4, as seven fourths -- -- and then proceeding by dividing each of the fourths in half to arrive at an answer of 14/8. 245 Rather than depicting the operation, 1 3/4 + 1/2, this picture displays 1 3/4 (or rather, 7/4) as the equivalent fraction, 14/8. This confounding of division with fractions was noted by Ball (1988) in her interviews with prospective teachers. She writes -- In responding to my questions, the prospective teachers tripped over their interpretation of the problem as essentially “about fractions” and their limited conception of and repertoire for representing fractions (Ball, 1988, p.121) A part of this confusion is undoubtedly connected with the overlap between the part/whole interpretation of fractions -- often represented by (dividing a single object into some number of parts and “taking” a portion of them -- and the partition interpretation of division -- where division is typically portrayed as the splitting up some quantity (the dividend) into some number of groups (the divisor). Informal Sense for the Partition Model of Division The partition interpretation of division has been noted to be the dominant -- and most cases, the exclusive -- way American prospective teachers think about division (Graeber, Tirosh, and Glover, 1986; Ball, 1988; Ma, 1999). This interpretation of division, typically learned and understood in terms of whole number examples, is conceptualized as dividing some quantity (i.e. the dividend) into a certain number of parts (i.e. the divisor) to find the quantity that comprises each part (i.e. the quotient). For example, 246 6+3 Mr. Salling has 6 pens to distribute among 3 students — how many pens will he give each student? Although this interpretation of division is correct and works well with whole number divisors, it is often inadequate for making sense of problems involving fractions (as mentioned in the section, Meaning Making). Liping Ma (1999) discusses the shift in thinking that is required to account for fractional divisors in the partition interpretation of division: Finding a number when several units is known and finding a number when a fractional part of it is known are represented by a common model -- finding the number that represents a unit when a certain amount of the unit is known. What differs is the feature of the amount: with a whole number divisor, the condition is that “several times the unit is known,” but with a fractional division the condition is that “a fraction of the unit is known.” Therefore, conceptually, these two approaches are identical. This change in meaning is particular to the partition model (p.75). This broader understanding of the partition interpretation of division is often unfamiliar to prospective teachers, involving connections they have yet to make and parallels they have yet to draw. Most of the prospective teachers who attempt to use the partition interpretation for the 1 3/4 + 1/2 Problem are not successful at expanding their use of the partition interpretation to accommodate the fractional divisor. Instead, they tend find alternatives such as converting the problem into 1 3/4 x 2 or, mistakenly creating a story problem for 1 3/4 + 2. There are also some who draw upon a second interpretation of division, known as the measurement interpretation. Although this way of thinking about division has typically received less attention in the United States and may not be the first meaning of division that comes to mind, it is, nevertheless, helpful for considering what it might mean to divide by a fiaction. In the measurement interpretation, division involves 247 dividing some quantity (i.e. the dividend) by a certain unit or quantity (i.e. the divisor) to find the number of groups or portions that can be made (i.e. the quotient). For example, 6+3 Mr. Salling has 6 pens to distribute and each student must receive 3 pens — to how many students will he be able to give pens? This way of thinking about division requires little adaptation to use with fractional divisors and prospective teachers can often use the measurement interpretation to find sensible solutions to the 1 3/4 + 1/2 Problem — for example, 1 3/4 hours divided by half- 73” hour “sitcom time blocks, 1 3/4 cups of flour divided out by a half-cup measuring cup, 1 3/4 yards of ribbon divided into half-yard strips. The work that is done on the 1 3/4 —:- 1/2 Problem can serve to bring these various meanings for division to the foreground, allowing them to move beyond vague, informal, example-based understandings to clearer distinctions and defined concepts. Episode Description: Reflection on Discrepancies I assigned the 1 3/4 + 1/2 Problem for homework at the end of the previous lesson. There were many difficult and interesting mathematical issues embedded within the seemingly simple task of writing a story problem for 1 3/4 + 1/2; I expected that some of the prospective teachers would not have been able to come up with a story problem on their own and many others would have developed a story for an alternative expression rather than the one assigned. We could have certainly used the entire two- hour class session to work on the problem but we had a full agenda and, at most, we could only spend an hour doing mathematics. If necessary, we could continue our work in 73 “Sitcom” is a contraction of the phrase “situational comedy,” a type of broadcast television programming that is typically a half hour long. 248 the next class session that was to be devoted, almost entirely, to mathematical explorations but this hour would allow us a good start. Preparation for the Whole Group Discussion I quickly launched us into the problem of the day, reminding the class of the homework assignment they had been given. Backpacks zipped and pages ruffled as the prospective teachers got out their work. I suspected many of them found the assignment somewhat challenging and would appreciate some time to confer in small groups before beginning the whole class discussion; however, I was also mindful of the short amount of time we had available to us. It would have been my preference to simply begin with a whole group discussion, going over students' solutions, but I left the decision with them. A quick show of hands revealed a clear majority wanted to a few minutes to work on the problem and discuss it with others seated nearby. "If you need time to work by yourself, go ahead and work... on the problem," I said, addressing the class. "If you want to confer, confer -- and maybe one thing (to do) while you're talking with somebody is to try to understand what's going on with their story..." The students immediately broke into small groups and worked intently for about 25 minutes. As I read over shoulders or listened in as students read their stories aloud, none stood out in any particular way -- none seemed to appropriately represent 1 3/4 + 1/2 and all seemed to confound division by one-half with another operation. At times, I'd point this out to a student or a group: "You said, shared equally between two people? That seems like dividing by two rather than dividing by one-half. Why is that?" At other times, I'd let it go as group members proceeded to point out differences among 249 stories or raise questions about what others had written. It seemed they could have easily continued to work productively for much longer but I was anxious to get to at least one or two of the story problems before we switched to the next activity. So, I proceeded to ask for a couple people to share what they had so far. There was no response at first. The prospective teachers seemed unusually reluctant to share their solutions. Finally, Kathy said she'd be willing to share hers but warned, "This is pretty confusing this morning..." There were no other volunteers as Kathy made her way to the whiteboard. I waited. Eventually, Tasha said she'd be willing to write hers up on the whiteboard if she could have Beth share it. As Kathy and Tasha wrote up their stories for the class, I had the rest of the students wrap up what they were currently working on. After a couple of minutes, I called the class together and started the discussion by asking about the quotient for, 1 3/4 + 1/2: "First, I'd like to make sure we have some shared understanding about (what) one and three-quarters divided by half equals. Kathy has three and a half. Did anybody get something different...?" A few shook