DOCUMENT RESUME

ED 368 698 SP 035 125

AUTHOR Kirsner, Steven A.; And OthersTITLE Changing Practice: Teaching Mathematics for

Understanding. A Professional Development Guide.INSTITUTION National Center for Research on Teacher Learning,

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

Washington, DC.PUB DATE 93NOTE 136p.; A product of the Teaching Mathematics for

Understanding Video Project.AVAILABLE FROM National Center for Research on Teacher Learning,

Michigan State University, East Lansing, MI48826-1034 (video and guide).

PUB TYPE Guides Classroom Use Teaching Guides (ForTeacher) (052) Audiovisual/Non-Print Materials(100)

EDRS PRICE MF01/PC06 Plus Postage.DESCRIPTORS *Change Strategies; Class Activities; Educational

Practices; Elementary School Teachers; ElementarySecondary Education; Faculty Development;*Mathematical Concepts; *Mathematics Instruction;Secondary School Teachers; *Student Role; *TeacherRole; Teaching Guides; *Teaching Methods; VideotapeCassett_s

IDENTIFIERS *NCTM Curriculum and Evaluation Standards; ReformEfforts

ABSTRACTihe materials provided in this package include a

24-minute video and a professional development guide. The videoportrays several different approaches to teaching mathematics in waysthat are consistent with the National Council of Teachers ofMathematics (NCTM) Standards. The methods employed differ fromconventional mathematics teaching in respect to the student's role,the mathematics content, and the teacher's role. The guide isorganized into the following sections: (1) Introduction to the Video(the aims and rationale of the video); (2) Using the Video andProfessional Development Guide; (3) Orienting Questions for Viewingthe Video; (4) Analyzing Classroom Segments on the Video (raisesquestions and issues in light of NCTM Standards); (5) AnnotatedBibliography; (6) Craft Papers (for further study); and (7) List ofPublications from the National Center for Research on TeacherLearning (NCRTL). The craft papers in section 6 are: (1) "ChangingMinds" (Educational Extension Service/Michigan Partnership); (2)"Could You Say More about That? A Conversation about the Developmentof a Group's Investigation of Mathematics Teaching" (H. L.Featherstone, and others; and (3) "Assessing Assessment:Investigating a Mathematics Performance Assessment" (Michael Lehman).(LL)

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CHANGING PRACTICE: TEACHING MATHEMATICS FOR UNDERSTANDING

A PROFESSIONAL DEVELOPMENT GUIDE

Developed by

Steven A. Kirsner, John S. Zeuli,Linda Alford, and Michael J. Michell

The Teaching Mathematics for Understanding Video ProjectNational Center for Research on Teacher Learning

Michigan State UniversityEast Lansing, Michigan

Summer, 1993

U S DEPARTMENT OF EDUCATIONOffice of EducationalResearch and Improvement

EDUCATIONAL RE SOURCES INFORMATIONCENTE R (ERIC)

Th)s document has been reoroduLed asreceived born the Person or 0.2a ni/MiOnonoinaling it

I: Minor Changes have been marle In rnpt ow.reproduction Qualify

PO"S of view ni opinions stated in th)sdrxment do not necessarily represent MN, ialOE RI position Or policy

This notebook contains materials that may be duplicated and used without the expressed permission of thedevelopers.

The project was conducted under contract with the Office of Educational Research and Improvement, UnitedStates Department of Education. The opinions expressed are those of the authors and do not necessarily reflectthe position or policy of the Office of Educational Research and Improvement.

2 am Cm van

CREDITS

Changing Pructice: Teaching Mathematics for Understanding, a product of the EducationDevelopment Center for The National Center for Research on Teacher Learning, MichiganState University, is funded by the Office of Educational Research and Improvement, U.S.Department of Education, and the Educational Extension Service, Michigan Partnership forNew Education.

Project Directors: Steven A. Kirsner & John S. Zeuli, Michigan State UniversityProject Consultant: Linda Alford, Michigan Partnership for New EducationProject Assistant: Michael J. Michell, Michigan State University

Special thanks are due to:

Deborah Ball, Alyjah Byrd, Sylvia Rundquist, and the students of Spartan VillageElementary School, East Lansing, MichiganAnita Johnston and the students of Napoleon High School, Napoleon, MichiganElizabeth Jones, Jane Hannah-Smith, and the students of Dwight Rich MiddleSchool, Lansing, MichiganMichael Lehman and the students of Holt High School, Holt MichiganPaul Sherbeck and the students of Beaubien Junior High School, Detroit,MichiganLamar Alexander, Secretary of EducationShirley Frye, Past President, National Council of Teachers of MathematicsMary M. Kennedy, Director, National Center for Research on Teacher Learning,Michigan State UniversityThomas A. Romberg, Director, National Center for Research in MathematicalSciences Education, University of Wisconsin-Madison

Special thanks are also due to:

Daniel Chazan, William Fitzgerald, Glenda Lappan, and Perry Lanier, and BruceMitchell, Michigan State UniversityCarole B. Lacampagne and Joyce A. Murphy, U.S. Department of EducationDavid R. Nelson, Producer/Director, Education Development Center, Newton,Massachusetts

DEDICATION

The Changing Practice: Teaching Mathematics for Understanding professionaldevelopment guide is dedicated to Steven A. Kirsner, Project Director, who died before theguide was completed. Steven's vision and energy were responsible for the creation of thisguide. Hopefully, uses of it will reflect his spirithis belief that open-ended discussion andexperimentation are the foundation for most forms of human improvement.

John S. ZeuliProject Co-Director

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CONTENTS

SECTION 1: Introduction to the VideoOutlines the aims and rationale of the 24-minute video, Changing Practice: TeachingIffs.thematics for Understanding.

SECTION 2: Using the Video and Professional Development GuideProvides suggestions and options for use of the professional development guide.

SECTION 3: Orienting Questions for Viewing the VideoProvides discussion questions about the content of the video. Me questions willorient viewers to the provocative ideas in the video and will facilitate closer analysisof classroom segments featured on it.

SECTION 4: Analyzing Classroom Segments on the VideoFor classroom segments on the video, raises questions and issues in light of theNational Council of Teachers of Mathematics (NCTM) Standards documents. Thesegments include:

1. Ms. Ball's Third Grade Classroom - Using a Task to Introduce a Unit onFractions.

2. Ms. Jones's Seventh Grade Classroom Acquiring a Conceptual Understanding ofDecimal Numbers.

3. Mr. Sherbeck's Classroom Using Writing to Elicit Students' Thinking aboutMathematical Ideas.

4. Mr. Lehman's Algebra II Assessment - Assessing Students' Understanding ofAlgebraic Concepts.

SECTION 5: Annotated BibliographyProvides a select, annotated bibliography in order to promote Arther study of theideas and issues raised in the video and professional development guide.

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SECTION 6: Craft PapersIncludes three papers for further study. The first two papers focus on what teachersare thinking, including the struggles they experience, as they attempt to teachmathematics in ways that will enhance students' understanding. A third paper,written by a teacher featured on the video, describes one teacher's attempt tointroduce innovative assessment practices in mathematics.

1. Educational Extension Service/MPNE. (1990, Summer). Rethinking mathematicsteaching. Changing Minds, 1(1).

2. Featherstone, H., Pfeiffer, L., Smith, S.P., Beasley, K., Corbin, D., Derksen, J.,Pasek, L., Shank, C., and Shears, M. (1993). "Could you say more about that?" Aconversation about the development of a group's investigation of mathematics teaching(Craft Paper 93-2). East Lansing: Michigan State University, National Center forResearch on Teaching Learning.

3. Lehman, Michael (1990). Assessing assessment: Investigating a mathematicspetformance assessment (Craft Paper 91-3). East Lansing: Michigan StateUniversity, National Center for Research on Teacher Learning.

SECTION 7: List of Publications from the National Center for Research on TeacherLearning (NCRTL)

SECTION 1: INTRODUCTION TO ME VIDEO

Teaching methods have to be more for solving problems, reasoning, figuringthings out rather than only for memorizing, drilling, and doing things by rote.And that's simply because that's what the world needs today, that's whatstudents have to know.

Lamar Alexander, Secretary of Education (1992)

Despite this statement and similar ones by members of the educational community,reasoning and problem-solving are not currently the focus of most mathematics instruction.The results of recent national and international surveys of mathemaiics achievement showthat students in the Uaited States have great difficulty understanding mathematical conceptsand applying their mathematical knowledge and skills in problem solving situations.Although reformers advocate changes in teaching practice in order to improve mathematicsinstruction, it is often not made clear what these changes might look like in real classrooms.

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What is HaRoening in the Video?

The video that you are watching portrays several different approaches to teachingmathematics in ways that are consistent with the National Council of Teachers ofMathematics (NCTM) Standards documents.

The mathematics teaching portrayed in this video differs from conventionalmathematics teaching in three ways: the students' role is different; the mathematics contentis different; and the teachers' role is different.

A. Students' Role as Learners of Mathematics

Students are active learners. Young children enter school with a natural curiosityabout and some intuitive I-nowledge of numbers and mathematics. Students should notpassively absorb new knowledge acquired in school, but should actively draw on theirinformal, intuitive knowledge as they internalize new meanings and understandings. Theclassrooms shown on the video have become, over time, places where students, workingindividually and in groups, are actively involved in making sense of mathematics, drawingupon, confronting, and expanding their knowledge of mathematics learned in less formalsettings.

This portrayal of active mathematics learning is considerably different from what moststudents have experienced and come to expect in conventional mathematics classrooms. Inconventional classrooms, students often work alone and their work consists of practicingroutine exercises rather than reasoning or arguing about ideas.

What is proposed in the Standards documents is very different. Teachers create alearning environment in which students are engaged in a variety of activities and discussions.Students work through genuine mathematical problems. They reason about mathematics andmake connections among important mathematical ideas. A typical mathematics classroom atall grade levels should be one in which students learn to "examine," "transform," "apply,""prove," "communicate," and "solve" mathematical problems.*

Masters have been provided of the curriculum standards in the "Overheads" section ofthis professional development guide for use as overheads or as handouts.

B. Mathematics Content

The teachers highlighted in the video think of mathematical content according to thestandards in the National Council of Teachers of Mathematics (NCTM) Curriculum andEvaluation Standards for School Mathematics (1989). The vision put forth in the Curriculumand Evaluation Standards is that the curriculum serves to build students' mathematicalcompetence and confidence and should represent what mathematicians actually do.Consequently, the curriculum at all levels of schooling focuses on:

(1) Mathematics as Problem-Solving. The curriculum centers on students workingthrough problem situations that help them to investigate and understand mathematicalcontent. Focusing on computational skills and practicing algorithms does nsg precedestudents' engagement with problems. Students develop computational skills as theyinvestigate and eventually solve a wide variety of interesting problems appropriate forthem.

(2) Mathematics as Reasoning. Students make conjectures, offer evidence, and, in short,build an argument for why they think as they do when working through problems.Students draw on and create models or cite other mathematical facts and relationshipsto justify their thinking. The focus is on students' reasoning about a problem, notsimply getting the right answer.

(3) Mathematics as Communication. Students learn to read, write, and speak thelanguage of mathematics in order to interpret and evaluate mathematical ideas.Students recognize how important it is to use language and symbols to communicatetheir thinking about mathematical content.

(4) Mathematical Connections. Many opportunities exist in the curriculum for students tosee relationships between different mathematical concepts as well as the connectionsbetween mathematics and other school subjects.*

Masters have been provided of the curriculum standards in the "Overheads" section ofthis professional development guide for use as overheads or as handouts.

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C. The Teacher's Role

The video also portrays teachers as taking on a very different role. This new role ismore difficult, but ultimately more rewarding for both teachers and their students.Specifically, the teachers in this video have rejected conventional classroom practices thatoften focus on telling students what to do and then telling them whether they did it right ornot. They have slowly moved away from a role that defines them as the only source ofauthority for right answers and in contrast, have made ongoing progress toward incorporatingthe fundamental changes advocated in tl,e National Council of Teachers of Mathematics(NCTM) Standards documents.

(1) The teachers pose worthwhile mathematical Tasks that determine, and do not simplyinfluence, what students have the opportunity to learn.

(2) The teachers use classroom Discourse as a strategy for helping students make sensefor themselves of the mathematics they encounter.

(3) The teachers create and nurture a le:uning Environment that supports students'reasoning, problem solving, communicating, and connecting ideas in the curriculum.

(4) The teachers engage in systematic Analysis of their teaching and what students arelearning as a consequence. Based on the analysis of their teaching, teachers adapttheir questions and tasks accordingly.*

Teaching in ways consistent with the Standards documents is notably different fromwhat most teachers have been prepared to do and it is not simple to accomplish. Teachersmust be committed to rethink their mathematics teaching. They must be willing to rethinkwhat it means to teach mathematics for understanding. Learning to teach for understandingalso requires teachers to have colleagues' support and encouragement.

In summary, the mathematics education community has reached consensus aboutrecommendations that call for fundamental changenot simply improvementin whatmathematics students learn and how they learn it. The National Council of Teachers ofMathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics (1989),and its companion document, Professional Standards for Teaching Mathematics (1991),explain what these important recommendations mean in the areas of mathematics curricula,teaching practices, and teacher preparation. Most importantly, the Standards' vision ofmathematics teaching and learning is meant for each and every student. The educationalcommunity recognizes the power of this vision to enhance the education of jj students.

Masters have been provided of the curriculum standards in the "Overheads" section ofthis professional development guide for use as overheads or as handouts.

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SECTION 2: USING THE VIDEO AND PROFESSIONAL DEVELOPMENT GUIDE

Introduction

The teachers whose classrooms are highlighted in this video are learning to teachmathematics in ways that are consistent with the National Council of Teachers ofMathematics (NCTM) Standards. None of the classrooms that appear is meant to representideal practice. Instead, we see teachers who are involved in the long-term process ofre-orienting their teaching along the lines suggested by the Standards documents.

The video and the professional development guide are designed to raise questions andstimulate conversations among educators as well as help teachers to try alternativeapproaches to conventional practice. Raising questions, experimenting, and engaging insustained conversations about mathematics teaching and learning are essential if we are torealize the vision of a high-quality mathematics education for all learners.

The video and the accompanying guide will be useful to both teachers who are juststarting to rethink their mathematics teaching and those who already possess some knowledgeabout teaching mathematics for understanding and are now beginning to change the way theyteach. The video and guide are appropriate for teachers in many different contexts, such as amathematics study group, a mathematics department, or at a school in-service. The materialscan also be used to introduce other viewers (e.g., prospective teachers, policy makers, andparents) to the Standards' vision of mathematics teaching and learning.

In any of these contexts, the showing of the video will be most useful within alen el h v. i rifer t n h r h in h 'r h minstruetio.

When planning to show the tape, it is important to create a setting that is conducive toviewing and discussing the video. Keep in mind that:

the video is 24 minutes long;

the room should be comfortable with seating and table arrangements that facilitatethe give and take of dialogue; and

participants will likely need a notebook or handouts to jot down their thoughts asthey watch the video and to record their responses to the questions in the guide.Depending on how discussion of the video is orchestrated, discussion leaders alsomay need chart paper, a blackboard, and a overhead projector and screen.

Uses of the Video and Professional Development Guide

We recommend the following sequence for using the video and professionaldevelopment guide. These recommendations are intended as helpful suggestions. Discussionleaders may develop imaginative variations that also offer a good foundation for a long-rangeplan that supports teachers at the local level.

1. General Orientation

Viewers observe and reflect on the video using the "Orienting Questions for Viewingthe Video" provided in Section 3. The questions are designed to help a wide range ofviewers (teacher% policy makers, parents) focus on important differences betweenconventional mathematics teaching and the teaching highlighted on the video. Discussionleaders should allocate roughly 11/2 to 2 hours to completely work through the orientingquestions with the participants. (We have included in the "Overheads" section of this guidemasters of all the orienting questions for discussion leaders to reproduce as overheads or ashandouts.)

2. AnalysiisgSdass_oo_le,g,_s_ts_m_tbs_skallr m mn Vi in light of the Standards documents)

Once viewers are oriented to the provocative themes in the video, they will benefitfrom a closer analysis of the classroom segments featured on it. In Section 4 we havedevised discussion questions for four of the classroom segments shown on the video.

We think that each classroom segment will be of interest to teachers no matter whatgrades they teach. For example, the discussion questions for "Mr. Lehman's Algebra IIAssessment" are appropriate for teachers at all grade levels who are interested in innovativeassessments of their students' understanding of mathematics. Also, in "Mr. Sherbeck'sClassroom," student writing is used to elicit thinking about and encourage discussion ofmathematical ideas. Analysis of this segment could lead to broader discussions about usingwriting to promote meaningful learning of mathematics at any grade level. "Ms. Ball'sThird Grade Classroom" could stimulate discussions about using real-life problems inmiddle-school and high school mathematics classrooms. Consequently, teachers will benefitfrom working through the discussion questions for each classroom segment.

D13cussion leaders may find it useful to show again the classroom segment on thevideo before or after asking teachers to discuss the questions related it. Also, leaders maywant to focus first on the classroom segment that best fits teachers' grade level and thenfocus on other segments that are also relevant to teaching mathematics for understanding.Discussion leaders may find some overlap between teachers' responses to questions inSection 3 and Section 4. This overlap will help teachers revisit and refocus their attention onsignificant issues. (Once again, we have included in the "Overheads" sections of the guidemasters of the questions for each classroom segment. Discussion leaders can reproduce thesemasters for handouts or overheads.)

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Many discussion leaders have found that one hour (or more) is a useful block of timeto allot for discussion of the questions related to each classroom segment. (The number aftereach of the segments indicates the timed place on the tape where these segments are located.Also, included in the folder on the white pages are the questions for each of the classroomsegments.)

Count TimeBegin End

1. Ms. Ball's Third Grade Classroom 0:48 10:542. Ms. Jones's Seventh Grade Classroom 10:55 14:383. Mr. Sherbeck's Classroom 14:39 16:394. Mr. Lehman's Algebra II Assessment 16:40 20:25

3. Long-Range Planning

After teachers have had extended opportunities to discuss the video in-depth, it isessential that they participate in a long-range plan that revisits, reinforces, and supporls themas they change their mathematics instruction.

We have included a select, annotated bibliography in Section 5 of the professionaldevelopment guide. The annotated bibliography focuses on a wide array of issues related toteaching mathematics for understanding, including teaching and assessing problem solving inmathematics, using writing as a tool to learn mathematics, and helping all students to reasonand communicate mathematically. We have also included the National Council of Teachersof Mathematics (NCTM) Standards documents in the bibliography (including information onhow to order them). We recommend the Standards documents as particularly valuableresources for a long-range staff development plan. The NCTM Curriculum and EvaluationStandards for School Mathematics provides a detailed vision of what the content, priorities,and emphases of the mathematics curriculum should be for grades K-4, 5-8, and 9-12. TheNCTM Professional Standards for Teaching Mathematics provides a vision of the kind ofteaching needed to support curriculum changes in mathematics. It includes a number ofannotated vignettesdrawn from actual classroomsthat will help teachers at all gradeslevels understand and become more involved in the process of re-orienting their teachingalong the lines suggested by the Standards.

