DOCUMENT RESUME

ED 384 568 SP 036 041

AUTHOR Featherstone, Helen; And OthersTITLE Expanding the Equation: Learning Mathematics through

Teaching in New Ways. Research Report 95-1.INSTITUTION National Center for Research on Teacher Education,

East Lansing, MI.SPONS AGENCY Office of Educational Research and Improvement (ED),

Washington, DC.PUB DATE Jan 95NOTE 35p.

AVAILABLE FROM National Center for Research on Teacher Learning, 116Erickson Hall, Michigan State University, EastLansing, MI 48824-1034 ($7.72).

PUB TYPE Reports Research/Technical (143)

EDRS PRICE MF01/PCO2 Plus Postage.DESCRIPTORS Case Studies; *Elementary School Mathematics; Faculty

Development; Grade 2; Grade 3; Higher Education;Inservice Teacher Education; InstructionalImprovement; *Knowledge Base for Teaching;Mathematics Curriculum; *Mathematics Instruction;*Mathematics Teachers; Primary Education; *TeacherCompetencies; Teacher Improvement

IDENTIFIERS NCTM Professional Teaching Standards; *TeacherKnowledge

ABSTRACTAs reformers urge elementary and secondary school

teachers to teach mathematics in new ways that highlight problemsolving and engage students in important mathematical ideas,researchers have been pointing out that few public school teachersknow mathematics in the ways that they would need to know it in orderto teach in these new ways. These researchers point to deficienciesin teachers substantive knowledge (their understanding of the "stuff"of mathematics), in their syntactic knowledge (their understanding ofwhat mathematicians do and of the nature of mathematical evidence),and in their attitudes towards the subject matter; they raisequestions about the possibilities for addressing these difficultiesthrough school-based staff development or university-basedmathematics courses. The present study explores the possibility thatchanges in teachers' own teaching practices may provide opportunitiesfor learning of and about mathematics. The study examines the casesof three primary teachers who, influenced by the National Council ofTeachers of Mathematics (NCTM) "Standards," made significant changesin the way that they taught second and third grade mathematics andwho also reported significant changes in their understandings oftopics in elementary math, their attitudes toward the subject matter,and beliefs about what it means to do math. The conclusion looks atsome of the reasons that teaching math in new ways may helpelementary teachers to learn some of what reformers say they need toknow of and about mathematics. (Contains 31 references.)(Author/ND)

Research Report 95-1

Expanding the Equation: Learning MathematicsThrough Teaching in New Ways

Helen Featherstone, Stephen P. Smith,Kathrene Beasley, Deborah Corbin, and Carole Shank

NationalCenter for Researchon Teacher Learning

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Research Report 95-1

EXPANDING THE EQUATION: LEARNING MATHEMATICS THROUGH

TEACHING IN NEW WAYS

Helen Featherstone, Stephen P. Smith, Kathrene Beasley, Deborah Corbin, and Carole Shank

Published by /National Center for Research on Teacher Learning

116 Erickson HallMichigan State University

East Lansing, Michigan 48824-1034

January 1995

This work is sponsored in part by the National Center for Research on Teacher Learning, College ofEducation, Michigan State University. The National Center for Research on Teacher Learning is fundedprimarily by the Office of Educational Moorland Improvement, United States DepartmentofEducation.The opinions expressed in this publication do not necessarily represent the position, policy, or endorsementof the Office or the Department

0 1995 by the National Center for Research on Teacher Learning

Ii

NATIONAL CENTER FOR RESEARCH ON TEACHER LEARNING

The National Center for Research on Teacher Learning (NCRTL)' was founded at Michigan StateUniversity in 1985 with a grant from the Office of Educational Research and Improvement, United StatesDepartment of Education.

The NCRTL is committed to research that will contribute to the improvement ofteacher education and teacherlearning. To further its mission, the NCRTL publishes research reports, issue papers, technical 110613,

conference proceedings, craft papers, and special reports on contemporary issues in teacher education. Formore information about the NCRTL or to be placed on its mailing list, please write to the Publications Clerk,National Center for Research on Teacher Learning, 116 Erickson Hall, Michigan State University, EastLansing, Michigan 48824-1034.

Directors: Robert E. FlodenG. Williamson McDiarmid

Study Directors: Linda Anderson, Deborah Ball, Daniel Chazan, HelenFeatherstone, Sharon Fe iman-Nemser, Mary Kennedy, G. W.McDiarmid, Barbara Neufeld, Kenneth Zeichner

Director of Dissemination: Debra Peterson

Publications Clerk: Tamara D. Hicks Syron

Office Manager: Linda Quint

Many papers published by the NCRTL are based on the Teacher Education and Learning to Teach (TELT)study, a single, multisite longitudinal study. The researchers who have contributed to this study are listed below:

Marianne Amarel Monica MitchellDeborah Loewenberg Ball Harold MorganJoyce Cain James MosenthalSandra Callis Gary NatrielloBarbara Camilleri Barbara NeufeldAnne Chang Lynn PaineDavid K. Cohen Michelle ParkerAda Beth Cutler Richard PrawatSharon Feiman-Nemser Pamela SchramMary L. Gomez Trish StoddartSamgeun K. Kwon M. Teresa TattoMagdalene Lampert Sandra WilcoxPerry Lanier Suzanne WilsonGlenda Lappan Lauren YoungSarah McCarthey Kenneth M. ZeichnerJames Mead Karen K. ZumwaltSusan Melnick

'Formerly known as the National Center for Research on Teacher Education (1985. 1990), the Center was renamed in 1991.

Abstract

As reformers urge elementary and secondary school teachers to teach mathematics in new ways thathighlightproblem solving and engage students in important mathematical ideas, researchers have beenpointing out that few public school teachers know mathematics in the ways that they would need toknow it in order to teach in these new ways. These researchers point to deficiencies in teachers'substantive knowledge (their tmderstanding ofthesnffofmathematics), in their syntactic knowledge(their tmderstanding ofwhat madiematicians do and ofthe nature ofmathematical evidence), and intheirattitudes towards the subject matter, they raise questions about the possibilities for addressing thesedifficulties through school-based staffdevelopment or university-based mathematics courses. Thepresent study explores the possibilitiy that changes in teachers' own teaching practices may provideopportunities for learning of and about mathematics. The authors examine the cases ofthree primarygrade teachers who, influenced by the NCTM Standards, made significant changes in the way that theytaught second and third grade mathematics and who also reported significant changes in their under-standings oftopics in elementary math, their attitudes toward the subject matter, and their beliefs aboutwhat it means to do math. The authors conclude by looking at some of the reasons that teaching math innew ways may help elementary teachers to learn some of what reformers say they need to know of andabout mathematics.

EXPANDING THE EQUATION: LEARNING

MATHEMATICS THROUGH TEACHING IN NEW WAYS

Helen Featherstone, Stephen P. Smith, Kathrene Beasley, Deborah Corbin, and Carole Shank

Over the past decade, various reports have iden-tified serious deficiencies in mathematics educa-tion in the United States (Mathematical SciencesEducation Board, 1989 &1990; McKnight, etal., 1987, National Commission on Excellence inEducation,1983). TheNational Council ofTeach-ers of Mathematics (NCTM) responded to callsfor reform with the publication ofthe Curriculumand Evaluation Standards (1989) and the Pro-fessional Standards for Teaching Mathemat-ics (1991). In these documents the NCTMpresents a vision of mathematics educationgrounded in three areas: cognitive psychology,philosophy of mathematics, and how mathemati-cians do mathematics. In the classrooms theStandards describe, teachers are:

Helen Featherstone, an associate professor ofteacher education at Michigan State University, isa senior researcher with the National Center forResearch on Teacher Learning.

Stephen P. Smith, a doctoral candidate in teachereducation at Michigan State University, is a re-search assistant with the National Center forResearch on Teacher Learning.

Kathrene Beasley is a third grade teacher at AverillElementary School in Lansing, Michigan.

Deborah Corbin is a third grade co-teacher at PostOak School in Lansing, Michigan.

Carole Shank is a second grade tncher at AverillElementary School in Lansing, Michigan.

Selecting mathematical tasks to engage students'interests and intellect;

Providing opportunities to deepen their understand-ing of the mathematics being studied and its applica-tions;

Orchestrating classroom discourse in ways that pro-mote the investigation and growth of mathematicalideas;

Using, and helping students use, technology andother tools to pursue mathematical investigations;

Seeking, and helping students seek, connections toprevious and developing knowledge;

Guiding individual, small group, and whole-class work(NCTM,1991,p.1).

Many teacher educators have argued that el-ementary teachers aspiring to meet such stan-dards would require subject matter knowledgethat differs as much in kind as in degree from thatwhich most now appear to possess (Ball, 1992;Ball & Wilson, 1990; McDiarmid,1992; Shulman,198P. They assert that teachers' knowledge issuspect in three areas: knowledge ofthe contentof mathematics, knowledge about the nature ofmathematics, and attitude toward mathematics.

First, some mathematics educators question whatprospective teachers learn ofthe content of math-ematics as students in elementary and secondaryschools. As Ball (1990a) points out, their under-standing of mathematics is often procedural andfragmented. The prospective teachers that Ballstudied were able to solve problems by followingstandard algorithms. However, they were unableto explain meanin4fully the mathematical reason-ing which lay behind those algorithms. They didnot seem able to connect in a coherent way thevarious bits ofmathematical knowledge they hadaccumulated. Evidence from the National Center

Mlehlaan State University, Ernst Lamina, Michigan 41524-1034 RR 9S-I Page I

for Research on Teacher Education's (1991)Teacher Education and Learning to Teach studysupports Ball's contention. For example, re-searchers found that most prospective teacherswere able to solve the following problem: What is1% divided by 1/2? However, very few coulddevise a context in which such a problem wouldmake sense. Some proposed problems whichinvolved dividing 1% by 2. Others, while recog-nizing that sharing pizza between two people didnot represent the problem accurately, were un-able to create one that was.

Not only do teachers need connected, sensibleunderstandings of the content of mathematics,they ris o need to understand the nature of math-ematics. Ball and McDiarmid (1990) argue thatknowledge about mathematics includes knowing:(1) distinctions such as convention versus logicalconstruction, (2) relationships among mathemati-cal ideas, and, (3) the nature of the fundamentalactivities ofmathematicslooking for patterns,making conjectures, justifying claims, etc. (pp. 9,10).

McDiarmid (1992) presents three reasons for in-creasing the subject matter knowledge ofteachers.First, teachers need to know mathematics in order"to help their learners develop similarunderstand-ings." Second, the teacher's stance toward thesubjectmatter commtmicates aviewofthe nature ofthe discipline to her students. Finally, McDiarmidargues, there are "critical ties" between subjectmatter knowledge and pedagogical content know'.edge(Shulman,1986). Shuhnaninventedthispluaseto describe the knowledge that enables a teacher to"build bridges between learners from a variety ofbackgrounds and the subject" (McDiarmid, 1992,p. 9). Ball (1990b) discusses the ways in whichknowledge of subject matter enables her to create avariety ofrepresentations ofnegative numbers forthird graders.

Mathematics educators also worry about teach-ers' attitudes towards mathematics. The litera-ture on math anxiety, while generally not scientific,is extensive. Cross-cultural studies (Stigler andBaranes, 1988) suggests that American, morethan Japanese or Chinese, adults, attribute suc-cess (or failure) in learning math to "ability" ratherthan effort or opportunity to learn. The belief thatthe ability to think mathematically is predeter-mined can influence teachers' interpretations of

their own math history. It can also influencepedagogy: If some people just can't do math, thenteachers cannot expect all students to understandwhat they teach.

So, how can teachers learn what they need toknow of mathematics in order to teach in newways? An obvious suggestion is that they return touniversities and take math courses. However,most teachers avoid such immersions. We sel-dom see elementary teachers (either in- or pre-service) in college calculus courses. AndMcDiarmid (1992) has argued that, if they didtake college math courses, the kinds of experi-ences they would encounter would not promotethe kinds of knowledge of or about mathematicsthat math educators advocate. Moreover, suchcourses, with their pre-constructed syllabi andemphasis on coverage, are unlikely to alleviatemath anxiety. Thus teacher educators have lookedfor alternative approaches. Some of these involvegiving teachers opportunities to be learners invery different settingssettings similar to thosethe NCTM Standards advocate for K-12 class-rooms.

Educators taking these alternate approaches of-ten start with the same social constructivist per-spectives on learning that drive much of thecurrent effort to reform public schools. Drawingon the work of Vygotsky, various authors (e.g.,Harre, 1989; Wertsch, 1985) have argued thatlearning takes place in social interactions: Stu-dents must have opportunities to make publictheir thinking, thus making it available for criticismand re-formulation. Ball has suggested (1990b)that student conversations that include conjectur-ing about mathematical problems and ideas helpstudents to develop an understanding of the na-ture of mathematics. Cobb (1989) has arguedthat "each child can be viewed as an activeorganizer of his or her personal mathematicalexperiences and as a member of a community orgroup [which continually regenerates] taken-for-granted ways of doing mathematics . . . Childrenalso learn mathematics as they attempt to fit theirmathematical actions to the actions of others andthus to contribute to the construction of consen-sual domains (p. 34)." Wilcox, et al., (1991)extend this reasoning to prospective teachers,urging that if we want teachers to develop theknowledge ofmathematics they will need in orderto teach in new ways, we need "powerful inter-ventions that challenge and yet are safe situations

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in which students can take mathematical, emo-tional, and intellectual risks. Creating a commu-nity of learners with shared responsibility forlearning holds the promise of providing such anenvironment" (pp. 1,2). Because few universitymath classes create such communities, someteacher educators have attempted to deepen andbroaden teachers' knowledge of mathematics byalternate approaches (e.g., Shifter & Fosnot,1993; Duckworth, 1987). Duckworth and hercolleagues met with a group ofteachers bi-weeklyover a period of a year to explore the learrfing ofmathematics among other topics (Duckworth,1987). The members of the group reported thelearning of content, a better understanding of thenature of mathematics, and a changed attitudetoward the subject.