their heads, "No.". "So everybody got three and half as their answer...?" I asked. "Yeah," someone responded, and a couple people nodded in affirmation. It seemed there were no contentions about the answer to 1 3/4 + l/2; so I turned to see how Kathy and Tasha were progressing. Kathy's Story for 1 3/4 + 1/2 Kathy finished writing before Tasha and said she would go first. She read her story: 250 ' There were 2 groups of 4 people. One group of four people, and three people from the other group got together and wanted to form a separate group. They formed their own group of 7, which was 1 3/4 of the original 8 people. Each of the members of this new group chose one other friend to join. This made a very large group of 14 people. They all decided to split the group back up into small groups of 4. They did this successfully with three of the groups, but the fourth group only had 2 people. So the result was: 4 people = 1 group 4 people = 1 group 4 people = 1 group 2 people = 1/2 group She began her explanation saying, "Okay, this is kind of confusing but, Anyway--" One thing I had been coming to appreciate about Kathy was how willing she was to put her ideas out in public for consideration. The ideas didn't have to be polished; she didn't have to be certain they were correct. There was a quality of openness about her -- a willingness to share, an eagerness to take in others' thoughts and questions -- coupled with a sense of humility about herself and her work that was admirable. She seemed always ready to be proven wrong or shown a better way. Her claim that her story was quite confusing seemed a fair assessment. I didn't even see a "1/2" in the story: only at the end, as a part of the results. And there were numbers in there that I wasn't expecting (e. g. 14, 7, 8). I would've liked to have started off with a simpler, more straightforward story -- there were certainly many to choose from had others been willing to volunteer -- but it was Kathy I had called upon. Indeed, I had not noticed any appreciable differences among most of the story problems I read as I walked around the room and did not suspect Kathy's solution would take such a complicated form. 251 i Kathy continued, "I put that there were two groups of four people -- which makes eight people altogether." It was not unusual to start with two wholes (two pizzas, two candy bars, two yards of fabric); students often feel a need to begin with two wholes rather than one and three- quarters of something. However, Kathy's use of groups was not so typical. The use of " groups of things" (dozens of eggs, 10-packs of pens, groups of four people) introduced a complexity that isn't present when working with single wholes: Groups are wholes made up of other wholes. For example, a group of four people is comprised of individual people who are whole and countable all by themselves. It can be tricky to consistently talk about the group as the whole and the components that make up the group as only a part -- some fraction -- of the whole. Things can get confusing. This was certainly the case with Kathy's story: There were two groups-of-four; there were eight people. Already, the complexity of having different "wholes" was creeping in. "One group of four people and three people from the other group got together and wanted to form a separate group on their own. So that's seven people -- they'd leave one behind." Here, as people often do, Kathy tried to account, in her story, for having only one and three-fourths wholes rather than having two wholes. But it is a mess: There is one group-of—four plus three people. A listener or reader would need to do a small conversion to get the "units" consistent: Three people = 3/4 a group-of-four. Therefore, 1 group-of-four + 3 people = l group-of-four + 3/4 a group-of-four = 1 3/4 groups-of-four 252 This may or may not be a trivial calculation for those in her audience but she went on to confound the matter more saying, " So, right there, that's one and three-fourths of the original eight people." One and three-fourths of eight is quite different than one and three-fourths of a group-of-four. A small calculation to convert the units would no longer do the trick. Kathy then went on to read, "Each of the members of this new group chose one other friend to join." With the introduction of this "new grou " -- the group-of-seven -- and more people, the story's complexity increased. Kathy elaborated, "That's going to make the group increase by twice the amount" (emphasis added). The size of the group, the number of individuals, and the "whole" seemed, to me, incredibly jumbled. Kathy paused, turned toward the class, and asked, "Does everybody understand that?" Someone said they understood, so Kathy proceeded: "This made a very large group of fourteen people. They all decided to split the group back up into small groups of four. They did this successfully. . .. So the final result was: Four people equaled one group, four people equaled another group, four people equaled another group, and then the two people (remaining) equaled half of a group.... So there's the three and a half groups" (emphases added). And, she declared, "That's my story!" I wasn't sure what to say about Kathy's story or her explanation. There was so much I wasn't sure about, so many questions running through my mind. Where was the division (for 1 3/4 + 1/2)? Or, for that matter, where was the 1/2? It seemed she had transformed the equation from 1 3/4 + 1/2 to 1 3/4 x 2 but what did Kathy understand about this transformation? Was it only a rule she had learned in school or was it something she could justify if pushed to articulate it? There was also the issue of the 253 ever-changing group-size. Two groups of four switched into one group of seven (or, possibly, a group of eight), then became a "very large group of fourteen " and, finally, returned to groups of four. Did Kathy see the "whole" as changing in size? Was it just a limitation — a loose use -- of language? Did Kathy not see "group-size" as being a crucial component of working with fractions and division? There were many questions I wanted to ask in order to develop a more complete picture of what Kathy understood. But I was eager to hear what questions and comments other students had. Their questions would allow me to get a better feel for how the class, in general, was thinking. It would also provide them with opportunities to try to get at Kathy's thinking: As prospective teachers, this capacity to probe another's way of making sense of something is an important skill to develop. I asked, "Do people have questions or comments for Kathy about what she did?" Discussion of Katly's Story "How did you get," began Heidi, "The one and three—quarters of eight people?" This was a good question with which to begin: It directed us to a key point in Kathy's explanation where the "whole" was not at all clear. It was also a place where Kathy's words, taken literally, were quite misleading: One and three-quarters of eight is NOT equal to seven. Kathy emphasized how she had started off her story saying that there were two groups of four people. And, re-reading what she had written, she said "One group of four people, and three people from the other group got together and wanted to form a separate group, so --" Heidi filled in the blank, "So one and three-quarters groups?" 254 "Yes," confirmed Kathy. And they continued on to the next question. Heidi seemed satisfied with this clarification but I was not. I wanted to press Kathy to say more about how she was thinking about the shifting group size. One reason I wanted to pursue this was to get clearer about what Kathy was thinking but another reason had to with the mathematical importance of the "unit", coupled with the cognitive challenges of learning to attend to the "whole", when working with fractions. This work of finding and examining "big" mathematical ideas in the elementary curriculum, as well as identifying things that are (or might be) difficult for students were central to the goals of the course and, more importantly, fundamental to the work of teaching mathematics. The conversation quickly moved on, however. Chris jumped in asking Kathy, "What happens to that one that's left out?" The two of them then began discussing possible scenarios to account for the departure of the eighth person. This was not what I had intended. I had no interested in developing the story line further. I stepped in to redirect the discussion: One thing that's really challenging as a teacher is listening hard to your students and trying to figure out what -- how are they thinking about the problem... One thing that would be good for you to try to develop are skills that it would take to ask questions of Kathy.... Why don't we try to work on that?... I would like to ask more questions about that (Kathy's story problem) but I would like to give you the opportunity to try to think really hard: How can I ask Kathy a question that will help me better understand what she's thinking about this problem? There were no volunteers so I asked Kathy to help us see a few things in her story problem: I'm having a little bit of trouble seeing... the numbers. Where's the one and three-quarters? Where's the one half? And where's the division that's going on? Could you kind of highlight for me where those things are in your story? 