DOCUMENT RESUME ED 431 630 SE 062 694 AUTHOR Nelson, Barbara Scott TITLE Building New Knowledge by Thinking: How Administrators Can Learn What They Need To Know about Mathematics Education Reform. INSTITUTION Education Development Center, Newton, MA. Center for the Development of Teaching. PUB DATE 1999-01-00 NOTE 23p. AVAILABLE FROM Education Development Center, 55 Chapel Street, Newton, MA 02454. PUB TYPE Reports Research (143) EDRS PRICE MF01/PC01 Plus Postage. DESCRIPTORS *Administrators; *Educational Change; Elementary Secondary Education; *Mathematics Education ABSTRACT The mathematics education reform movement is built on ideas about the nature of learning, teaching, and indeed mathematics itself that are very different from the views that have prevailed in American schooling for many years. This has implications for administrative practice, since an enterprise that exists to support rigorous thinking on the part of students requires administrative supports different from those for approaches that exist to transmit accepted knowledge from teacher or textbook to student. Understanding new ideas about mathematics, learning, and teaching, and exploring the implications of these ideas for administrative practice require conceptual change on the part of many administrators. This paper describes the pedagogical principles that underlie a program designed to provide opportunities for such conceptual change for administrators. (Contains 48 references.) (Author) ******************************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ********************************************************************************
Transcript
DOCUMENT RESUME
ED 431 630 SE 062 694
AUTHOR Nelson, Barbara ScottTITLE Building New Knowledge by Thinking: How Administrators Can
Learn What They Need To Know about Mathematics EducationReform.
INSTITUTION Education Development Center, Newton, MA. Center for theDevelopment of Teaching.
PUB DATE 1999-01-00NOTE 23p.
AVAILABLE FROM Education Development Center, 55 Chapel Street, Newton, MA02454.
PUB TYPE Reports Research (143)EDRS PRICE MF01/PC01 Plus Postage.DESCRIPTORS *Administrators; *Educational Change; Elementary Secondary
Education; *Mathematics Education
ABSTRACTThe mathematics education reform movement is built on ideas
about the nature of learning, teaching, and indeed mathematics itself thatare very different from the views that have prevailed in American schoolingfor many years. This has implications for administrative practice, since anenterprise that exists to support rigorous thinking on the part of studentsrequires administrative supports different from those for approaches thatexist to transmit accepted knowledge from teacher or textbook to student.Understanding new ideas about mathematics, learning, and teaching, andexploring the implications of these ideas for administrative practice requireconceptual change on the part of many administrators. This paper describesthe pedagogical principles that underlie a program designed to provideopportunities for such conceptual change for administrators. (Contains 48references.) (Author)
Can Learn WhatThey Need to KnowAbout MathematicsEducation Reform
Barbara Scott Nelson
January 1999
EDC
The Center for the Development of Teaching (CDT) is a research and developmentcenter within the Center for Learning, Teaching, and Technology (LTT) atEducation Development Center, Inc. (EDC). The goal of the Center for theDevelopment of Teaching is to learn, with teachers, how teachers' practice can betransformed so that it supports students' construction of knowledge. The Center isnow focusing on mathematics and science teaching, but will expand to includethe teaching of history and/or language as well. The Center carries out a coordi-nated program of research and action projects that address the issues involved inteacher change at three interacting levels: (1) teachers' beliefs and knowledgeabout their subjects and about learning and teaching; (2) teachers' classroom prac-tice; and (3) the complex social system that extends from the school and schooldistrict to the society at large.
This CDT Paper Series is intended as a vehicle for discussion of research on teach-ing and teacher development as they relate to education reform. Publications inthis series will contribute to the community's understanding of the practice andprofession of teaching, as well as to the process of change. It is our editorial policyto have papers reviewed by a panel of four readersthree named by the authorand one chosen by CDT.
Core support for the Center for the Development of Teaching, and for CDT'sPaper Series, has been provided by the DeWitt Wallace-Reader's Digest Fund,whose mission is to foster fundamental improvement in the quality of educationaland career development opportunities for all school-age youth, and to increaseaccess to these improved services for young people in low-income communities.Support for the editing and production of individual papers is provided by thegrant that supported the work described in the paper.
If you would like to be in direct contact with the author of this paper, please writeto:
Barbara Scott NelsonEDC, 55 Chapel Street, Newton, MA [email protected]
Building New Knowledge by Thinking:How Administrators Can Learn What They Need to Know
About Mathematics Education Reform
Barbara Scott Nelson
The mathematics education reform movement is built on ideas about thenature of learning, teaching, and indeed mathematics itself that are verydifferent from the views that have prevailed in American schooling formany years. This has implications for administrative practice, since anenterprise that exists to support rigorous thinking on the part of studentsrequires administrative supports different from those for approaches thatexist to transmit accepted knowledge from teacher or textbook to student.Understanding new ideas about mathematics, learning, and teaching andexploring the implications of these ideas for administrative practicerequire conceptual change on the part of many administrators. Thispaper describes the pedagogical principles that underlie a programdesigned to provide opportunities for such conceptual change for admin-istrators.
Uhe mathematics education reform effort currently under way suggeststhat children, teachers, and administrators develop deeper and more
considered ideas about what mathematics learning and indeed the intel-lectual character of school more generally might be. That is, the veryideas about the nature of learning, teaching, and mathematics on whichtypical educational practice has long been based are changing. Ratherthan viewing (1) mathematics learning as the absorption of a series offacts and mastery of procedural manipulations and (2) teaching as theprovision of conditions for absorption and practice (here called the"transmission" view), teachers and children are now meant to viewmathematics as a subject that can be reasoned out and make sense.Reformers want mathematics classrooms to function as mathematicalcommunities in which students have opportunities to reason mathemat-ically, communicate about mathematical ideas, and make connectionsamong mathematical ideas and between mathematics and their owndaily lives (here called the "socioconstructivist" view).
Both teachers and children have begun to reconstruct their sense ofthe learning and teaching enterprise, and consequently what their
Nelson, B. S. (1999). Building New Knowledge by Thinking: How Administrators Can Learn What TheyNeed to Know About Mathematics Education Reform. Newton, MA: Center for the Development ofTeaching, Education Development Center, Inc.
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work together is. These shifts in meaning andpractice have been well documented for teach-ers (cf. Fennema & Nelson, 1997; Franke et al.,1997; Schifter, 1996a, 1996b; Schifter & Fosnot,1993) and to some degree for children (Cobb etal., 1992; Lester, 1996; Soucy-McCrone, 1997).The possibility that similar changes in beliefand practice are indicated for school and districtadministrators is beginning to be investigated(Nelson, 1997; Spillane & Halverson, 1998;Spillane & Thompson, 1997), and there is anemerging body of research on the ideas thatadministrators construct about the reforms(Nelson, 1997; Nelson & Sassi, 1998; Spillane,1998).