The SummerMath Program at Mt. Holyoke Col-lege has created the opportunity for in-serviceteachers to experience learning in the kind ofclassroom situation that the NCTM Standardsasks them to provide for their students. Theprogram gathers groups of practicing teacherswho meet for two weeks in the summer to exploremathematics from elementary and secondary cur-ricula. The teachers also interview children toprobe their mathematical thinking, design a lessonbased on that knowledge, and teach it. Theyexplore the mathematics in groups of three orfour. The teachers that Shifter and Fosnot (1993)describe experienced reduced levels of mathanxiety in the supportive atmosphere of the groups.This relaxed atmosphere provided them an op-portunity to think about mathematics without fearof evaluation. The members of the !pups hadopportunities to experience success in thinkingabout and solving mathematical problems.

The results of these interventions seem promising.The authors report that many of the participantshave developed more positive attitudes towardsmathematics. Furthermore, their understanding ofthe nature of mathematics changed. Many partici-pants no longerviewmathanatics as a set ofrules tobe memorized, which= beyond their understand-ing. SununerMathteachers like Sherry Sajdak have"developed anew understanding of mathematics"(Shifter & Fosnot, p. 112) while those like GinnyBrown have lam -Wmathematics in theirown class-rooms, from their students (p. 158). No longerconfined to the state of ignorance they resignedthemselves to as children,they are expanding theirmathematical horizons.

Projects such as SummerMath, while immenselyhelpful for participants, and possibly for theircolleagues, can handle but a tiny fraction ofteachers. So, how can the reforms called for bythe NCTM be successful? This paper explores,through the experiences of three primary gradeteachers, the possibilities that teachers who beginto teach mathematics in new ways may growsignificantly in their knowledge of and aboutmathematics through their teaching.

The teachers are members of Investigating Math-ematics Teaching (IMT), a project of the Na-tional Center for Research on Teacher Learning(NCRTL). This group of seven teachers andthree researchers started meeting in the fall of1991 to explore a multi-media collection of ma-terials documenting teaching as learning ofmath-ematics in two elementary math classes, one ofwhich was taught by Deborah Ball. ' During thatfall, Helen Featherstone, Lauren Pfeiffer, andStephen Smith structured activities around watch-ing videotapes of Deborah Ball's third grademathematics class and looking at Ball's journaland those of her students. They also visited theseven participating teachers' classrooms and in-terviewed each teacher on a regular basis. InJanuary of 1992, the focus of discussions in themeetings began to move toward conversationsaround individual teachers' practices. The grouphas continued to meet on a bi-weekly basis duringthe school year. Helen, Lauren, and Steve con-tinue to visit classrooms and interview teachers.

Helen, Lauren and Steve first began to thinkabout the possibility that teachers who begin toteach math in new ways might learn subject matterfrom their students in February, 1992, as aresult of a conversation between Carole Shankand Helen, in which Carole spoke eloquentlyabout changes in the way she saw mathematics(see below). As they analyzed data from the earlyphase of the study, they began to suspect thatother teachers had also made significant changesin their understandings of the math they taught,their perceptions ofmath and what is involved indoing math, and their perceptions of themselvesas learners of mathematics. In the fall of 1992,when they invited teachers in the IMT group tocollaborate in looking at IMT data on this issue,Carole, Debi Corbin, and Kathy Beasley wereparticularly interested. The three cues that fol-low are the result of a collaborative effort to telltheir stories.

Michlipm 5th University, East Leming, MIchipn 41114.1034 RA 93.1 Pap 3

Although the other four teachers in the IMT grouphail from four different districts, Debi, Kathy, andCarole all teach at the same urban elementaryschool. Carole and Kathy have been colleaguesthere since 1983; Debi is a relative newcomer,having been assigned to student teach in Kathy'ssecond grade classroom in 1990 and having thenstayed on in the school as a "co-teacher" as-signed, as patt ofAverill's Professional Develop-ment School effort,2 to provide restructured timeto a team of four primary grade teachersa teamthat includes Kathy and Carole. In the fall of1991, when Helen, Lauren, and Steve were re-cruiting teachers for the IMT group, Kathy andCarole had just moved out of the grade levelteams in which they had both been teaching sincethey arrived at Averill in order to follow theirsecond and third grade students for two years.The new structural arrangements and shared in-terests in new ideas about teaching led the threeteachers to spend many school lunch periodstalking about teaching and to join the IMT grouptogether. Ali three were interested in lamingmore about Deborah Ball's math teaching. Butbecause none of the three felt at all confidentabout her own knowledge of mathematics, allwere also somewhat nervous aboutthe announcedfocus on a unit from the 1989-1990 school yearin which Deborah had introduced her third grad-ers to operations involving negative integers.

CAROLEI thought math was very, very individualisticand dry. There was a process (algorithm], youhad to learn it, and you got through it. And Ihad trouble memorizing processes often. Iwasn't good at math in high school, I wasn'tgood at math in college, and so I avoided itbecause I wasn't good at memorizing what todo and how to do it.

I never really got what it was about, and nevereven really realized that even memorizationwould get me through it. I just kept trying tofigure out .. . and it just never made any sense.I knew my facts, my addition, subtraction,multiplication and division facts, and wherethat didn't work, I avoided it. My husbandwould measure stuff and he'd ask me "How fardo you think it is?" and I'd just say, "I have noides."

Like many, perhapsmost, American adults, Carolehas felt inadequate in relation to mathematics fora very long time. This sense of inadequacy startedat least by the time she was in high school. Itcontinued through her college years and was wellestablished by the time she became an elementaryschool teacher. Unlike Debi, who memorizedformulae and felt able to do problems that lookedlike the ones she had practiced, Carole had littleconfidence even in her memory. Unable either tomemorize formulae or to make sense of thematerial, she felt that she was without tools fordealing with mathematical problems.

Because she felt incompetent mathematically, shekept math out of her life as much as she could.And the example she offers suggests just how farout of her life it was possible to push it.

Over the past year and a half, as conversationsabout mathematics problems have come to oc-cupy an increasingly prominent place in her mathteaching, she has come to a different view aboutwhat mathematics is, about what it means to domathematics. She has also come to feel verydifferently about math. These changes in herperception ofwhat mathematics is and how onedoes math, and in het feelings about math, havebrought about changes in the way she sees mathoperating in her own life.

Twenty Years of Traditional Math TeachingFor the first twenty-three years of her teachingcareer, Carole taught third grade in the sameelementary school and "teamed"taught andplannedwith the same colleague.

I pretty much followed the book, followed thecurriculum guide. It was pretty much Me ori-ented, I've com. to believe. Once in awhilewe'd use manipulatives to show them some-thing, but they weren't really tools that kidsmanipulated. I wouldn't really refer to it as atool, it was more like a demonstration. It wasmore my tool, to show them something. Itwasn't for them to use. I didn't know how tohave them use them. Or even how to watchthem to see what they could do.

And there was always the routine assignmentof problems involved. Problem after problem.And then we got so you don't have to do everyproblem, you can do every other problem. Andthat wu a big step. And then we got to the

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point where you didn't have to do every page,you could skip some. And those were all bigsteps. It sounds ridiculous, but that's the wayit was.

Given her own experiences with math, it is per-haps not surprising that even moving from assign-ing all problems to assigning only those with evennumbers seemed like a major step. In 1989, at thePDS Summer Institute, she saw Deborah Ballteaching math for the first time:

I guess I started to see something new whenwe went to our first summer work shop, and[Deborah] Ball was there. And the thingsthat she was saying and doing; wereunbelievable. . . . She had brought in somekids, she showed videotapes. She gave thekids some problems, and asked them whatthey were doing and they would explainwhat they were doing and she would askthem to explain what they were doing. Welooked at their journals and we saw thosekids in real life.

And the things that she was doing were sodifferent from anything I was doing. The con-versation, the discourse, the discovery, theaccepting of answers, kids listening to eachother, the teaching from one child to another.And she was pulling from the conversationthings to move to the next step. I was amazed.

That was my initiation, my first real experienceof something different. Then, there were differ-ent times, throughout the three years after thatthat things happened. And then, somewherealong the line, we found Marilyn Burns, andshe seemed to have all the answers. And thatgave me an avenue of experimenting with mathdifferently. We started doing some of her ac-tivities, but the discourse wasn't there: it wasthe activities that we were looking at. So theactivity became the thing that sort of got at theunderstanding of what multiplication was. Butwe were still doing the drill, and all the multipli-cation facts and all the addition facts, and allthe timed tests, and we were still doing all thewritten stuff. Lots of homework.

During 1990-91 she described her math teachingas a mix of Marilyn Burns activities and drill andpractice worksheets. "I did some multiplicationstuff from Marilyn Burns, but then I'd slip backinto the workbook."

In 1991 several things happened:

Having joined the team that was setting direc-tions for her school's Professional Develop-ment School effort, she was thrust into aseries of conversations with several of hercolleagues about the role of teachers, womenand power, and open communication. Sheexplains:

We had spent hours and hours and days learn-ill how to talk to one another as members ofthis management team who didn't know whatit was about and what we were supposed to doand what we really wanted to do and why wewere even there.

These conversations led her to think moreabout the role she was taking in the school,and about changes she needed to make if shewas to grow professionally.

Because she decided to follow the samegroup of children for two years, she moved tosecond grade and stopped planning with thecolleague she had teamed with for the previ-ous 23 years.

She began lunching regularly with Kathy andDebi and talking with them about teaching.Kathy had taken a course over the summer onthe NCTM Standards and many of theselunch time conversations focused on mathteaching. In late September, when Kathy andDebi began to think about joining the IMTgroup, they suggested that Carole join themand she agreed to do so.

October, 1991Carole had embarked on an adventure when shemoved to a new grade and out of a comfortableand familiar teaching team. She joined the IMTgroup because she wanted to learn more aboutthe Standards and because she had been in-trigued by what she had seen of Deborah Ball'steaching two summers before. From the very firstIMT meeting, she made connections betweenwhat happened in the group meetings and whathappened in her classroom. She also took activesteps to create a group. Lauren's journal accountof our first meeting notes:

MIchlpn Sum Unlvsnity, E 1 Lansing, MId I an 41114.1034

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Carole broke the linear format of "reporting"and made it more of a converstion. She askedquestions of the other teachers. She spoke tothe teachers and not to us!

When other teachers raised questions about thedifficulties the third graders in Ball's class werehaving understanding why 200-190 was not 190,she related the children's difficulty to an observa-tion about her own students: students this agehave difficulty understanding 0 as a placeholder.

Just as she carried what she saw in her third Fadeinto the IMT group, she took what she saw in thefirst IMT meeting back to her classroom. At thatfirst meeting we showed videotape ofBall's thirdgraders discussing number sentences they hadgenerated in response to her request that they"write number sentences that equal 10." Beforewatching this tape, members of the IMT groupdid the task themselves. For Carole, writing herown number sentences, examining the numbersentences that Ball's students had written in theirmath notebooks, listening to their discussion ofthese sentences, bringing the task to her own thirdgraders, and observing what they did and saidbecame both a pivotal event and a metaphor forchanges in her own viewofmathematics that grewout of observing and listening in anew way to herstudents as they grappled with math.

Here's what Carole wrote in her notebook thatfirst night. The first column responds to Helen'srequest that everyone write number sentencesequal to 10; the second that they write numbersentences equal to ten that they thought a thirdgrader might write:

Write number sentences to ten

3+2+56+3+14+4+23+1+2+2+1+111-1

6+45+52+81+9

(}+1

4+6

3+77+311.1

All of Carole' s number sentences except "11-1"involved additive combinations of integers be-tween 1 and 9. She was, she remembers, sur-prised at the much wider variety of sentences that

we found in the notebooks of Ball's studentswhen we looked at them later that eveningequations like 100 + 10=10 and 200-190=10.

A few days later she gave her own third gradersthe same task; like Ball's students, they rangedadventurously across the numeric territory theyknew. A year Pad a half later she remembered thescene this way:

When I looked at that problem, "Write all theways you can write 10," I thought, "Hey, I cando this, I can do this, I can do this, I can dothis." . . . Then when my kids did it . . . all thedifferent ways to look at ten, I thought, "Wow,I didn't realize ten was out there all thosedifferent ways."

But that happens a lot.

And I took my first directions from m kids, Ithink. And then it was encouraged, I think, byconversations with colleagues, in [the IMTgroup], with Kathy, with Debi. They were get-ting excited about it, too. I wanted to knowmore: I wanted to figure out what else the kidsknew.