255 Kathy began by rereading the first two sentences of her story: "There were 2 groups of 4 people. One group of four people, and three people from the other group got together and wanted to form a separate group." At that point she started to explain, "Seven out of eight equals one and three-fourths." This was a rather odd statement outside the realm of Kathy's thoughts; we usually think of "seven out of eight" being "seven-eighths." Tammy began to make suggestions about how Kathy might improve the wording of the story. "Maybe write, 'Which was one and three-quarters groups of the original two groups.’ Maybe write it like that? Would that clear it up?" Puzzled, Kathy asked, "To do what?" "... Maybe say, 'Which was one and three-quarters groups of the original two groups'?" suggested Tammy. "Like write that on, maybe, so it's easier to read?.... If you say, 'Of the two groups,‘ then we know where you're getting the three-quarters. See what I mean? 'Cause (when) you think of 'one,‘ you're thinking of four over four. And then the three-quarters of something else." "Um. Oh. I didn't understand that," Kathy said. "Okay," Tammy started again. She pointed to the white board and read, "'Which was one and three-quarters groups of the original eight people'?" "Yeah," said Kathy. Tammy continued, "... It wasn't -- You weren't talking originally about eight people. We were talking about groups of..." "How about," Kathy jumped in, "'Of the original two groups equaling eight people'?" 256 "Yeah," said Tammy, "that would probably, I think, clear it up." Although Tammy seemed satisfied, I believed she was noticing something about the importance of the unit -- of the groups-of-four -- which Kathy was not seeing so I pressed Tammy asking, "Why is that important, Tammy?" Tammy responded saying that "when you're reading it, you see eight people... you don't think of it as two separate groups 'cause you're seeing eight people." It seemed the issue of "units" was finally being raised. I pressed further, "What is one and three-quarters of eight people?" Tammy replied, "It's seven which answers that--" This surprised me; I had assumed that Tammy was seeing what I was seeing: That seven people was NOT one and three-quarters of eight people -- it was one and three- quarters of a group-of-four. Despite being taken aback, I continued with my press: " Is it? What does one and three-quarters of eight equal?" A number of people began talking at once. "Let's just pause for a second." I said loudly, exerting my voice over the din. "If eight is a whole... what would one and three-quarters equal?" There was silence. A few began to scribble out equations. "Would it be more than eight or less than eight?" I asked. I could hear a whispered, "Oh!" as some people came to realize what I was getting at. "It would be more," students said. Shelly was the first to calculate fourteen as the answer to one and three-quarters of eight. This seemed good to me. Not only were we able to raise the issue of the "unit" being unclear in Kathy's story, this was a perfect demonstration of why it is so important to attend to the "whole." One and three-fourths of eight (i.e. fourteen) people is clearly, as 257 we had just seen, not the same thing as one and three-fourths of a group-of-four (i.e. seven) pe0ple. However, this "perfect demonstration" was not as perfect as I believed nor were things as clearly seen as I had hoped. Shaya entered the conversation, voicing a sentiment shared by others -— I understand what you're saying, that that's not one and three-quarters of eight people, but it is one and three quarters of two separate groups. And you're supposed to know that from the beginning if you read it. I don't see what the problem is with her writing eight people. Tammy, as well, despite her earlier efforts to help revise Kathy's story to account for the groups, explained -- I didn't have a problem with it. I'm just trying to say maybe that would help for where other people might be confused at. Because I could see where that would clear it up. Kathy began to tinker with her story at the whiteboard to address the potential confusion Tammy had raised. She underlined the "2" and the "4" in her first sentence changing it to, "There were 2 groups of 4 people," in order to "make them stand out more." Elizabeth, however, joined the conversation reinforcing the idea that Beth’s story needed no revision-- I think that you explained it quite clear that it's one and three-fourths by saying one group of four people -- that's one group -- and three people from the other group, which is three-fourths... I don't think we need to change anything... I think it's fine. Discussion of Our Discussion The conversation continued a little longer: Some students made further efforts to find out more about why Kathy did the things she did; others persisted in searching for 258 better wording. Eventually, however, the side-conversations increased in frequency and volume. There had been times in the past when the whole group discussion fractured into numerous small conversations but I had never before had this much difficulty holding the class discussion together. Something was going on and I needed to find out what it was. Perhaps Kathy's story was just too much for us to take on and we needed some time to discuss it in smaller groups or move on to something simpler for now. I raised my voice above the din -- I'm having a hard time concentrating with all the side conversations going on today.... Sometimes it's that you are talking about the problem because you're trying to work on it and, in some ways, I want to encourage that, but in other ways it's also hard to follow (the conversation). Is it that you that you need more time to just kind of think about it a bit? Someone immediately and assertively replied, "No!" Chris added, "We need to stop talking about it." It was clear that quite a few of the students were frustrated and wanted to stop what we were doing but I couldn't tell, from what was said, where the problem lay. "I'm not quite sure," I said, "If that means you really don't understand and you're feeling upset or --" A voice broke through saying, "No. We get it." A flood of students began to comment along similar lines -- Megan: We all get it; it's going on too long... We all figured it out. Tasha: Everyone's sick of rehashing (this). Tess: They keep drilling her (Kathy). Tasha: If she's ready to sit down, she's ready to sit down an hour ago!74 7‘ We had been discussing Kathy's story for approximately twenty minutes at that point. 