Discussion leaders may find more immediately helpful the three papers included inSectior: 6 of the professional development guide. The first two papers include a focus on ateacher featured on the video, Deborah Ball. Both papers further describe several teachers'attempts to grapple with the questions raised by the provocative ideas and practices featuredon the video. A third paper, written by a teacher shown on the video, Michael Lehman,describes his attempt to introduce innovative assessment practices in his Algebra II class. Itmay be helpful for discussion leaders to share these papers with participants for discussionfollowing a viewing of the video and extended discussions of the orienting questions and thequestions for each classroom segment.

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Also included in Section 7 is a list of publications from the National Center forResearch on Teacher Learning (NCRTL). Through its research focus on learning to teach,the NCRTL works to help teachers learn how to create worthwhile learnh g activities forstudents, increase students' active involvement in learning, foster students' responsibility forlearning, and promote mutual respect and inclusion in the classroom. The Center'spublications may also be used to enrich teachers' conversations about the video and mayserve as an excellent resource to support a long-range plan for teacher learning.

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SECTION 3: ORIENTING QUESTIONS FOR VIEWING THE VIDEO

The following discussion questions are offered i a order to stimulateconversation after a viewing of the video. The questions should help theviewer focus on important differences between conventional mathematicsteaching and the teaching highlighted on the video. It might be useful forteachers to jot down notes while Jiewing the video. These notes may includeany questions that come to mind during the viewing, as well as responses tothe set of questions below. Masters of the orienting questions are included inthe "Overheads" section of the guide.

Questions about the Siudents' Role

(1) When you observe the students in the video,

(a) What are students saying that you find particularly interesting orconsiderably different from what students typically say inconventional mathematics classrooms?

(b) What are students doing that you think is different from what onetypically sees during mathematics instruction?

Questions about Mathematical Content

(1) The NCTM Standards suggest that students should engage inworthwhile activities. How would you define the content thesestudents are thinking about and how does this content differ from thecontent offered in your textbooks?

(2) The NCTM Curriculum and Evaluation Standards recommends thatmathematics curricula at all grade levels focus on (a) mathematics asproblem-solving; (b) mathematics as reasoning; (c) mathematics ascommunication; and (d) mathematical connections. What examples ofeach curriculum focus are shown on the video?

(3) How do teachers and other educators in the video think about the roleand relevance of basic skills and procedures?

Questions about the Teacher's Role

(1) How do the teachers in the video describe what students in theirclassrooms are saying or doing differently as they learn mathematics?

(2) What kinds of questions do teachers pose and how do they get allstudents involved in thinking about these questions?

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(3) How do teachers guide students' responses and how do they respondto students' unexpected ideas?

(4) Teachers in the video use and describe different approaches toteaching mathematics (e.g., using manipulatives, alternativeassessments, student writing, group work).

(5)

(a) Could teachers use these different approaches during mathematicsinstruction, yet remain tied to traditional mathematics teaching?For example, could a teacher use group work without teachingmathematics for understanding, or use manipulatives without anyfocus on helping them reason about mathematics?

(b) If so, what inakes these different approaches consistent withteaching mathematics for understanding?

What difficulties do teachers and other educators describe in learningto teach mathematics for understanding?

(a) What do they describe as the benefits of trying to overcome thesedifficulties?

(b) What obstacles might you confront if you want your students toengage in these activities? How might you manage theseobstacles in order to help students focus on understandingmathematics?

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SECTION 4: ANALYZING CLASSROOM SEGMENTS ON THEVIDEO (in light of the Standards documents).*

MS. BALL'S THIRD GRADE CLASSROOM

In 1980, Ms. Ball had been teaching third grade for five years whenshe began to have doubts about the success of her mathematics teaching.When she moved from third to fifth grade, she found the math morecomplicated. While she felt that she was still able to explain mathematicalideas clearly, her students did not get the right answers with the sameconsistency as in previous years. When Ms. Ball talked with the fifthgraders about their mistakes, she found that they were not clear about whatthey were doing and why. She began to talk with her colleagues at theUniversity who steered her toward articles and books about what was wrongwith a lot of mathematics teaching and how to do it better. She slowly beganto rethink what it meant to learn mathematics and experimented withalternative ways of teaching it. Her interest in mathematics grew. Althoughshe had already started a master's program in reading in Michigan State'sCollege of Education, she changed her focus to mathematics and thenpursued advanced graduate studies. She is now an associate professor in theMSU Department of Teacher Education and continues to teach mathematicsat Spartan Village Elementary School.

In this third grade classroom, Ms. Ball is about to introduce a unit onfractions. None of her students had formally studied fractions; however, Ms.Ball was certain that they, like most eight-year-old children, had somefamiliarity and experience with fractions. Therefore, the tasks she selectedfor this lesson consisted of problems that would serve not only as anintroduction for her students, but also as an opportunity for her to assesstheir informal knowledge (including misconceptions) of fractions.

The video segment begins with Ms. Ball in a classroom posing thefollowing problem about a number of loaves of bread that are to be shared bya given number of people.

Problem: Dena had six loaves of bread. How can she share theloaves equally among twelve people?

As the video continues, students are given time to work on the problem.Students first work individually and then, if they choose, they may work inconsultation with the teacher or with other students. Ms. Ball then guides awhole class discussion centering on how students thought about solving theproblem.

Masters of the questions for each classroom segment are included inthe "Overheads" section of the guide.

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Discussion Questions:

(1) According to the NCTM Curriculum and Evaluation Standards,

"Students need to experience genuine problems regularly. Agenuine problem is a situation in which, for the individ-al orgroup concerned, one or more appropriate solutions have yetto be developed. The situation should be complex enough tooffer challenge, but not so complex as to be insoluble...Learning should be guided by the search to answerquestionsfirst at an intuitive, empirical level, then bygeneralizing, and finally by justifying (proving)." (p. 10)

(a) Do you think that the students in this class were trying to solve a"genuine problem"? What features of the problem make itgenuine or not genuine?

(b) How did Ms. Ball use this problem to engage students and elicittheir mathematical reasoning and communication?

(c) What could be a genuine problem for students at your gradelevel?

(2) During the whole class discussion, the teacher rarely gave studentsany hint about whether she thought they were right or wrong, or evenif they were on the right track. Why do you think she behaved thisway? What effect did her behavior have on the students' reasoningabout math?

(3) One of the purposes for this lesson was to diagnose what studentsknew about fractions as they began a unit on fractions.

(a) Why might it be important for a teacher to learn what herstudents know about a topic they had not yet studied?

(b) What did you find out about what students knew? Did anythingsurprise you?

(c) Contrast this teacher's diagnosis of her students' knowledge offractions with a more traditional pre-test approach.

(4) Ms. Ball accepted a student's definition of the unit as "cutted bread."Would you do this? Is this good practice? What is your reasoning?

(5) According to the Professional Teaching Standards, "The teacher ofmathematics should orchestrate discourse by:

posing questions and tasks that elicit, engage, and challenge eachstudent's thinking;

listening carefully to students' ideas;

asking students to clarify and justify their ideas orally and inwriting;

deciding what to pursue in depth from among the ideas thatstudents bring up during a discussion;

deciding when and how to attach mathematical notation andlanguage to students' ideas;

deciding when to provide information, when to clarify an issue,when to model, when to lead, and when to let a student strugglewith a difficulty;

monitoring students' participation in discussions and decidingwhen and how to encourage each student to participate." (p. 35)

Orchestrating discourse depends "on teachers' understandings ofmathematics and of their studentson judgments about the things thatstudents can figure out on their own or collectively and those for which theywill need input." (p. 36)

(a) In what ways did Ms. Ball attend to these suggestions fororchestrating discourse?

(b) Do you see places where a teacher might have made differentdecisions from those Ms. Ball made? What do you think herdecisions were based upon? On what are your decisions based?

(6) The students in this class seemed comfortable reasoning aboutmathematics in a whole group setting, including at times disagreeingwith their classmates. How might a teacher establish these norms ofcommunication in her class?

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MS. JONES'S SEVENTH GRADE CLASSROOM

Ms. Jones has taught mathematics to middle school/junior highstudents for 16 years at Dwight Rich Middle School in Lansing, Michigan.After having taught for 11 years, she had become dissatisfied with the wayshe taught math. Ms. Jones recalls teaching the operation of fractions. Shewould provide children with a rule to compute a quotient or product;carefully demonstrate the correct procedure to follow, and expect students toduplicate what she had done. Year after year, the procedure never changed,and just as predictably, children would get to the first or second step in theprocess, and then forget what to do next. Ms. Jones would reteach theconcept and hope students would catch on.

Though Ms. Jones was afraid that she did not know enough about thesubject of mathematics, Ms. Jones wanted to learn more. In 1985, she hadthe opportunity to participate in a local mathematics project. As a participantin the project, she learned more about teaching and learning mathematicsthan she had learned in all her years of teaching. Though she had to face herown lack of knowledge and misconceptions about some fairly basicmathematics concepts, she learned that relying on learning rules andprocedures was not enough. She learned that she needed to ask why, and tocontinue asking why, until she understood the mathematics that she wasexpected to teach. In the end, Ms. Jones found that she "was completelytransformed into what it meant to be a teacher and learner of mathematics."

In this segment, Ms. Jones is teaching a seventh grade generalmathematics class. Having learned that her students do not have a goodconceptual grasp of decimals, she is using manipulatives to help her studentsdevelop decimal number sense. (Ms. Jones knows that these seventh gradershad been exposed to decimals in previous years; she also assumes that theirprevious experience with decimals likely consisted mainly of paper and pencilcomputational procedures.) In the class segment that appears on the video,students are asked to use base ten blocks to find different ways ofrepresenting a specific decimal number. They work in groups before Ms.Jones guides a whole class discussion.

Discussion questions:

(1) The video segment shows students representing numbers with base tenblocks. According to the Curriculum and Evaluation Standards,

"Students need to experience genuine problems regularly. Agenuine problem is a situation in which, for the individual cgroup concerned, one or more appropriate solutions have yetto be developed. The situation should be complex enough tooffer challenge, but not so complex as to be insoluble...Learning should be guided by the search to answerquestionsfirst at an intuitive, empirical level; then bygeneralizing; and finally by justifying (proving)." (p. 10)

(a) Do you think that the students in this class were trying to solve a"genuine problem"? What features of the problem make itgenuine or not genuine?

(b) How could the problem be re-cast to make it more genuine or lessgenuine?

(c) How does this method of learning decimals compare with yourtextbook's treatment of decimals?

(2) The students in this segment worked in groups, as they do regularly inMs. Jones' class (and as is encouraged by the Standards).

(3)

(a) What is the rationale for group work? What benefits do youthink students derive from working in groups?

(b) This class was filmed in the middle of the year. What norms ofclassroom discourse and behavior do you think Ms. Jones had tocultivate throughout the school year for students to be able towork together as they did in this segment?

(c) During the class discussion Ms. Jones consistently pushedstudents to elaborate when they responded to her questions. Whydo you think she did this? Compare and contrast how she askedquestions with the more traditional questioning that seeks a rightor wrong answer from students.

(d) What other questions or different prompts can you think of tohelp students develop decimal number sense?

The Professional Teaching Standards ask teachers to attend tostandards in four areas: selecting worthwhile tasks; creating alearning environment; orchestrating discourse; and engaging inthoughtful analysis. How did this teacher appear to be attending toany of these standards during the lesson?

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(4) Four standards in the Curriculum and Evaluation Standards arecommon to all grade levels: mathematics as problem-solving;mathematics as communication; mathematics as reasoning; andmathematical connections. How did these students appear to beattending to any of these standards during the lesson?

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MR. SHERBECK'S CLASSROOM

Mr. Sherbeck has taught for 14 years at an urban school, BeaubienJunior High School (grades 7-9) located in Detroit, Michigan. Mr. Sherbeckhas always considered himself a teacher who focuses on eliciting and probingstudents' thinking about subject matter. His undergraduate teacherpreparation as a mathematics and science teacher helped him rejectconventional classroom practices and oriented him toward a more ambitiousview of teaching. Mr. Sherbeck describes the changes in his mathematicsteaching over the years as "evolutionary rather than revolutionary." He hascontinued to learn more about subject matternot only learning new subjPetmatter but also coming to understand the subject matter he teaches trylzdeeply. He has also had to learn more about innovative teaching practiceswhich help students understand mathematics, such as how to include writingin a math class. When Mr. Sherbeck first started this practice, he asked hisstudents to write about mathematics and then showed their work to hisEnglish department colleagues who encouraged and supported his efforts.Mr. Sherbeck quickly saw the value in creating a context where studentswrite about and explain their thinking about mathematics.

In this segment, Mr. Sherbeck has asked students to write flown "in aparagraph or two how to construct a 45 degree angle." We see on the videostudents discussing their reactions to what their classmates have written.

Discussion questions:

(1) The students in this segment worked in groups.

(a) What is the rationale for group work? What benefits do youthink students derive from working in groups? What are thedrawbacks?

(b) This class was taped in the middle of the year. What norms ofclassroom discourse and behavior do you think Mr. Sherbeck hadto cultivate throughout the school year for students to be able towork together as they did in this segment?

(c) If you wanted your students to talk to each other like the studentsin Mr. Sherbeck's class, how would you model this kind ofconversation? How would you give feedback to students thatwould encourage constructive comments and discourage off-taskcomments?

(2) Why might it be important for students to write about theirmathematical thinking?

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(3) A teacher could ask students to write about mathematics withoutnecessarily contributing to their problem solving, reasoning, orcommunication skills. How can teachers use students' writing inways that promote meaningful mathematical learning?

(4) The Professional Teaching Standards ask teachers to attend tostandards in four areas: selecting worthwhile tasks; creating alearning environment; orchestrating discourse; and engaging inthoughtful analysis. How did this teacher appear to be attending toany of these standards during the lesson?

(5) Four standards in the Curriculum and Evaluation Standards arecommon to all grade levels: mathematics as problem-solvingmathematics as communication; mathematics as reasoning; andmathematical connections. How did these students appear to beattending to any of these standards during the lesson?

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MR. LEHMAN'S ALGEBRA II ASSESSMENT

Since 1990, Mr. Lehman has worked closely withcolleaguesincluding Michigan State University professors and graduatestudents as well as his fellow teachers at Holt High Schoolas part of HoltHigh School's transition to becoming a "Professional Development School."As a result of this work, Mr. T eh man has made a concerted effort to changehis teaching. He realized that his Algebra II students were learning tomechanically manipulate symbols reasonably well, but were not learning toreason, make connections, or communicate about algebraic concepts. Hewas aware also that traditional paper and pencil tests did not allow him toanalyze the kind of teaching and learning for understanding he was trying topromote. Consequently, he incorporated into the semester examination anassessment process that would allow students to demonstrate theirunderstanding to a panel of judges. A segment of this assessment activityappears on the video.

Students had three days of preparation time, in class and on their owntime, to work with each other to create solutions to six problems. Studentswere randomly assigned one of the problems when they met with the panels.The problem to which students on the video responded is on the followingpage.

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Problem: Tom and Patrici: Wainwright have decided to start raising HungarianLonghair V ite Rabbits to si as pets. The profit is greater if they sell their rabbits to awholesaler in quantities of 1000 instead of individual rabbits to individual pet stores.They therefore plan to sell their rabbits in groups of 1000. Tom and Pat plan to start outwith 25 rabbits. They know the rabbits will approximately quadruple every month. Theywill count the next month's populations after they have subtracted the rabbits they sold.The Wainwrights have several questions they need answered before they start thisbusiness. Your job is to try to answer them for the Wainwrights. They must be carefulnot to sell too many rabbits otherwise they would end up having no income for a monthor more after they start selling rabbits.

Part 1: The Wainwrights would like to know when they will be able to start makingsales and how many groups of 1000 they can sell each month. Caution: Selling andreproduction from one month effect the total population of the next. When figuring thenext month's population, do not count the rabbits sold during that month. Could yougive them a forecast of their first year of business?

(a) During which month will they be able to start selling rabbits?

(b) Is there a month when they will have to hold back more than 1000 rabbits frommarket in order to maintain a breeding stock? If so, which month?

(c) During which month will tneir income even out to a steady income?

(d) If under their current contract, they make $2.50 for each rabbit sold, what willtheir income be when it finally evens out?

Part 2: The Wainwrights start up cost where $15 for each rabbit they started with,$5,000 for the building to house the rabbits, $250 for the proper permits to run this typeof business, and $300 for food for the rabbits.

(a) In which month will they finally b:eak even?

(b) The Wainwrights' monthly expenses are approximately $24,000 per month. Howmuch must they sell their rabbits for in order to continue to average $2.50 perrabbit profit?

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Members of the panels which assessed students included teachers,administrators, university faculty, community or school board members, andadvanced (i.e. Calculus or Pre-Calculus) high school students. Mr. Lehmanensured that each panel of three contained at least one person who specializedin mathematics. Panelists were encouraged to ask conceptual, sense-makingquestions instead of procedural ones. Each student had approximately twentyminutes to defend individually his/her solution. Students were encouraged toshow panelists written work that supported their solutions. Panelists ratedeach student according to criteria developed by Mr. Lehman; they had theopportunity to discuss with each other each student's performance and adjust

numerical rating if they were so persuaded. (See the enclosed copy ofthese criteria and the grading scale.)

Discussion questions:

(1) What are the advantages and disadvantages of this type of assessment,where students individually demonstrate their knowledge andunderstanding by orally defending their solutions to problems?Contrast what students must know and be able to dr, for this type ofassessment with what they must know and be able to do for moretraditional pencil and paper tests.

(2) Although the students prepare their solutions while working in groups,they were required to explain their solutions individually. What mightbe some advantages, or disadvantages, of allowing groups todemonstrate their solutions collaboratively in this type of assessmentactivity?

(3) If classroom instruction is not consistent with the type of sense-making, conceptual questions that panelists ask students, then studentscannot be expected to do well on the assessment. If you knew yourstudents would be assessed by a panel like this, how would youprepare them? Where would you find problems for them to work on?How would you organize classroom discussions to model this kind ofreasoning about mathematics?

(4) What logistical or technical problems might be associated witharranging this type of assessment? What alternatives might be usedby teachers who might have difficulty gathering a large number ofpanelists, but who would like to give students opportunities todemonstrate orally their mathematical sense-making and problemsolving abilities?

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Algebra HPerformance Assessment

Name

Mathematics:

(1) Making sense of problem 1 2 3 4 5(Understanding concepts)

(2) Problem solving strategies 1 2 3 4 5(Methods used)

(3) Accuracy of results 1 2 3 4 5

(4) Interpreting results 1 2 3 4 :(What do the results mean?)

Presentation:

(1) Ability to communicate results(Clarity, use of charts/graphs)

(2) Explanation(Able to answer questions)

Grading Scale

26 30 A22 - 25 B18 21 C15 - 17 D

Comments:

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1 2 3 4 5

1 2 3 4 5

Overall Score

SECTION 5: ANNOTATED BIBLIOGRAPHY

CHANGING PRACTICE: TEACHING MATHEMATICS FOR UNDERSTANDING

Ball, D. L. (1990). Halves, pieces, and twoths: Constructing representational contexts inteaching fractions (Craft Paper, 90-2). East Lansing: Michigan State University,

National Center for Research on Teacher Learning.