However, professional development programsfor administrators that provide support as theywork to understand reform ideas are just begin-ning to appear. It is the premise of the workdescribed in this paper that such programs needto provide administrators with the opportunityto explore new ideas about mathematics, learn-ing, and teaching, but that they also need toaddress the fit between ideas of school adminis-tration and ideas about learning and teaching.That is, there is a mismatch between the prin-ciples that underlie much educational adminis-tration and the principles that underlie reformedmathematics instruction. To manage reformedinstruction effectively, administrators need toreexamine both their ideas about mathematics,learning, and teaching and their ideas aboutschool management.
Some forms of instructional management arealigned with transmission forms of instruction;other forms align better with reformed instruc-tion. For example, bureaucratic forms of man-agement and control assume, with transmis-sion views of teaching and learning, that stu-dents are sufficiently standardized that theywill respond to instruction in predictable ways,that teaching tasks are routine enough to beconverted to procedures, and that classroomscan be viewed as similar, self-contained units tobe organized by a common schedule and com-mon rules (Campbell et al., 1987; Rowan, 1990).Entire lines of research and practice in themanagement of instruction have been built onthe transmission view of learning and teaching:direct instruction in basic skills as one of thehallmarks of effective schools (Edmonds, 1979),
process-product research on teaching (Brophy& Good, 1986), and the consequent processes ofteacher supervision (Darling-Hammond & Sclan,1992). Other administrative practices, such asthose deriving from human relations theories ofmanagement (Campbell et al., 1987), commu-nity metaphors for schooling (Sergiovanni,1994), or views of administrators as learners(Barth, 1990), are more likely to emphasize therelationships between professional colleaguesin a school and the growth of individuals thanthe standardized delivery of instructional ser-vices.
When fundamental ideas about learning andteaching that form the center of the enterpriseof schooling begin to shift, as now, the eclecticarray of existing management ideas and prac-tices is thrown into relief. Inconsistencies be-tween what is being managed and the nature ofthe management become more apparent, andconsideration of what administrative practicesare now most appropriate becomes relevant(Rowan, 1995; Sykes, 1995). It is also possiblethat in the effort to support new modes ofteaching and learning, new administrative prac-tices will emerge.
The disjunction between what is being man-aged and the models available for managementis not just an abstract matter; it plays out in theideas and daily actions of many school anddistrict administrators. Most administratorswere educated at a time when the transmissionview of learning and teaching prevailed. Theirpersonal teaching histories, often based on thisview, inform their administrative practice. Ad-ministrators have specific images of classrooms,teaching, and learning in mind as they makeadministrative decisions that they intend assupportive. These images, along with the im-ages of management that they have acquiredover the years, ground their sense of what it isthat is being managed and how it can be sup-ported.
To make this connection clear, consider theexample of teacher evaluation. Many adminis-trators have images of mathematics classroomsin which the lesson is presented, students dowork at their seats or in groups, and homeworkis assigned, with the expectation that the lessonshould be tied up neatly by the end of the class
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periodthe facts of the lesson and the home-work assignment clear. This image of class-rooms is based on the notions that knowledgecan be unproblematically transmitted fromteacher or textbook to students in discrete chunksthat can fit neatly into a 42-minute class periodand everyone's mind can then shift cleanly tothe next subject on the agenda. Such an imageleads many principals evaluating a teacher'sperformance to expect "closure" to the lesson.But evaluating classrooms in which knowledgeunfolds through discourse and in which inter-esting questions are not all answered by the endof the class period requires a different image ofwhat knowledge is and therefore what shouldgo on in classrooms and how the lesson mightend. Traditional notions of closure may beinappropriate. Administrators whose expecta-tions of how a lesson should end are based on atransmission view of learning will be out ofalignment with contemporary ideas about in-quiry- or discourse-based teaching and are likelyto perform the administrative function of teacherevaluation in a way not attuned to the intent ofthat teaching. One principal with whom wehave worked described how his expectation forclosure to lessons changed:
From a traditional observation's point of viewand a paradigm of the past, most principals, Iassume, would go in and look for a total lesson,that the closure would be there . . . that youwouldn't leave any cliff hangers to be carriedover. [In the lesson on this videotape] there's ashift to saying, "I'll carry it on another day."Kids go home and do their follow-up assign-ment. . . . I look for closure in most lessons andI'm not seeing [it] any more and it doesn't upsetme as it would [have] in the past.
This is not to say that there is now no sensibleway for lessons to end, only that the traditionalidea of closure may be inappropriate for a class-room environment in which important andhard ideas are being discussed over a long pe-riod. "Closure" may need to be redefined.
A premise of our work with administrators isthat their ideas about the nature of learning andteaching matter. That is, if one is responsible foradministering a school or school district, it isnot sufficient to employ management tech-niques without regard to the degree to whichthey are appropriate to the nature of the pro-cesses being administered. Understanding the
nature of the organization's basic processesinthis case, teaching and learningis a prerequi-site for appropriate management.' And so, inour view, it is necessary for administrators tounderstand contemporary ideas about the na-ture of learning and teaching in order to take acritical stance toward their own administrativepractice and modify it where appropriate.
In this paper, I will first describe the context andtheoretical underpinnings of the work my col-leagues and I did with school and district ad-ministrators on the ideas of mathematics, learn-ing, teaching, and management. Three peda-gogical design principles for this workwhichtook the form of a monthly seminarare iden-tified. I then present three vignettes that showthe design principles at work. I use each vi-gnette to illustrate the nature of one of thepedagogical design principles, the kind of con-ceptual change it supports, and the relationbetween new educational ideas and issues ofmanagement.
Methodology
Context
To understand how administrators' ideas aboutlearning, teaching, and mathematics affect theirwork and to explore the possibility that theymight ground their administrative practice onnew and different ideas about the nature oflearning and teaching, we worked for threeyears with a group of administrators who wereinterested in developing a deeper understand-ing of mathematics education reform and inconsidering its implications for their work. Fortyschool and district administrators from districtsin metropolitan Boston, including Boston, par-ticipated in the program: 32 elementary schoolprincipals, 5 district-level elementary mathemat-ics coordinators or supervisors, and 3 assistantsuperintendents of curriculum and instruction.While the total enrollment was 40, administra-tors participated for the years during whichteachers from their schools took part in a relatedteacher enhancement program. Therefore, to-tal administrator enrollment each year was about20. On average, 15 administrators were inattendance at any given monthly meeting. Mostadministrators participated for two years, sevenparticipated for three.
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Of the group, one was concurrently a highschool mathematics teacher. The others hadlittle formal mathematics training; the elemen-tary principals would have taught mathematicsas part of their elementary school teachingexperience. Two had Ph.D. degrees in liberalarts subjects; most had master's degrees in edu-cation. All had been administrators for at least5 years; many, for 20 or more.
While a few of the administrators in the groupconsistently expressed views that indicated atransmission perspective on learning and teach-ing, most held mixed positionscombinationsof transmission and socioconstructivist views.Two administrators consistently expressed ideasabout the development of children's knowledgethat could be characterized as socioconstructivistand had for many years participated in profes-sional associations that supported these views.