259 Chris, trying to smooth things over, suggested, "Maybe if we talk about the other problem (Tasha and Beth's story) for a while --" I was, however, determined to discuss this issue. There would be other times in the future when mathematics problems would take longer than they believed necessary and times when the mathematics might initially seem "elementary" or trivial to them. We needed to find a way to deal with this. I interrupted Chris and, turning to the class said, "Wait, no, I want to address this.... I think this is a really important issue... So, why don't we just take a minute and talk about this." "It's not the length of the problem," began Elizabeth. "If it was a difficult problem... then we'd feel like we were shooting towards a goal but we all understood the answer to the problem even before she started." It was a bit surprising to hear Elizabeth's claim that they "all understood the answer to the problem even before" Kathy had started. I wasn't surprised that they understood how to find the answer to 1 3/4 -:- 1/2 before Kathy went up to the board -- indeed, I would have been concerned if they couldn't do this calculation -- but I was surprised Elizabeth thought the problem was merely to find the answer. I had been so intent on getting at Kathy's reasoning and opening up various mathematical issues embedded within her story that calculating an answer seemed remote, distant from what we had been doing. Elizabeth continued, "It would be different if we were second graders but to go over this problem for how long it has been, I personally feel like it's way too long..." This belief that the problems we did were actually for elementary children and not adults had come up before and, although I had tried to emphasize that I believed they 260 were challenging for adults, I had not understood how powerful -- how tightly affixed -- this view could be. Likewise, had not I appreciated how this interpretation could contribute to seeing the problems as simple and therefore not worth much time or effort. Tammy joined in, voicing similar frustrations. "It's like we understand it, you know? I'm feeling now like I'm going off in my own world... I understand what she did originally when she first went up. I knew what she was doing. It just like seems like we've got it. It's not like we're totally clueless." Like, Elizabeth, Tammy felt she had quickly apprehended what there was to understand. However, for Tammy, it wasn't only understanding how to calculate 1 3/4 + 1/2: Tammy also understood what Kathy was "doing." Amidst the frustrated and heated voices, JoAnn -- who rarely spoke class -- entered the conversation. "What we're trying to do is not for us to understand this problem -- we're all adults and we all know how to do this problem -- but it's to get into the minds of other people and see ourselves as teachers, not as learners." I wasn't sure what I thought about her statement that, "What we’re trying to do is not for us to understand this problem" and her claim that the goal is to see themselves “as teachers, not as learners”: There was much I wanted them to learn about the mathematics in this problem. She did, however, introduce the possibility that there might be more going on in our mathematics discussions than met the eye. Her comment directed us to consider what there might be, beyond calculation, we could learn fi'om our work together. Kathy built on JoAnn's comment saying, "This is like a good model for when we have our own classroom because you're always going to run into situations where you have students who understand it but you have someone at a different level where they 261 don't understand it. And I think that having students lead discussions and discuss together is allowing them to get involved more. And so that's helping people who possibly don't understand." For Kathy, the discussions were also "modeling" a way of teaching -- one that addresses some of the challenges of teaching students "at different levels." Like others, however, Kathy continued on to echo the sentiment, "I know that we all do understand that problem but, in situations where students are at different levels, I think that's a good way to teach it" (emphasis added). "I agree with JoAnn," Tess chimed in, "Because... I completely understood what she (Kathy) did on the problem. But then other people started to bring up things and... (I thought,) 'Oh, you know, I didn't see that. I didn't look at that.‘ Because I understood -- I thought I understood what she was doing." For Tess, the discussion helped her to "see" and "look at" more than she had before, despite her sense that she already "completely understood." JoAnn and Kathy had tried to point out how the discussion might generally allow for more than meets the eye; Tess, however, was the first to talk about how the discussion allowed for more than met her eyes. The conversation continued on for a while, drifting into other aspects of our class discussion. However, Elizabeth raised a question that returned us to JoAnn's statement about "what we're trying to do" as we work through these mathematics problems. "JoAnn's comment was very good," began Elizabeth. "So, why, when we're doing the problem, I feel like I'm being asked every five minutes, 'Do you guys understand?’75 And we do understand... I mean, I have a problem -- does (Kara) really not understand what we're talking about? or is she talking about, 'If I was a student?‘ You know?" 75 It was actually the students, sharing their solutions and stories up at the whiteboard, who used the phrase, "Do you guys understand?" I usually asked for comments and questions from the class and I often asked the person sharing to say more. 262 Something switched in the conversation, with Elizabeth's question, and the students began to discuss my teaching practice as their instructor. In some ways it was bizarre; they were talking about me as if I weren't in the room. In other ways, it was exciting; they were behaving like students of teaching, questioning and examining a practice to make sense of it. Deidre: I just wanted to make a comment to Jennifer that... she said, that she (Kara)... didn't get the problem. But (Kara) said, "If I don't get it, how could somebody explain this to me so I can get it?"76 Megan: But did (Kara) really not get it? or was she... a child? Elizabeth: I mean, I know Kara understands. I'm not... They seemed genuinely puzzled in a way that I had not seen them before: If Kara understands the mathematics, why does she say she doesn't understand? Is it an act? Is she pretending to be a child who doesn't understand? Or could it be that she really doesn't understand something? I raised my hand, partly in jest but also from a sense that, for the moment, I had become just another participant in the discussion: not the leader of it. It felt — and almost sounded — as if the instructor had left the room and we were now discussing her teaching. Elizabeth paused as my hand went up and declared, "You do understand!" Her emphatic statement was directed straight at me. It seemed a plea for clarification, an incomplete sentence I was supposed to fill in: "You do understand (so why do you say you don't?!)" 7" This was not something I had said. This statement (or something similar) was not on the video or audio recordings of this class or any prior class. In this case, I was only having difficulty accessing what Kathy had done and what she understood. It would have been uncharacteristic of me to say, in such circumstances, " lfl don't get it." 263 There was much I wanted to say in that moment -- things I had been thinking about all through this discussion of our discussion. I began by trying to address Elizabeth's unspoken question -- I understand a lot about the math going on in this problem and it's very complicated -- not just for children but also for adults. It's a complicated problem. And, part of what I see (as) my job... is trying to help you see and appreciate some of the complexity in this math.... I do know a lot about this problem and I know a lot about what's going on in it. So, Elizabeth, in that sense, you're right. I wanted to confirm Elizabeth's claim that I did indeed "understand" but I wanted to also communicate that there was a lot of mathematics there to understand -- more than may meet the eye. Their assumption that the problem was trivial, something only for children, was not productive. I hoped to plant a seed of doubt in the firm conviction that they could all indeed see, at glance, all that there was to see about the mathematics we were doing. In addition to the complexities of the math, there were also the complexities of Kathy's understandings and ways of thinking. I continued-- I really didn't know what Kathy was thinking about this problem... Were the “groups” slipping for her somehow? Or was there a good reason why, in her head, (she thought) you should divide that fourteen by four? I have some good reasons about why that fourteen should be divided by four but I didn't know if Kathy had good reasons... (she had said) something like, "Ah, and then I just sort of divided it by four because the answer needed to come out to three and a half." I suspected that Kathy had other reasons... about why it