Learning to teach mathematics for understanding is not easy. First, practice itself is complex. Second,many teachers' traditional experiences with and orientations to mathematics and its pedagogy are additionalhindrances. This paper examines the territory of practice and reviews some of what we know about thosewho would traverse itprospective and experienced elementary teachers. In analyzing practice, the authorfocuses on one major aspect of teacher thinking in helping students learn about fractions: the constructionof instructional representations. Considerations entailed are analyzed and the enactment of representationsin the classroom is explored. The term representational context is used to call attention to the interactionsand discourse constructed in a classroom around a particular representation. The author provides a windowon her own teaching practice in order to highlight the complexity inherent in the joint constructionwithstudentsof fruitfulrepresentational contexts. The paper continues with a discussion of prospective andexperienced teachers' knowledge, dispositions, and patterns of thinking relative to representing mathematicsfor teaching. The author argues that attempts to help teachers develop their practice in the direction ofteaching mathematics for understanding requires a deep respect for the complexity of such teaching anddepends on taking teachers seriously as learners.

0 Ball, D. L. (1991, September). Beginning a conversation about the NCTM ProfessionalStandards for Teaching Mathematics: Improving, not standardizing, teaching. TheArithmetic Teacher, 39(1), 18-22.

The first in a series of "professional conversations" about the National Council of Teachers ofMathematics (NCTM) Professional Standards for Teaching Mathematics, this article begins with a generaldiscussion of the standards, explaining that they do not reduce teaching to recipes; rather, they expressa set of shared professional values and goals for mathematics teachers and help guide teachers' effortsto teach mathematics well. The author provides an overview of the document's first section, "Standardsfor Teaching Mathematics," including a discussion of the terms tasks , discourse, environment, andanalysis. Then, focusing specifically on the standard for selecting and using good learning tasks forstudents, she analyzes the strengths and weaknesses of three sample problems to help teachersdetermine what a worthwhile mathematical task entails.

Ball, D. L. (1991, November). What's all this talk about "discourse"? Arithmetic Teacher,39(3), 44-48.

The author defines "discourse" as intended by the standards. Using a discussion drawn from her ownthird-grade-class and entries from her teaching journal, the author illustrates how thoughtful discoursecan help students learn to discuss mathematics.

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Baroody, A. J. (1987). Children's mathematical thinking. New York: Teachers CollegePress.

In this book, the author explains how young students (K-3) think and learn about mathematics. Heargues that what we know about how children think mathematically is inconsistent with what he termsthe "absorption theory of learning." Instead of passively absorbing new knowledge, children activelybuild on their intuitive, informal knowledge as they produce new meanings and understandings. Theauthor also suggests teaching practices which are more consistent with how students learn mathematics.

Charles, R. I., & Lester, F. (1982). Teaching problem solving: What why and how. PaloAlto, CA: Dale Seymour Publications.

This book addresses what mathematical problem solving involves, why problem solving is so important,and how teachers can make problem solving an integral part of mathematics instruction. After definingand discussing what is and is not a problem, the authors present many useful guidelines and suggestionsfor teachers who want to make problem solving the focus of their mathematics programs.

Charles, R. I., & Silver, E. A. (1988). The teaching and assessing of mathematical problemsolving (pp. 371-375). Reston, VA: National Council of Teachers of Mathematics.

This research-oriented collection of essays is directed to all who play a role in the reform ofmathematics teaching (math educators, teachers, administrators, etc.). Selections by Resnick,Schoenfeld, Lester, Silver, Carpenter, Noddings, and others represent perspectives from several fields.All are concerned with the teaching and evaluation of problem solving. Student collaboration,alternative assessment, motivating students, student-centered instruction, the use of a constructivistapproach to teaching and learning, and problem solving for all students are just some of the issueshighlighted.

Chazan, D. (1992, May). Knowing school mathematics: A personal reflection on theNCTM's teaching standards. Mathematics Teacher, 85(5), 371-375.

Chazan talks about his struggle to think about math and rnath teaching in new ways. The question ofwhy the quadratic formula works to identify the solutions of quadratic equations was student initiated.The question inspired Chazan to pursue understanding the "whys" and not just the "how to's" inlearning and teaching mathematics. Chazan shares the process of his realization that if educators areto teach in a meaningful and non-manipulative way, helping students to recognize the uses of math intheir daily pursuits, then teachers need to know math in that way as well. "A way I was not taught in,"says Chazan.

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Cocking, R. R., & Mestre, J. P. (Eds.). (1988). Linguistic and cultural influences on learningmathematics . Hillsdale, NJ: Lawrence Erlbaum Associates.

A major premise of the Standards is that all children can learn, regardless of cultural background. Thisbook ersplores issues critical to educating all children. Some of the topics addressed are languages ofminority children, bilingualism, and studies of specific cultural groups such as Asian-Americans,Mexican-Americans, and Native Americans.

Connolly, P., & Vilardi, T. (Eds.). (1989). Writing to learn mathematics and science. NewYork: Teachers College Press.

The Standards emphasize writing as a tool for learning mathematics. One of the video segmentshighlights one use of writing in a middle school setting. This book contains several chapters thatillustrate a spectrum of options and rationales for using writing at all.levels of mathematics teaching.

Cooney, T. J., & Hirsch, C. R. (Eds.). (1990). Teaching and learning mathematics in the1990s: 1990 Yearbook . Reston, VA: National Council of Teachers of Mathematics.

This highly accessible volume is rich with practical ideas for teachers of mathematics at every level. The28 essays are organized in seven sections: the relationship between research and practice, effectivemethods for teaching mathematics, assessing students' mathematical understanding, cultural factors inteaching and learning, contextual factors in teaching and learning, technology, and professionalism andits implications for teaching and learning. Each essay summarizes relevant research and includespractical suggestions for teachers at both elementary and secondary levels.

Cramer, K., & Bezuk, N. (1991, November). Multiplication of fractions: Teaching forunderstanding. Arithmetic Teacher, 39(3), 34-37.

The authors use the "Lesh translation model with five modes of representation" to illustrate how amultiplicity of approaches help students to better understand mathematical concepts. Many activitiesfor teaching multiplication of fractions are used as models for different modes of representation.

Grouws, D. A., Cooney, T. J., & Jones, D. (Eds.). (1988). Effective mathematics teaching(Vol. 1). Reston, VA: National Council of Teachers of Mathematics.

This collection of papers reports on what research has to say about what effective mathematics teachingis and how to foster it. The papers discuss a variety of issues that must be addressed in building aframework for effective mathematics teaching.

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Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems inpractice. Harvard Educational Review, 55(2), 178-194.

The author is a university mathematics educator and an elementary school teacher. As a practitionerstriving to improve her craft and the learning environment of her elementary school classroom, she viewsthe teacher as a dilemma manager turning conflict into teachable moments. In this article she analyzestwo pedagogical dilemmas. The first case is drawn from her own fifth grade class. Lampert addressesher discovery of how the misbehavior of her male students created a situation of gender inequity in herclassroom. The second case focuses on a conflict over the nature of knowledge arising in a fourth gradescience lesson.

Maher, C. A., & Martino, A. M. (1992, March). Teachers building on students' thinking.Arithmetic Teacher, 39(7), pp. 32-37.

This article touches on several important issues related to teaching math for conceptual understanding.The authors use dialogue between two girls working "independently yet cooperatively" on a real worldproblem. The classroom teacher enters into the conversation to extend the girls' problem solvingdiscussion by adding a new dimension to the problem. The authors illustrate that holding back theanswer can be an effective instructional strategy.

Mathematical Sciences Education Board. (1989). Everybody counts: A report to the nationon the future of mathematics education. Washington, DC: National Academy Press.

This report issues a comprehensive call for reform in mathematics education, from kindergarten throughgraduate school. It explains the seriousness of the problem, charts a general course for the future, andoutlines a national strategy for pursuing that course. Discussing mathematics as the key to economicopportunity, the link between equity and excellence in education, the nature of mathematics, newstandards for curriculum and assessment, new approaches to teaching, and the impact of calculators andcomputers, Everybody Counts provides an overview of the current situation, necessary reforms, andobstacles to those reforms. Arguing that high expectations yield greater achievement, the authors callfor all students to learn a significant core curriculum. They also propose changing from a "transmissionof knowledge" model of teaching mathematics to "student-centered practice" and from inculcation ofroutine skills to "developing broad-based mathematical power."

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Mathematical Sciences Education Board. (1990). Reshaping school mathematics. A

philosophy and framework for curriculum. Washington, DC: National Academy Press.

This booklet proposes a framework for reform of school mathematics curricula in the United States.It is intended to complement Everybody Counts and the NCTM Curriculum and Evaluation Standards,and it emphasizes two fundamental issues discussed in those documents: (1) changing perspectives onthe need for mathematics, the nature of mathematics, and the learning of mathematics; and (2) changingroles of calculators and computers in the practice of mathematics. Philosophical issues are treated inmuch greater depth than in either of the other two documents and extensive reference is made tocurrent research on topics such as technology in education and teaching for higher-order thinking. Thebooklet also proposes a set of fundamental principles, specific goals, and enabling conditions for the newmathematics curriculum.

McDiarmid, G. W. (1992). Kathy: A case of innovative mathematics teaching in amulticultural classroom. Fairbanks: University of Alaska Fairbanks, Center forCross-Cultural Studies.

This case study presents a clear portrait of a math teacher who bases her instruction on achievingmathematical understanding rather than learning mathematical rules. Her methods are challenged byan African-American parent who wants her daughter to learn mathematics and do well on standardizedtests. Kathy's story illustrates how all students, especially children of color, benefit from the innovativeinstruction she uses in her classes.

McIntosh, M. E. (1991, September). No time for writing in your class? Mathematics

Teacher, 82(6), 423-433

The author distinguishes between math teachers teaching writing, in the spirit of writing across thecurriculum, and using "writing to learn." The article provides clear descriptions of four written forms(logs, journals, expository, creative) that can help students learn, help the teacher assess student learning,and put math in more "human terms."

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National Council of Teachers of Mathematics. (1989, March). Curriculum and evaluationstandards for school mathematics . Reston, VA: Author.

Reflecting a broad consensus within the mathematics education community, this document establishesa comprehensive framework to guide reform in school mathematics in the 1990s, providing a detailedvision of what the content, priorities, and emphases of the mathematics curriculum should be. The mainbody of the document consists of curricular standards for grades K-4, 5-8, and 9-12, and evaluationstandards. The curricular standards articulate the following goals for all students: (1) that they learnto value mathematics; (2) that they become confident in their ability to do mathematics; (3) that theybecome mathematical problem solvers; (4) that they learn to communicate mathematically; and (5) thatthey learn to reason mathematically. Some of the standards address these goals directly; others addressa specific topicsuch as whole number computation or statisticsin the context of the goals. Allarticulate a vision of the classroom as a place where students learn important mathematical ideas byactively engaging in conjecturing, reasoning, and exploring interesting problems; using manipulatives,computers, calculators, and other hands-on tools; and working often in cooperative groups. Theevaluation standards address assessment strategies, judgements about students' progress, and evidenceabout the quality of mathematics programs. They stress using assessment to guide instruction, aligningassessment with the curriculum, and taking advantage of multiple methods of assessment.

To order, call (800) 235-7566 or (703) 620-9840.

National Council of Teachers of Mathematics. (1991, March). Professional standards forteaching mathematics. Reston, VA: Author.

As part of the NCTM's broad framework for reforming school mathematics, these standards providea vision of the kind of teaching necessary to support the curriculum proposed in the Curriculum andEvaluation Standards. This document rests on the assumptions that teachers are key figures in changingmathematics education and that teachers need long-term support and adequate resources to effect theproposed changes. It includes sections on teaching mathematics, evaluating teachers, professionaldevelopment, and institutional support for teachers. Annotated vignettes, drawn from actual classrooms,are used throughout to illustrate teachers' interactions with colleagues, students, supervisors, anduniversity faculty. Running through the entire document are characterizations of five major shifts to bemade in the environment of mathematics classrooms: (1) toward classrooms as mathematicalcommunities (away from classrooms as collections of individuals); (2) toward logic and mathematicalevidence as verification (away from the teacher as sole source of right answers); (3) towardmathematical reasoning (away from just memorizing procedures); (4) toward conjecturing, inventing,and problem solving (away from mechanical manipulation and symbols); and (5) toward connectingmathematics, its ideas, and its applications (away from viewing mathematics as isolated concepts andprocedures).

To order, call (800) 235-7566 or (703) 620-9840.

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Parker, R. E. (1991, September). Implementing the Curriculum and Evaluation Standards:What will implementation take? Mathematics Teacher, 82(6), 442-448, 478.

The author describes the kind of instruction implied by the standards in the context of two problems,emphasizing the importance of "good content" to creating rich problem solving experiences. Studentwork on these problems is included to illustrate the approach taken and results achieved. The articlealso includes a brief discussion of the kind of support teachers will need in order to change theirpractice.

Peterson, P. L., & Knapp, N. F. (1992). Inventing and reinventing ideas: Constructivistteaching and learning. In G. Cawelti (Ed.) (in press). The 1993 yearbook of theAssociation for Supervision and Curriculum Development (ASCD) . Washington, DC:ASCD.

The authors begin this chapter by "unpacking" the constructivist view of knowledge and learning. Twocase studies of constructivist mathematics teaching and learning in elementary classrooms are presented.Through the case study approach, the perspectives and voices of the two teachers have been preserved.Four critical questions are pursued here: What is the teacher's role in constructivist learning? How canthe teacher honor both student-constructed knowledge and traditionally-accepted knowledge? How canteachers involve diverse students in community problem-solving? How can teachers help students handlethe risks of publicly sharing and debating ideas?

Resnick, L. (1987). Education and learning to think. Washington, DC: National AcademyPress.

This monograph addresses the question of what schools can do to promote the teaching and learningof what are called "higher-order skills." The author presents a working definition of higher-orderthinking skills and discusses these skills in the context of various subjects, including mathematics.Resnick challenges the common assumption that there is a natural learning sequence that moves fromlower level activities ("basic skills") to higher level ones. This assumption, which is used to justify yearsof drill on the basics before thinking and problem solving are introduced, is contradicted by the cognitiveresearch that Resnick discusses. She says that "the most important single message of modern researchon the nature of thinking is that the kinds of activities traditionally associated with thinking are notlimited to advanced levels of development. Instead, these activities are an intimate part of evenelementary levels of reading, mathematics, and other branches of learning." Moreover, failure tocultivate aspects of thinking, she argues, may be the source of major learning difficulties, even inelementary school. Resnick also offers suggestions for teaching and organizing instruction in higherorder thinking.

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Resnick, L. & Klopfer, L. E. (Eds.). (1989). Toward the thinking curriculum: Currentcognitive research. 1989 Yearbook of the Association for Supervision and CurriculumDevelopment . Alexandria, VA: ASCD.

This collection provides an overview of recent research by cognitive psychologists on the mentalprocesses underlying thinking and learning in the classroom. Seven of the nine chapters focus on thethinking processes that contribute to learning in mathematics, science, reading, and writing. Thesechapters include recommendations for curriculum and instruction in these subject areas. Of particularnote are Chapter 1, an overview of cognitive research findings, and Chapter 9, a discussion of theimplications of research for instructional practices.

Romberg, T. A. (1992). Assessing mathematics competence and achievement. In Berlak,H., Newmann, F. M., Adams, E., Archibald, D. A., Burgess, T., Raven, J., &Romberg, A. (Eds.), Toward a new science of educational testing and assessment (pp.23-52). Albany, NY: SUNY Press.

In this article, Romberg reviews the rationale for reforming the teaching and learning of mathematics,and analyzes current testinj procedures. He concludes that current testing is based on an outdated setof assumptions about mathematical knowledge, learning, and teaching, and that its continued use willbe counterproductive to needed reform. He presents principles upon which new assessment instrumentsshould be based and gives examples of tasks that are consistent with these principles.

Romberg, T. A., & Carpenter, T. P. (1986). Research on teaching and learningmathematics: Two disciplines of scientific inquiry. In M. C. Wittrock (Ed.),Handbook of research on teaching: A project of the American Educational ResearchAssociation (3rd ed., pp. 850-873). New York: Macmillan.

In this chapter, the authors review the recent shift in direction of research on students' mathematicallearning and thinking and discuss what this shift implies for research on mathematics teaching. Theybegin by examining the current state of instruction in mathematics, and they challenge the assumptionsabout student learning on which current instruction is based. After analyzing more recent research onmathematics teaching and learning, the authors discuss the new directions that they think research inmathematics education should take.

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Schoenfeld, A.H. (1988). When good teaching leads to bad results: The disasters of "well-. taught" mathematics courses. Educational Psychologis4 23, 145-166.

The author conducted observations over an entire school year of a "good" geometry class, taught by awell-regarded teacher whose students received good grades and continued on to college. He shows thatmuch of what students learned in this traditional setting is inconsistent with the type of learning that isadvocated by the standards. This article raises important and provocative questions about what kind ofmathematical knowledge students really learn, even in traditionally "well taught" mathematics classes.

Silver, E. A., Kilpatrick, J., & Schlesinger, B. (1990). Plinking through mathematics:Fostering inquiry and communication in mathematics classrooms. New York: CollegeEntrance Examination Board.

This booklet explains changing perspectives on what it means to learn and do mathematics and exploreshow these perspectives can be incorporated into the teaching of secondary school mathematics. Usingnumerous examples of classroom situations, the booklet aims to stimulate teachers to connect thinkingand mathematics, moving toward classrooms in which students construct their own mathematicalunderstanding, justify conjectures, solve open-ended problems, and create their own problems. Theauthors suggest ways in which teachers can encourage greater communication and inquiry in their classesby modifying textbook exercises, incorporating non-routine problems, using group work and written logs,and making other minor but thought-provoking changes in the ordinary curriculum. The bookletcontains much practical advice for teachers beginning the difficult, but rewarding, process of changingthe mathematics classroom, including advice on goal setting, identifying where to make changes, andcollaborating with other teachers.

Steen, L. A. (1990). On the shoulders of giants: New approaches to numeracy. Washington,DC: National Academy Press.

Written for educators, curriculum developers, educational policymakers, and parents, this bock it:intended to stimulate creative approaches to mathematics curricula and prompt a redefinition of whatis basic or fundamental in mathematics education for the twenty-first century. The volume includes fiveessayseach written by a different mathematician presenting mathematics ns the language of patternsand offering imaginative ways of developing mathematical ideas from kindergarten through college. Thetopicsdimension, quantity, uncertainty, shape, and changeprovide examples of rich mathematical ideasaround which the mathematics curriculum might be reorganized. Themes such as measurement,symmetry, visualization, and algorithms connect the essays. The topics covered, which are at theforefront of current mathematics research, may be unfamiliar to many teachers, but are adaptable toclassroom teaching. The authors bridge the gap between school and real mathematics.

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Stenmark, J.K. (Ed.). (1991). Mathematics assessment: Myths, models, good questions, andpractical suggestions Reston, VA: National Council of Teachers of Mathematics.

This booklet, written for teachers, provides a collection of assessment techniques that focus on studentthinking The booklet stresses the importance of linking assessment closely to instruction and the needfor authentic assessment tasks that bridge the gap between school and real mathematics. It describesthe advantages for both students and teachers of assessment alternatives; identifies and counters anumber of myths about assessment (for example, that there is always a single answer to a mathproblem); reviews current assessment practices and identifies needed changes; and offers a sample planfor developing a new approach to assessment. At the heart of the booklet are three sections onalternative assessment approaches: performance assessment; observations, interviews, conferences, andquestions; and mathematics portfolios. Each of these sections contains guidelines for developing andusing the assessment techniques including classroom management and logistics; sample tasks, questions,and forms; and evaluation criteria. The booklet concludes with a section on implementing new modelsof assessment, that focuses on documentation and demonstration of validity, along with guidelines forstudent self-assessment.