It surprised me that those kids would figurethat out. . . . When Jason' came out with "It's200.190," I thought, "Look at what he's doing!And I was just copying something that[Deborah Ball] did."

Then when my kids did it, and when [DeborahBall]'s kids did it I thought "Wow, there are allthese different ways to look at 10 ?"

Three months after giving her students this task,reflecting in a conversation with Helen on thechanges she had made in her math teaching,Carole referred to this experience and the way itseemed to represent for her the way a world thathad seemed to be tightly sealed was beginning tocrack open for her:

What has been a real awakening for me,1 think,as much as dnything, is the relationships innumber. I really never saw much relationshipbefore. I mean, addition's addition and carry-ing was related to addition and borrowing wasrelated to subtraction. But now the world ofnumber is really exciting forme. When I see thecombinations of numbers that [the students]get with the mini-computer or the combina-tions that they got with the problem that[Deborah Ball] gave, "What's 10?" and someof the things they are coming up with. And I

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always thought 10 was 6+4 and that there werecertain facts.... But it's a huge world of 10 outthere, it's a whole world of all different num-bers and I always looked at it as a very narrowthing. . . .

And that is really growing for me this year. It'sexciting. It really is.

"I wonderwith my limited math backgroundif I can do it?"Even though she was intrigued by the numbersentences her students wrote during this lesson,Carole felt very uncomfortable about her ownknowledge of mathematics. On October 15, afterattending two meetings of the IMT group andspending some time reading Ball's journal shewrote:

Reading [Ball's] journal is helping me to seethe process she goes through deciding what toteach, how to follow one piece of the curricu-lum to the next. Making these connections isessential. I wonder with my limited math back-ground if I can do it.

Subject matter knowledge seemed critical to thekind of teaching she saw Ball doing in the twovideotapes we had watched. Indeed, even fol-lowing Ball's thinking seemed a bit of a reach toher. Because she was also feeling very short ontime, she wondered whether it made sense tocontinue in the group. She concluded her Octo-ber 15 journal entry with this question:

I'm finding it difficult to have time to thinkabout my own curriculum and what their mathunderstandings are, let alone trying to follow[Ball's] thinking. Maybe [the IMT group] isn'tthe place for me right now?

As soon as she read this journal, Helen calledCarole to talk and to say that she hoped Carolewouldn't drop out of the group. Carole reportedimmediately that she was feeling much better. Shewas very excited about that day's math discus-sion:

We were talking about ways to equal 12. Onelittle boy had written "100-100+12=12." Someof the other kids were confused. Another littlebuy explained it, saying,"It' s like I -1=0.' and0+12=12" The other child understood!

I was so excited because I had taken the time totalk and listen to them! . . . I haven't done thisbefore, but I'm just so excited!

Once more it washer own students thy' ideas,their success at explaining their ideas to othersand to herthat had made the difference. Shewas excited both by what she was learning abouther students"I had taken the time to ask them,and I'm hearing their ideas"and about themathematics that was surfacing in the room.

Carole was still troubled, however, about thechasm she saw between her own knowledge ofmathematics and Deborah Ball's "I don't knowthe math that she does"and for the same rea-son: "Where do we go tomorrow?"

And the IMT meetings were not always easyeither. Reflecting back on the year during thefollowing summer she recalled that she had oftenhad trouble following what Ball's third graderswere saying in the videotaped discussions that theIMT group watched:

I really had trouble. . . . I would get halfwaythrough the conversation and think, "what arethey talking about?" I had completely missedthe whole thing. But I'm better. It's not easybeing a listener. So I don't bring in my interpre-tations but just listen . . . for what they aresaying, you know, and try to interpret whatthey are saying.... There is a fine line there, inbetween how I construe it so that it makessense to me but so that I could communicatewhat they're actually saying.

Hearing and celebrating the students' ideasAn important theme in what Carole wrote andsaid during the fall of 1991 was the pleasure shegot from hearing the children's ideas in math. Atour first meeting she had written that a centralconcern for her was to get the students to listen toone another: "They all want to share their solu-tions but they don't want to listen." As the weekswent by she referred to this concern from time totime, but what was most dramatically clear wasthe pleasure she was getting from listening to themand learning about their ideas.

In January she talked about the central rolepleasure and satisfaction needed to play in teach-ing and learning. And she connected her ownsatisfactkins to what she was learning about whather students knew:

Mkhisan Slats Univwsity, EOM Lansing, Michigan 48814.1034 RR 951 Pap 7

I'm amazed at how much knowledge kids reallyhave. I'm always amazed. I don't think a classgoes by that I'm not amazed. They know a lot.There are a lot of them who know a lot about alot of things. And you don't discover thatunless you let them share and let them talk.

It goes back to being open to change. I keepthinking, I'm struggling so hard right now, butI'm not preparing for something: this is liferight now. This is pan of it. If I can't get whatI want out of it right now, I've got to make somechanges, because I'm not preparing for any-thing. I've got to appreciate what this is. Forsome reason that really hits for me. Because Ihear so many peorle say, "I'm preparing forretirement," "I'm preparing for this." But youaren't going to do it then either. THIS IS LIFE.You aren't preparing to graduate, you aren'tpreparing for anything. This is it. What canyou get out of this right now, and what can wedo with this?

This emphasis on the importance of living life inthe present suggests another connection: In Oc-tober she saw her lack of subject matter knowl-edge as an impediment to planning, to using thechildren's comments and insights as a basis formoving the curriculum forward. In October sheseemed to be saying, as she reported on theexcitement of good discussions, that she feltcompetent to orchestrate the discussions thatwere occurring in the present but worried aboutlong-range planing. She saw Ball's knowledgeof math playing a central role in he: planning, in thedecisions about "how to move from one piece ofcurriculum to the next." In January, as she af-firmed the importance of the present, perhaps shewas also reminding herself of the importance ofwhat she was able to do.

Summer ReflectionsLooking back over the year in the summer of1992, in a conversation/ interview with Steve,Carole touched on some new changes in herthinking about math and the learning of math:

What I thought was understanding is no longeranywhere near where I see my students, whatthey talk about, how they talk about math. . . .

I've really developed the confidence that theycan figure this out. Where before I never reallythought about it, I guess. ..I just thought mathis Just (writing down a problem) and spitting itback out on the paper. You know, it was justkind of pushed through this hole. I don't talkabout it very eloquently, but it's just so differ-

ent: Before it was just pushing out problemsand pushing out . . . the right answer. And it'snot there at all anymore. And it's outstanding,I marvel at that. I really do.

I don't know how to do it well, ... but it just feelsright.

When Steve asked aboutwhether she saw changesin her own view of mathematics, Carole an-swered, "I'm not afraid of it because I can figureit out, too." She was still unsure, she continued,about negative numbers, "and I'm still not surewhere the next step is when my kids are strugglingwith a concept . . . but I feel I have people I cango to for help."

Six months later, when we all wrote individuallyabout changes in the way we viewed math, Caroledescribed her thinking this way:

Math always seemed a pretty black and whitesubject before. You followed a procedure, youfollowed a process, you got an answer. It wasindividualistic, not shared except with theteacher or the checker or whoever was in-volved with it, and you moved on.. Not evenmuch relevance to the real world. Except sub-traction and my checkbook.

Since joining this group, I guess, and learningabout the NCTM Standards and [Ball's) tapesand having discourse about what's importantand how to do it this way or that way orwhatever, math no longer is an isolated thing.It's communication, it's a discovery, it's anadventure. All answers are different and var-ied. It's about how we think and about hownumbers work and about how it all works to-gether. Math now has life, it has many ques-tions and lots of answers and a wonderful wayof manipulating all different numbers. When Ithink of the teaching, like, of place value in myclassroom and watch how hard kids are work-ing to figure out what tens, hundreds, and onesmean and what does it all have to do withaddition and subtraction and multiplication,and see the emerging discourse and the prob-lem solvin; that is going on, I find it just reallyreally exciting.

Math has become very obvious in my life u faras in the outside world. I can't get specificabout those, I'd have to think more aboutthose, but I see it much more as part of my life.And I feel like I've only just begun. I'm no-where near the end; I think it is just an ongoingprocess that I've started and I'm really excitedabout.

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When Helen asked her why she thought thesechanges had occurred she answered:

Partly because it's open. It's not ,just oneprocedure that I had to memorize 'out its waysof discovering how different numbers work.That's the release for me, that's the thing thathas opened it up as much as anything.

And trying to see how other people [childrenin the class] are thinking. . . . When I look atsome of their solutions, I think, "That can'twork. No, wait a minute, that does work!" It'sshocking sometimes the way things that theydo that are right there in front of me that I wouldnever have picked out in a hundred years, theysaw it that way. It's a whole different way ofthinking about it.

I have faith that my kids are going to come upwith the answer, or with some way of thinkingabout this.. . .(In theput) I wu looking for ananswerthe number at the bottom. I may noteven have known the answer myself. I lookedin the teacher's guide and checked off theanswersnot having any concept that therewas anything beyond the rote process.

So it's very different for me.

Where is Carole today, in relation to SubjectMatter Knowledge?

It's still an issue. I'm taking more cues from thekids. But I try to know where I want to go next.

From the start Carole has seen subject matterknowledge as intimately connected to issues ofplanning and knowing where to go next, seemslike a bit of a resolution. But clearly she does notfeel that she knows all she needs to know aboutmathematics in order to teach math well.

Reflecting on Carole's StoryCarole had arrived at a dead end in relation to thelearning of math well before she even enteredcollege. She had a view of math which made ithighly unlikely that she would ever learn anythingshe did not already know. Math presented asmooth closed surface to her: it was "black andwhite," it was solitary, and it was irrelevant to herlife. When something that looked like math ap-peared in the doorwaywhen, for example, herhusband asked her to estimate a distancesheshut the door, declaring firmly, "I don't know."Because she and math had agreed to live separatelives, nothing much changed in their relationship.

=MENNEN,

All this has now changed; math beckons to herboth in the classroom and outside of it. Herstudents' mathematical insights intrigue her; shetries to follow their thinking and sees the world ofnumber expanding. Her story recalls the momentin the Wizard of Oz in which Dorothy opens thedoor to her black and white house and realizesthat she is not in Kansas anymore.

KAThYKathy has always taken her own learning seri-ously. So seriously, indeed, that she changed hermajor from elementary education to French in hersophomore year of college because she felt thatshe wasn't learning much of value in her teachereducation courses.

After graduating from college in the late 1960sshe moved to northern Florida where she spentone year as a VISTA Volunteer and anotherteaching high school French. She then moved toMichigan with her husband where she left teach-ing to begin raising a family. After her secondchild was born, Kathy obtained her elementaryeducation certification; she began teaching atAverill elementary school in Lansing Michigan inthe mid-1980s.

During her first couple of years at Averill, Kathytaught second and fourth grades using many ofthetraditional methods she had learned. Then, how-ever, she and her team teaching colleague be-came interested in new approaches to teachingreading. Over the course of the next few yearsthey abandoned basal readers and ability-basedreading groups, to move toward a "whole lan-guage" approach to literacy. Although thesechanges were rather unsettling at first, Kathy andher colleague were reassured and very excited bytheir students' responses to the whole languageinnovations. As they became more proficientusing the whole language approach, they begin toexplore the idea o'ewhether it would be possibleto make changes in their mathematics teachingthat paralleled the changes they had made in theteaching of the language arts.

In the summer of1988 Kathy became very inter-ested in the ideas that were presented in a "MathTheir Way" workshop, sponsored by the schooldistrict. Following the workshop she began to trynew approaches to teaching math for example,

MIchIsan 511. UnIvuelly, It Lavin& Mlohlsan 488241034 t RR 95.1 Pap 9

she found the books ofMarilyn Burns very help-ful and to seek out others who were makingchanges in their prac ze. This led Kathy and hercolleague to attend, in the summer of 1989, aMichigan State University (MSU) ProfessionalDevelopment School (PDS) summer institute thatincluded a workshop on mathematics.

They saw video tapes of Deborah Ball andMagdalene Lampert teaching math and watchedDeborah work with several ofthe eight- and nine-year-olds she had taught the previous year. Muchintrigued, Kathy beganto further experiment withnew approaches to teaching math.

In the fall of 1990 Kathy and MSU teachereducator Sharon Feiman-Netnser initiated a PDSprojectthat involved long, searching weekly con-versations about teaching. These conversationsencouraged Kathy to experiment in a reflectiveway in her approach to teaching. Kathy says,"She helped me understand the joy and intellec-tual work that is what teaching is all about."Sharon put Kathy in touch with people at Michi-gan State University who were interested in alter-native approaches to teaching mathematics.

In the s of1991,1Cathyjoined a study groupthat De Ball and Janine Remillard had orga-nized for student teachers they had taught in amath methods class. The following summer shetook a graduate class on the NCTM Standards.In the fall of 1991 Helen approached her aboutjoining the IMT group. Kathy hesitated: she hada number of other outside commitments; in addi-tion, she recalls, "I thought I was going to beexpected to be more knowledgeable about maththan I knew I was." She warned Helen of thisworry on the phone, saying, "Helen, I have to tellyou that, when you say negative numbers, itmakes me feel very anxious.' However, she wasso much attracted to the idea of learning moreabout Ball's teaching methods that she decided totake the plunge.