We and the administrators explored the ideasabout learning, teaching, and mathematics thatunderlie the mathematics education reform ef-fort and related them to specific areas of admin-istrative practice of concern to the administra-tors. We did this work through a monthlyseminar, or "inquiry group," for which projectstaff assigned readings, developed activities, andfacilitated group discussions to give administra-tors the opportunity to more deeply examinetheir fundamental ideas about mathematics,learning, teaching, and the intellectual cultureof schools. Project staff intended these discus-sions to encourage administrators to articulateand examine their own understandings of learn-ing, teaching, mathematics, and school cul-tureideas that for many administrators func-tioned as assumptions and were no longer criti-cally examined. The project's goal was to pro-mote administrators' reflection on the degree towhich those ideas were helpful guides for prac-tice in the current reform climate.
Data collection and analysis
Ethnographic field notes were taken at all ad-ministrator inquiry group meetings. Thesemeetings were also audiotaped, and the tapeswere transcribed. In-depth interviews were con-ducted with all administrative participants atthe beginning and end of the program. Theseinterviews were audiotaped and transcribed.
The data analyzed for this paper consisted of theplanning notes for all seminar meetings, tran-scriptions of the audiotapes of seminar meet-ings, and transcriptions of all interviews withadministrators. Seminar planning notes werereviewed to identify the major design principlesthat underlay the work. This resulted in theidentification of three primary design principles.Seminar sessions in which those principles wereparticularly well illustrated were identified.Transcripts of those sessions were analyzed toidentify events that could serve as vignettes forthis paper.
Pedagogical Design to SupportAdministrator Learning
For many administrators, developing new ideasabout learning, teaching, and mathematics en-tails fundamental conceptual changes (Nelson,1997). In particular, most of the administratorsin the group we were working with viewedlearning, teaching, and mathematics in trans-mission or mixed ways, and therefore the op-portunity to think through these ideas, as repre-sented in mathematics education reform, wasindicated. While there is little research onconceptual change on the part of administra-tors prompted by the discipline-based reformmovement, there is substantial literature aboutthe conceptual changes required of teachers(e.g., Carpenter, Fennema, Peterson, & Carey,1988; Fennema et al., in press; Lampert, 1987;Schifter & Fosnot, 1993; Schifter & Simon, 1992;Thompson, 1991; Wasley, 1990). As a startingpoint for our work with administrators, weadapted the theoretical position that underliesour work with teachers, which is designed toprovide opportunities to explore transmissionand socioconstructivist ideas about learning,teaching, and mathematics (Nelson &Hammerman, 1996; Schifter & Fosnot, 1993;von Glasersfeld, 1990).
When working with teachers, we are very awarethat, by and large, they have been the recipientsof the traditional form of mathematics educa-tion that is currently under revision. Therefore,they need opportunities to experience math-ematics differently, to deepen their mathemat-ics knowledge in ways that will strengthen theirteaching, and to consider new ideas about howchildren's mathematical thinking develops. All
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of these will have an impact on what and howteachers teach. Moreover, just as we now knowthat children do not develop mathematicalunderstanding solely by being told mathemati-cal facts and practicing mathematical proce-dures, neither do teachers develop new ideasabout mathematics, learning, and teaching sim-ply by being told new facts and practicing newtechniques. In our experience, teachers learnbest by actively working on intellectually inter-esting mathematical and pedagogical problems,resolving dissonances between the way theyinitially understood the situation and new evi-dence that challenges that understanding, anddeveloping new understandings of mathemati-cal and pedagogical ideas (Nelson & Hammerman,1996; Schifter & Fosnot, 1993). As Thompsonand Zeuli so compellingly put it in the context ofstudent learning,
Such thinking is generative. It literally createsunderstanding in the mind of the thinker.Thinking is to a student's knowledge as photo-synthesis is to a plant's food. Plants don't getfood from the soil. They make it throughphotosynthesis, using nutrients and water fromthe soil and energy from sunlight. No photo-synthesis, no food. Students don't get knowl-edge from teachers, or books, or experiencewith hands-on materials. They make it bythinking, using information and experience.No thinking, no learning. At least, no concep-tual learning of the kind reformers envision.(Thompson & Zeuli, 1997, p. 9)
In planning our work with administrators, wetook it as our task to determine how to providethe contexts in which administrators could buildnew knowledge by thinkingnew knowledgeabout mathematics, about learning, about teach-ing, about their own administrative practice.
We expected that administrators' learning about(1) mathematics and (2) children's thinkingabout mathematics would be similar to teach-ers'. However, administrative work has a rela-tion to children's mathematical thinking that isdifferent from that of teaching. Teachers havethe opportunity to observe student thinkingevery day in their classrooms, reflect on whatthey see, and base the next teaching move on anassessment of what their students might pro-ductively think about next. Teachers' work isvery close to and intertwined with students'thinking. Administrators' work is distanced
from the classroom, and it was not clear at theoutset how ideas about mathematics andchildren's mathematical thinking could be maderelevant to administrators' own work, nor howaspects of their administrative practice mightchange once administrators' ideas about thenature of mathematics, learning, and teachingbegan to change.
In this paper, I sketch out three aspects of thedesign of our work with administrators that wefeel are fundamental in helping administratorsthink deeply about learning, teaching, and math-ematics, and consider the difference to theiradministrative practice of grounding it in newviews. Each addresses the relationship betweenthe ideas of mathematics, teaching, and learn-ing embedded in mathematics education re-form and administrators' work in a subtly differ-ent way. Together they offer a powerful oppor-tunity for administrators to think through therelationship between educational ideas and ad-ministrative ideas.
The first design principle, called "layering," is astructure for the seminar as a whole and forindividual sessions that points out the directlinks between children's mathematical think-ing, teachers' mathematical and pedagogicalthinking, and administrators' thought and prac-tice. This structure enables administrators toconsider the possibility that reformed math-ematics classrooms have norms, values, andpractices that might be valuable for the schoolas a whole and therefore for their own adminis-trative practice.
The second design principle is to situate theconceptual work of thinking about mathemat-ics, learning, and teaching in areas of adminis-trators' own work. That is, new ideas aboutmathematics, learning, and teaching affect notonly the nature of instruction but also adminis-trative functions that are related to instruc-tionclassroom observation and teacher super-vision, curriculum adoption, selection of stu-dent assessment instruments, communicationwith community stakeholders, and so on. Think-ing about new ideas in the context of theseadministrative tasks motivates administratorsto do the hard work of reconceptualizing funda-mental ideas.
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The third design principle is to operate theseminar according to the pedagogical principlesthat inform reformed mathematics classrooms.This gives administrators the opportunity todirectly experience "thinking in order to learn"for themselves and to analyze together what hashappened in the seminar to make such learningpossible. Implementing this design principlehas many ramifications, only two of which arediscussed here: (1) doing mathematics togetherand (2) functioning as an analytic and reflectivecommunity.
These design principles are not mutually exclu-sivethat is, most sessions of the inquiry groupwere layered, grounded in an area of practicalaction, and functioned as an analytic and reflec-tive community. The distinction made here isanalytic. More than one design principle can beseen in each of the vignettes below.