makes sense to divide by 4. I didn't know what they were. So I pushed her more on that. Although I was not able to articulate it well, there was much more to knowing "what Kathy was thinking about this problem" than simply the task of "following" her line of reasoning. Rather, it was the pedagogical work of locating areas of confusion and fragile understandings —- finding out where, out of all possible points of confusion and mathematical challenge, Kathy might be getting muddled. This is a pedagogical effort 264 that involves, in part, an attempt to appreciate a way of reasoning quite different from one’s own — an empathic projection -- but it also involves an attempt to critically discern what is understood relative to what there might be to understand. There may have been ways in which the students "understood" Kathy's words and decisions, but there was more to explore about what Kathy did and did not understand. I asked Elizabeth if these comments addressed what she had raised. She replied, "Yeah," however, I wasn’t sure what sense she (or other students) made of my words. Nevertheless, we had already used up the time allotted for the mathematics work and I needed to get them started on the next activity. It seemed clear that nothing had been "settled." There would be other times when the discussions seemed trivial or exceedingly long. There would be other times when people felt like they had " gotten" whatever there was to get and were impatient to move on. I wanted us to have some ways of productively working through such times -- One main issue that still remains unresolved is this whole thing about a math problem just taking up a lot of time.... You (may) feel like you get it and you zone out or something like that. Okay, that's reasonable. There might be a way in which, yeah, you really do get it. But one thing other people seemed to be raising is that there's a lot of... interesting (things) to be paying attention to.... I'd really like to get to the point where you're asking a lot of the probing questions to get at what someone's thinking. So, that's something to try to develop more... You can practice. You know, try out some questions... try some stuff on and see if you can get at what a student is saying. Try it... This is a good place to practice. I had hoped this class would help them develop dispositions and skills for accessing others' ways of reasoning. This was, in many ways, the point JoAnn made when she claimed that an aim of the class is “to get into the minds of other people and see ourselves as teachers...” I continued on to add -- 265 (And) ask yourself, "Is there more in the mathematics... to be learned that I'm not really seeing right now?" Perhaps, like Tess, students could come to see new things despite feeling they already "completely understood." Finally, I wanted to create some productive direction for these feelings of frustration to go when they surfaced. Rather than building and blowing up, it seemed possible to use them to move us forward in our discussions or, at least, to help us see when (and maybe why) things became so frustrating -- Maybe, when you start hitting your frustration point... we can start raising (that fact) and keeping tabs on (it). So if you start getting frustrated, why don't you just raise your hand and say, "I think I get it —can we move on?" Or, "Is this a place where we can move on because I feel like I'm getting it and I'm starting to zone out" Or something like that. Why don't we experiment with that and see how that goes? Okay? ' Perhaps, if we couldn't prevent fi'ustration from surfacing, we could prevent it from mounting to a point where it would grind us to a halt. With that, I moved the class on to the next activity. Episode Analysis: Differences the Make a Difference Significant differences surfaced during this lesson between the prospective teachers’ understanding of the mathematical work we were doing together and the intended purposes, meanings, and goals for the work. In earlier class sessions and for much of this lesson, these differences went unnoticed. The pre-service teachers had come to play apparently appropriate roles within the unfamiliar terrain — shaping their words and actions to patterns that had developed within the class — without necessarily challenging or changing their fundamental understanding of mathematics or what it might mean to do mathematics. At the same time, intently focused on cultivating the classroom 266 context, I only saw and attended to the overt signs of change — the shared expectations, language, and routines that appeared to be developing nicely — without seeing or seeking out signs regarding the prospective teachers’ unspoken understanding of our mathematical work together. However, differences came to the foreground, amplified through a positive feedback loop, as we tried to communicate from our very different perspectives during this class session. The difference-amplifying feedback began subtly — a moment or two in the discussion when I felt the conversation moved on too quickly, switching topics without adequately considering 3 students’ mathematical idea that had been raised or sufficiently unpacking a significant piece of mathematics in front of us. In response, I attempted to slow the pace and pressed for more discussion. However, the slower pace felt unnecessary to the prospective teachers who believed they understood what there was to understand. In a variety of subtle and not-so-subtle ways, these students responded by displaying an eagerness to move on, conveying their sense that there was really nothing more to discuss. This, however, only served to further convince me that they were missing the very things I hoped they would get from this work and I responded by slowing the pace even more. This, in turn, only increased the pre-service teachers’ desire to move on and stepped up their efforts to bring quick closure to the discussion. However, the more they tried to speed things up, the more I worked to slow things down. Eventually, as tension and frustration grew, it became quite obvious that important differences existed that could no longer be overlooked. In section below, I examine some of these differences. 267 Description of Differences Although a variety of viewpoints could be glimpsed in the prospective teachers’ comments and questions, there were several recurring ideas that surfaced about the fundamental nature and purposes of our work together on the 1 3/4 + 1/2 Problem — ideas that were particularly significant in shaping people’s experiences and driving the positive feedback loop in this lesson. In the sections that follow, I examine three sets of ideas that arose during the class discussion and contrast them with the intended meanings and goals for the work. The Nature of the Task One theme that surfaced early in the discussion was the interest in centering the work on crafting a well-written story — to make sure the storyline is easy to follow, the descriptions clear, the sequences logical, the motivations of its actors plausible.77 For example, Chris’ early question about “what happens to the one (person) who is left out” (i.e. from the original eight people) focused our attention upon elaborating the storyline to account for why the person left. Likewise, Tammy’s efforts to help Kathy revise her wording of the problem - from emphasizing 1 3/4 of the original eight people to emphasizing 1 3/4 of the original two groups of four — were largely efforts to make it “easier to read,” to “help where other people might be confused.” Rather than addressing a mathematical problem she perceived with shifting the units from individual people to groups-of-four, Tammy was attempting to address a potential problem in the exposition that might cause readers difficulty. 77 This may have been influenced in part by their earlier work during the Literacy/Language Arts Block. 