Fall, 1991: Feelings aboutMath and Math TeachingThe first paragraph of the journal that Kathy keptfor the IMT group captures some of her feelingsabout math and math teaching:

Did I say I hated math at our last class? I feelbadly about that. I don't really hate it anymore.Maybe I never did. It is far more accurate to sayI fear math. But it feels more powerful to say Ihate it. I guess that's why I said that. I know Iwould be offended if someone said that theyhated literature. What I really want to do isunderstand math so that I won't be tense andworried about it. Mostly, I never want mystudents to fear or hate math. They all seem tolove math. And really I do like teaching it.Teaching math has helped me understand math.

A few days later Steve visited Kathy's third gradeclass and watched her teach a math lesson in-spired by Marilyn Burns. As she had noted in herjournal, students seemed to be thoroughly enjoy-ing themselves and their task.

On the board Kathy had written lists of itemscosting $3, $4, and $5 in preparation for animaginary shopping trip. She told students thatthey can "spend" $10, that they should decidewhat they want to buy and why, figure their totals,and explain how they arrived at the figure theydid. They worked on this task alone or in smallgroups, devising a variety of methods for keepingtrack of their purchases. After the class recon-vened, groups shared lists, methods of computa-tion, and totals. Kathy concluded the class byasking the students to look for patternsonenoted that "All the numbers in the tens place are1"and telling them that the next day they wouldtalk about what the totals would add up to.

I'm Jealous....Kathy's first journal entry also highlighted twoissues that compelled her interest across the nextyear. The first was listening:

I am also trying to think about children listen-ing and learning from one another. I want thisto happen in my classroom. I've been thinkinga lot about listening. My children listen to eachother best in the morning during sharing time.Each child shares one thing, anything theywant. They can't talk when someone else istalking and they really observe this. I am not incharge. A different child is each day. Michelleis inch arge ofkeeping track of who is in chargeand making sure everyone gets a chance. Any-way, they listen to each other: They talk toeach other. It seems to me that the reasonDeborah's children listen to each other is thatthe questions they are discussing are theirs.This is a really clear example of responsivecurriculum I think. How is this different thanwhat I do in math?

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The second was the language Ball used to de-scribe her third graders mathematical thinkingand the knowledge of mathematics that stoodbehind this vocabulary:

She had labels for children's thinking like"decomposition of numbers." I was interestedin her note about the "compare meaning" ofsubtraction. "I know from experience that it isthe harder meaning for children to understand.Using pictures and comparing the amountswith manipulatives seemed to be the only waywe could make sense of that meaning.' I'vereread her entry a couple of times and I'm notsure what she is saying. What does she meanwhen she writes "This problem is interesting inpart because their ability to reason mathemati-cally depends on their understanding of thecompare meaning of subtraction"?

At the group's next meeting she spoke of feeling"jealous" of Ball's command of a vocabulary formaking these mathematical distinctions. Althoughno one else in the group took up this topic, Kathycontinued to show considerable interest in fol-lowing Ball's thinking when she ventured moredeeply into mathematics and when she used unfa-miliar mathematical vocabulary. In early Novem-ber, for example, when we examined a chart thatBall had made to compare the advantages anddisadvantages ofvarious representations for ex-tending her students' understanding of negativeintegers, Kathy asked more questions than any-one else. Even though she remained less thanconfident in the numeric territory below zero, shecontinued to try to make sense of operations withnegative numbers and to use her own efforts atsense making to assess what she saw in video-tapes of Ball's third grade.

The Attack of the Killer ElevensAlthough still uncomfortable in the realm ofnega-tive numbers, Kathy was making changes in theway she taught mathematics. In late January, inorder to explain the character of these changes,she described a recent math lesson to Helen.Kathy had begun, she explained, by asking herstudents to compare 30 and 19.

But I didn't stop there. I said "use your mini-computer to figure this out, and then explainhow you figured it in your journal. And then,if you finish, here are some other problems," Iknew I had to have something else for them todo while some of the others finished the prob.

km. So I gave them a whole series of problems.And as I got to creating them I thought, "Oh,I'll do a pattern iu Id they'll all come out thesame and I'll see what they do with it."

After the students had worked on these problemsindependently for a while they reconvened andlooked at the first problem together. They agreedwithout much difficulty that the answer was 1 1 .

So, we finished with it, and everyone wasfeeling pretty good about it, except that one ofmy students, Lisa, tried to talk about how 30take away 19 and 50 take away 39, which werethe only two problems she had done, were thesame.

In trying to articulate this Lisa came to the boardand wrote

50 30

:32 1211 11

And she's seen a pattern! . . . Which I thoughtwas interesting. So, I decided to pursue thatwith the kids.

The next day, Kathy gave her students a series ofproblems like this:

30.19 -50-3960.49-90-79.40.29-

asking the third graders to work individually onthe problems and to look for patterns. Severalstudents responded immediately that they knewthat all the answers were 11. Kathy said that thiswas fine; they could just write down 11 and thenstart looking for patterns.

When her class reconvened, students talked ex-citedly about the patterns they saw: they noticedthat they were adding first 10, then 20, etc. toboth the top and the bottom number, that all thetop numbers ended in zero and the bottom num-bers ended in 9, etc. Then they got interested inwhat would happen if they added some numberthat was not a multiple of 10 to the top and thebottom numbers. They tried 7 and were surprisedto see that the difference was still 11.

Michigan Slaw University, Ent Lway, Michipn 41524-1034 It* 95-1 pugs I I

"Then," Kathy reported, "someone said, `Well,this is the attack of the killer elevens, we have toget rid of these elevens!' And I said, 'Well, howwould you get rid ofthe elevens? What would youadd?' The third graders became much engagedwith this question. They tried adding severaldifferent numbers with, of course, no luck. ThenNathaniel called out excitedly, 'Eleven! Let's try11." There was a murmur of excited approval.As a group, the third graders appeared to beconvinced that ifthey added 11 to both the 19 andthe 30, the difference could not continue to be 11.They were again astonished by the results of theirarithmetic:

19 30 41

±11 ±LL :III3C 41 11

Finally Cindy got Kathy's attention: "Mrs.Beasley, I've had my hand up for an hour and younever call on me." After making her way to thefront ofthe room, she turned triumphantly towardher classmates: "You're all wrong. 60-49 isn't11: It's 29! ! See [writing it on the board]: 0 takeaway 9, you can't do it so you write 9. 6 takeaway 4 is 2.29! So, if you add 30, you get 29, not11!"

The third graders stared at the numbers on theboard, and then many agreed! Some didn't. Allthis despite the fact that Kathy had workedextensively with regrouping only a few monthsearlier and, with most of the students (she hadtaught second grade the previous year), the yearbefore as well. Gregory disagreed. He said thatyou could take 9 from 0, and that, when you didthis, you got -9.

After telling this story, Kathy returned to the issuethat had been puzzling the third graders beforeCindy took the floor:

I started wrestling with. "How am I going tohelp children understand that the space, theamount between those two numbers, remainsconstant as long as you're adding the sameamount to each of them?" I don't even know ifI understand this really clearly. So I don't evenknow what to do with that, Helen. I think that'sthe whole issue. I'm really in a bind here.

The students were clearly exploring unfamiliarpatterns and asking challenging questions of thenumbers. Kathy was very excited about theirextended exploration; the lesson felt quite differ-ent from the one Steve observed in early Octo-ber.

Discussing this lesson with Helen four monthslater, Kathy identified the way in which she waslistening to the students as the key differencebetween this lesson and the ones she had beenteaching earlier. A comparison ofthe two lessonshelps us see what she meant. In the Octoberlesson Kathy clearly listened with interest to thechildren's lists and observations. However, thereis far more to listen to and for in the Januarylesson.

Revisiting her comment in January, 1993, Kathyexplained, "It's not that I didn't!:=-4 Odom, Wsthat I didn't let them say anything." Continuingthis line of reflection she went on to say that shethought that she had been so focused on correctanswers that there wasn't that much interestingfor the children to say. Then she stopped herselfin mid-sentence to revise: "You know, I probablydidn't listen. The whole structure didn't allow forthe children to say anything, so there just wasn'tanything to listen to.' She shook her head disbe-lievingly, "What a weird way to teach!"

In January, the children were giving the lessonnew direction all the time. They were posingmathematical questions that had not been explic-itly on their teacher's agenda when she put theproblem on the board. As Kathy pointed out, it isone thing to listen for expected answers andsomething quite different to listen for, and to,unexpected questions.

This would seem to be the "responsive curricu-lum" that Kathy had seen in Ball's videotape inOctober. At that time she conjectured that Ball'sstudents attended because the questions camefrom them. She was attracted by what she sawand because she believed that if her students'were pursuing their own questions, they too wouldlisten more closely to each other and learn moremath. Her "killer eleven" story suggests that shehad achieved, at least in this series of lessons,what she set out to do: Her third graders were

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0 1995 by the National Can* for Mardi on Tonchor Lambs

pursuing answers to their own questions; theyseemed to be listening carefully and thinkingabout what they see and hear. Their teacher wasequally excited and attentive.

Near the end of this January conversation, Kathysuggested to Helen that she would like to followup on Gregory's observation"You can take 9from 0"by teaching her third graders aboutnegative numbers, "If I could have someonecome in my classroom to help me do it." Helenwas dumbfounded: all fall Kathy had declared herdiscomfort with the idea ofbelow zero numbers;only two months earlier she had declared in class,"It's hard for me to imagine pursuing anything todo with negative numbers.' Helen tock the pro-posal as evidence that Kathy was as stronglycommitted to expandin4 her understanding ofmathematics as to altering her pedagogy. In-trigued by the possibilities for learning more aboutthe challenges of this kind of teaching, Helenagreed, after several more conversations, to co-plan a unit with Kathy and spend two to threemath periods a week in her classroom.

Teaching about IntegersOverthe course ofwhattumed out to be five weeks,Kathy and Helen worked together with Kathy'sthird graders. They revisited regrouping becausemany ofthe children seemed confused about whento regroup when doing subtraction. And, using athermometer, they explored addition and subtrac-tion with negative numbers. While children workedon problems alone or with others, Helen and Kathycirculated, asking questions and listening to ideas.Kathy orchestrated full class discussions; Helenwatched enthralled and made occasional su4ges-dons during class; in the evenings they debriefedextensively and planned next steps together. (SeeBeasley and Featherstone, in preparation.)

Kathy and Helen were exhilarated by the work,by the children's delight in their own discoveries,and by the richness and diversity of the theoriesthe third graders generated as they explored thethermometer and wrote number sentences that"ended below zero." In addition, Kathy herselfwas learning to navigate this new numeric terri-tory. On February 23 she wrote in her journal:

rwlith this negative number stuff I am learninghow to think about it right along with the kids.I wu very excited when I understood thestrategy Janine and Violet and Jonathon had

all been talking about and that Cindy posed theconjecture for: "If you have a problem that islike Violet's (11.9) and the answer is abovezero, if you switch it around (9-11) you'll havean answer below zero." When I realized Thurs-day night hr, w well that works and just felt nowI can "do" negative numbers I decided I wantedeveryone to understand that. (2-23)

A week later she described to Helen a full classdiscussion of students' efforts to start with anegative number and write a number sentenceequal to zero.

They all gave examples of how to get tozero. . . . They said it was really easy. ThenNoah and Justin gave theirs: "- 10 +- 10-0." Iwasn't sure whether it was right or not.

Another student disagreed with Noah and Jeff'sformulation, pointing out that [someone else] hadshown on the thermometer that -10+10=0 andthat, if this were true, -10+-10 could not also beequal to zero. As she listened to the discussion,Kathy saw that -10+-10 would have to be -20.She asked Noah and Justin to ponder the follow-ing question: "What if it were -10 degrees inAnchorage and the temperature fell another 10degrees?'

A year later, writing to Helen about what shelearned as she taught the unit, Kathy recalled that,before teaching it, "I had absolutely no confi-dence in my understanding of negative numbers."As evidence she pointed to a slip she had made informulating the problem with which she and Helenhad planned to launch the unit.'

think that first problem is evidence of howlittle I understood (when we started). What didI learn? I think I learned that I could do reallyhard math by teaching it. The fact that I under-stood negative numbers, how to add and sub-tract them, was very helpftil to me. To this dayI know that if I stop and think, "draw a ther-mometer," I can always understand how to addand subtract negative numbers. I think I dohave a mental block when it comes to this, butI know it can't really block me anymore, I knowI can understand this. I feel like I should say Ido understand negative numbers, but I hon-estly don't believe I do enough yet, I can sayI can understand them and that for me ismonumental.

Mlohlsan Slats UnlvanIty, East Lambs" MIshIpn 45124.1034 PA 93.1 Fags 13

Yes the children's explorations helped me. Itwas their thinking that taught me. They couldget up there and say out loud my misunder-standings. As soon as they said them I wouldunderstand. They unpacked the concepts, thethinking, the parts of negative numbers that Ineeded and they needed in order to under-stand. I remember every time someone putforth a wrong answer and explanation I wouldhave to think and test the idea, I didn't justknow it was wrong.