Stigler, W. S., & Stevenson, H. W. (1991, Spring). How Asian teachers polish each lessonto perfection. American Educator, 15(1), 12-20, 43-47.

This article, which is based on a forthcoming book entitled Cultural Lessons: A New Look at theEducation of American Children, illustrates how mathematics teaching in Asian elementary classroomsis more engaging than in American classrooms. Asian teachers seem to "polish each lesson toperfection." Lessons are coherent and the teacher acts more as a "knowledgeable guide, ratherthan . . . dispenser of information and arbiter of what is correct." Of the many differences between whatthe authors observed in Asian and American classrooms, the discussion of mathematical concepts inAsian classrooms is of particular interest; questions posed in Asian classrooms are designed to stimulatethought, not simply to get a correct answer. Most relevant to American concerns of educational equityis how Asian teachers deal with diversity of students backgrounds. Asian elementary children are nottracked or grouped by ability. In a whole-group instructional setting, Asian teachers vary theirapproaches to teaching and presenting materials within a lesson to better engage students who cometo the setting with diverse past experiences.

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Willoughby, S. S. (1990). Mathematics education for a changing world. Alexandria, VA:Association for Supervision and Curriculum Development.

Willoughby, a mathematician and educator, discusses ways teachers and supervisors can change themathematics education of children, focusing on the most striking differences between current andsuggested practice. He maintains that mathematics instruction should move from the concrete to theabstract, neglecting neither; that unreal problems do more harm than good; and that children shouldbe taught to use calculators properly. Throughout the book, Willoughby provides examples of specificactivities and curriculum suggestions. For instance, to illustrate making "verticalconnections" throughoutthe curriculum, as recommended in the NCTM Curriculum and Evaluation Standards, he suggestsactivities for teaching functions at every grade level. These range from kindergartners creating a"computer" out of a cardboard box, to third graders using circle-and-arrow diagrams, to sixth gradersusing graphing computers and calculators. The book concludes with a discussion of ways to supportteachers in bringing about the kinds of changes advocated.

Summer 1990(Special Double Issue)

.

RethinkingMathematics Teaching

When teachers in the Profes-sional Development Schools be-gan to talk to one another aboutwhat they wanted to rethink andrevise in their schools and class-rooms, the subject of mathematicsteaching came up over and overagain. Some winced as they re-membered the math classes oftheir childhood and worried thatthey were doing little better thantheir own teachers had done. Theywere not sure that their own stu-dents either enjoyed or fully un-derstood the mathematics thatthey did in school. The depth andbreadth of this concern led teach-ers in several schools to organizemath study groups.

This first issue of ChangingMinds describes some particularways of teachingmath, some of thethinking that PDSteachers, undergrad-uates, and graduatestudents in educationare doing with regardto math teaching, andsome of the thingsthat teachers are try-ing as they attempt toteach mathematics inways that will en-hance their students'understanding. Wehope that theseglimpses of studentsand teachers grap-pling with the ques-tions raised by thissort of teaching willhelp others see whatis involved in changing the teach-ing of math, and will tempt someteachers to look again at their ownteaching, and to join these ad-venturers as they explore andexperiment.

The issue begins with a look attwo new publications of the Na-tional Council for the Teaching ofMathematics which call forsweeping changes in mathematicsteaching. It then focuses on Deb-orah Ball, a teacher for 15 years atSpartan Village ElementarySchool in East Lansing, an assist-ant professor at the Michigan StateUniversity College of Education,and an important voice in the na-tional conversation about theteaching of mathematics.

After an intToduction to Ball'sclassroom and to her students, herpractice, and her thinking, we lis-ten in on two groups as theyreflect on what they have seen ofBall's teaching. The first group is aclass of college students who are

ChangingMinds .

This is the inaugu-ral issue of ChangingMinds, a publication ofthe Michigan Educa-

tional Extension Service (EES). The title is intended toconvey the several senses in wlich contemporary ed-ucation in Michigan is a matter of changing minds.

First, we are changing our minds about the kindand level of education required for today's students toprosper in a dramatically changed economy, in ascientifically and technologically advanced democ-racy and in a society that is growing more diversealong just about every imaginable dimension ethni-cally, culturally, linguk ically, in family patterns, inthe integration of handicapped peoplç into main-siTeam institutions, and in numerous other ways.

Continued on page 2.

40

considering careers in teaching.These young men and womenhave read a chapter of Active Math-ematics Teaching by Thomas Good,Douglas Grouws, and HowardEbmeier and are comparing the

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Changing Minds continued from page 1.

To build a robust twenty-first century economy, we need peoplewho can go beyond the rote application of standard procedures andbureaucratic rules. We need people who can think, who can solvenovel problems on the job, who can work in teams with diversemembership, who can communicate clearly both orally and in writ-ing, who understand the mathematical and scientific ideas underly-ing complex systems, who take responsibility for themselves and forachieving results, and who want and know how to go on learningthroughout their lives. Many of these same forms of knowledge andknow-how are also required for active citizenship at all levels, as wellas for life as a consumer, a family member, and in other phases of ourlives.

Second, we now know that learning itself is a matter of changingyour mind. Real learning, learning of the sort required by the emerg-ing economy and society, consists not in the passive accumulation offacts and skills, but in active construction and reconstruction ofideas, of mental models of how things work, of what written materialreally means, and of the mental tools we refer to as skills, such as theskills involved in reading, writing, and mathematics. Learning ascientific idea the notion that when we push on an object the objectpushes back, for example frequendy means revising our common-sense or intuitive understandings of the world around us, not justmemorizing a Newtonian law. The same is true of learning about his-tory, about social systems, about literature, about mathematics.

Third, for most experienced as well as new teachers, learning toteach in a way that fosters such constructivist learning involveschanging their minds about how learning really takes place and howthey can stimulate, support, and orchestrate it. For a teacher who un-derstands that learning is construction rather than absorption, theclassroom is a room full of minds. A room full of people who bringdifferent strengths, preconceptions, learning strategies, languages,values, and perspectives to the material to be learned and thereforesee it in different ways. A room full of people who may learn indifferent ways, but all of whom can learn. At some level, all teachersknow this, but too few of us are aware of the conflict between what wesay we believe and how we actually behave. We have to re-examineour beliefs, and our practice in light of our beliefs. Changing our prac-tice involves changing our minds.

Fourth, we have to change our minds about schools and universi-ties as institutions and about the relationships between them. If stu-dents are to be active thinkers and learners, then we must rethinkteachers' work, as well. We cannot view teachers as industrial age pro-duction workers, carrying out routine tasks under close supervision

Continued on page 23.

traditional style of teaching al-gorithms recommended by theauthors with what they see Balldoing. They feel comfortable withthe predictable, teacher-con-trolled teaching Good and his col-leagues describe; they react with

a mix of feelings to Ball's verydifferent approach, with its em-phasis on the reasoning of eight-year-olds. As one of them says,"We felt she could give them moredirection, lead them to something,because they were just spending a

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lot of time discussing zero: Thcsecond group is the Math StudyGroup in Elliott ElementarySchool in Holt: five experiencedteachers, two -interns- who haverecently graduated from MichiganState and are beginning their pro-fessional careers by teaching mathin this professional developmentschool, one graduate student, andtwo professors from the College ofEducation.

We then move back to thcclassroom and watch RuthHeaton, a veteran teacher and agraduate student, as she begins tochange the way she teaches math-ematics. Ruth is working withMagdalene Lampert, a colleague ofDeborah Ball both at the Collegeand at Spartan Village Elementary,and also a nationally recognizedexpert on mathematics education.As they discuss and examineRuth's changing practice, Maggieand Ruth are learning about chal-lenges that traditional teachersface as they try to teaLh mathe-matics in this unfamiliar way

The last two articles, "SpecialCircumstances: Rethinking Mid-dle School Math" and "Math Pro-jects at Holt High Schoor take usinto math classes in two second-ary schools: Holmes MiddleSchool in Flint and Holt HighSchool.

All the teachers described inthese pages have studied mathe-matics in traditional classrooms.All have taught in traditional ways,explaining the rules for conven-tional mathematical proceduresand helping their students to mas-ter these algorithms through re-peated practice on a variety ofproblems. As thoughtful studentsand teachers encountering an ap-proach to teaching which differsdramatically from the images ofmathematics teaching they carryin their heads, they are looking attheir own teaching in a differentway, and grappling with what itmight mean to try to change theirpractice.

A Call to Change the Teaching ofMathematics

A year ago the National Coun-cil of Teachers of Mathematics re-leased a report calling for swep-ing changes in mathematicscurricula in American schools andin the ways in which we assessour children's knowledge ofmathematics.'

The Council asks us to imag-ine ways of teaching and learningmath which are quite differentfrom those that most of us en-countered in our own schooling,and very different from the picturethat develops in our minds whenwe hear a teacher telling her classthat the time has come for math.Most teachers, both now and inthe past, have first taught childrenalgorithms e.g. "to divide by afraction, invert and multiply-and then assigned them numericalcalculations on which to developskill and speed in using this rule

before moving on to the tokenhated) "story- problems at the endof the chapter.

The Council proposes revers-ing this strategy, arguing that-knowledge should emerge fromexperience with problems: In oth-er words, if children encounter anInteresting problem befOre theyhave learned the anthmetic opera-tion which adults would use tosolve such a prohletn. they w ill in-\ ent lor themselves I it is theright problem at the nght time a

variety of ways to to solve theproblem. "Knowing' mathematicsis -doing' mathematics,- says theCouncil. "A person gathers, dis-covers. or creates knowlege in thecourse of some activity having apurpose:

But although the Council, intheir 1989 volume on curriculumand evaluation standards. envi-

sions elementary schoolers ex-panding their mathematicalknowledge through investigationsof interesting real world problems,neither popular elementaryschool math textbooks nor experi-ence have prepared most teachersfor this sort of math teaching. For-tunately the second volume struc-tures its suggestions around par-ticular vignettes of classroomteaching which allow teachers toec more concretely what the

Louncil envisions, and to begin totry to imagine themselves andtheir own students -doing- mathe-matics in new and quite differentways. rhis excerpt from a workingdraft of Professional Standards forTeaching Nlathematics- suggests therichness of the descriptions andthe commentary:

A class of primary students has been working on the following problem:

If we make 49 sandwiches for our picnic. how manv can each child have?

These students have informal ideas about division buthave not yct learned conventional procedures kn- dividing.They also have begun to develop some understandings offractions, connected to their ideas about division. The teach-er has selected this problem because she knows that it islikely to elicit alternative representations and solution strat-egies. as well as different answers. It will also help the stu-dents to develop their ideas about division and fractions.

After thcy have worked for about 20 minutes. first aloneand then in small groups. the teacher asks if the children areready to discuss the problem in the whole group. Most, look-ing up when she asks, nod. She asks who would like tobegin.

4 2

The teacher provides and structures the use of time so thatthe children have opportunities to tlevelop their solutions inde-pendently. with a few others, and then in the whole group. BVasking who would like to share their solution. she encourages thestudents to take intellectual risks. She shows that she respectstheir work by consulting them before she moves to the wholegroup discussion.

Two girls go to the overhead projector. They write:

49-2822

One explains. "There are 28 kids in our class and so ifwe pass out one sandwich to each child, we will have 22sandwiches left and that's not enough for each of us, sothere'll be leftovers:'

The teacher and sn.idents are quiet for a moment, think-ing about this. Then the teacher looks over the group andasks if anyone has a comment or a question about thesolution.

One boy says that he thinks their solution makes sense.but that he thinks that "9 minus 8 is 1, not 2, so it shouldbe 21, not 22:' The two girls ponder this for a moment.The class is quiet. Then one says, "We revise that. 9 minus 8is 1:'

Another child remarks that he had the same solution asthey did one sandwich.

"Frankie?" asks the teacher, after pausing for a momentto look over the students. Frankie announces, "I think we cangive each child more than one sandwich. Look!" and pro-ceeds to draw 21 rectangles on the chalkboard. "These arethe leftover sandwiches:' he explains. "I can cut 14 of them inhalf and that will give us 28 half-sandwiches, so everyonecan get another halE'

"1 agree with Frankie:' says another child.

"Each child can have 11 sandwiches:'

"Do you have any leftovers?" asks the teacher.

"There are still 7 sandwiches left over:' says Frankie.

"What do the rest of you think about that?" inquires theteacher.

Several children give explanations in support ofFrankie's solution.

"I think that does make sense says one girl,but I had another solution. I think the answer is1 + 1 +);:'

Students expect to have to justify their solutions, not just togive answers.

The teacher solicits other students' comments about the girls'solution, instead of labelling it right or wrong herself She expectsthe students, as members of a learning community, to decide if anidea mahes sense mathematically She respects and is interestedin their solutions, but this does not mean that "anything goes:'The ncla step is to verify the girls answer

Students respectfidly question one another's ideas. The girls"revise" their solution because they have been convinced bv theboy's point. There is no sense here that being wrong is shamefid.The teacher, listening closely, lets the students talk; she does notget into the middle of their interchange.

Listening to a variety of solutions. the students work together

to solve the problem.

Students expect to have to justify their ideas.

The teacher expects students to reason mathematically

Students seem willing to take risks by bringing up different

ideas.

"I don't understane the teacher says. "Could you show The teacher cxpects students to clarify and justify their

what you mean?" ideas.

How does a math teacher be-gin to think about the changes thatthe Council is urging upon her?How do we prepare new teachersto think about teaching math inways quite different from the onesthat they experienced for manyVears as students? Teachers in pro-fessional development schools aregrappling with these ideas, andtheir efforts may provide somehelp and inspiration for otherMichigan educators who want tobegin to make changes in theirways of thinking about "doingmathematics:'

ational Council ol Teachers of Mathe-matics (19891 Lurriculum and Evaluation`-tandards for School Mathematics. Reston

P NCTM.

National Council of Teachers of Mathe-matics (1989) Professional Standards forreaching Mathematics: Working Draft, Re-ston Va: NCTM (This working draft can beordered direcdv from the Council 11906Association Drr.e. Reston, VA 220911 at nocost until the final draft is printed. TheCurnculum and Evaluation Standards canbe ordered for $25.00 a copy)

A Community of YoungMathematicians

In 1980. when she began tohave doubts about her mathemat-ics teaching, Deborah Ball hadbeen teaching at Spartan VillageElementary School in East Lan-sing for five years. The crisis arosewhen she moved from third tofifth grade. She had always taughtmath pretty conventionally: "Ithought I should present it. and Ishould present it in ways that wereboth engaging to kids and concep-tual, and that the math just wasn'tthat difficult to understand:' Herstudents got right answers on theirmath papers. and so she assumedthat thc processes made sense tothem. But in fifth grade the mathwas more complicated, and al-though she felt she was explainingnew ideas with the same clarity,students did not get right answerswith the same regularity. Whenshe talked to them about theirmistakes they seemed less clear onwhat they were doing and why.Troubled, she began to talk to Per-ry Lanier and Bruce Mitchell, atthe University. They steered hertoward books and articles about

mathematics, about what waswrong with math teaching, andabout how one might do it better.She began to rethink what it meantto learn mathematics, and to ex-periment with different ways ofteaching. She also enrolled in col-lege math courses in algebra.calculus, and number theory.

From the courses she took andfrom the articles she read, Deb-orah learned more about the disci-pline of mathematics, and about

Studentspropose, ar-gue, concur,dispute, and

theorize.

alternate ways to think aboutteaching it. The ComprehensiveSchool Mathematics Program(CSMP), which she worked within her fifth grade, also helped. Shehad already started a master's pro-gram in reading at Michigan State's

4 4

College ol Education, and as herinterest in the math grew. shechanged her locus from reading tomathematics. In 1988 she com-pleted a Ph.D. and became an as-sistant professor in the MSU De-partment of Teacher Education.She has continued, however, toteach math at Spartan Village Ele-mentary School.

Math as a Conversation

In 1990 Deborah's third-grademathematics class is a communityof mathematicians. Students pro-pose, argue, concur, dispute, andtheorize. Deborah explains that itis her view of mathematics, andher ongoing observations of theseeight-year-olds, which shapeswhat she tries to do there. Shegives over much of the agenda tothe children's thoughts and com-ments, because to some extent shemodels her mathematics class-room on a vison of the commu-nity of adult mathematicians: shewants her students to see the en-terprise of mathematics as a con-versation, an ongoing communityinquiry. But Ball is a teacher asmuch as she is a student of mathe-matics, and, she says, "There's aback and forth between readingabout the work of mathematiciansand watching and listening toeight-year-olds and trying tofigure out what makes sense:' Shepoints to a couple of concreteways in which her classroom dif-fers from the communities of adultmathematicians: as a teacher sheis concerned about the participa-tion of all individuals she mustworry and take steps if some stu-dents do not understand themathematics others are discus-sing; she also tries to keep com-petitiveness out of her third grade,even though it certainly has aplace in the universities wheremathematicians do their work.

Deborah's work both em-bodies and inspires the hopes

for professional developmentschools. Every day she works withstudents, both in elementaryschool and in the university, try-ing to understand and extend theirthinking. Her work with eight-year-olds informs her research;her reflections on her own prac-tice, on the work of others, and onher academic discipline informher work with children. And shecontinues to ask questions.

Visiting her classroom on acold afternoon in late January, vis-itors from the University get achance to see Deborah and herstudents at work on one of hernewest interests: what can the ideaof mathematical proof look like ina community of eight-year-oldmathematicians?

Although math period has notyet begun, these third gradersclearly feel that they are in themiddle of something. As theytrickle back into the classroomfrom lunch, still laughing andtalking, they reach into their desksfor their math notebooks, openthem up. and inspect yesterday'swork all without prompting.

"Conjecture" Is an EverydayWord

"Are you guys ready?" asksDeborah, "I'd like to put up theconjectures from yesterday:' "Con-jecture" seems to be an everydayword for these eight-year-olds.T to the chalkboard are threep. ,es of orange construction pa-per, each stating a conjecture:

Lindiwe's conjecture : Nomatter how many even num-bers you add together, theanswer will always be even.

Betsy's conjecture: If youadd an odd number of oddnumbers, the sum will alwaysbe an odd number. If you usean even number of odd nurn-bers, you will get an evennumber.

"Sean Numbers":Eight-Year-OldsDeveloping NumberTheory

About a week and a half earlier,during a discussion of the propertiesof zero, a student named Sean hadproposed that 6 is "both even andodd" because when six objects aregrouped by twos, there is an oddnumber of groups. Neither his class-mates nor his teacher agreed. Ofalaargued that their working definitionof an even number said only thatwhen you grouped by twos, nothingwould be left over it said nothingabout the number of groups of two.Deborah asked Sean "why it wouldbe useful to have definitions suchthat some numbers would end upboth even and odd?" (Sean respond-ed that it wasn't necessarily useful,but that he was just thinking that itcould be.) And Mei pointed out withan edge of exasperation that if sixwas "both even and odd:' then sowas 10. Sean considered this argu-ment calmly and then agjeed, withgracious formality "I didn't think ofit that way. Thank you for bringingthat up I see 10 can be an odd andan even number.'