268 Undoubtedly, these types of considerations are important for teachers when writing story problems. Indeed, I planned to spend a little time at the end of our work on the problem examining not only issues of plot and clarity but also issues of creating stories that make sense given the numbers chosen, are sensitive to the audience — appropriate for children and the context school, and account for the interests and experiences of young students. However, these considerations were not meant to be the main focus of the task. Instead, the task was meant to engage prospective teachers in the mathematical work of giving meaning to the operation of division (particularly, the poorly understood instance of division~with a fractional divisor), identifying and analyzing errors, and developing better conceptual distinctions and definitions. This mismatch in understanding was one of the first I perceived and to which I responded by attempting to slow down and refocus the conversation. It was one of the important differences underlying the positive feedback loop. The Nature of the Problem Another theme that came up later in the lesson was the idea that the prospective teachers “got it” and were more than ready to move on. One thing the pre-service teachers seemed to mean when they said they got it was that they knew how to find the quotient for 1 3/4 + “2. These sorts of calculations are common in the school curriculum and familiar to adults. As children, students typically learn an invert-and-multiply algorithm for solving problems involving division by fractions: Convert mixed numerals or whole numbers into (improper) fractions Invert the divisor and change the division to multiplication Multiply “across” 269 Simplify the answer (if not already in simplest terms) For most of the prospective teachers, this sort of calculation was indeed quite trivial. Although it was necessary to find the quotient in order to come up with a complete solution for the 1 3/4 + 1/2 Problem, it was not intended — or expected — to present a problem for adult learners. Nevertheless, there were some for whom the task appeared to be focused upon the unproblematic problem of finding an answer to 1 3/4 -=- 1/2 = _. This is best seen in the explanation Elizabeth gave for her own feelings of impatience: “If it was a difficult problem... then we’d feel like we were shooting towards a goal but we all understood the answer to the problem even before she (Kathy) started” (emphasis added). Her comments about the simplicity of the problem — involving a single answer (the answer) that could be grasped-even before the first student presented her story problem — seem to indicate an interpretation of the problem, focused upon calculating the quotient. Such an understanding would justifiably lead her to feel that she, and the others in class, have “got it” and are more than ready to move on. The Work of Understanding Another A final theme that came up was the idea that the prospective teachers “got” Kathy’s story for the 1 3/4 + 10 Problem and were ready to move on. For some, this may have meant they were able to follow her storyline: They were able to understand the logic and details of the story as a story and could see the scenario she created. This seemed to be the case, for example, with Shaya who told Kathy — I understand what you’re saying - that that’s not one and three-quarters of eight people but one and three-quarters of two separate groups... (but) you’re supposed to know that from the beginning (sentence) if you read it. I don’t see what the problem is. 270 For others, this may have meant they were able to follow Kathy’s reasoning as she attempted to write a story to represent the mathematical equation, 1 3/4 -:— 1/2 = 3 1/2. Despite my own struggles to understand how Kathy’s story mapped onto the mathematical expression, 1 3/4 + 1/2, there did seem to be a sense in which other prospective teachers really could understand what Kathy was doing with a clarity and sameness-of-view that I did not share. Although I had been pressing the class to try to " get inside of" Kathy's reasoning, for many of the pre-service teachers who understood division by fractions exclusively in terms of the standard algorithm taught in schools (see above) this task may have presented no real challenge -- and no possibility for new insight. Indeed, Kathy’s story maps quite well onto the algorithm described in the previous section: mm The 5‘6 S ofthe A1 with!“ There were 2 groups of 4 people. 0 O O O (2 groups-of-four) One group of four people, and three people a. Convert mixed numerals/whole numbers from the other group got together and into (improper) fiact ions wanted to form a separate group. They O O O 0 formed their own group of 7, which was 0 O O 1 3/4 of the original 8 people. (7/4 groups-of-four) 271 b. Invert the divisor and change the division to multiplication (7/4 x 2) Each of the members of this new group C Multiply across chose one other friend to join. This made a O O O 0 very large group of 14 peOple. O O O OO 00 OO 0 (14/4 groups-of-four) They all decided to split the group back up into small groups of 4. They did this successfully with three of the groups, but the fourth group only had 2 people. So the 0 ob result was: 0 O :.| 8? d. Simplify the answer (if not already in simplest terms) 4 people = 1 group 4 people = 1 group 8 O O 4 people = 1 group 2 people = 1 /2 group (3 1/2 groups-of-four) Several prospective teachers, like Tess, commented that they “completely understood what she (Kathy) did on the problem.” For some, like Tammy, persisting in our efforts to understand Kathy’s solution seemed almost so trivial as to be insulting. She commented, “I understand what she did originally when she first went up. I knew what she was doing. It’s just seems like we got it. It’s not like we’re totally clueless.” Although these were important ways to understand another’s work on this problem there was a way in which I, as the instructor, was also trying to understand where Kathy was confused or where her understanding was fragile. I was actively seeking ideas that were potentially muddled for her or concepts that were confounded. As her teacher, it was not enough to only follow her - I was also looking for new directions to take her, ways to expand upon her current understanding. Perhaps this was not possible for the prospective teachers to do at this point in the class but it was, 272 CHAPTER 5 Epilogue In the weeks that followed, there were students who continued to feel frustrated at times and, although they did not bring up during class, they would write in their mathematics notebooks, voicing their objections to the pace of the discussions, concerns about the content we were studying, and confusion about the goals and purposes of the course. I followed up on each frustrated note in writing or, in some cases, by meeting with the student individually. Two weeks after the class discussion regarding our work on the 1 3/4 + ‘/2 Problem (Chapter 4), the expressions of frustration had almost entirely ceased but Deidre, who had missed several class sessions during the intervening time, seemed to grow more agitated. Afier class on November 17, Deidre wrote a message to me in her notebook, expressing her annoyance with the mathematics work and questioning its value given the many things going on in her life. She wrote, This type of thing is really annoying to me. I really don’t like going on and on with something that will never be solved! I have a child and other classes where within the next two weeks I have two 10-12 page papers due among other things going on. This just seems like a waste of my time. We will just continue to go on and we will end up right back where we started. Kara, maybe you can help me with this. I’m getting frustrated with this whole class. I have other things that are important to me in my life and I just don’t feel like I’m learning anything from these discussions. . .. I just don’t know where we are going with all of this! (Deidre) Soon after this entry was written Deidre and I met to discuss her concerns. She asked many questions about the work we were doing and the purposes of the class. I 274 nevertheless, something for which I had been pressing in my own attempts to slow the pace of the discussion and probe Kathy’s (and other’ 5) ideas. New Information Flows The discussion regarding our work on the 1 3/4 + 1/2 Problem brought attention to the existence of differences in how people understood the unfamiliar class context and opened up other possibilities for interpreting our work together. This was significant for the class, allowing us to initiate a larger conversation about the mathematical work in which we were engaged. Additionally, I made a suggestion at the end of class that also served to open new possibilities for thought and discussion. Anticipating that things were far from settled, I suggested that we keep tabs on and begin to call attention tothose instances when the students felt fi'ustrated about the pace of the class. As it turned out, no one raised such instances of frustration during class however many of them did write to me in their notebooks and I followed up on these in different ways. One such instance is described in the epilogue that follows. 