Design Principles
Layering
If administrators are to consider the possibilitythat the mathematics education reform effortimplies a very different intellectual culture forschools and different behavior on their part,they need opportunities to explicitly think abouthow central elements of this new culture wouldbe enacted at different organizational levels,including their own.
The first design principle, layering, is designed toprovide such opportunities. In general, weconstructed class sessions so that it would bepossible to consider the way in which impor-tant aspects of the mathematics education re-form effort would be enacted at several levels ofthe educational system: the level of students,the level of teachers, and the level of administra-tors. To illustrate what layering is and whatadministrators can learn from it, we describe thedesign of a session in which administrators werelearning about "a different kind of listening."
One important tenet of reformed mathematicsinstruction is the need for teachers to listencarefully to students' mathematical thinkingand to base their instruction on what they learn(Franke et al., 1997; Schiffer & Simon, 1992).Such listening is an essential element of themove from a didactic mode of teaching to a
more facilitative one. Of course, teachers alsoneed to know enough mathematics to be able tointerpret the mathematical significance of whatchildren say, and have good ways of interven-ing to help children's mathematical thinkingmove forward. But a fundamental differencebetween traditional and reformed teaching isteachers' stance of listening itself. The layeredsession described below was designed to pro-vide administrators with the opportunity tolisten to a child's mathematical thinking, tothink about what teaching would be like if itwere built on knowledge of children's thinking,and to consider how the kind of respectfullistening that teachers are being encouraged todo in their classrooms could be part of a newintellectual culture for the school as a whole.
Layer 1: Listening to a Child's MathematicalThinking. Administrators watched a 17-minutevideo clip of a clinical interview with Genevieve,a fifth-grader. They listened to Genevieve'sexplanations of her mathematical thinking, notwith an attitude of judgment, or searching forwhat needed to be fixed, but in an effort tounderstand the mathematical world she inhab-ited. They developed conjectures about whatGenevieve understood, pointed to evidence thatsupported or disconfirmed a conjecture, andacknowledged that they were unsure about somethings.
Genevieve had been asked the following ques-tion: "You have 15 cookies and there are 6 kidsat your party, total. How many cookies wouldeach child get?" Genevieve set up the standardlong-division algorithm, 6M, did the calcula-tion, and answered, "Two and a half." Then theinterviewer said, "Now, suppose your motherate one of the cookies before she served them toyou. Then there would be 14 cookies to shareamong you and your friends. How many cook-ies would each of you get, provided you weresharing equally?" Genevieve correctly executedthe long-division algorithm, 6Y14. She wrotethe answer as 2.333 and said the answer was"two and one-third." Genevieve and the inter-viewer then discussed the relation between deci-mal and fractional representations of one-third,and Genevieve tried to represent the problem inother waysrepresenting the cookies in adrawing, using uni fix cubes, and making
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manipulatives from torn pieces of paper. Ingeneral, Genevieve had trouble making alter-nate representations of how the cookies wouldbe apportioned, and trouble relating these rep-resentations to the long-division algorithm. Inworking on the second problem, she arrayed the"cookies" into 3 rows of 4 cookies each, with 2left over, and spent most of her time trying tofigure out what to do with the remaining 2. Shedid not seem to understand that the reason shehad divided 6 into either 15 or 14 was that shewas trying to find out how many cookies each ofher 6 friends would get and therefore, in orderto represent the problem with a diagram, neededto draw or make with the unifix cubes 6 piles.The problem to be solved was, How many cook-ies would be in each pile? Her inability torepresent the problem diagrammatically indi-cates that while she could execute the long-division algorithm, she had only a fragile under-standing of the idea of division.
In their discussion after viewing the videotape,the administrators were interested in Genevieve'sperseverance and her willingness to keep ex-ploring the problem. They talked about thetrial-and-error nature of her guesses at one point,and about how she didn't seem to use herknowledge from the first problem to help hersolve the second one. In general, they werestruck by the difference between Genevieve'sfacility with the algorithm and her hesitancyand confusion when trying to explain how itworked "in cookies." They developed conj ec-tures about thisfor example, noting that shedidn't seem to have an internal sense of whatputting a group of things into groups really is."So, where's the sixness?" one asked. The ad-ministrators were listening to Genevieve muchas teachers listen when they are trying to under-stand the mathematical thinking of a child intheir classroom. They were experiencing "adifferent kind of listening" at the first levelthelevel of listening to students.
Layer 2: Teaching Based on Listening. In thenext part of this session, the administrators readand discussed excerpts from the journals ofteachers who had done their own clinical inter-views with children for the first time. At thislevel, administrators were exploring what itmeant for teachers to engage in "a different kind
of listening" and the impact that such listeningwas likely to have on their ideas about teaching.
The teachers had been asked to write about whatwas powerful for them in the experience oflistening to children's mathematical thinking.Sample excerpts from their journals included:
I was most struck by how much I learned aboutwhat [the child] knew. I had him as a studentand had no idea about his system of thinking . . . orhis conception of zero, or even which numbers he"liked." The wealth of information I learned isamazing to me, once I know the right questions toask. It never occurred to me to ask a child howthey knew that a 10 is ten, or why they carry itover to the next column in the depth I nowknow how to do.
There were a couple of times when I thought[she] understood a concept or was really firm inunderstanding a strategy. But when I would askher similar questions, it would turn out thatshe was not really as sure as it had appeared.She could do it that time but it wasn't some-thing she was really solid on. This happeninga number of times made me aware of how youhave to ask about the same concept in differentways to ensure a clear understanding of theinterviewee's knowledge.
I enjoyed the "bonding" during the interviewand how much I could see the value in it forgaining understandings of how kids see math.I'm so caught up in "teaching" math. I can seehow I've been missing the boat. Thinkingabout their thinking is the key to success for thechild and the teacher. I also enjoyed it becauseit was nonthreatening and friendlyyet Ilearned a lot about what he knew. Tests makekids uptightand you wouldn't get half asmuch out of it!
In the discussion of the full set of excerpts fromteachers' journals, administrators noted that allthe teachers were amazed that they could havetaught all this time and never thought to ask thechildren about their thinking. They noted thatwhen they had been teachers, they too had beenoriented toward simply teaching the content.One said:
We're taught what math is and how to teach it,and the strategies for teaching it, and neverreally think about the end product [i.e., whatthe students understand].
They noted that at least one of the teachers saidthat she was learning as well as the students, and
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commented that this is something administra-tors sometimes forget. They talked about howteachers' continual learning connects to theshift in the role of teacher, from being didacticto being a facilitator, and how it is clear from thejournal entries that the teachers were thinkingabout this shift in role. One administratornoted:
In some ways . . . you begin to see a shift in howthey see their role as teachers. That is, ... there'sa lot of talk about facilitating learning, about adifferent kind of role that they're going to play.And perhaps a different approach to what theydo with kids.