Reflecting on this discussion af-ter the end of the teaching day, Deb-orah wrote in her journal:

(I'm wondering if I shouldintroduce the idea that Seanhas identified (discovered?) anew category of numbersthose that have this property

Keith's conjecture : I f you

add an odd number to aneven number, the answer willalways be an odd number.

Sheena's Question:Would it work with sub-traction? Odd minus evenequals odd?

45

he has noted. Maybe theycould be named something.Or maybe this is silly willjust confuse kids since it'snonstandard knowledge i.e.,not part of the wider mathe-matical community's sharedknowledge. I have to thinkabout this.)

(It has the potential toenhance what kids are think-ing about "definition" and itsrole, nature, purpose in math-ematical activity and dis-course, which, after all, hasbeen the substantive point ofspending so much time onthis this week.) What should adefinition do? Why is itneeded?

For example, a definitionof even numbers that says it'severy other (whole) numberstarting with 0 (or 2) is prettyuseless for dealing with Lucy'sexample of 1,421. Even thegrouping by 2 definition (cor-ollary to Ofala's odd numbeTdefinition) is not too helpfulfor 1,421. But a definitioncentered on divisibility is (theyall relate to this, of course, to agreater or lesser degree) be-cause then you can show that1,421 is odd because 1,000 iseven, 400 is even, 20 is even,but 1 is odd. That leads to the

Continued on page 7.

(The variety of names on thepapers around the classroom sug-gest the diversity of the students:more that half were born outsidethe United States; many learnedEnglish in this elementaryschool.) "Numbers that end in ze-ro may always be even;' volunteers

one child. carefully emphasizingthe "may.' Deborah writes tins onthe chalkboard. Torv reports thatshe is trying to figure out whathappens when two -Sean num-bers.' are added together. ("Seannumbers- are even numbers thatresult from doubling an odd num-ber. Their history - they wereidentified by a member of thisclass two weeks ago - suggests abit about the way Deborah and herstudents approach mathematics.See "Sean Numbers: Eight-year-olds Developing Number Theory:in this issue) A third student re-ports that he is investigating to seewhether the sum of an odd num-ber and an even number is always

an odd number: a fourth is look-ing at the results of adding twoeven numbers.

As these students have pro-posed conjectures. classmateshave been volunteering to assisttheir investigations. Everyonenow seems to know what theywant to work on, but before theyadjourn to their groups Deborahasks the class, "What did we proveyesterday?"

"An odd plus an odd equals aneven.- replies Jeannie. Tembacomes to the board. writes

7+ 7

14

"Sean Numbers" continued from page 6.

proof of Lucy's conjecture that"you have to look at the lastnumberr (Many adults knowthis rule but do not know whyit works.)

A second aspect of defini-tier is that it facilitates dis-course. That was where westarted, because people weremeaning different things by"even numbers" and that wasgoing to make debates over thefour conjectures difficult -impossible.

Another, I think, is logicalpartitioning? Well, in a case likethis, maybe, but certainly notalways, or there wouldn't beintersecting sets in numbertheory (e.g., prime numbers,odd numbers). Thaes whereSean's discovery fits, maybe.

Deborah turned the matter overin her mind over the weekend. OnTuesday, she wrote:

I decided to let Sean's ideaof numbers that can be botheven and odd have legitimacyby pointing out that he hadinvented another kind of num-ber that we hadn't known beforeand suggesting that we call

them "Sean numbers:' Heseemed quite pleased, theothers interested. I said thatwhen mathematicians inventnew ideas or make new conjec-tures, sometimes their ideas arenamed after them and, after all,we had "Ofala's conjecture'"Nathan's conjecture etc.

Commenting on her decisionlater, Deborah observed:

You could have interpretedthis very differently - youcould have simply seen Seanas a kid who was confusedabout even and odd numbersand just looked for a way tocorrect him. This is a verydear case of my view of math-ematics shaping what I did.An observer who focused onchildren's feelings and self-esteem would probably havesaid, 'Here is a kid who ishaving trouble at school, andhere is a teacher who is find-ing a way to make him feelgood about himself: But I atleast as much saw it as anoccasion for [the class] to seesomething about how theseideas are invented by people.

4 6

and explains. pointing to thenumbers. "Seven is odd. 7 is odd,and is even:* Definitely an ex-ample, but not. he agrees. a proof.Betsy offers a proot: "If you groupan odd number by twos (she dem-onstrates by making seven lines.and circling three pairsl.

you have one left over. You canmake a pair with the two left overones."

"I agree that this is true, but itwasn't the way we did it Yesterday:Mei demurs. Yesterday, she asserts.they had defined an odd numberas "an even number minus one:'The two girls discuss whether thisdifference is important.

Mei argues that we cannot besure that this statement holds truefor really big numbers - "numbersin the thousands, or numbers thatwe can't say' because we haven'texamined them. Jeannie disagrees."All odd numbers have one leftover. And you'd die before youcounted every number. Evenmathematicians can't count themall!" Arms wave. The third gradersdebate whether it is possible toprove anything about numbersthat are too big to count.

Students Investigate

With no need for an organ-izational intermission, groupsconvene quickly at tables andchalkboards. Students make cal-culations in their notebooks andon the board: the few voices thatrise above the busy hum are dis-cussing math: "Yes, but does it al-ways work?" "But that's a Scannumber .. :' Two girls arc investi-gating operations with Sean num-bers by adding 22 and 22, making

1111111,

r`

41

S. -or

3.

Students in Deborah Ball's math class work together on a problem.

a diagram which shows them thenumber of pairs in 44. As Deborahnotes in her journal, they havesome trouble deciding whether 44is or is not a Sean number. Whenthey finally agree that it is not,they conjecture that a Sean num-ber plus a Sean number wrII al-ways equal an even number that isnot a Sean number.

Nearby a boy is drawing lineson the black board with coloredchalk. A girl asks with indignant

incredulity, "Are you playing?"Deborah moves from group togroup, finding out exactly whatevery one is doing, and howthey are thinking about the prob-lem they have set themselves. Aboy explains to her, "We only wentup to 12,000. That doesn't proveit: what about all the othernumbers?"

After 20 minutes of groupwork, Deborah brings the classback together and asks for a report

4 7

on the conjecture that "an oddplus an even equals an odd:' Markexplains that, "At first, we weregetting lots of answers, but weweren't getting proor Then, hecontinues, Betsy joined them,"and she had prooC' When Deb-orah invites him to show one ofhis examples, Mark defers to Na-than, whose examples are "better:This turns out to mean that theyinvolve larger numbers.

After consulting his notebook,Nathan writes on the board

100000000000001+ 98124

98125

explaining as he does so that "Bet-sy said that when you add a shortnumber to a long one, you have tomatch up the last numbers, andthe next to last:' He adds that sinceLucy says that you only have tolook at the last number to tellwhether it is even or odd. he con-cludes that this number is odd.

Ignoring the gasp from theback of the room which greetedNathan's addition, Deborah letsBetsy explain her "proof' that anodd number plus and even num-ber will always equal an oddnumber.

Later that afternoon, Deborahreflects in her teaching journal onher decision not to correct theaddition.

I did not comment, eventhough it was obviouslywrong. At this particularpoint, we were concentratingon the issue of whether or notodd plus even equals odd,and, within that, the functionof examples and other kinds ofargument in proving thatsomething was always true. Inaddition, getting them tounderstand what it means (oreven what to do) to add thesenumbers is not a trivial matter. . Nathan had followed arule - that Betsy had said that

when you add a big numberto a smaller number, you haveto add the ones first (or addthe ones together?) - and nperspective was, what goodwould it do to say anotherrule? . .

In tact there is little dangerthat Nathan's error will go unno-ticed. Just before the class ends.Jeannie raises her hand to disagreewith the addition, and Betsymoves to correct the computation.

Accompanying the visitors tothe car, Deborah remarks that"eight-year-olds really get hookedon number theory They love it. it'shard to get them off it, to move onto other things: She explains thatthe unit on odd and even numbersbegan a few weeks earlier, out ofquestions that arose from the fol-lowing problem: "If pretzels cost 7cents and gum costs 2 cents, whatcan Mark buy if he wants to spendexactly 30 cents?" (See "Mean-while at Elliott . . :' in this issue)

\Vorking on this problem, stu-dents saw some patterns involvingeven and odd numbers Iheirquestions have led them in a vari-ety of directions.

As eight-Year-olds discussideas about numbers, measure-ment, shapes, and probability.Deborah studies their learning andher own teaching and carries herthoughts and observations toundergraduate and graduate stu-dents at MSU, to 'her colleagues atSpartan Village Elementary andother professional developmentschools. and, through articles inprofessional journals, to mathe-maticians and teacher educators atother universities. Some studentsand colleagues visit her classroom.Others watch videotapes ol herteaching and discuss them incourses or study groups. Used indifferent ways by different people.Deborah's classroom and her vid-eotapes catalyze important think-ing about teaching, learning, andmathematics. 3

Sharing a ClassroomWhen teachers join the Profes-

sional Development School effort,they open the door of their class-rooms to outsiders - people fromthe University and the world be-yond. What does this feel like? Whatsacrifices must they make? Do thedemands and intrusions overwhelmany benefits they and their studentsmay get from the association?

Sylvia Rundquist began her col-laboration with Deborah Ball twoyears ago, when Deborah asked Jes-sie Fry, the principal of Spartan Vil-lage Elementary, to help her makearrangements to teach mathematicsdaily in someone else's classroom.At that time Sylvia had been teach-ing elementary school for two years.She did not feel very confident about

her math teaching and the opportu-nity to learn from watching some-one else attracted her. Shevolunteered.

Reflecting on the arrangementthis spring, Sylvia reports that shehas loved observing her studentsregularly in a situation where shedid not have to teach: through thedual perspective of teacher and ob-server she has come to know thechildren better than she did whenshe was their only teacher. And shenotices that at parent conferencesshe has more insightful things to sayabout them.

Just as she hoped, she has alsolearned a lot about the teaching ofmathematics. Her earlier assump-tions about math teaching, she re-

Continued on page 10.

48

The Viewfrom theUniversity

When college teachers try tointroduce new ideas about mathe-matics to the young adults in theirclassrooms, they get a variety ofreactions. Those reactions reflectthe fact that for the past 13 yearsthese students have been con-structing ideas about what teach-ing does and doesn't mean as theywatched their own teachers assignworkbook pages, write on theblackboard, guide discussions.and collect milk munev Becausemany of these college studentswill be the teachers of tomorrow, itis interesting to listen in as theywrestle with their mixed feelings.

Margery Osborne, a doctoralstudent at Michigan State Univer-sity's College of Education. workson a research project which in-volves her in filming DeborahBall's third grade math class sev-eral times a month. She also teach-es a section of "Explonng Teach-ing' CIE- WI). a course designedto help sophomores to examinetheir assumptions about teaching,learning, and schools. In late Janu-ary. Margery's TE-101 studentswatched a video ot one of theclasses in which Deborah and herclass investigated the behavior ofeven and odd numbers and theproperties of zero. Margery thenbroke her class into groups tothink about the similarities anddifferences between Deborah'steaching and that recommendedby Good. Grouws, and Ebmeier inActi'x Mathematics Ti aching. To-

day the groups are reporting theirideas back to the rest of the class.

Similarities and Differences

The reporter for the first grouprelates what she sees Deborah do-ing to the teaching of Vivian Pa-ley', whose book about her kin-dergarten class students have readearlier in the term. Most TE-101students like Paley's informal.conversational kindergarten, feel-ing that five-year-olds will learnsocial skills and gain self-esteemfrom exploring their ideas with ateacher who seems never to con-tradict them. Most doubt, how-ever, that you could learn the in-formation and academic skills thatolder children need without muchmore adult guidance. "There weretimes when we felt they went offthe track and hit a blind spot. \Vefelt she could give them more di-rection, lead them to something,because they were just spending alot of time discussing zero:'

She turns then, with apparentrelief, to the more familiar ap-proach of Good and his col-leagues, who direct teachers toteach algorithms directly andbriskly, assigning independentseatwork after 10 minutes of ex-planation and 10 minutes of oralquestioning or "controlled prac-tice": "Whereas in the book theyhad a step-by-step procedure inteaching math: the teacher ex-plains and demonstrates, andgives them work and checks it."

Teaching Unsettling

The routines of Good,Grouws, and Ebmeier recall thethird grades where these sopho-mores studied arithmetic 11 yearsago. Deborah's teaching strikes adiscordant note and raises someunsettling feelings. Janine fum-blingly defends her own teachersagainst the unspoken criticisms ofthis new approach.

'Paley, V (1981) Wally's Stones. Cam-bridge: Harvard University Press.

IC5

Talking about math re-minds me that in elementaryschool, and in high school,and even here at State lastyear, I had math teachers whohave said, "You're not going tounderstand this, you don'tknow why this happens, I justwant you to do it:' And italways bothered me, but Ilook at this now and I want tosay "I love what Ball is doing.The kids are disagreeing andthey're talking!" But I didlearn a lot of math growingup, and I didn't understand it.

But I did it. And I know howto do it now. Maybe I don'tunderstand it. but I'm kind ofeducated because I do knowthat math. And I wouldn'thave understood the princi-ples. I still don't.

Janine pauses and Margerytries to help her clarify her point."So what you are saying is, whyunderstand math?"

Janine nods, "I see what she isdoing and I think that's great, but Ijust don't see how . . . The lastsentence dangles as she tries to

Sharing a Classroom continued from page 9.

calls, mirrored those of her own ele-mentary school teachers: theyseemed to believe - incorrectly, shenow thinks - that "the more calcula-tions I did, the more I would come tounderstand." As she discusses theways her ideas have evolved throughthe last two years, she jots down fiveother things she has learned:

- how much discussion therecan be about math

- finding answers in differentways is OK - [children]don't all have to get an-swers in the same way

- that math is not all rightand wrong - some issuesare not resolved

- a great deal of day-to-daymath involves estimating

- using concrete objects andmanipulatives to find solu-tions is OK beyond lower[elementary school grades]

Like outsiders who observeDeborah's math classes, Sylvia is of-ten amazed at the problems her stu-dents can solve, and the level of theirthinking. As she watches her thirdgraders attack difficult problemswith energy and sophistication, sheraises the expectations she has forthem in other areas. And she usesthe vocabulary of the math class -estimating, predicting, making hy-potheses - to extend her students'thinking across the curriculum.

49

But there are costs as well asbenefits to her collaboration withDeborah, Sylvia admits. She and herstudents are locked into havingmath at a particular time each day,so that even if they become en-grossed in another project they mustturn their attention towards mathat five minutes before one everyTuesday, Wednesday, Thursday, andFriday.

And there are some benefitswhich, the listener guesses, probablydid not come without pain: Sylviasays that through her associationwith Deborah she has learned to bemore flexible. "I was very rigid aboutmy teaching:' She felt possessiveabout her students and her class-room. Now Deborah's name is nextto Sylvia's on the classroom door,and they do parent conferences to-gether and cosign all communica-tions that go home to parents. Be-cause she shares her classroom withDeborah (and Deborah's entourage -- often a film crew as well as visitorsfrom other schools and universities)she has to endure intrusions and re-arrange her own schedule frequent-ly. She is managing to do all this, andeven to feel increasingly relaxedabout the changes. She is also learn-ing, she says, that she can speak upand set some limits: After a bit of si-lent suffering she decided that itreally would be all right to ask thevisitors not to sit on her desk.

make sense of her thoughts. mem-ories, and feelings. .Maybe with,-;omething else, but with math?"Fier voice trails off uncertainly.

A classmate dissents:

The problem with that isthat ... once you stop usingwhat you were just told to do.you forget it ... I mean, youdon't really learn it. And peo-ple say. 'Well. I'll never usemath. But there are a lot ofconcepts that, if we hadlearned them so we under-stood them, maybe we woulduse them.

To most of these college soph-omores. math is a long list of pro-cedures and algorithms. Theywonder if it is possible to spend anhour or more struggling with oneproblem and still "cover" the cur-riculum. But even if it were, thespokesman for the next group re-ports. many would prefer to seethe teacher run the discussionwith a firmer hand:

[Good. Grouws. and Eb-meierl suggested using thefirst 4-5 minutes for review.Then you could let them dis-cuss for 20 minutes. but notgo on the whole hour. After awhile it seemed like it wasjust the same four or fivestudents . . Plus, she nevertold students they were right.or gave them any encourage-ment and I think that thatkind of leaves students hang-ing. I mean. I think youshould tell kids the right an-swer after a certain point.

"What is the right answer?"asks Margery. Students' concernabout the length and form of thethird grade discussion has crowd-ed out the mathematical contentso completely that a puzzled si-lence falls. "About zero:' Margeryreminds them. No one seems to besure that they really know, but nei-

ther do they pause to discuss thequestion or to wonder at the in-terest that these eight-year-oldsdisplay in an abstract point ofnumber theory. "Nothing: callsout one. "It's not a number:

think thatwhen you

are listeningto other

people pre-sent ideas,

whether theyare right or

,mrong, you3re !ea-minasomething:

But as some students identifythe ways in which Deborah'steaching troubles them, others arebeginning to look at their own dis-satisfactions with her teachingmore skeptically to questiontheir own questions. Judithvolunteers:

I think a lot of our prob-lem is that a lot of us. proba-bly all of us. we've beentaught in the traditional way:"Here is what this is. Here'show we are going to do it. Goahead and do the problem:'And it's hard for us to try tounderstand: looking at thisother form of teaching, wethink, "How can kids do this?It's so hard for them:' Butmaybe if they do it all along,maybe it will be better forthem.

This student is worried, how-ever, about confusions generatedby different teachers' expecta-tions: "What if they go to fourth

°arade

and the teacher doesn't dothis? And the kids are challenging

50

and saving .1 disagree: .1 agree'?And:' she laughs. "the teacher tellsthem to shut up. You have to haveconsistency:.

Alicia voices a different con-cern: she doubts that mathematicsis logical enough to support thescrutiny that these eight-year-oldsare giving it.

They were learning thetheory behind all this math,but I mean, there are so manyexceptions anyway that at firstthey'll think, 'Oh. great: butonce she adds in. 'Oh. thisisn't right: this is an excep-tion: ultimately they won'twant to know everythingbehind it.

Re-examining Beliefs

In applauding the way thatDeborah's students work togetheron a problem over a considerableperiod of time, a young man as-serts. "The teacher can't just tellthem the answer. I think it is goodthe way that they try to come upwith it by themselves:'

A groupmate disagrees:

I don't think that's right. Alot of times the teacher justhas to give you the answerand you have to accept it.There are lots of things in lifeyou just accept. At least, I'vebeen taught that way.. .. And.I mean, discussion with allthe little friends, to me itseems useless. Because thereare kids who never talk. Iwould never talk. whether Iunderstood it or whether Ididn't.

After a thoughtful silence, an-other student points out that she isassuming that those who do notjoin the discussion learn little:

There's a problem insaying that if you aren't saying

anything, you aren't gettinganything out of the discus-sion. That would mean thatpeople in here who haven'tsaid anything this period, thatthe whole hour has been awaste for them, and theyhaven't learned anything. But Idon't think that's true. I thinkthat when you are listening toother people present ideas.whether they are right orwrong ... you are learningsomething A lot of timesthe people who don't speaklearn the most, because theyhave the observer's point ofview, and they get to see thewhole of the discussion. Likeright now I'm talking, so I'm

not listening in the same wayfiguring out what makessense.