Teaching FractionsAfter integers, Kathy moved on to fractions; thisunit proved unexpectedly difficult. At the end ofthe first week, she wrote in her journal:

I am feeling bewildered by math. This way ofteaching is difficult. This is the first time I havedone this on my own. By that I mean thatworking with Helen I started really under-standing this way of teaching. As long as I hadHelen to consult with I was doing great. (1didn't realize how much she was helping me.Not that I didn't appreciate and deeply valueher presence and being able to talk to her) I justdidn't realize how lost I would feel without thatsupport.

Helen was puzzled by Kathy's assertion that shewas teaching differently from the way she hadtaught before their five weeks of collaborativework: In the lesson involving the attack of thekiller eleven she had been listening carefully tochildren and allowing their questions and conjec-tures to guide the direction of the conversation;what was different now?

Her visits to the classroom and Kathy's journaldescriptions of some of these classes suggestedan answer that connected directly to the divisionof labor Kathy and Helen had established whenthey worked together. During those lessons Kathyhad orchestrated all full-class discussions; Helen,who was thus freed from the responsibility formanaging the minute-by-minute interactions ofacla ss of eight-year-olds, attended carefully to themathematical ideas and theories that childrenwere sailing into and around. As a natural conse-quence of this division of labor, Helen had takencharge of suggesting a task for the journal writingwith which Kathy and her third graders usuallyclosed a class discussion.

Before Helen and Kathy worked together,Kathy's journal assignments followed no set pat-tern. Some focused on a piece of mathematics:After the "killer eleven" lesson, for example, thechild= wroteabout whetherand why they thought30-19 was equal to 11 or 29. Others were quitegeneral: After a discussion that centered on re-grouping, students addressed the question, "Whatdid you learn in math today?" During her time inthe third grade, Helen tried to capture someimportant mathematical disagreement that hadbeen embedded in the preceding conversation.Circulating around the room, reading over stu-dents' shoulders and talking to them about whatthey were writing helped Kathy and Helen tolearn more aboutchildren's conceptions and mis-conceptions and to push their thinking. Not want-ing to lose this piece, Kathy added Helen's "job"to her own: From the first day of the fractions unitall her journal assignments required the thirdgraders to "do math." An excerpt from a journalentry that Helen made after visiting the class onApril 9 suggests both how hard Kathy was listen-ing for and to the children's mathematical ideasand how complex was the task she had setherself. The students had been working on aproblem involving dividing 10 brownies among 4people and had done some very nice reasoningboth about the answer and about how they mightexpress that answer:

Must as it was getting to be time to break forsnack, Jonathon, who appeared to have im-pressed everyone in the class with his com-mand of the division, said he wanted to askthe class a question: "Do you think that 1. 3divided by 4 is the same as 4 divided by 10?"Some oneor maybe a few peoplesaid no."Yes," said Jonathon with great authority,"it is. My friend told me. No matter whichway they write it, the number of cookies isalways the big number, the number of peopleis always the small number. You alwaysdivide the small number into the big num-ber." I think he wanted us to write this downas a conjecture. I realized that we hadn't hadany wrong conjectures beforeperhapsbecause we had played a role in encouragingkids to formalize their promising ideas intoconjectures. I'm not sure why. Anyway, Iwas wishing that this wasn't happening: Idid not want to end the class by writing upa wrong conjecture. ... He or someone elsesaid something else about how you couldn'tmove the numbers around in subtraction,but you could in addition and division. Histone of authority was impressive. I did notknow how we should respond. . . . I felt that

RR 95.I Pass 14 0 1995 by the National Center ax Rama on Teacher Lambs

he had brought in an authority from outsidethe classroom, and that the information ofthis authority would be accepted becausethe authority was not there to debate with.I looked over at Kathy, and instead of look-ing as puzzled and worried as I felt, she waswriting notes in a notebook. When he wasdone she stood up (looking equally authori-tative) and told the kids that she had ajournal assignment for them: They were toget their snacks and then they were to thinkabout these three ideas and write down whichones they agreed with, which they disagreedwith, and why. The ideas were: (1) that youcan move the numbers around in addition,(2) that you can't move them around insubtraction, and (3) that you can move themaround in division and that you always di-vide the small number into the big number.Students adjourned eagerly to their seatsand began to write. Kathy wrote the conjec-ture up in orange [we had instituted theconvention of writing conjectures in orangewhen they were formulated and student,were working to "prove" or "disprove" them;we recopied them in blue when everyonewas convinced that they were true] asJonathon insisted. As I walked around look-ing at notebooks and talking to children I.saw that just about everyone was disagree-ing that you could move the numbers aroundin division, and agreeing with the other twoassertions. I thought that the assignmentwas wonderful: the kids got refueled, theysaw that it was up to them to really thinkabout this idea, and they did. Whoopee. Ijust lacked faith, I guess. ! thought, at thatmoment, if we were doing a musical aboutthe work Kathy and I did together (a differ-ent way to tell our story), this would be theculminating moment: I am completely super-fluous.

Kathy was clearly listening to what students weresaying for issues of mathematical substance. Be-cause she was both orchestrating the discoursemoment-to-moment nowand creating journalassignments that would begin with what childrenwere saying and use it to push their thinking asthey worked more individually, she had to thinkconstantly about the mathematical issues that thediscussion was raising, and to decide which oneswere important enough to pursue. The notebookin which she had begun recording the representa-tions students used and what they said during thegeneral discussion helped her to keep track ofthecontributions of individuals; equally important, itwas also a tool that allowed her to tease outmathematical issues both for herself and for thechildren. She explains:

1 had to engage in the thinking, the mathemat-ics, not just identify the correct answer, butlook at all answers in a new way, not whetherthey were correct, but what they told me aboutthat child's understanding of math. Many timesthe model or answer illuminates a mathematicalconcept that is a piece of the mathematics that1 have just missed.

Sometimes just by writing down what they sayI get more clear on what the mathematical ideais. Some days 1 just don't and we end with afizzle, but I don't worry about that so much anymore because I know that during the rest of theday and the evening it will usually come to mewhat problem to present or where to start thediscussion the next day.

In a summer conversation with Lauren, Kathytalked about the skill she had learned over thecourse of the year: "It's learning the right questionto ask. It's asking the question that will synthesizethe discussion and knowing the question that willpull it together and challenge the kids in a way thatwill move them forward."

In the IMT GroupEventhough interesting things were happening inthe classroom, Kathy reported at the next meet-ing of the IMT group that she was feeling over-whelmed by the number and complexity of thequestions the third graders were raising. Movingfrom fractions of a whole to fractions of a set hadintroduced unexpected confusions. For example,Marianne rejected the claim of a classmate thatone plate was 1 /8 of a set of 8 plates, asserting,"You can't have fractions when you haven't cutsomething up." Kathy added, "They are reallypushing on this." Debt said, "What I think is, it'sneat that they are pushing." After some furtherdiscussion, Kathy announced, apparently onlyhalf in jest, "I want to go back to negativenumbers!"

However, an event that occurred less than anhour later in the meeting suggests that she wasbeginning to gain new confidence in her ownability to address mathematkal questions. Laurenand two other members of the IMT group weredescribing what they had seen on a visit they hadmade to Ball's classroom earlier that day. (Al-though we had been watching videotapes ofBall's teaching on and off all year, this was the firsttime the teachers had actually visited the class-room.) Lauren mentioned a question Ball hadposed to one of the students: Why do you get an

Mlehlpn State University, Ent Lensing, Michigan 4024-1034 1Ut 9S-1 hp IS

even numbe- wit= you add two odd numbers?She added that she was still puzzling over itherself. Kathy explained to her why this wassothe first time she had volunteered to explaina mathematical idea in the IMT group. Althoughher explanation was clear and cogent, she re-ported later that she had felt uncomfortable aboutthe exchangeperhaps because she was so un-accustomed to taking the role ofmathematicsteacherwith another adult.

Reflecting on FractionsTwo months later, reflecting back on what shehad learned and how she had changed over thecourse of the previous year, Kathy noted that shethought that she had become clearer about theconcepts she was trying to teach and had startedto think more in terms of concepts and less interms of activities: in the past she would begin toplane unit by looking through books for activitiesand then organizing them into a logical sequence.She now start* with the ideas she wants thechildren to explore. This change had not occurredjust over the course of the previous year butrather as part of an agenda on which she had beenworking for several years. However, in the pre-vious months she felt she had made a quantumleap forward. In the fractions unit, for example,she had focused on helping the students to under-stand what the to and bottom numbers in afraction mean and on the connection betweenfractions of a whole and fractions of a set.

She continued this reflection six months later, M aconversation with Helen, this time talking aboutwhat she had learned about fractions as shetaught this unit:

The kids were having trouble understandingwhat the top number meant and what the bot-tom number meant. I had never wondered, andI saw it would be important to understand it. Ithink, like Debi said, you learn something solid,like it just is, it's hard to pick apart. Like afraction, 1 /2 or 2/4: To me it was clear that youcould have 2/4 of a pie or a rectangle. I knewyou could have 2/4 of a set.

I remember doing the crayon box problem, with48 crayons, and the kids struggling with thatand as they struggled I began to ask, "Oh, OK,what does the 2 mean? what does the 4 mean?

11111IN

I did sit down and I went to math booksliketheStandards and I have this brown book thatI usesI don't know why because it neverhelps me and I think I looked at Burns andTank.

The kids teach me how to teach. I don't con-sider that I have a strong grip on mathematics;I was surprised, but maybe not shocked, thatthere was more to fractions than I had seen.

"The kids teach me how to teach." Clearly theydo this in part by helping her locate thecentral andinteresting ideas in a problem, by asking ques-tions, and by showing her which of her ownmathematical ideas she has not probed deeply.When the children raise the questions, Kathylistens carefully, pushes her ideas hard, talks toother people, and comes to new understandingsof her own as well as new ways to teach.

DEBIHow to really teach for understanding?

How do I know if the students are really under-standing?

N hat does it mean to know?

How to get students thinking and talking aboutmath?

How to create lessons in which there is dis-course and students have tools and strategiesto search out solutions and talk about theirsolutions?

How do I find time and people to talk aboutmath this way and not the more traditionalway?

How do I learn to question students and push,their thinking in math and all areas?

How can 1 learn to create my own curriculumwhen I am not strong in my math skills?

How can I learn more about math so that ! knowhow to take advantage of teaching situations(teachable moments)???

RR 95.1 Pop 16 6 0 1995 by the Nalco& Center for Rosin* on Tudor WWII.

In October of 1991, having been a paid teacherfor just one month, Debi attended the first meetingofthe IMT group. Afterwards she recorded theseseven questions about math and math teaching inher journal. The depth and range ofthe questionssuggests how hard she was thinking about whatshe needed to know in order to teach math inways that were different from those she hadexperienced as a student. Her last question soundsa particularly interesting note, for it supfests akind of confidence about the possibilities forexpanding her knowledge of mathematics whichis relatively rare among teachers who have beendefeated by math in elementary school and stillconsider themselves weak in this area

But ifDebi was optimistic, in the fall of1991, thatshe could "learn more about math," it was notbecause she had experienced more success inelementary and secondary school math classes.On the contrary: The path to this moment hadbeen long and difficult. Five years earlier she wasunsure about her ability to take college courses orcontribute anything to a conversation. She had aparticularly low opinion of her own capacity tolearn math.

Elementary school, high school, and collegeIn February, 1992, Debi recalled her school andcollege experiences oflearning math this way:

When I was a student in elementary and highschool I didn't understand math and as a resultI hated it. I was taught how to do the process[algorithm) but I had no clear understanding asto why I was doing it. The teacher would giveout the page numbers in our math book thatwere to be done and if it was a new concept shewould explain the one or two examples at thetop of the page. Then each child would com-plete the problems. The students didn't talk toeach other or share ideas. The only time astudent interacted with the teacher was if sheasked a question or wanted an answer to theproblem as we were checking the pages forcorrect answers later. I don't remember a teachersaying "I don't think you all have a clearunderstanding of this or you seem to be havingproblems and so we'll go back and look at thisagain." The next day, no matter how we did onthe previous pages, we'd be off to the nextpage in the book. I think I knew that each pagein the book was going to be covered that yearand by the end of the year you could alwaysfeel a push from the teacher to "finish" thebook.

111111

I always struggled to keep up and never felt Ihad a good understanding of math. It took mea long time to catch on to the algorithm that wasbeing taught and so I was always behind andonce you get behind in math you never seem tocatch up. At least that was how I felt. Theseexperiences created a strong cue of "mathanxiety" and ! made every effort to avoid takingany math classes that weren't absolutely re-quired to graduate from high school. I eventu-ally decided that some people could "do" mathand some couldn't do math. This was rein-forced by a society that suggested that boyswere better suited to study math and sciencethan girls and by a father that steered metoward literature and history type courses be-cause he thought I was better suited for thosetype of courses.

This avoidance of taking math courses fol-lowed me into college and I was always search-ing for majors that didn't require any mathcourses. Naturally this eliminated a lot ofchoices in my college career. I eventuallydropped out of college to get married andfound myself doing basic accounting work inmy job. I discovered I could understand and domath that was related to accounting, such asadding, subtracting; and percentages. It wasmore real world math and it seemed to makesense to me. I decided that this was the "kind"of math I could do. The other "kind" of math(and I am not sure what I would have includedin that category perhaps intellectual math) Icouldn't do.