And the administrators began to explore theimplications for their own action when theyconsidered the possibility that professional de-velopment for teachers might be based on lis-tening to their mathematical and pedagogicalideas. As one administrator put it:
This type of teacher reflection will cause agreater degree of introspection on the part ofthe teachers in order to . . . come to grips withwhat they do or don't know. . . . The impli-cation that would have for teacher trainingis that you might not have to teach everybodyall the same kind of thingpeople might beable to do some self-diagnosis and figure outwhat it is they want to know more about.Because we always try to put people through aseries of events and everybody may not needthe same thing.
These administrators were beginning to see thatlistening to children's mathematical thinkingcould inform teaching and that teaching basedon such listening would be a different kind ofteaching. Further, the design of professionaldevelopment for teachers would be similarlyredefined if it were to be based on listening toteachers' mathematical and pedagogical think-ing.
Layer 3: Listening Among Administrators.The final part of this session on listening pro-vided an opportunity for administrators to get ataste of what it felt like to do one of theseinterviews, or be interviewed in this way. Thepurpose was not to understand each other'smathematical thinking in order to teach, but toappreciate the kind of intellectual work in-volved in listening in order to truly understandanother's thinking and to see how the opportu-
nity to articulate one's ideas can clarify one'sown thinking (or uncover confusions).
In pairs, administrators interviewed each otheron topics designed to provide them with theopportunity to learn about each other's think-ing in areas related to mathematics education.Administrators were asked to listen carefully toeach other's thinking and to try to understandit as thoroughly as possible. The purpose, theywere told, was not for the interviewer to focuson whether she or he agreed or disagreed, or totell of events in her or his own experience thatconfirmed or disconfirmed what the other hadsaid. The purpose was to "put yourself insidethe head of the other" and to look at the issuefrom his or her point of view. Administratorsthen discussed this experience. They discov-ered that the interviewee's train of thoughtmight be quite different from their own andthat it took real work to stay with theinterviewee's thinking and explore it deeply. Asone administrator described the role of inter-viewer,
I found that I really had to pay attention to notasking questions about what interested me fromwhat my interviewee was saying in answer tothe question, but sticking with what she wassaying and trying to frame questions that camefrom what she was saying in response to thequestion.
Another administrator/interviewer said,
I was really focused and trying to go with himin the line of thought that he had, and my nextquestion came from where he left off, not froma question that I had already prepared. . . . I wasthinking very hard.
A third reported failure:
I learned that I'm not a good interviewer. Myquestions lacked depth. I kept asking questionsthat were things of real interest to me. I wantedto hear ideasthings I could use. It was almostselfish, now that I look at it. Like, what are youdoing with this, and kind of with an ear to, howcan I use that back at my school. And so, I guessI'm a failure tonight.
One administrator/interviewee reported:
She [the interviewer] was really listening care-fully and responding with another question[from within my frame of reference.] And I hadthe experience that ... I articulated some things
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that maybe I hadn't before. So I learned some-thing [about my own thinking].
From this layered activity, the administratorslearned about listening from the child's point ofview, noticing how respectful it was to be lis-tened to nonjudgmentally, how affirming itwas to have someone be interested in what youthink, and how much you can learn from theopportunity to explain your thinking. Theyalso came to understand listening from theteacher's point of view and came to appreciatethat when teachers are listening to children'smathematical thinking so as to understand whatthe child understands, they are engaged in ardu-ous intellectual work.
These administrators were also struck by howuseful it would be to listen in this way toteachers in their buildings and to administra-tive colleagues. Such listening would make a bigdifference in the culture and "feel" of schools,they felt. The systemic implications of thelayered structure to this session were not lost onparticipants. As one administrator said,
I love the way you structured the sessionstudents, staff, us . . . I'd like to be able to usethe architecture of the session in my own workwith department headsmoving from class-room to school to system wide parallels in aprocess or problem.
Layering, as a pedagogical design strategy, ap-pears to help administrators think about theway important values and norms of the math-ematics education reform movement can ex-tend beyond the classroom and come to charac-terize the profession of teaching and the intel-lectual culture of schools and districts moregenerally. It also connects a fundamental as-pect of reformed mathematics instruction witha skill (active listening) that is often used byadministrators in their administrative practicebut that they may not have seen as related to thecore of instruction itself (perhaps because thecore of instruction had not involved listening).As the core ideas of instruction change, the wayinstruction aligns with management practicealso changes and particular management skillsmay have new relevance. As one administratorput it,
This is really wild! My wife is after me to listenbetter, and I take these courses in active listen-ing that the district wants me to take. So, here
I come to math class, where the answers aresupposed to be cut and dried, and we're talkingabout listening!
Situating conceptual work in the consideration ofpractical action: The case of teacher supervision
When we began our work with administrators,our aim was to familiarize them with new ideasabout mathematics, learning, and teaching sothat they could help othersso that they couldunderstand changes teachers were trying tomake in their practice and provide appropriatesupports. However, we quickly learned thatfocusing on topics that were salient in adminis-trators' own work was far more effective. There,there were real puzzles for administrators tothink through in order to know how to act on adaily basis. Focusing the discussion on issues inadministrators' own work made the effort torethink fundamental ideas seem worthwhile tothem. There would be real consequences thatmattered to them.
Among such functional arenas are supervisingand evaluating teachers, understanding the cri-teria for good professional development forteachers, dealing with the impact of students'standardized test scores, and communicatingabout mathematics education reform with suchstakeholders as parents and school boards. Eachrepresents a compelling and problematic aspectof administrators' work for which the ideasembedded in mathematics education reformare significant. To do their work in each of theseareas, administrators need to understand whatmathematics education reform is fundamen-tally about and how the practical situation athand might need to be understood in a newway. In the process of puzzling about thesepractical topics, administrators in our groupwere thinking through the implications of oldand new ideas about math, learning, and teach-ing. They were "thinking in order to learn"(Thompson & Zeuli, 1997, p. 8)that is, strug-gling to solve problems or resolve dissonancesin order to come to a new understanding of theissues embedded in the task.
We focused most of our work with administra-tors on these functional areas. At their request,we devoted an entire year to thinking aboutclassroom observation and teacher supervision;that work serves as the basis for this vignette.
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Teacher supervision is a powerful lever forchange, lying precisely at the intersection be-tween administrative practice and teacher prac-tice. What supervisors see when they look atclassrooms, what they think those classroomsshould be, and what views they hold of thenature of their responsibility to both the teacherand the school system are central elements ofsupervision and very much affect what teachersare encouraged to do.
The history of ideas about teacher supervisionparallels the history of ideas about the nature oflearning and about how to ensure high-qualityperformance in organizations. What adminis-trators see when they look at classrooms isshaped by prevailing views about the nature ofmathematical knowledge, learning, and teach-ing (Bolin & Panaritis, 1992; Darling-Hammond& Sclan, 1992), and, relatedly, their views of thenature of the supervisory relationship with theteacher are shaped by prevailing views aboutwhat it means to assist teachers, and what itmeans to assess their practice (Rowan, 1990;Tracy, 1995). The current education reformeffort provides a new turn to the kaleidoscope.New ideas about the nature of mathematicsknowledge and what it means to learn (andteach) are resulting in classrooms that operateaccording to a new and different logic; newpedagogical ideas imply that supervision itselfbe interpreted as the facilitation of teachers'knowledge construction (Nolan & Francis, 1992).