As the class continues to reactto what they have liked and dis-liked, students raise questions andchallenge one anothers' assump-tions. Do you learn more from ateacher who comes on strong anddisplays more "personality"? Doyou learn for the "wrong reasons"with such a teacher? Will Deb-orah's students, even if they do notoutscore others on standardizedtests in third grade'. carry withthem, into high school and col-

'In fact. Deborah's students do exceedinglywell on these measures.

Meanwhile, at Elliott...Like the sophomores in

Margery Osborne's exploringteaching class, the five classroomteachers in the Math Study Group(MSG) at Elliott ElementarySchool in Holt remember themath classes of their childhoodvery well. Unlike many of the col-lege students, they are sure thatthey want something different fortheir own students. All have beenteaching for at least four .ears:none are satisfied by the way inwhich they have been workingwith their students on mathemat-ics. Several joined the PDS effortspecifically in order to look forways to move beyond what theyknew about math and mathteaching.

In an effort to deepen under-standing of the teaching andlearning of math, the group(which includes, in addition to thefive classroom teachers, two "in-terns" who graduated from collegeonly last year, one graduate stu-dent and two professors fromMSU) has read articles, inter-

viewed students, shared teachingjournals, worked together on mathproblems. discussed personalmath anxieties, and analyzed cur-ricula. Shortly before Christmas,the group watched a video of Deb-orah Ball teaching her class atSpartan Village about negativenumbers. Impressed by the thirdgraders' command of a languagefor discussing these abstract ideas(in the interviews, teachers hadfelt that their students seemed tolack a language for expressingtheir mathematical understand-ings), and curious about the strate-gies Deborah used to initiate st-u-dents into this sort of discourse,the teachers welcomed Deborah'soffer to come to Elliott and teach alesson. Karen Dalton volunteeredher third-grade classroom and thegroup arranged to videotape thelesson, watch the tape together,and discuss it with Deborah.

The film they view on Febru-ary 12 shows Deborah giving theElliott third graders a problem shehas investigated with her class at

511

lege, the energetic enthusiasm formathematics that they displaythis class?

Talking about their responsesto Deborah Ball's teaching, thesestudents lay on the table somefundamental beliefs about teach-ing, learning, -id mathematics. Inthe next few \yeeks they will get achance to visit Deborah's class, toask her questions about her teach-ing, and to re-examine, both indiscussion and writing, the no-tions that have surfaced today.Some ideas will change; otherswill not. But nearly everyone willleave their introductory educationclass seeing some new possibili-ties and new complexities inteaching and learning. 3

Spartan Village three weeks earli-er: "If pretzels cost 7cents each,and sticks of gum cost 2 centseach, what are the possible pur-chases a child who wanted tospend exactly 30 cents couldmake?" In Deborah's class, thisproblem has led to several weeksof interesting work on odd andeven numbers including that de-scribed in "A Community ofYoung Mathematicians" and "SeanNumbers" (in this issue of Chang-ing Minds). After some discussion,the eight-year-olds break intogroups and start experimentingwith number combinations, talk-ing animatedly about prices ofgum and pretzels.

The voices are those of trea-sure hunters. As a boy in a bluesweater counts by rwo's "... 18, 20,22 . . .;' other children hiss excit-edly, "I got one;' "Three pretzelswon't work," "Try two." Karenhelps a group which has gottenconfused enroute to a workablecombination: "You had 2 pretzelsand 7 pieces of gum and it was 28

.4ffAP'

'4

II del IN

t...741&

1

Thy Elliott Muth Study Group at work.

cents. 1 iow would you make it 2cents higher?" . . . "Would that doit? Try it" Deborah crouches nextto a pair who are trying to decidewhether they can buy 7 pretzels.

After 15 minutes the class re-convenes to discuss their findings:a girl who proposes 4 pretzels andone piece of gum explains, withthe help of several friends, howshe figured out. by repeated addi-tion, that 4 pretzels would cost 28cents. Another youngster proposesthat 15 pieces of gum will cost ex-actly 30 cents.

"Why: asks Deborah."Because: volunteers a class-

mate, "15 plus15 equals 30.""But I don't see what that has

to do with gum: probes Deborah."I thought gum cost 2 cents?"

Another little bo- explainswhy 15 plus15 is the sante as 15groups of 2.

Not all answers are on themark: someone has forgotten thatgum cost 2 cents and proposesthat the purchaser can get 23 piec-es of gum and one pretzel. But thiserror leads eventually to the inter-esting conclusion that it is impos-sible to buy one pretzel and comeout even. As their 40-minute mathperiod ends, Deborah is asking

4

students to think about whetherthey have found all the possiblepurchases.

A Question of Equity

As the screen on the monitorgoes dark, the Elliott teachers turnfirst to questions of equity: howdoes a teacher meet the needs of adiverse population of students?After some discussion aboutwhether a chart might help someof the slower kids to ce the pat-terns involved, a teacher puts thequestion more generally: "Thatwas one of my questions, watch-ing the variety of kids in there:How do you keep the kids whohave already got all your solutions... challenged, and still going; andthen how do you get those slowkids to come along?"

In responding Deborah turnsfirst to the lesson they have justwatched, and then to the moregeneral curricular strategies thatshe uses to address diversity.

Well, it didn't happen inher class that anyone got allthe solutions. [Two boys]thought they had found them

52

all, and I asked them if theywere sure and they said theywere, but in fact they hadn't.That was one of the ways Iused the group discussion -to help them see that theydidn't get all the solutions, sothat next time they get such aproblem they'll try to push alittle harder, to prove to them-selves that they arc done.

I guess part of it, to me,lies in trying to figure outproblems that arc rich enoughthat it's not so likely that somepeople are going to be totallydone and other people aregoing to be totally stuck - thatthere are just different thingsthat you could think about ordo with the problem. Andthen, I usually try to havethought through differentkinds of extensions that kindof relate to the problem.

Part of my goal would bethat by this time of year, inmy own classroom. I wouldwant kids to have questionsthat the data raised for them.So partly I would be trying toget them to a place where theystart raising questions forthemselves, saying "Hmmm,

I'm noticing something hereor "Hmmm. I see a pattern:'

And at this point, in myown classroom, the:e's somesign that they do do that.Whereas earlier in the yearI'm showing them that that'sone thing that you do, whenyou think you are done. Yousay, "Am I sure this makessense? Am I sure I'm done?And is there anything inter-esting going on here that goesbeyond this problem?" I try toget them used to the idea thatthat's part of what you dowhen you do mathematics.Traditionally when a teachergives you a problem, you doit, and then you are done andyou just wait until she tellsyou what to do next.

A few minutes later a teacherwho taught some of these studentswhen they were younger makes anobservation which goes to theheart of the equity question: "Thekids who were contributing themost were not the math stars?'"No," the third grade teacheragrees, "those kids held back.They were afraid to take risks witha kind of math they weren't usedto:' She had noticed one of herbest students whispering ideas tothe boy in front of him, and wait-ing to see how Deborah would re-ceive his second-hand sugges-tions. "This is really important,"emphasizes the first teacher. Wehave a lot of needy kids. The tapeis evidence that this teachingwould work for them:'

Members of the Study Groupcomment that students respondpositively because Deborah ac-cepts all comments and sugges-tions. "But it's a particular versionof accepting everything': Deborahreminds them, "because it's ac-cepting everything but also ex-pecting that everything has to bebacked up:' It is, she explains, dif-ficult to respond to all commentswith the same neutral interest.

I think the hardest thingfor me has been to learn to actneutral when kids are right.Most of us are pretty inclinedto be at least nice to kids whenthey are wrong, so it's notsuch a big leap to learn that if

"I guess partof it, to me,

lies in trying tofigure out N:

problems thatare rich

enough thatit's not solikely that

some peopleare going to

be totallydone and

other peopleare going to

be totallystuck."

something's wrong you aregoing to say -why?" andsound kind of neutral about it.But what's harder is when it'sright, because we are so muchin the habit of saying "goodjob:' But it's so key to do itboth ways. If you only ask"why?" when they are wrong,then you have just gone to anew system of cuing them.

How to Justify

Like the college sophomores.the group wonders how Deborahjustifies to parents, colleagues, andadministrators spending so muchtime on a few problems. She ex-plains that she shows people whoask this question the district ob-

jectives in mathematics for thegrade she is teaching. "And most oithe problems that my kids are do-ing address a surprisingly largerange of these objectives? The uniton even and odd numbers that shehas just finished, for example, in-cluded work on more than half ofEast Lansing's conceptual andcomputational objectives forthird-grade math.

Instead of seeing the curricu-lum as "little bites that come oneright after another?' Deborah usesthe model of a sandwich. With agood problem "you are workingon a bite that is kind of deep andhas a lot of layers you want to getat:' And although her students donot use flashcards or practice add-ing or subtracting in isolation,they do extremely well on teststhat require knowledge of numberfacts because they do many com-putations as they probe a problemor conjecture.

New Vocabularies:Conjecture and Revision

Everyone wonders how Deb-orah helps her students make thetransition from a traditional math-ematics classroom to one in whichall answers are considered anddiscussed. Deborah explains thatshe introduces new words like"conjecture" and "revision- as theneed arises. The new vocabulary,she thinks, helps to build a mathe-matical community; it also drawsattention to activities that she val-ues like proposing "conjectures"and "revising" answers in light ofpoints made by others. In addi-tion, she coaches students in be-havior that contributes to goodconversation. When they come tothe board, for example. she tellsthem to stand to the side of theirdiagram and face the class as theyexplain their reasoning. She setsup explicit rules about respondingto the comments of others: no oneis to interrupt, because students

need to think as they talk, and tobe able to revise ideas as they pro-pose them: when your classmatefinishes what he has to say youmay (1) ask for clarification orrepetition: (2) agree or disagree.giving reasons: or ( 3) explain howyou got the same answer by adifferent process.

New classes seem to pick allthis up easily in the fall. but stu-dents who enter later in the yearneed more explicit guidance. Shetells of a newcomer who laughedat someone whose English waslimited. Upset. she stoppea theclass to discuss the incident. "Whydid I get upset?" she asked. "Be-cause: one student answered. -ifomeonc gets laughed at. then he

won't want to contribute to thediscussion: "And why would thatmatter?" asked Deborah. "BecauseII someone is too nervous to go tothe board:. explained the new-comer, "then he won't learn any-thing and he won't be able to showthe teacher what he knows: Look-ing puzzled. another youngsterraised his hand. "I don't think of it

that way. I think if someone isafraid to go to the board, we alllose out. Because that might be theonly person who has an idea thatwe ail need in order to solve theproblem: The eight-year-old hadsummarized elegantly Deborah'sgoals as a math teacher: a class-room in which a community ofyoung mathematicians take oneanother's ideas seriously andwork together to construct newunderstandings.

At three o'clock Deborah ex-cuses herself apologetically, andheads for another meeting. As thedoor closes behind her, a groupmember sighs, "I just feel so inade-quate. in terms of my subject mat-ter knowledge: Another nodsagreement: "I just can't imaginebeing able to be this thoughtfulevery day. Maybe one day aweek ..

As experienced teachers whoNere already questioning theirown teaching and asking hardquestions about their students'understanding of mathematics,the Elliott Study Group came to

Deborah's lesson with differentconcerns than Margery Osborne'sTE-101 students. They believedthat what Deborah was doing wasdifficult and, as someone had saidin the December meeting, "Not thekind of thing you can be -trained'to do in a one or two day mser-vice: They were certain that thissort of teaching required a lot ofmathematical knowledge and adifferent and more demandingsort of planning. They seem to un-derstand immediately what Deb-orah means when she talks aboutachieving many objectives withthe same problem. Because theyknow their students, they can seehow Deborah's teaching reallydoes address questions of equity.They sense where the discussionis going, and why Deborah needsto respond to students' contribu-tions with low-key, neutral ac-ceptance. They bring to the videoan appreciation of children andcurriculum that the TE-101 stu-dents are just beginning to go outin search of.

"This Teaching is So Hard":A Teacher Learns on Her Feet

As interest in teaching for un-derstanding has grown over thclast few years. teacher educatorshave startcd to ask what skills, re-sources, and knowledge experi-enced teachers would need tolearn in order to teach mathemat-ics in a way that has students rea-soning rather than memorizingrules. Like Deborah Ball, MaggieLampert has been developing thiskind of math teaching at SpartanVillage School. During her firstYear of graduate studs' at MSU,

Ruth Heaton, who had taught ele-mentary school for nine years buthad no special background inmathematics, became interestedin the work that Maggie and Deb-orah were doing. In the winter of1989. she and Maggie began to talkabout the possibility of Ruth's tak-ing over the math teaching in an-other class at Spartan Village sothat Ruth could learn more aboutteaching for understanding andshe and Maggie together couldstudy what would be involved in

Ruth's changing her practice.Since Ruth aims to become ateacher educator herself, shehoped to learn valuable lessonsfrom this experience.

In September Ruth began toteach math in the fourth-gradeclassroom of Jackie Frese. EveryWednesday and Thursday untilChristmas, Maggie watched Ruth'sclass, made notes in a journal shekept on Ruth's teaching, andtalked to Ruth about what she hadseen: Jim Reineke, a graduate stu-

dent at the College of Education,studied her learning and filmedher class on Monday andWednesday.'

Clearly, Ruth doesn't havemuch opportunity to make mis-takes in private. On September 27,after she had been working on thisteaching less than three weeks. shewrote in her teaching journal:

This teaching is so hard.Maybe [another professor atthe College] was right. Itwould have been better tohave some time to messaround on my own beforehaving people observe. As hesaid, you need some time to

Not havingrealized that

"wholenumbers"

would puzzlethe children,she had not

thought abouthow she

would explainthe term.

make an ass of yourseLf. Well.I have. It really is hard onething that is hard is havingthings pointed out to me that Ialready recognize. It's alsodifficult to have things point-ed out that I don't necessarilyrecognize. I am taking enor-mous risks.

Tickle Frese took over the math teachingduring winter quarter while Ruth taught atMSU. In late March Ruth returned to Spar-tan Village where she will continue toteach fourth grade math for the remainderof the year.

16

What Was So Hard?

Most of the time, however,Ruth talks cheerfully about herstruggles - for example, she sharedthem with the faculty aE SpartanVillage in November. And in Feb-ruary she volunteered to watch avideo of the September 27 lessonand talk about what was "so hard"as she first began to change hermath teaching.

As the TV screen lights up, wesee the math problem printed onthe chalkboard:

-What whole numberscould be put in

the boxes?"

26

A small hand shoots up:-What is a whole number?"

"Like one, or two. Not ex-plains a classmate.

"Not 1.73," adds another. Quite afew children volunteer examplesof whole numbers:

"A million.""Zero:."That's not a number:' a girl

objects."What is it?" Ruth asks."It is nothing," responds the

10-year old. After a brief but spir-ited discussion of zero (these stu-dents had Deborah Ball for mathlast year), students return to nam-ing large whole numbers.

"10E'"60,000:'"Is negative 13 a whole num-

ber?" someone asks."Yes: answers Ruth. She calls

on another child with a raisedhand.

"I don't understand:'Ruth stops the VCR and turns

to me, laughing: "This sort of thinghappened a lot in the beginning. Iwouldn't anticipate problems withwords." Not having realized that-whole numbers" would puzzlethe children, she had not thoughtabout how she would explain the

term, or help the children to in-vestigate its meaning. As Ruthswitches the tape back on we seeher gesturing toward the line ofnumbers running across the top ofthe chalkboard, saying "Wholenumbers are the ones that are onthe number line:' Mathematically,this is not precisely correct, asMaggie, who was observing thatday, explained in a journal shewrites on the lessons sheobserves.

[T]he numerals writtenbelow the red dots on thenumber line are what arebeing referred to as "wholenumbers" i.e. 64 is certainlyon the number line althoughthe numeral "64" is not writ-ten on the chart that repre-sents the "number line:' Acomplicated and abstractpoint which reminds us ofhow much mathematics isinvolved even in the problemsthat elementary schoolersundertake.

Issues Identified

Maggie uses the journal tohelp Ruth identify mathematicaland pedagogical issues:

It's hard to always knowhow to handle these ambigui-ties, even when you are awareof them, but the issue of howto use the number line inmath teaching is a compli-cated one. Within mathematicsthe importance of the numberline is that it represents conti-nuity. . . . that is, it representsthe idea that there are alwaysmore numbers in between theother numbers.

By now the children haveopened their math notebooks andbegun generating number pairs forthe open sentence on the board.The screen shows Ruth crouching

next to a child who has just re-cently joined the class and speaksno English. Her head on a levelwith his. she gives him an exam-ple of two numbers which, wheninserted in the boxes, will com-plete the equation. Probably this isthe first time that this little boy hasencountered a problem of thissort. Nearby, another child ex-plains whole numbers to the littlegirl on her left: "All the numbersthat aren't like 13',

Ruth talks quietly with ayoungster who is trying numberswith nine or more zeros, and thenleaves him with a manageablechallenge: "Raise your hand whenyou have five reasonable ones."Another boy has generated quite afew appropriate pairs and tellsRuth that they did this last year.She gives him a somewhat moredifficult problem to work on:26 - < 10. He does not,however, pick up on this chal-lenge, and Maggie, in her journalnote, helps Ruth to think aboutwhat may be going on inside of hishead.

(1-1lis behavior makes methink of how much contentand social/emotional issuesare intertwined.

1. It seems to have been hardfor him to cope with hav-ing been singled out at histable. Right after youwalked away, the boyacross from him (who hadbeen listening) joked a bitwith him and then they alltook up fooling aroundwith rulers.

2. It may have been that hefelt "I've done my job here,why should I be 'rewarded'by being asked to do extrathings, when these otherguys at my table haven'tgot nearly as many possi-bilities down in their note-books as I have in mine.

These two issues arerelated to the problem ofmultiple levels of ability andtrying to develop a classroomculture that supports every-one working at his/her ownlevel. In certain social groups(maybe the one [this studentlis in) it is probably not verypopular to be really good at aschool thing, like math: not toshow up your buddies.

A little girl consults with Ruthand then carries her math bookover to the number line postedabove the blackboard. After a fewmore minutes, Ruth calls thegroup together, congratulatingthem on the way in which theyhave worked on the problem,while noting that a few did notstick to math and "that's not okay."Then, before we can blink, teacherand students are off on a second,quite different, problem: they areconstructing an "arrow road:'

time, seemed reasonable L., her toproceed in this way, she realizesthat this was part of her smuggle tofigure out how to use the mathtextbook in her teaching: the booklinks the two problems together inone lesson. She laughs as shepoints out that she had ploughedahead without ever finding outwhether the students have evenunderstood what they have donebecause that's the way the text-book had laid out the lesson.Showing me the teacher's edition,which provides a complete scriptfor teachers' questions and stu-dents' answers, she observes witha chuckle, "When I ask 'why?' itthrows everything off'

As Ruth and I turn back to thescreen, we see the arrow road di-recting students to figure out whatoperation they will need to per-form to get from 65 to 36. A littlegirl comes to the board, but as sheexplains her reasoning, she forgetsto "borrow" in the tens column

-29 +36 +28Wassorm÷masrerasmom+rwassmosesswisms±0

65 36 72 100

Ar-row Road. Ruth put the part in black on the chalkboard. Students added Ow numbers in pink t<

show the operations which would get them from one number to the next

Struggle with the Text

Ruth turns from the screen,shaking her head in disbelief.Why, she wonders aloud, had shemarched onto a new problemwithout even discussing the an-swers the students had generatedas they worked independently?Now, she says, she would spendthe whole math period exploringwhat they had found and thesense they had made of their an-swers. She explains that as shepuzzles about why it had, at that

56

and a classmate gasps audibly. "Ifyou are going to disagree: Ruthurges. "just raise your hands qui-edy: A boy corrects another stu-dent's slip of the tongue, and Ruthcommems to me.