Returning to CollegeIn 1988, having concluded that she needed to beable to make more money than she could earn asa secretary or a bookkeeper, Debi decided toreturn to college. She needed to take some mathclasses as part of her program:

This meant I would have to take a placementtest with the math department to see where Iwould have to begin. I dreaded this. I knew thatI had very little background in math and didn'twant to make a fool of myself. I went to thebookstore and bought a pre-algebra book thatwould allow me to teach myself math. The booktook me, step by step, through different mathprocesses with great examples. I felt like is wasa challenge for me and I loved doing it. I spentthe entire summer doing every problem in thatbook and when I took the test I was able to gointo a beginner's algebra instead of pre-alge-bra. I felt successful in math for the first time inmy life.

Mlohlpn SUW Unlvenity, Bad Lansing, MIchlpn 45524.1034 4 RR 95.1 hp 17

Her success in the first math course boosted herconfidence still further: the course was self-paced;each student worked independently in a booksimilar to the one she had used during the summer.There were weekly meetings scheduled for thosewho wanted help, but Debi found that she did notneed to attend them. She was delighted: "I didn'teven need to go to the classes. I could do this onmy own!" She got a 4.0 in the coursea furtherboost to her confidence that she could learn mathafter all.

She felt, however, that being able to move throughthe material at her own pace was essential to hersuccess as a learner of math.

And at the beginning, especially during thatfirst summer, it was a very slow pace. BecauseI remember that when I finally hit that first classthat you had to take as a classbecause whenyou hit algebra and trig you are back in the realworld againI dropped the first one I took. Ilasted about two or three weeks. I could notkeep upor didn't think I could keep upwiththis guy's pace.

I was really upset, dropped the course, waited,took it again next time with a different instruc-tor. And that helped, but I still had to move attheir pace and that was harder for me. I did it inthe end, but it was harder.

Difficult as it was to step back into a math classwhere she had no control over the pace, she wasconvinced that she needed to prove to herself thatshe could do it.

I could not let it get the better ofme. I just wouldnot let t. . . I took two terms of economicsbased on the same thing: I had flunked them incollege the first time around and I was notgoing to let it get the better of me. So eventhough economics did nothing for me as far ascredit toward something, I took them.

It had hung over my head all those years andI had to beat it.

And after I conquered a little bit of math, andeconomics, I think I realized that I could doanything I wanted to. And then watch out!

These victories over the old demons of schoolmathematics seemed as crucial to her in retro-spect as they seemed in prospect. When shewrote and talked about her own learning, her

prose rings with the accents of celebration. Butwhen Helen asked her, in February of 1993,whether she had always felt this way about learn-ing she shook her head:

I love to learn. I really do. But I still strugglewith feeling dumb.

But this celebration of learning came when Irealized I could do it, which was when 1 did themath. And at the same time I was doing themath, I took a psychology class. And I got a4.0!

I can remember, in the orientation, on the firstday at [the community college), they asked,"Why are you here?" And I remember sayingsomething like, "I just need to see if I can dothis, and I m just so excited to be here." And Igot a 4.0 and I wasn't dumb! I shook in myboots the whole time, but

The other piece was, when I was in my teachered. program, I was in a cohort [taking all teachereducation classes with the same 29 people)and I finally became really comfortable withsharing my ideas. That's when I began to feel,"Well, I'm not so dumb." It was like my opinionwas worth something.

It always takes me a long time to warm up andsay something. I am always extremely quiet atfirst in a new class.

I'm still in the mode of thinking I'm dumb. I'llbe glad if I can ever get put that.

In the Elementary ClassroomBy the time she began to spend substantial piecesoftime in an elementary classroom in her last yearofcollege, Debi had come along way in defeatingher image of herself as someone who couldn't domath. She knew that, with hard work and time,she could get a 4.0 in a college math class.Looking back, however, she believes that shewas still entirely reliant on memorization for thisacademic triumph. In ajournal written in the fall of1991 she explains a bit about the way she hadthought about numbers, for example, in the earlymonths of the 1990-91 school year:

When I began teaching subtraction to thesecond graders, I had a process [algorithm]firmly in my mind. However, I knew that Iwanted to teach them for understanding. Iwrote my unit with that !is al in mind. Onpaper it was teaching for understanding.

RR 95-1 Pop IS 0 1995 by this Notional C.ontor fOr Research on Teacher Looming

However when 1 began teaching the unit itbecame process-oriented because it waswhat made sense to me. When Sharon andKathy were talking about this with me I gotvery frustrated. They kept talking aboutnumbers in parts. 4 and 3 are parts of 7. Butto me it was a solid number 7 that couldn't bebroken apart. Because of this idea I had ofnumbers I was having trouble teaching sub-traction the way I knew I should because itdidn't make sense to me. Eventually duringthat unit it began to make more sense tome . . . maybe I learned with the kids. I no-ticed this year that numbers don't seem sosolid and I'm thinking of them as parts ofnumbers that can be put together.

Another quote, this time from a conversation withHelen in 2-8-93:

That first year, watching Kathy, I couldn't getpast, "Well, she is doing addition." And Ididn't know what addition meant, really. . . .

When I looked at 23 +23,1 saw the 23 as solid:It wasn't 2 tens and 3 ones.

Throughout the year Debi struggled to teach math"for understanding." There were intriguing mo-ments in which children managed to explain theirideas:

I got started with what the equal sign means. Idon't know if I helped or confused them. Ithought it was interesting to watch them tryingto think through what it meant to them. I foundit interesting to try to see how they werethinking. We came to a shared definition thatthe equal sign means that both sides are thesame. (December 12,1990)

Over and over, however, her journal recordedher frustrations. On the one hand, there were hergoals: "I want my students to understand whatthey are doing. I don't want them to just memo-rize procedures to follow." But on the other handthere was what she saw herself actually teaching:"I do think I've been concerned with the teachingof one strategy to use. If I teach one strategy itseems to become a process"(January 1991). Sheconnected this difficulty to the way that she hadbeen taught math and to the fact that numberswere still "solid" for her.

Malian Stets Univstsity, East Lansing, Michlpn 411524-1034

Fa11,1991In the fall of 1991, Debi took her first paidteaching job: She was a "co-teacher" at theelementary school in which she had student taught.Instead of working with one group of children allthe time she taught four different primary classes,spending two hours with each group each week.She also joined the IMT group.

In the Second and Third Grades. Her mathjournal is a kind of celebration of her own learn-ing: Over and over again, as she watches one ofthe teachers she works with teach a lesson, shesees math,;matical concepts embedded in thelesson that she is certain she would not have beenable to see a year earlier . For example, a weekafter school started she wrote:

When I planned the Stars and Circles lesson[an activity designed by Marilyn Burns], I sawso many concepts. I could see the concept ofadding groups of numbers, learning to repre-sent numbers on paper, multiplication con-cepts, putting numbers in groups, and thatnumuers can be broken apart. . . . Last year Iwouldn't have seen this.

She was also able to design a math unit ofher ownto give her students the kind of "feel" for metricunits that had been so lacking in all other school-based encounters with mathematical topics. In ajournal written for the IMT group she explains:

I work as a co-teacher and have the opportu-nity to choose what area of curriculum I wantto focus on and for how long. After observingthe lack of time spent on measurement last yearand that often it was just memorizing I decidedto choose this area for my first unit.

I planned my first lesson around helping thechildren learn about the metric system andespecially the decimeter. I gave them a "mea-suring stick" and asked them to go around theroom and find objects that were about the samesize. I didn't tell them that it was a decimeter.They were to find an object, draw the object lifesized and identify it.

Later as we processed the lesson I identifiedthe name of the unit. We compared it to inches.They were sent home and asked to find fiveobjects at home that were that size. I wantedthe students to have a ch.ice to measure andbegin to recognize what a decimeter size lookedlike with familiar objects. By having them drawthe objects it reinforced the recognition.

2RR 95 -I Pais 19

From here I'llhelp them discover that 10 dm-1meter and, if you divide, a dm -10 centimeters.

1 wanted metrics to make sense to them, forthem to be able to use it and identify familiarobjects as a certain length so it became a partof their experiences and knowledge. (Not justmemorizing).1 am attempting to make this a wayto better understand metrics and measure-ment.

Debi's subsequent journal reflections on the les-sons in this unit indicate that she felt she hadsucceeded in achieving many of these goals. Andher reflections on her own learning contain aconsistent note of celebration as she talks aboutthe math concepts she had seen embedded inlessons she taught.

The IMT Group. Debi's reflections on the firstmeeting of the IMT group reflected the sameexcitement about her own learning: As she satdown to write after the meeting, she contrastedwhat she had seen in the videotape that night withwhat she had seen when she watched a tape ofBall teaching two and a half years earlier in herfirst teacher education course.

I first saw [Ball] on tape when I was takingTE101. 1 was impressed with how she "let herstudents" teach themselves and didn't seem tohave much input in the lessons except to set upthe problem. She never seemed to tell themthey were right or wrong. . . .

When I was watching the tape Thursday nightit was through more experienced eyes (thoughstill very much the novice). ...Where the firsttimes I saw her tapes I thought, "What a greatteacher," and couldn't go any flirther, this timeI was able to watch to see what she was doingand ask myself why she wu doing it. I was ableto think about what the kids were saying andthen try to decide why they said it and howthey were thinking. I was also able to look at thelesson and see the many directions it wasgoing and not that it was just a subtractionproblem that they were having problems with.I was also noticing how she set the originalproblem up in a way that would bring outdifferent concepts (she asked them to make amathematical sentence that equals 10), such asaddition, subtraction, multiplication, and divi-sion and probably many others that could havebeen brought out.

Delighting in the knowledge that she is seeing andhearing far more than she had three years earlier,Debi identified these areas of change: She wasnow pushing Ix self to make sense ofthe studentsthinking; she was now also noticing all the differ-ent mathematics embedded in the lessonseeingmore than a "subtraction problem the kids werehaving difficulty with."

Although Debi continuedtluoughoutthe fall to feelgood about both her progress inteaching forimder-standingthe goal she had set forherself--andherown increasing ability to hear and see new math-ematical ideas m the lessons she was teaching, herinitial enthusiasinfortheIMT groupactivitiesquicklyturnedto dismay. Afterthe group' sthirdmm, i'n& inwhich we had worked, at first individually andthencollectively, onciesigning andthenevaluatingrepre-sentations forteaching third graders about negativenumbers, she wrote that she was feeling "veryfrustrated with Math class.' . . . These discussionsdon't seem interesting tome . . they seem to dragout and go no place." She went on to explain in herjournal that a part ofher frustration with the groupwas the focus on negative integers:

I'm also having trouble with the negative num-bers. To me they seem like non-numbers andhow do we teach them if they don't exist. . . .

When I look all think about negative numbersI think about the number line and it makessense because there are rules. If you have twonegatives you add them. If you have a negativeand a positive and the positive is higher theanswer becomes positive. (1 think that is therule). But you can see if I forget the rules I'mlost because I have no idea why it's true.

In her wuricwith Kathy and Sharon, and then in herworkwith children hithe classroom, Debi had foundnew understandings ofnumbers. She felt thatforthefirst time she was beginning to understand numbersand mathematical operations: "I noticed this yearthat numbers don't seem so solid and I'mthinking ofthem as parts of numbers that can be put together."She was excited by the fact that she had come tohave some understandings of things she had previ-ously learned by rote. She was determined not toslip back into the memorization mode:

Mt 95.1 Page 20 250 1995 by the National Center for Research on Teacher Leming

So somehow I ha re to gain a better under-standing of what negative numbers are if I'm toreally be able to take part in this class. Why wasthis class focused on such a difficult conceptas opposed to something more "normal"?

The IMT group confronted her with an area ofmathematics that did not yield very well to effortsto connect it with concrete reality. It did notprovide her with the tools or the environment thatwould help her to make sense of this area ofmathematics. Thursday night meetings becamemore and more painful. In December she re-flected back on her experience of the group:

I desperately wanted to understand [negativenumbers'] purpose and not just a process bywhich to use them. I still haven't discoveredthat yet. But I have been struck by the struggleI was going through and the sense of frustra-tion I was experiencing. I didn't want to go toclass. I didn't want to do the project. I reallyjust wanted to stop coming. I began to tuneout. . . . It was an old feeling (the feeling ofwanting to just drop trying to understandnegative numbers and feeling like a failure)that I haven't experienced in a long time. I'vebeen used to accepting the challenge since Ireturned to school and not getting discour-aged. But this time I did become discouraged.(December 1991 journal entry)

A year later she interpreted the encounter withnegative numbers this way:

I just stepped right back into that old mode. Itwas a real gut reaction. When I look back,1 ask,"Why did you do that, Debi?" I don't quiteunderstand it except to say that must be a realpowerful thing in me. It was a 35 year experi-ence, and it was sitting there underneath thesurface, and, for whatever reasons, it justjumped up and grabbed me and for a briefperiod I was back in that dumb mode.