All these issues were on the table for the admin-istrators in our group. They knew that thestandards for mathematics instruction werechanging and that they needed to learn whatgood mathematics learning and teaching wouldnow look like, if they were to do the work ofteacher supervision. As we listened to theirdiscussions about teacher supervision, it seemedthat the pedagogy of the supervisory relation-ship was also an issue. When they observedclassrooms, they tended to identify particularteaching problems and then give teachers ad-vice (albeit often very indirectly) about how tosolve those problems. One administrator de-scribed a typical scenario:
So, you go to Mrs. Smith and start talkingabout, well, what are some strategies that you'veused in the past that could be used? And youstart brainstorming [together] and develop a
list of strategies that would work for the kids.Then, the question is, the directive is, well,which of these do you think you might want totry? . .. So it's no longer [a discussion]. You'vepresented a choice.
Another administrator chimed in,
In other words, you [the teacher] are going tochange.
The pedagogical assumption that underlay theadministrators' practice was that they were ex-perts whose responsibility it was to identifyteaching problems and get teachers to producemore effective teaching behaviors. This is verydifferent from considering themselves to be thefacilitators of teachers' ongoing construction ofknowledge about mathematics, learning, andteaching. We expected that the pedagogicalpractices that supervisors would observe teach-ers using to help children construct rigorousmathematical knowledge in videotapes of re-formed classrooms would raise this issue. (Inthis respect, the work on teacher supervisionwas layeredthe same issue was considered attwo organizational levels.)
In the year-long seminar on teacher supervi-sion, our purposes were threefold: (1) to helpadministrators develop an eye for what reformedmathematics classrooms might look like, (2) tothink with them about the pedagogical rela-tionship between teacher and supervisor, and(3) to explore the characteristics of school cul-ture that seemed to hinder or support the super-visory process. At each monthly session, weshowed a 10-minute videotape of a classroom inwhich the teacher was in the process of trans-forming his or her mathematics instruction inaccord with reform tenets.2 At each session,administrators did and discussed the mathemat-ics that would be presented in the lesson. Theythen viewed the videotape twice, the first timediscussing the mathematics and the pedagogyof the lesson; the second time, discussing thekinds of things it would be important to talkabout with the teacher and the pedagogy of thesupervisor-teacher relationship.
Along the way, they learned many things. Asthey worked to understand what was happen-ing in the classrooms on videotape, they beganto question some of their former interpretationsof the basic elements of instruction and to make
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new interpretations. For example, administra-tors might formerly have thought it adequate tonote that the mathematics topic being taught(e.g., multiplication of fractions) was in thecurriculum and that the teacher had reached itat the appropriate time of year; over time, theybegan to attend to nuances and complexities inthe students' ideas about multiplication of frac-tions.' They might earlier have thought itadequate to observe that the teacher asked ques-tions of all students in the class evenhandedly;over time, they began to wonder whether theteachers' questions provided students with goodopportunities to explore important mathemati-cal ideas. (See Nelson & Sassi, 1998, for ananalysis of how administrators' eye for elemen-tary mathematics classrooms changed.)
They also began to reconceptualize the peda-gogy of the supervisory relationship. As theyviewed the videotapes and discussed how tohave a discussion with the teacher after doing aclassroom observation, we tried to build ontheir developing sense of the pedagogy of themathematics classroom and encouraged themto consider the possibility that the supervisormight ask questions designed to stretch theteacher's thinking, rather than giving advicethereby providing the teacher with the oppor-tunity "to think in order to learn."
At one seminar, the administrators consideredthe effect of asking questions that would promptreflection by teacherswhat they came to call"bingo" questions. This episode illustrates thebeginning of change in their ideas about thepedagogy of supervision. The excerpt from thetranscript provided below shows four adminis-trators and project staff exploring the likelyeffect of asking thought-provoking questions ofteachers.
Susan Jones4: I think it really does matterthe kinds of question that youraise. . . . There are so fewopportunities for colloquy atall in the teaching professionthat if the supervisor missesthe opportunity to have anhonest, authentic colloquywith the teachers, then thatperson's missed really agolden moment. And I thinkthe question that's raised thatweighs on somebody's mind
is a wonderful thing. Thequestion that makes you feela little uncomfortable is agood thing.
MFT staff: So, you can imagine a ques-tion I could raise that nags ata teacher in a pretty useful,pretty wonderful way, ... thatthey could . . . go home andhave this nag at them produc-tively.. . . How do you end upknowing that it's productive?
Ethel McGarry: Because we see change. Orthey themselves [the teach-ers] come back and say, "Youknow, we talked about this amonth ago. It's been bother-ing me. Can we talk about itbecause I'm thinking . . . youknow, I've been thinkingabout this."
Jim Parker:
Susan Jones:
MFT staff:
Steve Davenport:
Yeh, I think that's a good stan-dard, actually, in terms ofwhat you want to accomplish.It's authentic, that is, it's a realquestion, it's a meaningfulquestion.
If the principal or the supervi-sor, whoever it is, does thatconsistently, somebody'sgonna say, "You know, youreally asked a bingo questionfor me." But you cannot notuse that time to raise thosequestions.
Sounds like you're saying that,over time, that kind of bingoquestion is really kind of re-spect for the teacher.
You know, I feel successful ifthe person re-engages me afterone of those questions. . . . If
you raise a question with some-body and . . . he or she comesto you over and over again,not in a defensive way butjust in a "let's get this right". ..It is a respectful piece. Because ifyou just say it was a good lessonand you walk out, they don'thave a clue.
As a class exercise, the administrators developedseveral "bingo" questions that might be askedof a teacher on one of the videotapes they had
11
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been watchingquestions that would stretchthat teacher's thinking. Several were questionsabout instructional moves the teacher had made,designed to encourage the teacher to be explicitabout his or her mathematical or pedagogicalreasoning. For example, the questions, "Whydid you choose this particular pair of students topresent their work? How did you decide who topick?" would be asked not as though there werea correct answer that the supervisor knew but,rather, as the beginning of a mutually intrigu-ing discussion about the mathematical ideas atplay in the classroom and the pros and cons ofhighlighting this one or that one. Anotherquestion, "What would have happened if youhad asked the kids to make multiple represen-tati on s (i.e., using numbers, a drawing,manipulatives) of the way they were thinkingabout a mathematical problem?" would focusthe teacher's attention on the range of repre-sentations she or he was typically providingand the effect that had on students' opportu-nity to think through mathematical problems.Like the questions that teachers now ask ofstudents, these questions are open-ended, in-vite consideration of complex issues, and arechosen because thinking about them might beuseful next steps for the students (or teachers).