There's a lot of social stuffgoing on here, a lot of gettingreally picky, and they're notpicking at the mathematics,they are picking at each other.It was really hard to knowwhat to do about that. ...There's a fine line. You want

717

them to challenge each other,but what is it that they arechallenging?

A boy has volunteered that inorder to get from 36 to 72, hewould add 36. Ruth asks him howhe has arrived at this answer, butgets no substantial response.

"Did you just know it in yourhead?"

"Yes:'In this episode, Ruth com-

ments, she has entirely failed toprobe his thinking somethingshe would surely have done laterin the term. "I think some of thatreflects my own uncertainty atthat point, because I look at thisnow and I think, 'Why didn't youpush him?"

The class presses on, consid-ering how you decide whether toadd or subtract in order to get fromone number to another. One stu-dent explains that she had sub-tracted because if she added she"would get the wrong answer."Others describe their thinkingmore lucidly One little girl gets 38when she subtracts 72 from 100;when a classmate explains wherehe thinks she has gone wrong, sheasks some questions, ponders hisanswers for 30 long seconds, thenannounces with a satisfied nod, "Iagree?'

As the class adjourns, Ruthshares with her students her puz-zled concern about the nature ofthe challenges she is hearing.

I have to think moreabout this, but some of thechallenging that was going onhere today I thought was kindof picky.. .. More like youwere challenging each other.And that's not what thesediscussions are about. It's notyour chance to dig at oneinother, it's an opportunity toWink about mathematics.

Her tone is genuine, thought-ful, and puzzled.

Ideas Change Quickly

It has been hard. Ruth ex-plains as she rewinds the tape. butshe knew at the outset that shewould not start out an expert."The struggle was, in some ways,the point." The changes in herteaching have been large and im-portant, she reports. "It was ashard at the end of the term as itwas in the beginning, but differentthings were hard:' By Decembershe was worrying most about whatsort of math problems she wouldgive them. "And the math. Thatwas hard all the way through, butprobably it got harder as I recog-nized more what I didn't know"

Looking at tapes of her teach-ing, and reading old journal en-tries, was often painful, becauseher ideas were changing soquickly.

Doing this lesson now. Iwould have much more dis-cussion. It seems like I was socontrolled: I mean I'd put onelittle thing on the board andtalk about it. and then anotherlittle thing. And althoughthere were some places herewhere I pushed the kids totalk more, now I would dothis much more. But at thetime, that was really adven-turesome... . One thing thatstrikes me when I watch itnow is how serious and ner-vous I looked in the beginningof the class, walking around.That was all this stuff, likeabout whole numbers: Iwould think, 'What are theygoing to come up with next?'And so, when I would goaround and talk to people. Iwas really fearful of theirquestions.

Ruth's apprenticeship hasgiven everyone who has watchedher new insights about what ishard for a teacher who tries tochange her way of teaching math.

57,

Together she and Maggie arclearning more about how to orga-nize apprenticeships tor teachersand graduate students like herself.They are considering how a teach-er might begin to address limita-tions in her understanding ot asubject she is trying to teach.Equally important, Ruth hasshown colleagues in PDS and inthe University that, with help andhard work, a teacher can move in afew months to fruitful and excitingclassroom conversations aboutmatt, and towards an approachto teaching which feels moresatisfying.

SpecialCircumstances:

RethinkingMiddleSchoolMath

I guess overall the biggestthing [about the ProfessionalDevelopment School Project]is that it has really made methink . . Whereas before, Ijust did. I didn't think aboutwhat I wanted them to do,and maybe why. You startasking kids questions. Previ-ously, you didn't do that. Youweren't thinking yourself, sowhy would you ask kids tothink?

Three years ago. Patti Wagnertaught math to the seventh gradespecial education students atHolmes Middle School in Flint.

Math had never been her favoritesubject, but this time she felt total-ly frustrated - "I was angry at my-self, and angry with the students,because I knew they had notgained any understanding ofmath." Those who were havingtrouble with addition at the begin-ning of the year were still strug-gling in June. She had led themthrough the textbook, but neithershe nor the students had enjoyedthe journey. At the year's end shewas glad to hand math instructionover to another member of thespecial education team.

In 1989 she agreed to give itanother try. Hoping for inspira-tion, she sat in on the mathe-matics portion of the EducationalExtension Service's Summer Insti-tute: "I had never thought aboutteaching math differently. I hadnever put a whole lot of thoughtinto math, period. Pan of me al-ways thought, 'Well its just se-quential and it's very easy to teach:once you know a skill, you justbuild on it:" She was excited bythe introductory session in whichGlenda Lappan gave her audiencesets of one inch cubes and askedthem to explore area and perime-ter. "It was so interesting to see thatpeople did it all different ways,and all of them were right. In mathwe've been trained to think thatthere is only one right answer." Shealso attended a session in whichMaggie Lampert showed videos ofher class at Spartan Village Ele-mentary School and talked aboutthe ways she incorporated writinginto math instruction. She beganto think differently about her ownmath teaching, and to talk toSandy Wilcox, a faculty memberat MSU who was working with theHolmes PDS team, about new ma-terials and new approaches.

A Different Approach

This year she has taken a verydifferent approach to math in-

struction, choosing problems thatrequire thought rather than thesimple application of an al-gorithm, using manipulatives forthe first time, and encouragingstudents to talk about the ways inwhich they are making sense of

Ms. Wagnerfinds that asshe rethinks

the wayshe teaches,

she has toface basicquestions

about what toteach. -=7''s

math. She has learned a lot abouttheir thinking: "It just blew meaway" she recalls, "when I saw thatmy kids were coming to my classwith such a lack of understand-ing." And although she speaksmodestly about her accomplish-ments - "I'm just beginning. I havea long way to go" - she notes withsatisfaction that now all of her stu-dents enjoy math."And I think theyare getting a better understanding,though it's difficult to measure:'

This year, instead of followinga seventh-grade math textbook,she has focused on very simpleand basic things - "Things youwould think these kids wouldknow already - like 'what does 3plus 4 really mean? Now, write astory for that: It was very difficultfor them: Since September theyhave been working to understandcomputational processes that theyhave been encountering in schoolfor the last six years. Last monththey worked with one-inch tiles tofind ways to represent multiplica-tion visually; since then, they havebeen trying to make sense of

58

Today, Patti's small fifth-hourclass is creating pictures to repre-sent written division problemsvisually. After some cheerfulwarm-up conversation about ,Patti hands each student a piece ofpaper and asks them to write a di-vision problem on one side of it,and to draw a representation of theproblem on the other side.

After about five minutes ev-eryone is finished. Hands wave ea-gerly when Patti asks who wants toput a problem on the board. Shechooses a girl in a blue sweaterwho comes to the front of theroom and draws seven boxes, eachwith an apple inside. "Seven di-vided by seven:' calls a classmate;everyone agrees.

The next volunteer, however,leads her classmates into morecomplex terrain. She puts threecircles on the board, with fivelines (representing five sticks ofgum) in each one. "Five divided bythree;' announces the girl in theblue sweater. "Are we sure?" asksPatti. Four heads nod confidently."Are there just five sticks of gum?"asks the teacher? The girl recon-siders and revises her answer: "15divided by 3:' "Why divided bythree?" Patti wonders. "Because itwould be easier,- explains astudent.

When they have sorted thisout, and come to a more satisfac-tory reason for writing the prob-lem as 15 divided by 3. Patti turnsto the original answer: "Could weshow 5 divided by 3? Arms waveeagerly; everyone wants to try Atall boy draws three circles, withthree sticks in each, but everyoneelse agrees that this shows 9 ratherthan 5 divided by 3. A second boyoffers another suggestion: threecircles, with one stick in the firstand two in the second and third.The tall boy gazes thoughtfully atthis sketch; eventually he nods,"Oh, I get Patti puts the ques-tion into words: "If there are 5pieces of gum and 3 kids, howmany will each kid get?" "Two."

"One." "Two will get two, and onewill get one: explains someone,and everyone nods, apparentlysatisfied.

The problems are not easy forthese seventh graders, but every-one seems to be highly engaged. Agirl who entered the class lookingtired and indifferent strains for-ward, waving her left hand eagerlyas her classmate puts a picture onthe board. She is drawn into thelively inquiry all around her. Somestudents look puzzled when theirteacher disagrees with th( r an-swers, but no one withdraws fromthe conversation in discourage-ment, and no one wanders fromthe subject. A minute before thebell. Patti collects student papersand asks whether anyone watchedthe Grammy awards the previousnight. It is the first time that hourthat anyone has spoken of a worldbeyond mathematics.

New Problems Raised

Subjecting her math teachingto her own critical scrutiny hasraised new problems, even thoughit has been satisfying in someways. To begin with, Ms Wagnerhas found that as she rethinks theway she teaches. she has to facebasic questions about what toteach:

I'm still very much strug-gling with what my kids needto know, as learning disabledstudents, and as people whoin six years will be out ontheir own. I don't feel thatmonths of trying to conquerlong division will make themmore prepared. I try to thinkabout how I use math, andthen ask, why would theyneed to know this? SometimesI don't know. Yet for somewho continue struggling withconcepts and understanding,getting them to enjoy comingto math class is a beginning.

She also finds herself wonder-ing about common practices thatshe previously accepted matter-of- factly. "They really need toreach and reteach teachers:' shecomments, reflecting on a recentincident. A special educationteacher she knew had been tryingto help a learning-disabled stu-dent who was mainstreamed inmath and was confused about ahomework assignment involvingaddition of fractions.

Evidently the teacherwanted them to do it in aparticular way. And evidentlythe student did not get thisdown. Well, [the special edu-cation teacherl taught her todo it the way she thoughteasier and clearer. The [regu-lar] teacher marked them allwrong. The answers wereright, but she wanted them todo it in a particular way.

In the past I might havejust said, "Well, you shouldhave listened:' But now I won-der, should this have hap-pened? Should we be teachingchildren that there is only oneway to get the answer?

It is clear to teachers who arethinking about opening up theirmath teaching that they need toknow more math to teach studentswho ask questions and try a varie-ty of approaches and talk aboutthe ways that they are understand-ing a math problem. It is also clearthat teachers need to give mathteaching a great deal of time anddeep thought. Many elementary-certified teachers would arguepersuasively that because they be-lieve that the language arts (read-ing, listening, speaking, writing)are more important, they investtheir time and energies there andsimply do not have the time ortraining that innovative mathteaching would take. Patti does notfeel particularly confident abouther math knowledge, and even af-ter the exciting work she has beendoing in math she still gives thebulk of her out-of-school time tolanguage arts. Nonetheless, shehas made major changes in hermathematics teaching, and herspecial education students seem tobe understanding what they doand having a wonderful time. m

Projects at Holt Higb.School

When Sandy Callis Bethellstarted teaching at Holt HighSchool in 1984, she was, she nowrecalls, "a very traditional. execu-tive-style teacher:' But when shereturned to Michigan State's Col-lege of Education for graduatestudy three years later, she began aprocess of re-thinking schooling,learning, and mathematics.

On Friday, February 23. 1990,the visitor to Sandy's tenth-grade

59

Practical Mathematics class seeslittle sign of the teacher Sandy saysshe used to be. Within minutes ofthe bell, the 22 boys and girlspresent, about half of themofficially designated as havingspecial needs, (learning disabili-ties, emotionally impairment, orboth), have organized themselvesinto groups. Sandy hands eachgroup a piece of paper bearing anumber sentence involving two

equivalent proportions ( =3 ==:) telling students to work to-gether to represent their sentencevisually. A tall bov glances trucu-lently at the paper Sandy hashanded him. "I low am I supposedto do this?" "That s the question:she smiles. "Then I'm not going todo it: He lays his pencil on thedesk with emphasis. "You can do

Sandy reassures him, withoutmissing a heat. As she heads off toanswer another question, he re-trieves his pencil and reexaminesthe paper.

Jean Tomlinson. a special edu-cation teacher who team teacheswith Sandy. moves along the otherside of the classroom, offering sug-gestions and encouragement. Asthe groups finish. each one sends adelegate to the blackboard: afterabout 10 minut^s there arc sevennumbered drawings on the board.Sandy directs the students to putthe numbers on their papers andwrite next to each one the equa-tion they believe the picture at-tempts to represent.

The next two problems aremore straightforward, showingidentical rectangles with identicalshaded portions, divided up intodifferent numbers of pieces. Butthe fifth one provokes cries of out-raged confusion:

"There's only one picture!" "Itdoesn't make any sense:' No onehas a correct guess, so a studentfrom the group who generated thepicture ,comes to the board andwrites = . "A great picture:comments Sandy "Why did youdecide to do it that way?" "It wastoo simple: replies the tenthgrader.

Sandy is pleased with thisclass. especially when she com-pares it with the basic math classshe taught before she went tograduate school. She is struggling

Ten minutes later the classgoes over the answers together.Students immediately identify onedrawing that is incorrectly made:it shows ",-c equal to z;. Another dia-gram, which shows three shadedcircles followed by three shadedcircles divided into thirds, stumpseveryone but the group that hascreated it (see figure above).

"Does anyone want to take astab at it?" Sandy asks. = ?"

ventures a brave soul. "-9,, ?" sug-gests another. Together they workthrough these hypotheses untilthey arrive at a solution that, aftersome discussion. makes sense toeveryone: 3 =

to develop a curriculum which en-gages the students with math con-cepts and not just with practicalapplications of arithmetic skills.The atmosphere is comfortableand cooperative: "They are re-sponding very positively," saysSandy. "They love the class. . . .

There are none of the power strug-gles and behavior problems:'

Puzzled About Evaluation

She is puzzled. however, abouthow to interpret what she ob-serves and how to evaluate learn-ing. When she watches group

60

work she sees students talkingabout math and coming to agree-ment, but some other observersdescribe the same scene as stu-dents copying from one another.How, she wonders, does one get anaccurate and objective assessmentof what is going on? Her students'poor literacy skills prevent herfrom testing them on the sorts ofword problems she believes theycould solve. However she ispleased by some of the insightthey display when they come upto the overhead projector to ex-plain their solutions visuaPy. Atpresent she uses group assess-ment: students work on the prob-lems together in small groups andthen collectively evaluate the con-tribution each member has madeto the task.

Five Colleagues Collaborate

Unlike most teachers trying toinnovate, however, Sandy has fivethoughtful colleagues collaborat-ing with her to understand herteaching. At Holt High School, thePDS math team includes: MikeLehman, who teaches full time atHolt and developed a probabilityand statistics course that Sandy isteaching this year; Bill York. chair-man of the Holt High Schoolmathematics department: JanetWilson, a guidance counsellorwho coteaches Algebra 2 withMike: Perry Lanier and Pam Stra-hine from Michigan State; andSandy herself, who teaches in Holtin the morning and at MichiganState in the afternoon. The groupis looking closely at the two cours-es Sandy teaches Probability andStatistics and Practical Math andat Lehman and Wilson's Algebra 2.They observe classes regularly,and have interviewed students inpractical math in order to learnhow these adolescents are undcr-standing the math discussed inclass; they meet weekly to talkabout what they see and hear.

Bill York reports that his inter-views with the students in BasicMath strongly confirm Sandy'simpression that the students are"responding positively":

Students seem to reallylike the course. They say theyare able to learn more becauseof the groups, and because ofthe way Sandy teaches thecourse.... They say they likemath, and they seem to enjoybeing there. I think absentee-ism is lower:

He says that watching Sandyand interviewing her students hasprovided him with some real sur-prises: "I'm a very traditionalteacher, so it has been interestingfor me to see this. I teach a lot ofhonors sections. and I expect Istu-dents to be more on task. But it isimpressive to see that they learnanyway:

Like Sandy, Bill doesn't feelthat he has the evidence to claimthat these students are learningmore than they would in a moretraditional class, but he has foundthat in interviews in which he hasasked them to work in a variety ofmath problems, -they did muchbetter than you'd expect them todo:' The problems required seri-ous thought. One, for example.asked them to find the measure ofthe angles of a triangle in whichthe second angle is twice the first,and the third is three times thefirst. Another problem asked themto think about how fast two paint-ers working together could paint aroom if one took two hours andthe other took three hours work-ing alone. He was impressed bythe strategies they used, and theway they thought aloud about theproblems: -They are more free,more willing to share what theyknow:'

Sandy and Mike are also inno-vating in the courses they teach tocollege-bound students usingcooperative groups and emphasiz-

ing the importance of explainingsolutions. Mike team teaches Alge-bra 2 with guidance counsellorJanet Wilson. Sandy and Mikework together on a probability andstatistics course which Mike de-veloped and which is beingoffered this year for the first time.Although the students in Probabil-ity are older than those in Algebra2, Sandy and Mike mention manyof the same challenges and satis-factions when they describe theirtwo courses.

Inflexible Students

The biggest difficulties relateto the somewhat inflexible ideasabout teaching, learning andmathematics which successfulstudents have developed over 10or more years of schooling. For ex-ample, Mike, like many of his col-leagues in other departments inthe High School, reports thatmany teenagers resist working in

Classes thatmake

unfamiliardemands dis-please some

students whohave alwaysgotten Ns bymemorizingformulas for

the test.

),.\

A.

:ti.;_',

groups, arguing that teachers oughtto lecture to them and tell themhow to solve the problems in thebook. Recently, says Mike, a stu-dent put aside her work to tell himangrily, "This is your job:' She in-sisted that groups could never beas effective as a teacher's lecture.As she berated him, Mike recalls,

61

her partner shouted excitedly,"Oh. I get this: and began to ex-plain his new idea to a thirdstudent.

Mike had expected students tofeel some initial frustrationwhich is one of the reasons he hadwanted to coteach with a guidancecounsellor but he had hopedthat they would work it throughmore quickly. Sandy gets the samemsage fr3m the students takingProbability: "They feel I'm not do-ing my job because I don't validatetheir answers:'

Classes that make unfamiliardemands displease some studentswho have always gotten A's bymemorizing formulas for the test.'I am trying to force them into un-derstanding;' says Mike emphati-cally "They don't want to under-stand. This isn't [in their minds]the way you learn math:' But oth-ers thrive in an environmentwhich values reasoning through aproblem and explaining yourthinking to others. One of Mike'sstudents is taking Algebra 2 for thethird time. He is, says Mike, a goodthinker, and he has now becomethe class expert. Although hesometimes has difficulty justifyinghis solutions in writing on tests, hecan always make his ideas clear tohis group, and he contributes reg-ularly to full class discussions. "Itis:' says Mike, "a big change in rolefor him:'

Conversation Important

Conversation plays a leadingpart in the effort to rethink mathe-matics teaching in this profession-al development school. Studentstalk about mathematics in order tofigure out ways to apply what theyknow to mathematical problems,and teachers open up their classesto one another and talk aboutwhat they hope for, and what theysee and hear. Some of the biggestchallenges relate to the quality ofthese conversations.