273 Chapter Five Epilogue In the weeks that followed, there were students who continued to feel frustrated at times and, although they did not bring up during class, they would write in their mathematics notebooks, voicing their objections to the pace of the discussions, concerns about the content we were studying, and confusion about the goals and purposes of the course. I followed up on each frustrated note in writing or, in some cases, by meeting with the student individually. Two weeks after the class discussion regarding our work on the 1 3/4 + ‘/2 Problem (Chapter 4), the expressions of frustration had almost entirely ceased but Deidre, who had missed several class sessions during the intervening time, seemed to grow more agitated. After class on November 17, Deidre wrote a message to me in her notebook, expressing her annoyance with the mathematics work and questioning its value given the many things going on in her life. She wrote, This type of thing is really annoying to me. I really don’t like going on and on with something that will never be solved! I have a child and other classes where within the next two weeks I have two 10-12 page papers due among other things going on. This just seems like a waste of my time. We will just continue to go on and we will end up right back where we started. Kara, maybe you can help me with this. I’m getting frustrated with this whole class. I have other things that are important to me in my life and I just don’t feel like I’m learning anything from these discussions. I just don’t know where we are going with all of this! (Deidre) Soon after this entry was written Deidre and I met to discuss her concerns. She asked many questions about the work we were doing and the purposes of the class. I 274 responded to some of her questions but, more than getting answers, Deidre seemed to want me to come to understand her situation within and beyond the class. After our meeting, I did not hear more from Deidre about the matter nor had I noticed any drastic changes in her class participation or work. However, the next couple of weeks brought many subtle changes to the class itself. By December 1, there were noticeable shifts in the how prospective teachers were using their mathematical knowledge and the ideas of others to develop deeper and more complex understandings of elementary mathematics. This could be seen both in the class discussions as well as in the prospective teachers’ notebook entries. On the evening of December 1, the night before our first and only mathematics quiz, I had provided students with a “Quiz Review” to help them prepare. Almost all of them had begun working through the review sheet in small groups during class, helping one another come up with solutions and refine their explanations. That evening, Deidre wrote the following in her notebook — Ok, it’s Thursday night. I’m going to try to work on these review problems. I also want to try to figure out a way to explain 2b to Rachel; by the end of class she was really frustrated so I told her I would go home and try to figure it out for her. I am working on problem 2a.78 I cannot even figure that one out... 1 2/3 +1/3 Here’s a picture: 78 Problems 2a and 2b are shown below— 2. Write and solve a story problem for the following. Illustrate your problem and it's solution. (a) 1 2/3 + l/3 (b) 1 2/5 + 1/2 275 So 5/3 = E’ so now how do I figure out what 5/3 + 1/3 is equal to. I know when I flip and multiply that it equals 5, but 5 what? How do I illustrate that? If I drew this: a / 2» an . tr a This shows what the answer should be... a_l \E; ‘\ l: \ 5/3 + 5/3 + 5/3 =15/3 = 5 but this still doesn’t make any sense to me... I still don’t really know why 1 2/3 + 1/3 = 5. How can I possibly teach a rationale for this when I don’t get it myself? (next page) Well, I work(ed) through most of the problems; they got easier as I went on. I think I did them right. I’ll double check tomorrow; it’s 12:00 now and I need to go to bed! Goodnight! (Deidre) A few things in this December 1 notebook entry seemed to represent a shift away from the unproductive morass in which Deidre found herself two weeks earlier, when the work seemed to go “on and on with something that will never be solved,” and she felt she was not learning anything. First, Deidre seemed to have greater connection to her role as a teacher — within our class and, perhaps, in a more general sense as well. She noted that she wanted “to figure out a way to explain (question) 2b to Rachel (another student in class)” and later, as she struggled to make a pictorial representation of 1 2/3 + 1/3 = 5, 276 she asked, “How can I possibly teach a rationale for this when I don’t get it myself?” Notably, in both these statements, teaching seemed to be closely connected in her mind with learning. Second, as Deidre sat alone in her home writing this entry, she found ways to build new conceptual connections and develop a deeper, richer understanding of mathematics, moving herself productively between pictures and equations, between what she knew and what she didn’t know. This shift in Deidre’s understanding and in her ways of working was representative of shifts that had been occurring more broadly throughout the class. In the weeks following the discussion of our work on the 1 3/4 + 1/2 Problem, the processes described by Schwab (1959/1978) for learning from and within the unfamiliar terrain could be glimpsed in moments. He explains - New competencies for taking thought are roused. The new actions change old habits of thought and observation. Facts formerly ignored or deemed irrelevant take on significance. Energies are mobilized; new empathies roused. There thus arises a new a fuller understanding of the situation and a better grasp of the ideas which led to it. A revised practice is undertaken. The cycle renews itself (p. 173) One such moment in the class took place on December 6 as the prospective teaechers discussed the results of the mathematics quiz that was handed back to them that day. flaya's Question (December 6) Most students had done very well on the quiz. There were a few however who continued to struggle with the distinction between the partition and measurement interpretations of division. Shaya, in particular, had difficulty with Problem #4, which involved creating both a partition and a measurement story problem for the equation, 277 34 + 5 = 6 4/5. Although she believed she had created one of each type for her response, she had actually created two story problems using the partition interpretation. When I invited comments and questions about the quiz, Shaya raised her hand and asked if she could share her two story problems and have the class discuss them. She explained that she understood the measurement interpretation as, "'How many groups... can go into the general mass? How groups will I get?” She then went on to share, what she believed to be, her measurement scenario: There are five divisions of basketball. I have thirty-four students who can play basketball. How many groups can play in each division? So (1) divided thirty-four by five and got six and four-fifths groups, or seven people in each group and one group with six people. Shaya's story problem was complicated. There were five divisions, thirty-four students, and the question was, “How many groups can play in each division?” It was not clear how groups and divisions were related. And, in the "answer" Shaya arrived at — "six and four-fifths groups or seven people in each group and one group with six people" — it seemed groups and people were being blurred together. Aside from these problems, Shaya's story involved, dividing the thirty-four students into five parts: it involved the partition interpretation of division, not the measurement interpretation. In many ways, this reminded me of Kathy's story for 1 3/4 + 1/2 that was shared and discussed a month earlier. The ambiguity of the "groups," the uncertainty about the meaning of the results, the confusion about