In the ensuing weeks, the administrators wenton to discuss the kinds of issues it might beimportant for teachers to think deeply about,the nature of the relationship between supervi-sor and teacher that would need to obtain forsuch question-posing to happen productively,and the character of the school environmentthat would support such open-ended inquiryabout learning and teaching. But the notion of"bingo" questions marked the beginning of achange from the stance of advice-giving expertto the stance of facilitator of teachers' knowl-edge construction. Working out a new peda-gogy for supervision provided these administra-tors with the opportunity to think through, ina practical domain with which they had regularexperience, an important element of the newpedagogy that also underlay reformed math-ematics classrooms.
Creating an analytic and reflective communityamong administrators
That classrooms should function as reflectivecommunities for students and teachers is ahallmark of the mathematics education reformmovement. Whether *expressed in Standardsand Frameworks that call for communicationand discourse or in new mathematics curriculathat advise teachers to ask students to explaintheir thinking, teachers are enjoined to havestudents discuss their ideas, listen to one an-other, and think critically about whether or notthey agree with another student's thinking.Often teachers must work out for themselveshow to do this (Lester, 1996).
The classroom as reflective community can alsobe easily misunderstoodfor example, in thefear on the part of some that the right answer nolonger appears to matter, so long as students canexplain how they did the problem. If adminis-trators are to support teachers and ensure edu-cational quality and public understanding, it isimportant for them to understand why class-room inquiry and discourse are important andwhat the criteria are for good inquiry and dis-course. However, if administrators are to de-velop a subtle and grounded sense of what aninquiry-based or discourse-based mathematicsclassroom is like, they need the opportunity toexperience it for themselves, for, in general,their own mathematics education was not con-ducted in this way.
In most class sessions, we tried to do mathemat-icsthe mathematics that would appear on avideotape, the mathematics that would appearin student work, the mathematics that would bein an article they read. While it was not possibleto organize this mathematics to systematicallycover the major ideas in the elementary math-ematics curriculum, it was possible to conductthe mathematics part of our seminars much asteachers would conduct a reformed mathemat-ics class. And so administrators had the oppor-tunity to experience something of what it wouldbe like to be in such a classroom, and theylearned for themselves some of the behaviorsnecessary for such a classroom to work: respectfor others' ideas; willingness to expose one'sown, often tentative ideas to the scrutiny ofothers; subjecting ideas to principled examina-tion in order to determine if they were correct.
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This was an especially poignant experience foradministrators who had not done any math-ematics for many years and were not particu-larly confident of their mathematical abilities.Doing mathematics together at the beginningof every session was a real test of their ability tosay what they thought clearly, even when theywere unsure; to listen to each other seriously,searching for the sense that the other was tryingto make; to not judge.
One evening, the group discussed angles inpreparation for viewing the videotape of aclassroom in which the students would be learn-ing to measure the angles of a triangle with aprotractor. Several figures were put on theboard, and the administrators were asked whichwere angles:
A
Figure 1
The consensus among all but the mathematicssupervisor present was that B, C, E, and F wereangles. When asked what properties they werelooking for when they decided which ones wereangles, several said they were looking for inter-secting line segments. There ensued an ex-tended discussion about Figure C and whetheror not it could be said to be an angle.
Sylvia Pendell:
MFT staff:
Sylvia Pendell:
Ellen Christianson:
Sylvia Pendell:
The curved line isn't constantin its relationship to the lat-eral line.
And you're saying that makesit not an angle?
I don't know. It makes itmany angles. At every pointit's a different angle. Isn't it?Every point on the arc.
C is a problem for me. . . . I
named it as one of my anglesand it's not because .. . I guessI would call that a potentialangle, a kind of moving angle.
A dynamic angle.
Ellen Christianson: Thank you.
Joanne Smith: It's measurable, but you haveto pick a point along that arc.If you pick a point along thearc [and construct a line tothe vertex], then it becomesmeasurable.
Ellen Christianson: It's almost an infinite possi-bility.
C was a puzzling figure. It met the part of thedefinition of an angle that they could remem-berintersecting line segmentsbut theyhadn't seen an angle with a curved side before.They weren't sure enough about their memoryof the definition of an angle to firmly excludecurved lines. However, they were willing to digin and think it through. They listened thought-fully to one another's ideas (it makes manyangles; at every point, it makes a different angle),puzzled about what kind of sense those ideasmade (there are infinite possibilities), and ex-plored the implications (if you picked a pointalong the arc and constructed a line to thevertex, you could measure the angle). Theywere not afraid to suggest unorthodox ideas;they trusted one another enough to take evenunorthodox ideas seriously and think themthrough; and they knew that, in the end, themathematics had to make sense.
In general, the administrators' inquiry groupwas conducted as a reflective community ofinquiry among participating administrators.Whether the group was discussing mathematicsproblems, a videotape of a mathematics class-room, or an assigned reading, interest in andrespect for each other's ideas was established asa group norm. Over time, the administrators inthe group developed a sense of what a reflectivecommunity was like and what it took to main-tain it. As one administrator described it,
I feel that we are intellectual colleagues here. Ifeel that we're kind of tacitly agreeing to dealwith difficult questions in a very open way,here. And I think that's remarkable.
Another noted that it had taken a while for sucha community to develop:
It would be interesting to have someone jointhe group at this point, because over the lasttwo years we really actually have grown intothis relationship with each other. We certainly
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didn't start there. . . . It's more than just intel-lectual and, while we haven't visited each other,I think there's . . . been enough personal shar-ing and enough interaction that we really knoweach other in that professional sense.
And they saw both the relationship betweenreflection and action and the relationship be-tween what they had been doing and whatteachers might do.
One administrator noted:
I think there's another part to [the] process that. . . is significant. . . . I see it in teachers in myschool. This is the action part, that is, practicethat is informed by reflection. There's a pieceof this that is cyclical. Reflection and actionand then reflection. And I think that teachingis strengthening to the degree that that's whatwe really are talking about.
Structuring the administrators' inquiry groupso that it functioned as an analytic and reflec-tive community gave these administrators theopportunity to experience important aspects ofreformed mathematics classroomsthinkinghard about significant issues, showing respectfor everyone's ideas, trusting that others wouldnot judge or make fun of one's tentatively heldideas, and committing to support and investi-gate each other's thinking. It put them in theposition to understand what the virtues of suchclassrooms might be and what it would take tomake them work well.
It also raised issues for them about how admin-istrative work in general was structured. Ad-ministrators' workdays are fragmented. Forexample, principals are physically on the movein their buildings for much of the day; theircontacts with teachers, students, parents, andother administrators tend to be short; discus-sions are about current and pressing situations(Fullan, 1991). Central office administrators areoften dealing with finances and politics, ratherthan educational issues (Fullan, 1991). Suchwork provides few opportunities for adminis-trators to deliberate together about importanteducational issues. As one administrator put it,
We do have these discussions but we don't havethem in the places where we should. We don'tvalidate those discussions and we don't havethem and say, "These are an important part ofour work places." People find informal ways tohave these discussions . . . [but] we don't as an
organization value them and, therefore, I thinkpeople keep these conversations outside theorganization.