When Helen asked her why she thought she hadresponded so differently to the challenge of mak-ing sense of negative numbers than she had when,a year earlier, her cooperating teachers had chal-lenged her to think differently,about subtractionand the decomposition of numbers, she answered:

For some reason it was much harder content.

Partly it wu the setting of a different group ofpeople. A larger group.

I was still in that concrete versus abstract and"I must be a concrete learner and I'm not anabstract learner." ... I think that was still in theback of my head. Because I knew that the mathI had learned easily in my [community college]courses was stuff that I could just memorize... . When it came to problems that I really had todo some delving into or thinking about how togo about doing them, those were the ones Ialways struggled with. And I knew that pieceof me, so, of course, that was the abstractpiece. So I knew that I still had some limi onmath and obviously negative numbers wasone of them. So I had a good excuse.

I think I'll always struggle with it, but I waillet it get to me. I'll just struggle harder. . . andI think of it as a challenge.

Winter 1992Although she was strongly tempted not to returnto the IMT group after winter break, Debi de-cided to give it another try. And with the focusshifted away from negative numbers, she foundthat she enjoyed the Thursday evening meetingsfar more. The most important developments,however, occurred in second and third gradeclassrooms. Unlike most teachers, she had twodifferent school-based sites for learning aboutmath and about teaching.

Teaching Division. The first, ofcourse, was herown classroom. Having finished the unit on mea-surement, she decided to teach the second- andthird-graders a unit on division which she de-scribed in a January journal entry:

The students began by hearing the story, TheDoorbell Rang, by Pat Hufcbins. This is astory where the mother makes 12 cookies forher son and daughter. They divide it betweimthemselves and then the doorbell rings. Thekids now have to divide the cookies between4 people and soon. The kids retold the story inplay form and physically divided the cookies(blocks). Then I passed out the cookies to eachchild from the tray of real cookies and ended upwith some left over. As they were eating theircookies I asked them to write.

The idea for this unit came out of a book. But afterteaching a few lessons she altered her plan; onJanuary 14 she wrote:

Wien I first muted this unit, I saw it as a 4-6week unit' that would end when I did the lastactivity in the book. I would do exactly whatthey told me (which is fine and a good place to

Michigan star University, East Linn% Michigan 411241034

26tut 93-1 Page 21

start) and then the unit would end. Now I'mtrying to think of a way to extend this and useit more as an introduction to division. I'm notsure how to go about this but I really want totry.

The events in the classroom led to conversationsoutside ofthe classroom:

I talked with [a graduate assistant working inthe school] later about (the different ways thechildren had found to share the cookies] andshe was telling me about two types of division.One type she called "pariative," which shedescribed as How many groups of somethingwill I get? The second type she called "dis-tributive," which was How many each box willget.

The decision to go further into division also ledDebi to begin to read and think more about whatdivision actually was:

As I was planning the unit focused aroundThe Doorbell Rang] I said to myself that thiswas about division. But I didn't think aboutwhat division meant. I began teaching the unitand it began to take on a life of its own. I startedthinking about going beyond the planned les-sons. Then I began thinking about how I wouldteach division for understanding. . . . I askedmyself the question What was division? Whatconcepts were embedded in division? I pulledout my math books' and began researchingand thinking about it. I've decided that divi-sion is the opposite of multiplication (in-verse)-42+67 and 6 x 742. Other conceptswere subtraction (12+3 can be figured out by12-3-3-3-3 or -4) You could count backwardsto get the answer 15+3 ... 15,12, 9, 6, 3, O. Fivenumbers. Multiplication is needed. Addition.Place value understanding. Fractions. Remain-ders. Decimals. There are so many things/ideas that go into division.

In March, summarizing what she saw herselflearning over the previous months, Debi wrote:

It's become very clear to me that the first stepin teaching a concept to children is for me to tryto understand the concept first. I get out mybooks and try to find out what mathematicalideas are embedded in the concept, talk withother teachers, and do actual problems myself.I'm also realizing that as I begin teaching theconcept I will probably learn more from thechildren as they try to solve problems.

During the summer, in a conversation with Steve,she recalled how her learning continued as shebegan teaching:

And (the students] taught me because whenthey were doing it themselves, somebody wastaking the original number and subtracting it,and immediately I thought, "you can't do that."And then I started thinking, "But it works!"

I was more open and then I started watchingdifferent ways they found to solve it.

Seeing connections among topics was exciting. Inaddition, it made the mathematics more interest-ing and accessible:

Even your negative numbers are really so con-nected to subtraction and trading. And I thinkthat is fascinating. Once I connected the nega-tive numbers to subtraction, it made a whole lotMOM sense.

Some months later she discussed the pedagogicalimplications of discovering connections betweendivision, subtraction, addition, and multiplication:

Last year, when I saw what the different piecesof division were and how connected all theseconcepts are, I began to wonder why do wehave to teach one before the other necessarilybecause in a way they are so [connected]. Inever knew all this stuff before. It was just, ifI couldn't memo& 3 the process, I couldn'tpass the class.... That's how I got a 4.0 in mycollege classes. (IMT, 1-93)

Kathy teaches about numbers below zero.Debi had a particularly strong connection withKathy's third graders because she had studenttaught in Kathy's room and Kathy was keepingthe second graders she and Debi had taughttogether for a second year, following them intothird grade. As a consequence, when Kathydecided to venture into the land of negativenumbers, Debi was particularly interested to seewhat happened. She was also quite astonished byKathy's decision to teach this material, since sheknew how little Kathy had enjoyed exploring thisnumeric territory in the IMT group:

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I kept popping in on Kathy's group u much asI could when she started this. I had to see howthis [would go]. I couldn't believe it wouldwork. (summer conversation with SS)

When Kathy announced her decision to do thisunit, Debi wondered, "What's the point?" just asshe had when she learned that Deborah Ball hadtaught this material to third graders. But as sheobserved Kathy's students, she began to findsome answers to this question:

I just kept remembering Nathaniel going, "I didit, I did it!" or something like that. And itanswered his questions.

And then I remember Lucas from the first year[when she and Kathy were working with thesame students as second graders] saying,"Can't you do that? Isn't there . .. a number?"And Kathy and I were going, "Nope. You can'tdo that." And we were both looking at eachother and going, "Well, what are we supposedto do?" And that year we chose to ignore it.

Kathy wouldn't do that anymore. .. I don'tknow if she'd pursue it, but she'd give them ananswer, some sort of explanation.... Whethershe would pursue it or not, who knows, but shewouldn't want to drop it like she and I did thefirst time it came up.

Watching the third graders that she had known foralmost two years delightedly creating numberssentences "that end below zero," she began to seereasons for introducing them to this numeric ter-ritory. In addition, she told Steve, "Watching thekids go through it helped me figure it out more."

Before long, the second graders she was teachingasked her if dlere weren't numbers below zero. Ayear earlier, she and Kathy had ignored a ques-tion that seemed to be headed in this direction;this time she addressed it head on: "I said, You'reabsolutely right. That's called a negative num-ber,' and I pointed to my number line. And thenI said, ' You'll get into that. Talk to [yourhomeroom teacher] about that one.'"

Fall, 1992Debi had come away from her experience withnegative numbers in the IMT group determinedthat she would never again let herself slide intopassivity and defeat as a learner. She had seen thedanger of "going right back into that old mode,"and resolved to guard against it. For that reason,

an event that occurred in the third IMT meeting of1992-93 stood out as a marker for her and forSteve, Helen, and Lauren. Another teacher wasexplaining an idea that had cropped up in her sixthgrade math class about the division ofdecimals; atthe IMT group's suggestion she had moved to theblackboard in order to make the idea clearer.Many of the rest of us were copying down theproblem she had put up in order to think moreabout it. Confused by what Lisa was saying, Debileaned over to Steve, who was seated next to her,to ask a question. As he explained what hethought Lisa's students had been saying, Debiwhispered with a triumphant grin, "It's like nep-tive numbers all over again. Only this time l' mchallenged!"

REFLECTIONS ON THREE CASESKathy, Carole, and Debi have much in common:They are all white women, all three teach primarygrades in the same urban elementary school, allstudied mathematics in highly traditional elemen-tary and secondary classrooms and all emergedfrom them between twenty and thirty years agowith a strong aversion to mathematics, with littleexperience of learning math conceptually, andwith a self-definition that had "not good at math"

on it in bold, apparently indelible, letters.sAtilulVe have worked hard to learn to teachmathematics more conceptually and to provideexperiences for their own students that will pro-mote deeper understandings of mathematics andmore enthusiasm for doing mathematics. In addi-tion, they spend considerable time talking to-gether about teaching, children, and the puzzles ofmathematics teaching. All have made importantchanges in their understandings of mathematicsand their stance towards mathematics.

Yet despite these important similarities and sharedconcerns, their stories, although overlapping, arequite different. They describe different paths tolearning and different outcomes. We want to lookhere at some of the key features of these stories.After that we will look at the learning they de-scribe and offer some conjectures about whatfeatures of their own primary grade classroomsseemed to have fostered their own learning ofmathematics so much more succeinad I y than dida college classroom.

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To us, the most dramatic feature of Carole 'sstory is the language excitement, celebration,and discovery. When she talks or writes abouther own learning or about her students' ideas andinsights, Carole's language fairly glitters with vividverbs and compelling images oftravel, awaken-ing, and discovery. Math, which used to besterile, `!dry," "black and white," now "has life, ithas many questions and lots of answers." Thechildren have knowledge and ideas she neverbefore imagined. Ideas and numbers that seemedisolated now connect in previously unimaginedways. Carole seems to be telling a story aboutdiscovering connections. She sees new connec-tions among mathematical ideas as, for example,she watches her students create number sen-tences equal to ten: "I didn't realize 10 was outthere all those different ways." She finds thatpeople can connect with one another as they domathematics and try to communicate their ideasto one another in her own classroom and in thoseofher colleagues. She discovers new connectionsbetween people and mathematicsit has be-come, for example, "very obvious" in her ownlife.

Her excitement about her students' ideas and theconnections they are forging seem to propel herforward. It makes her want to listen to her stu-dents, to hear more of their ideas and to work tounderstand and appreciate them. Her discoverythat strategies that she saw them using to solve aproblem and dismissed"That can't work"were fruitful"No, wait a minute, that doeswork!"leads her to listen to them with faith andcareful attention.

By creating an environment in which childrenexplore and articulate mathematical ideas, and bylistening to the ideas that the children then articu-lated, she has learned important things about thenature of mathematics and about what it means todo mathematics.

Debi 's story is somewhat different. Although shewas originally defeated by math in school andcollege, several later experiences with formalmathematics courses built up her eroded confi-dence. By the time she started student teaching in1990, she had earned 4.0s in several math courses.Although she believed that her knowledge of this

math was highly procedural and depended on hermemory of formulas in the book, she felt that shecould now handle what she called "concretemath."

The skill she had developed in her communitycollege courses did not, however, equip her toteach math in the way she wanted to teach it. Forthis she had to explore numbers and basic arith-metic operations more deeply. She had to findways to see numbers as decomposable ratherthan "solid." She had to learn more about theconnections among arithmetic operations. Someof this she accomplished in the classroom, listen-ing to children present ideas and stretching tounderstand what they were saying. But a goodpart ofher learning came outside the classroom asshe prepared to teach, as she thought through, forexample, a unit on division and tried to connect itto work her students had done earlier on multipli-cation. The work she did outside helped her tounderstand their understandingsthe things theysaid, the representations they created on thechalkboard and in their notebooks. It also led herto connect addition, subtraction, multiplication,and division in new waysand to raise questionsabout the practice of teaching them in isolationfrom one another.

If Carole's learning was fueled by her students'newly visible ideas, Debi seems to be propelled inpart by her sense of herself as a learner, hercelebration at continuing to learn. If Carole'sstory evokes images of Dorothy opening the dooron the technicolor world of Oz,, Debi's suggestssomeone who tasted both defeat and success atthe learning game and takes special delight in herown learning because of the journey that hasbrought her to it.

Listening is an even more central theme in Kathy 'sstory than in those of her colleagues. Her firstjournal entry examines why students in her class-room listen best to one another during sharingtime; in June she identified changes in the way shewas listening to students ascentral to the changesshe had made in her practice. It is listening to herstudents, in part, that carries her into the land ofnegative numbers. And it is listening to them thatconvinces her that negative numbers are not, infact, as difficult and abstract as she had thought.

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Over and over, as she watched the third gradersworking in this numeric territory, she shook herhead disbelievingly and whispered to Helen, "Ican't believe how easy this is for them."

Kathy's interest in listening connects closely toher delight in her own learning and in her students'learning. Like Carole, she is excited by theirideas, by the way that they make sense of math-ematicsu well as other things. And she isdelighted by the ways in which ideas in the class-room generate other ideas, by the ways in whicha good question generates intellectual discourse.

Her commitment to the adventure of her ownlearning carried her to the IMT group; three anda half months later it led her to propose that, withhelp, she would like to teach her third gradersabout negative numbers. To someone as uncer-tain as Kathy was about her own grip on thiscontent, this was a truly frightening undertakingshe recalls feeling quite panic stricken on the daywhen Helen took a wrong turn on the way toschool, leaving her to launch the unit on her own,totally without support. And yet, as she said toHelen five months later, "I was like Henrietta Hen:I couldn't wait to get to school in the morning toteach about negative numbers."