the meaning of the operation (division) was similar in both Shaya and Kathy's stories. However, the prospective teachers played a much larger role in this discussion. Heidi questioned Shaya about the "five" in her story, attempting to find out how Shaya divided the thirty-four students. JoAnn asked Shaya several questions, trying to get 278 f . clearer about how she was thinking about "groups" and "divisions." Kathy and Beth probed to find out what the result of "six and four-fifths" meant to Shaya. Through such questioning, the other students worked with Shaya to uncover some of the issues surrounding the use of "groups" and "divisions" and to surface underlying confusion regarding "groups" and individual "people." But still, despite seeing these problems, Shaya remained unclear about why this story problem was a partition one and not a measurement one. Earlier in the discussion, as people questioned Shaya about various aspects of her story, Julie had asked Shaya to draw pictures of her two story scenarios: The partition one and the measurement one. Shaya, however, had gotten distracted before completing them. For the partition scenario, Shaya had drawn on the board — oo oo 0 00 o o (o oo 00 000 oo oo0 o oo o oo o o oo oo 34 students divided among 5 desks (34 + 5) But the representation for the measurement scenario remained unfinished. As Shaya expressed her continued confusion about partition and measurement, Julie raised her hand and requested that Shaya continue drawing, adding, "Because I think once you finish your drawing (of the measurement scenario) it is going to look just like the other one (i.e. the representation of the partition scenario)" Although it was fairly common to request drawings during our conversations, Julie's request was unusual in that it was designed to help Shaya see something new. It was a pedagogical move. 279 Shaya returned to her drawing of the measurement scenario -- oo oo o. o o o .. o co. . .0 o oo '00 000 34 students and 5 basketball divisions (34+5) As Shaya completed her picture she exclaimed, "It does! It looks exactly (like the other drawing)" "You're not going to have six and four-fifths groups," Julie pointed out. "You're going to have just five (groups)." Laughing at the sudden insight, Shaya said, "Okay. Yeah. We do end up with just five. It's just like the tables (the scenario for her partition problem)" 280 CHAPTER 6 Conclusion The descriptions and analyses in the preceding chapters provided images of what it might mean to create an educational context for pre—service teachers where “the new is presented mainly in its new terms” and unpacked some of processes involved in cultivating such a context. Along with this, the earlier chapters also shed light upon the processes by which change can occur within an unfamiliar terrain as prospective teachers adapt to —— and adapt -— the context in which they find themselves. This chapter briefly reexamines the approach put forth by Schwab (1959/1978) in light of this effort to apply his ideas to an elementary mathematics methods course for pre-service teachers. In the discussion below, I highlight some of the findings and observations of this study that elaborate upon the processes sketched out by Schwab — filling in details not seen in Schwab’s broad strokes and extending the implications of his proposition to the context of elementary mathematics teacher preparation. The Struggle to Understand True to the initial process anticipated by Schwab, “From the first,” there was indeed a “struggle to understand” (p. 173). However, several notable things arose in the mathematics methods course that would not be apparent from Schwab’s brief comment about what can and does occur “from the first.” 1. The struggle takes on a variety of forms. There were some clear and undeniable signs of “struggle” in the first lesson — the most vivid being Shelly’s pained expression of feeling responsible for the delays and fi'ustration she sensed in the class. 281 However, much of the struggle to understand took less obvious forms and, rather than a clear conflict or conscious effort to overcome a well-defined obstacle, it was often experienced as frustration or confusion as routine efforts that had been successful in the past proved unsuccessful within the new context — or, even more subtly, it surfaced as a sense of uncertainty about the meanings and purposes of the work. The struggle is varied but patterned. Prospective teachers’ experiences within the unfamiliar terrain were diverse - indeed, no two appeared to be exactly alike. But nevertheless, there were specific occurrences that stood out, salient facets that drew attention, and clusters of common experiences that could be seen among the prospective teachers’ observations and experiences when the class was viewed as a whole. Such patterns imply the presence of important constraints. The Actions Undertaken Lead to Unexpected Consequences An important part of the process of learning from and within an unfamiliar terrain occurs as learners begin to act upon their nascent understanding of context. According to Schwab, As they translate their tentative understanding into action, a powerful stimulus to thought and reflection is created. . .. it creates new food for thought. The actions undertaken lead to unexpected consequences, effects on teachers and students, which cry for explanation (p. 173). This could certainly be seen in the methods course. For example, even as early as the first week — the first day — of the semester, as the prospective teachers tried to make sense of the mathematical work we did together, many made attempts to uncover the lesson-to- be-leamed or to conjecture about the possible intended benefits of the unusual work. 282 There was much that was unexpected and seemed to beg for an explanation. However, one thing that cannot be adequately appreciated from Schwab’s description of these early attempts to learn from and about the unfamiliar terrain is how unexpected consequences are created through the pedagogical efforts of the instructor, the design of the mathematical tasks, and students’ efforts to use the resources they bring within the new context. Constraints that are created for, encountered in, and brought to the unfamiliar terrain play an important part in shaping the learning opportunities that are available and pursued. There’s Reflection on Disparities Another important piece of the process, according to Schwab, occurs as learners begin to reflect on their efforts and experiences within the unfamiliar context. He writes, There is reflection on the disparities between ends envisaged and the consequences which actually ensue. There is reflection on the means used and the reasons for why the outcomes were as they were (p. 173). Such reflection was indeed significant for the class, allowing assumptions to be examined, difficult questions to be raised, and alternative ways of seeing and thinking to be made accessible. However, this study brought to light some of the challenges involved in becoming aware of and attentive to possible discrepancies. New Competencies Are Roused Schwab also sketched out a cyclical process that is initiated as people begin to develop new capacities for working within the unfamiliar terrain. He writes, At the same time new competences for taking thought are roused. The new actions change old habits of thought and observation. Facts formerly ignored or deemed irrelevant take on significance. Energies are mobilized; new empathies are roused. 283 There thus arises a new and fuller understanding of the situation and a better grasp of the ideas which led to it. A revised practice is undertaken. The cycle renews itself (p. 173). This study only provided a few glimpses of the type of shifts that occurred — among individuals and among the class as whole — as they developed an ability to work more comfortably and productively within what was once, an unfamiliar terrain. Capturing and examining such changes were beyond the data collected and the analytic methods used in this study. The subtle -— and often sudden - changes required a much more fine-grained examination than was within the scope of this work. “If the new is presented mainly in its own new terms” it seems there is indeed the potential for creating an “uncomfortable but productive” situation. 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