Listening and thinking together, rather thanimmediately moving to resolve issues, was hardfor these administrators. This was particularlyevident at one session, when an administratortalked with the group about the decision he hadto make about whether a new teacher would gettenure. The group talked about how hard it hadbeen for them to stick with the issues and not tryto solve the problem:
At times we were going through the processwith Sam, and at times we ended up wanting tosolve his problem for him. And I thought theconversation sort of seesawed back and forth,that way. It might be because it was such anemotional topic that we wanted to solve theproblem for him, rather than think about Sam'sprocess of going toward it, and giving himsome reflections on his process.
Sam later said that he had valued the discussionbecause it provided him with a number ofdifferent lenses through which to examine hisdilemma. "In fact," he said,
for decisions of this magnitude you should haveto go through that kind of reflective process, asa kind of check on your process, the ways inwhich we make our judgments. . . . So, I foundit helpfulyour probing and your questionsthat asked me to think about what I meant . . .
or what this process was.
The thoughtful reflection characteristic of re-formed elementary mathematics classes standsin stark contrast to the generally frenetic andunreflective nature of school life, in which ad-ministrators engage in their work isolated fromeach other and without much opportunity forreflection or discussion. The administrators inour group valued the group itself for providingthese opportunities and saw that their work,and the climate of their schools, would beimproved if there could be more of it on aneveryday basis.
Conclusions
Administrators are important actors in educationreform, not solely because of their influence onschool- and district-level policy but also becausethey enact, daily, a set of ideas about the nature oflearning and teaching, thereby influencing the
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intellectual culture of schools in particular ways.The ideas about mathematics, learning, andteaching that are embedded in the current math-ematics education reform effort are complexand subtle, an interwoven web of assumptions,attitudes, and orientations that are quite differ-ent from business as usual in most schools. If wewant these ideas to become a more prevalentpart of American schooling, we need to learnmore about what administrators themselvesneed to know about them, how they can cometo learn about them, and, in turn, how they canreflect on and perhaps change their own admin-istrative practice.
In the beginning of our work with school anddistrict administrators, we perceived that therewas often a misalignment between the ideasabout learning, teaching, and mathematics thatundergirded much of their administrative prac-tice and the ideas about learning, teaching, andmathematics embedded in the mathematicseducation reform effort. Often administrators'practice was built on transmission ideas aboutlearning and teaching and entailed assump-tions about the standardization of student learn-ing and the proceduralization of teaching prac-tice. However, the ideas about learning, teach-ing, and mathematics that underlie the math-ematics education reform movement are quitedifferent. Now learning is seen as the process ofthinking through puzzling and often difficultideas, and teaching as providing resources andguidance for such thinking. Learners and teach-ers are viewed as being engaged in creativeactivities requiring judgment; classrooms arenow the locus of intellectual debate and discus-sion that can stretch far beyond the classroomwalls and the prescribed 42 minutes.
To help these administrators consider new ideasabout mathematics, learning, and teaching andthink about how such ideas might influencetheir administrative practice, we undertook toprovide experiences that would give them theopportunity for fundamental conceptualchange. This required finding ways to make theideas embedded in mathematics education re-form relevant to their own work, so that theywould have the motivation to think throughcompelling and real problems in order to con-struct their own, situated knowledge of math-
ematics education reform. The three designprinciples described in this paper represent oureffort to do this.
In the process, these administrators learnedseveral substantive things about mathematicseducation: what it feels like to puzzle througha mathematical problem with others, and con-sequently, what both the certainty and theuncertainty of mathematical understanding feelslike; how children's mathematical thinking de-velops; what teachers need to know and knowhow to do in order to support the developmentof children's mathematical thinking; what cur-ricula, student assessment, teacher supervision,and communication with community stake-holders need to be if they are to support re-formed mathematics instruction.
But the administrators also considered the pos-sibi lity that embedded in mathematics educa-tion reform were norms and values that hadimplications for their own administrative prac-tice. Examples discussed in this paper are listen-ing respectfully to the thinking of another inorder to truly understand what he or she means;learning to facilitate teachers' construction oftheir own knowledge about mathematics,children's learning, and instruction; and creat-ing reflective and analytic communities forthinking through complex educational issues.Considering these raised questions about thenature of school administration itself and pro-vided the opportunity for these administratorsto think about how their own administrativepractices could be connected more directly tothe core of new instructional practice.
Acknowledgments
The work described in this paper was supportedby a grant from the National Science Founda-tion (ESI-9254479). Any opinions, findings,and conclusions or recommendations expressedin this paper are those of the author and do notnecessarily reflect the views of the NationalScience Foundation. My colleagues EllenDavidson, Christine Kaplan, and Annette Sassishared in the design and facilitation of theseminar. Davidson and Sassi helped with theconceptualization of early versions of this paperand in thinking about successive drafts. Thanksfor their comments on an early version are due
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to my colleagues Linda Davenport, David Ham-mer, Amy Morse, and Deborah Shifter. Review-ers who were most generous with their timewere Judy Mumme, Leslie Santee Siskin, andGary Sykes.
Notes1Elmore defines "the core of educational practice" as
how teachers understand the nature of knowledgeand the student's role in learning, and how theseideas about knowledge and learning are manifestedin teaching and classwork. The "core" also includesstructural arrangements of schools, assessment pro-cesses, etc. (Elmore, 1996).2Many of these videotapes were produced by the
Educational Technologies Department of Bolt Beranekand Newman, Inc., in a project entitled "Mathemati-cal Inquiry Through Video: Tools for ProfessionalGrowth," directed by Ricky Carter and Fadia Harikand supported by the National Science Foundation.3Students first develop their ideas about the basicarithmetic operations (addition, subtraction, multi-plication, and division) in the domain of wholenumbers. Later in their school years, they are intro-duced to a different kind of numbersfractions. Thebasic operations may work differently on fractionsthan on whole numbers. For example, if one multi-plies two whole numbers, the answer is larger thaneither of the original numbers. However, if thenumbers are fractions, the answer is smaller. Ifstudents are to truly understand this and not justmemorize the procedural rules for multiplication offractions, they need to understand multiplication asbeing about numbers of "groups." So the question,What is 2 x 3? is asking, How many will you have ifyou have 2 groups (of 3 each)? The answer is 6. Butthe question, What is 1/2 x 1/3? is asking, How manywill you have if you have 1/2 of a group (of 1/3 ofsomething)? The answer is 1/6.4Administrators' names are pseudonyms.
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BUILDING NEW KNOWLEDGE BY THINKING
Barbara Scott Nelson
Barbara Scott Nelson is Senior Scientist and Director of the Center for theDevelopment of Teaching at Education Development Center, Inc. She holds theB.A. degree in philosophy from Mt. Holyoke College, the M.A.T. from JohnsHopkins, and the Ed.D. from Harvard University. Currently, Dr. Nelson isworking with school- and district-level administrators on the implications ofmathematics education reform for the intellectual culture of schools and fortheir own work. Dr. Nelson's research focuses on the processes by whichadministrators develop new views of the nature of learning and teaching, andthe relationship between changes in their interpretations of these core processesof schooling and changes in their professional practice. Grants provided by theNational Science Foundation and the Pew Charitable Trusts support the devel-opment of instructional materials for use with administrators.
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