Inside the Primary ClassroomBefore they chose the unit on which the IMTgroup would focus during the fall, Lauren, Steve,and Helen realized that some teachers would feelintimidated by the focus on mathematical opera-tions involving negative numbers. They believed,however, that as members of the IMT groupwatched eight-year-olds work with the represen-tations that Ball used in her classroom and lis-tened to videotaped discussions, these fears wouldfade (see Featherstone, Pfeiffer, & Smith, 1994).They thought, in short, that watching these video-tapes and exploring the thinking of children wouldprove an effective way to learn mathematics. Infact, however, they were wrong: The teacherswho said that they felt uneasy and unsure in thismathematical territory in October still claimed tobe uncomfortable there in December. Althoughthe explorations of the M.A.T.H. materials seemsto have laid the groundwork for other importantdevelopments within the group (see Featherstone,Pfeiffer, & Smith, et al., 1993) and led to changesin the way some of the teachers taught math, it did

not appear to have altered the way the teachersthought about themselves as learners of math-ematics or about the specific subject matteroperations with integers.

Nonetheless, over the course of the 1991-1992school year, Carole, Kathy, and Debi did makemajor changes in the ways in which they thoughtand felt about mathematics and in their knowl-edge of the subject matter. They traced most ofthese changes to things that happened in andaround their teaching: They learned mathby teach-ing it.

Our explorations of these three cases suggest anumber of reasons why their own primary class-rooms turned out to be particularly good settingsfor learning mathwhy, indeed, they were bettersettings thanthe vastmajority ofuniversity classeswould have been. We consider these reasonsbelow.

The relationship between the learner and themathematical ideas. When teachers begin tocreate opportunities for their students to inventnew ways to solve math problems and to sharetheir ideas and evolving theories about math-ematics publicly, they are often very much excitedby what they see and hear. At least, this is theexperience of teachers in the IMT group, and it isan experience reported by other teachers andteacher educators as well (Schiffer & Fosnot,1993). A teacher has a special relationship withideas generated by her own students in her ownclassroom. This relationship includes a sense ofpride and curiosity and is different and moreintense than her relationship with the ideas gener-ated elsewhere. Thus, although Carole remem-bered that she sometimes lost the thread of themathematics discussions that we observed onvideotapes of Ball's third grade, she focusedcarefully on listening to her own students, trying tohear exactly what they were saying. All threeteachers' journals are filled with excited reportsof particular insightsrecall, for example,Carole's excitement when one of her studentsexplained 100-100+12=12 by saying "It's like1-1=0 and 0+12=12." Hearing and celebratingthese ideas is one of the rewards for all the workand uncertainty that efforts to teach in new waysentails. Again, Carole is eloquent on this point:

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I'm amazed at how much knowledge kids reallyhave. I'm always amazed. I don't think a classgoes by that I'm not amazed. They know a lot.There are a lot of them who know a lot about alot of things. And you don't discover thatunless you let them share and let them talk.(interview, January 1992)

If a teacher feels a special interest in the ideas towhich she has in a sense served as a midwifebycresting the aivironment inwhichtheywae bornshe may study them with special care, making agreater effort to understand them than she wouldinvest in a mathematical idea she encountered inanother setting. In addition, Kathy points out, asa teacheT she feels a professional obligation tomake every effort to understand her studentsmathematical ideas: "I am responsible for thechildren learning this. That is the bottom line andit is of utmost importance to me."

The expectations the learner brings to thesetting. When Carole, Debi, and Kathy learnedthat they were going to encounter unfamiliar andsomewhat abstract mathematics in the IMT groupmeetings, they did not feel very optimistic aboutunderstanding this math. Probably most Ameri-can adults would have felt the same pessimism:their school encounters with mathematics havenot encouraged them to believe that they willunderstand new mathematics even if they suc-ceeded in getting adequate grades in mathematicscourses. They bring the legacy of these schoolexperiences with them to any formal setting inwhich they are students and mathematics plays avisible role.

But elementary school teachers who arrive at auniversity mathematics or mathematics educationclass expecting to be confused may feel verydifferent in their own classrooms. In this setting,they expect to understand what is said. They donot expect their own students to formulate math-ematical ideas which are beyond their own ca-pacity to understand. And the expectation thatthey can understand what their students say maysupport their efforts to make sense of what stu-dents say and the representations that they cre-ate.

Once the effort has been made and the difficultnew idea understood, Kathy's story demon-strates that the experience may have immensesymbolic importance. " I have learned," Kathyreports after a year of listening hard to her stu-

dents mathematical ideas, "that I can do reallyhard math by teaching it." Carole makes a similarcomment: "I'm not afraid of it, because I canfigure it out too." And Debi notes, "I feel chal-lenged."

Nature of the learning opportunities. TheNCTM argues that a classroom in which childrenare working on real problems, explaining theirthinking, and generating multiple ways to look ata question creates a better environment for learn-ing mathematics than does a conventional math-ematics curriculum. Their arguments are based onrecent research in cognitive psychology and onsocial constructivist ideas about learning. Thisresearch applies as much to the learning ofadultsas to elementary school children; it follows thatteachers, ifthey are to construct understandingsofmathematical ideas, need a chance to engage inmathematical discussions and to play intellectu-ally with alternative representations. As mattersnow stand, they are unlikely to find these oppor-tunities in a university math class. Having workedhard to create them in their own primary class-rooms, Carole, Debi, and Kathy did find themthere.

In an analysis of what teachers need to know inorder to teach history in secondary schools,Wilson, Shulman, and Richert(1 986) argued thatin order to convey a concept to the diversecollection of students present in any secondaryschool classroom, teachers need to know theirsubject deeply enough to be able to represent it inmultiple ways. One representation simply will notwork for all learners. In the mathematics classesof Debi, Carole, and Kathy, both children andteachers have access to multiple representations,because classroom norms encourage students togenerate and present them. These norms generateopportunities for teacher as well as students tolook at an idea from multiple viewpoints.

What is taught. In the past decade, multiplevoices have pointed out that what students do intraditional elementary, secondary, and collegemathematics classes bears little relationship towhat mathematicians do (see, for example, Ball,1990b; Lampert, 1990; NCTM, 1989 and 1991):While mathematicians work, both alone and col-lectively, to solve mathematics problems for whichtheir disciplinary community currently has no so-lutions, students in math classes work to memo-rize or understand the results of the work of

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mathematicians who" have been dead for centu-ries. In the vast majority of math classesat anygrade levelstudents get no experience of doingmathematics. They do not learn that mathematicsis a human construction, that learning to do mathis, in large part learning to hear and make sense ofwhat other mathematicians think about how toapproach math problems. They learn, as Carolesays she did in her years as a student, that"[m]athematics was very individualistic and dry.There was a process, you had to learn it, and yougot through it."

In their own primary grade classrooms, Carole,Debi, and Kathy got to see peopleseven andeight year olds "doing mathematics." Day afterday, they saw young mathematicians workingtogether to find a way to solve a problem thatmade sense to all in the community. They workedon ways to communicate about mathematics thatwould foster shared tuiderstandinis. They workedto validate conjectures, or to find counter ex-amples. As they nurtured, presided over, andobserved the work of these communities, theyhad the opportunity to redefine what mathematicsis, and what it means to do mathematics. AsCarole said in January 1993, "It's communica-tion, it's discovery, it's adventure."

In their primary grade classrooms, Carole, Debi,and Kathy are having an opportunity that none ofthem had during the first 35 years of their lives,and that only a tmy minority of American adultswill ever have: they participated in aconununity ofmathematicians. We ought not to be surprisedthat this experience deepened their knowledge ofthe discipline.

What the learner must do in order to learn.Common sense tells us that few Americans wholeave college with little knowledge ofmadiemat-ics and little confidence in their ability to docomplex math will deepen or extend their knowl-edge of the discipline. It is hard to learn moremath as an adult, if you begin by feeling that youknow very little. There are several reasons forthis.

First, in the absence ofcompelling external incen-tives, most people avoid settings in which theyexpect they will feel uncomfortable or incompe-tent, and a history ofbad experiences with schoolmath will probably lead most people to expect to

experience a potpourri of negative emotions inany organized math class or even in an informalsetting where they are routinely expected to thinkabout math.

Second, in most settings, one has to ask embarrass-ing questions in orderto learn basic math. In April1992, Kathy asked to IMT group whether one plateout of a set of 8 was 1 or 1/8. A year later sherecalled, "I felt like I was taking a big risk when Iasked that question. And then, everyone else blew,everyone else said, "Both!" like it was really obvi-ous." Six months later she hesitated visibly beforeasking others in the group to explain a point thatcame up as we watched a videot ofaa%iiWonin Ban; classroom. It is one thing to say, "I'm notFood in math." It is something else to display yourignorance by asking a question that may turn out tobe, as Kathy says, "embarrassingly elementary." Inaddition, when you ask someone to teach yousomething, the spotlight focuses on you inaway thatmay turn out to be very uncomfortablethe would-be explainer will keep asking whether you under-stand. Sometimes you say that you dojust becausehaving yourunderstanding taken out, inspected, andevaluated at frequent intervals is intensely unpleas-ant

In her own classroom, Kathy came to new under-standings about fractions and ne ;ative numberswithoutdisplaying ignorance publicly or being puton the spot in the same way. She and Carole andDebi learned by listening to what children saidand by thinking carefully about their claims andtheir representations. They found these activitiesdeeply congenial.

The rewards for learning. Teachers in theIMT group are strongly committed to creatingthe best environments they can for their stu-dents to explore mathematical ideas and growin mathematical power. Had they believed thatthey could have accomplished this by enrollingin a college math class, many would havestruggled to overcome a natural reluctance toput themselves in a situation where they feltpessimistic about succeeding as students andenrolled. In fact, however, research does notsuggest that teachers can count on learningwhat they need to know about mathematics toimprove their teaching in such an environment(Ball & McDiarmid, 1990). Nor is this par-ticularly surprising: Just as Debi did not findthat taking algebra and trigonometry courses

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in collegeand getting top grades in them!helped her to think about numbers in ways thathelped her to teach subtraction to seven yearolds, most teachers will not find that the bow-ing acquaintance with limits that they get in aten-week calculus class helps them to teachdecimals or area. If a teacher's focus is on herown classroom, the rewards for learning mathin a college class are, at best, far removed intime from the effort expended; at worst, theynever materialize.

By contrast, the rewards for working hard tounderstand what your own students are saying orare likely to say tomorrow are immediate andsometimes immense. Consider, for example, thepleasure that Kathy felt when, after struggling fora moment with Noah and Jeff s assertion that "-10+- 10 =0," she understood both that it wasincorrect and why it was incorrect and managedto formulate a question to help them to look at theproblem from a new angle. (And imagine herfurther satisfaction when, a few minutes later, theone little girl who had previously failed to makemuch sense of numbers below zero came to theboard and explained, clearly and cogently, why -10 + -10 had to equal -20.) Outside of her ownclassroom it would be difficult indeed fora teacherto find such powerful and immediate rewards forher efforts to understand a mathematical idea.

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Nntes'Thus materials were generated by Madianades and Teach-

ing Through Hypennedia In the summer of 1989The National Science Foundation fimded the M.A.T.H. projectpermitting Deborah Ball, Magdalene Lampert, and colleagues todocument the toad ing and learning in their classrooms. Ova thecourse of the next academic year they videotaped many math-ematics lessons, u well as interviews with students, mathema-ticians, and mathematics educators. They saved and reproducedall student work, including the students' math journals, home-work, and tests. A team of graduate students kept fieldnotes onthe =damsels:al and pedagogical issues raised in each lean, andreproduced the teachers' journals, in which they recorded eachday their reflections on lessons. During the following two yearsBall and Lampert worked with teams of graduate students tocreate videodiscs that would permit prospective and practicingteachers outside ofthe college to explore some ofthese materials.

'Since 1989, their school, like a number of other publicschools in mid-Michigan, has been linked to the Michigan StateUniversity College of Education as a part of the College'sprofessional development school effort.

'The names used for students in this publication arepseudonyms.

'We had decided to ask the third graders to figure out whatthe temperature in Anchorage was at nightfall if it had been 2degrees in the morning and had idles by 6 degrees during the day.Instead, she asked them to figure out how much the temperaturehad fallen if it started at 2 degrees and ended up at -4 degrees.

'Although Steve, Helen and Lauren did not thhdrofthe groupas a math class, it is interesting to note that, at least on thisparticular day, this was how Debi described it. And, indeed, theactivity of nesting and evaluating representations for the opera-tion of subtracting a negative number clearly did require someunderstanding of mathematics.

'This view of negative numbers has a long and honorablehistory. As Barrow (1992) observes, "Negative numbers do notappear to have bean generally recognized u "numben" until thesixteenth century. Thus Diaphanous described as "absurd"equations with negative answers (p. 90).

'She was meeting with each group only once a week.

'In February, 1993, Debi elaborated on what shemeant here: "I always go to the Standards, plus I have acouple of other books I use. And so I went and read whatthey were saying about it and then I tried to list what I wasgetting out of them. And I also just did a couple ofproblems and tried to figure out what I thought I waspulling out of It. And I also probably talked to Kathy."

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