Changing Minds continued from page 2.

designed to ensure standardization. Instead, we have to think of teach-ers as genuine professionals who need and who model all of the sameskills as other workers in the knowledge age economy of the twenty-first century. Children are not less but far more complex and variedthan automobiles, computers. or financial systems. Helping themlearn is similarly complex. It requires knowledge, judgment, andteamwork. Schools must be organized and managed accordingly.Changes in teachers' work means reciprocal changes in the work ofprincipals and other administrators, making their roles more likethose of managers in high-tech organizations and less like those ofshop foremen in the oid industrial model factory.

To make all of these changes in teaching, learning, and schools asorganizations, we need new knowledge knowledge that is at oncetheoretically powerful and deeply practical. Generating such knowl-edge will require far closer collaboration between university-based re-searchers and public school teachers than the profession has seenover the past half century Professors need to spend more time in K-12schools, working alongside teachers as professional colleagues. Recip-rocally, teachers need to spend more time engaged in reflection andinquiry Only through such collaboration and crossover between theuniversity and the schools will we get the knowledge we need to re-new education.

The Educational Extension Service was created in the fall of 1988to support just the kinds of changes in minds and practices sketchedabove. It consists of two parallel sets of partnerships. First, partner-ships between MSU's College of Education and a small number ofpublic schools these are called professional development schools(see list on back page for specifics) where teachers and other practi-tioners collaborate with university faculty to (1) improve teaching andlearning for K-12 students, particularly students at risk of academicfailure, (2) improve the education of new teachers and other educa-tors, and (3) make supporting changes in both the schools and theCollege as organizations.

The second set of partnerships is designed to make the results ofwork in the partnership schools accessible to other schools all acrossthe state. Fortunately, a large number of organizations committed togetting new knowledge into educational practice already existthroughout Michigan. These include intermediate school districts,professional associations, and institutions of higher education. Overtime, we plan to form partnerships with many such organizations. Byworking with and through them rather than setting up an entirely newstatewide network, we can reach many schools, teachers, and admin-istrators, and we can do so more efficiently than we ever could if weworked alone. Such a statewide network of networks will take time tobuild. We have begun working with a few organizations in each cate-gory (see list on back page for specifics). Over the new few years, wewill gradually expand this set of dissemination partners to cover virtu-ally every corner of the state.

62

It is both difficult and impor-tant. says Mike, to find problemsworthy of extended group atten-tion problems that people needto discuss, problems where youcan't just find an answer by plug-ging some numbers into a formu-la. He cites the example of a prob-lem he posed on a recent test:"Why is = x'.1'?" The groupswere to try to explain this phe-nomenon, and failing that, to givesome examples they could use in aproof. Some groups managed thetask while others didn't, but alldiscussed their theories with oneanother. Mike continues to searchfor problems that provoke and re-quire this sort of discussion. "I'vehad lots of non-success here I

don't know if this is failure:'While students struggle to ex-

plain math problems to one an-other, teachers search for produc-tive ways to discuss teaching. "It ishard:' reflects Sandy, "to develop aculture wriere we can talk. Wetend to go either to the high morallevel 'We all want kids to learn'or to the real nitty gritty 'Whatbooks do you need by Monday?'PDS is about Jeveloping a waythat lies in between:'

Editor Helen Featherstone

Changing Minds is published asa nonprofit service by the Michi-gan Educational Extension Serv-ice which is funded by the Mich-igan Department of Education.Please address all inquiries to:Changing Minds, 116 I EricksonHall, Michigan State University,East Lansing, MI 48824.

Professional DevelopmentSchools

East Lansing

Flint Hoimes Middle School

Spartan Village Elementary School

Holt Elliott Elementary SchoolHolt High School

Lansing Averill Elementary SchoolKendon Elementary SchoolOtto Middle School

Dissemination Partners

Associations:

Michigan Education Association

Michigan Federation of Teachers

Michigan Association of School Administrators

Middle Cities Education Association

Michigan Association of Colleges for Teacher Education

Colleges fic Universities:

Lake Superior State University

Northern Michigan University and the Upper PeninsulaCenter for Educational Development

Northwestern Michigan College and the Northern LowerMichigan Leadership, Teaching and Learning Consortium

University of Michigan

Wayne State University

Intermediate School Districts:

Ingham Intermediate School District

ISD's Associated with the Northern Lower MichiganLeadership, Teaching and Learning Consortium and theUpper Peninsula Center for Educational Development

changin.g minds518 Erickson Halllichigan State University

Last Lansing, Michigan 48824-1034

Non-Profit Org.U.S. Postage

PAIDE. Lansing, MIPermit No. 21

Craft Paper 91-3

ASSESSING ASSESSMENT: INVESTIGATING AMATHEMATICS PERFORMANCE ASSESSMENT

Michael Lehman

Published by

The National Center for Research on Teacher Learning116 Erickson Hall

Michigan State UniversityEast Lansing, Michigan 48824-1034

February 1992

This work is sponsored in part by the National Center for Research on TeacherLearning, College of Education, Michigan State University. The National Center forResearch on Teacher Learning is funded primarily by the Office of Educational Researchand Improvement, United States Department of Education. The opinions expressed in thispaper do not necessarily represent the position, policy, or endorsement of the Office or theDepartment.

.g4

National Center for Research on Teacher Learning

The National Center for Research on Teacher Learning (NCRTL) 1 was foundedat Michigan State University in 1985 with a grant from the Office of EducationalResearch and Improvement, U.S. Department of Education.

The NCRTL is committed to research that will contribute to the improvementof teacher education and teacher learning. To further its mission, the NCRTLpublishes research reports, issue papers, technical series, conference proceedings, andspecial reports on contemporary issues in teacher education. For more informationabout the NCRTL or to be placed on its mailing list, please write to the Editor,National Center for Research on Teacher Learning, 116 Erickson Hall, MichiganStatc University, East Lansing, Michigan 48824-1034.

Director: Mary M. Kennedy

Associate Director: G. Williamson McDiarmid

Program Directors: Linda Anderson, Deborah Ball, G.W. McDiarmid

Director of Dissemination: Cass Book

Project Manager: Anne Schneller

Editor: Sandra Gross

Many papers published by the NCRTL are based on the Teacher Education andLearning to Teach Study (TELT), a single multisite longitudinal study. Theresearchers who have contributed to this study arc listed below:

Marianne AmarclDeborah Loewenberg BallJoyce CainSandra CallisBarbara CamilleriAnne ChangDavid K. CohenAda Beth CutlerSharon Feiman-NemserMary L. GomezSamgeun K. K wonMagdalene LampertPerry LanierGlenda LappanSarah McCartheyJames MeadSusan Melnick

Monica MitchellHarold MorganJames MosenthalGary NatrielloBarbara NeufeldLynn PaineMichelle ParkerRichard PrawatPamela SchramTrish StoddartM. Teresa TattoSandra WilcoxSuzanne WilsonLauren YoungKenneth M. ZeichnerKaren K. Zumwalt

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

65

Abstract

The author of this paper describes how he tried several different strategies toenhance his students' understanding of the mathematics he was teaching. He asked hisstudents to work in cooperative learning groups, to give presentations of problems they hadsolved, and to write about solutions they had found. With all these changes in teachingstrategies he began to question if traditional methods of assessing his students weresufficient for testing the type of understanding he was striving for in his students. In thispaper the author discusses how in answer to this question he developed a performanceassessment as a final exam for his high school Algebra II class. The author explains howhe organized the exam, how the students prepared for the exam, and then the results of theexam. He explains how some students surprised him as they performed very well and wereable to explain in detail the concepts they were dealing with. Given a chance or whenquestions were rephrased, students were able to do very well explaining a problem incontrast to drawing a blank on a traditional test and being doomed to a poor grade. Othersdid not do as well as expected. Even though they had done well in class and had a goodunderstanding of the mathematics he had taught, they were unable to explain the concepts.Instead they relied on memorized facts and simple computation. The author includes somecomments judges made about the students and some of the reactions of the students to thistype of final exam. The results of this assessments have direct consequences for the author'steaching. The author concludes by discussing the consequences of this assessment, how hewill use what he learned from this assessment to iLiprove his students' understanding, andquestions that were either unanswered or created by this assessment.

6 6

ASSESSING ASSESSMENT: INVESTIGATING AMATHEMATICS PERFORMANCE ASSESSMENT

Michael Lehman'

I have taught high school mathematics for 14 years. Over the past several years, Ihave become concerned with my students' understanding of algebra concepts and skills.Numerous research findings point to the lack of mathematical problem-solving skills andconceptual understandings on the part of our nation's adolescents (National ResearchCouncil, 1989, and National Council of Teachers of Mathematics (NCTM), 1989). After mylectures and whole-class discussions of the concepts, my students could perform thecomputations quite well, but seemed to have very little understanding of what they did andwhy. When I asked them to explain their reasoning, many seemed unable to go beyondsimply telling what they had done to get the answer.

To alter this situation, I have tried several different strategies to enhance students'understanding. For example: To help students discuss and share ideas, I used small-groupcooperative learning, oral group presentations of topics and problems, and writtenassignments about mathematics. My assignments seemed to help students by requiringthemto reasor about mathematics and justify claims and responses. Discussions in their smallgroups increased in length, frequency, and quality. I heard students say to each other,"That's fine, but why?" or "We need to put this into words the rest of the class will be ableto understand."

With all these changes in strategies, I began to wonder about whether my traditionaltests were sufficient for assessing student understanding. In trying to think about assessingstudent understanding differently I began to focus my attention on what the ProfessionalStandards for Teaching Mathematics (NCTM, 1991) says about assessment. An importantissue raised by the Standards is to "align assessment methods with what is taught and howit is taught" (p. 110). Faced with the problem of assessing what I speculated to be a newkind of mathematical understanding for my students, I began altering my tests both incontent and form to find out what my students did and did not understand. I wanted lesscomputation required on the tests and more written explanations about solutions toproblems. I have tried group assessments in the form of presentations of problems or topicsto the class as a whole. I assessed individuals based on their contribution to the group

'The author teaches mathematics at Holt High School. He is very grateful to Michelle Parker and Pam Geist for theirhours of editing and encouragement.

presentation. I also tried giving the groups a problem to solve and having them write abouttheir solution.

All of these changes provided me more in:,-)rmation (about my students as a groupand as individuals) than simple computational tests. However, the changes left me feelingthat I was not getting an accurate assessment of what my students understood. I keptwondering if my students understood more than I was able to give them credit for, orperhaps less. Furthermore, how could I "see" any lack of understanding in a way that wouldhelp both the student and me identify it and talk about it? My concerns were echoed in theCurriculum and Evaluation Standards for School Mathematics (NCTM, 1989), which states,"It is not enough for students to write the answer to an exercise or even to 'show all theirsteps.' It is equally important that students be able to describe how they reached an answer"(p. 140).

My concern for trying to understand what my students are learning in Algebra 2 ledme to design their final exam for the 1990-91 school year as a "performance assessment."Performance assessment can mean a variety of things. The California Mathematics Council(1989) in Assessment Alternatives in Mathematics provides alternative ways to think abouthow to assess student performance. The idea of performance assessment provided analternative to a traditional exam in which students solve a variety of problems, individually,using paper and pencil within an allotted exam time. The performance assessment offeredstudents the opportunity to discuss a limited number of problems representing a wide rangeof concepts and to solve them in cooperative learning groups prior to the exam. One of theprimary goals in this assessment was to have students be able to justify how they solved theproblems. During the first half of the exam period, four panels of adult "judges" eachlistened to a cooperative group discuss their problem solutions.

While half the groups were doing their performance assessment, in my classroomother cooperative groups wrote about solutions to three problems similar in style to thosedescribed above but differing in content. I asked these students to construct their solutionswith as much detail as possible, though they did not need to do the actual computations forthe problems unless they felt it lent validity to their methods. In the second half of theexam period, the groups changed places.

Background

Holt High School is a small Class-A school accommodating grades 10 to 12. Locatedin a blue collar and middle class suburb of Lansing, Michigan, our student population ispredominantly white. A 1984 survey of high school graduates shows that about 78 percent

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of the school's students take some college courses. Students are required to take twomathematics courses in grades 9 to 12 in our district, approximately 60 percent takeAlgebra 2.

Algebra 2 is a yearlong course in which students attend 60 minutes of class per day.In my Algebra 2 course last year, I had 28 students with a wide range of mathematicalabilities. I assigned students heterogeneously to small groups in the beginning of the yearbased on their Algebra 1 and Geometry scores and their reading scores on a nationallynormed achievement test.

After I assigned students to groups, I made changes only when I saw that a widerrange of knowledge and expertise would help the students better understand themathematics. Students were not allowed to change groups for social reasons. If a grouphad a social problem, a counselor,' who worked regularly with the class, helped resolveconflicts.

Preparing for the Exam

Three days prior to the exam, I gave my students six problems which drew on themain topics discussed over the year in class. I also gave them an explanation of how thefinal would be conducted, which read as follows:

You should find six problems included in this packet. You should work onthese problems as a group as well as on your own time. During the examperiod you will be asked to explain your results before a panel of judges.Each member of the group will be able to explain each problem bythemselves. Other members will be present but will not be able to offer ideason an individual's problem. You will not know which problem you will beasked so be sure to study all the problems. In your explanations includesamples of graphs you may have used, calculations you may have done, chartsyou made up and any other information you feel will help the judgesunderstand what you know. Do not write a script that you intend to read asthis would only prove you can read.

Each problem included some computation along with opportunities to explain andmake judgments based on the results of the computations. In these problems the studentswere either given a situation in which they had to derive data for the problems or they weregiven a set of data in which they had to decide how to analyze and expand the given

2The school counselor, Jan Wilson, and I participate in a classroom research project about high school mathematicsstudents' self-esteem and sense of mathematics efficacy.

information. Creating problems that would provide opportunities to show mathematicalreasoning in multiple ways, while also being interesting and worthwhile, proved to be amajor task. I worked to design the problems for several weeks along with asking anyone Icould find for ideas. With the help of many of my colleagues I was able to write suitableproblems.

I include three problems here illustrative of the range of questions students faced.

1. You and your partner have decided to go looking for a buried treasure described ona scrap of paper found in the basement of an old house. The only clues to thetreasure's location is the following:

"The treasure is buried in a spot that is the same distance from the boulder as it isfrom the railroad tracks. It is also . . ."

And the rest of the information is missing. But some other clues you may be wiseto consider are:

1) the distance from the track to the boulder is 11 yards.2) consider the tracks as the directrix.3) keep all of the units in yards or feet.

Keep in mind the distance of the treasure from the railroad track is interpreted asbeing the length of the perpendicular drawn to the tracks from the treasure.

2. Hooke's law states that the force F (weight) required to stretch a spring x unitsbeyond its natural length is directly proportional to x.

You have a spring hanging from the ceiling in the classroom whose hook is 8 ft.above the floor and you want to stretch it down to 3 ft. above the floor in order tohook it to Mr. Lehman's belt loop. Devise a plan to determine how much weightwould be needed to pull the spring down. What would you need to consider? Beas complete in your strategy as possible (What steps would be needed?). Howwould you know if the spring would lift Mr. Lehman or not? When wouldn't it lifthim at all?

Create a set of data to prove your conjecture.

3. Suppose you are a doctor doing research on cancer cells. You have found a certaintype of cancer cells are growing as follows:

Weeks 0 1 2 3

Number of Cells 4 16 I 64

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You experiment with different drugs and EUREKA! XI3V causes the cancer cellsto stop all further growth and the cells start disappearing at a rate of 10,000,000,000per hour with a maximum of 5 doses per day. More than 5 doses per day willdestroy the patient's liver and kidneys and the person will die.

If you have a patient with this type of cancer and you have estimated that they haveabout 2.814749767 x 10" cancer cells, how long have they had the cell growthoccurring?

How long would you prescribe your patient take XI3V in order to make sure thatall the cancer cells disappear? How many doses will this take? Give your answerin a reasonable unit of time (i.e., days, weeks, months, or years; which ever seemsto be the most useful).

If another patient comes in who has had this cancer for a year and is only given anestimated 5 years to live unless you can get the number of cells in her system below10,000,000 within the 5 years so her body can start to repair the damage the cancerhas done, would you put her on this medication and give her hope for a continuedlife? Be very clear in you explanation and have the appropriate figures to backupyour determination.

The students had three class days and one weekend to work in their peer groups onsolving the problems and constructing explanations that provided support for the solutions.When I first passed out the problems, I expected that students would waste a lot of time inthe beginning and get themselves into a time bind towards the end. Yet I was pleasantlysurprised; both the observers' and I noticed that students used their time very well duringthe three days. We also noticed that the conversations went beyond simple computation totalk about why different individuals solved the problems in certain ways and what alternativeapproaches were possible. The students worked very hard but did not seem to panic or beunder the tremendous pressure I was used to seeing in traditional reviews for finals. Theclass seemed to have an atmosphere of seriousness, but also of confidence. Students seemedto believe they could solve these problems in ways they could discuss.

The Judges

I organized judges into panels of three persons that included one person who knewthe mathematics subject matter necessary to solve the problem, one person who was not asstrong in inathematical knowledge, and one prospective secondary mathematics teacher

T'he observers included Pam Geist, a doctoral student studying mathematics educetion at Michigan State University, andJan Wilson.

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studying teacher preparation at Michigan State University. The mathematics person couldfocus on the computation and the logic of students' explanations and mathematicalunderstanding of concepts. The non-mathematics person could focus on the confidence levelof the student. As I explained to my students, those judges should have enough confidenceafter hearing the student's explanation to entrust the student to solve problems for them.I asked prospective teachers, scheduled to student teach in the fall, not only to help in theassessment of the computational algorithms used to solve the problems but also to judge themathematical conceptual understanding students had. I also wanted the prospective teachersto learn through experience that, even though a teacher may use many strategies to supportand develop student's learning and understanding, students may still not conceptuallyunderstand some content. I kept myself off the judging teams since I believed I would bringa set of preconceptions about students' abilities and understandings and thereby constrainwhat I could really "see" students doing and thinking.

Before the actual assessment I gave the judges guidelines and evaluation forms fortheir task (see Appendix). I developed the criteria used on the evaluation forms fromsimilar evaluations I had used during the school year for individual student and grouppresentations. I believed students would feel comfortable with these evaluation categoriessince they had seen them during the year. Furthermore, these six categories defined whatI was trying to assess about my students' mathematical knowledge without being tooburdensome to the judges. I also wanted a form that would not get in the way of students'and judges' discussions. I gave each student a copy of the evaluation form when I passedout the exam questions so they would know what they were being judged on as they werepreparing.

I allowed the judges to choose whether they wanted to use the evaluation form Iprovided or simply give me written comments with a numerical score within the range ofpoints allowed on the form. Most judges found the form very usable and stuck with it, justadding comments to the bottom. A few decided they could better reflect the students'understanding by using more extensive written comments. Both methods seem to work quitewell as far as helping me know what my students understood. I found I had no problemevaluating students with the combination of methods used by the judges.

I created two main categories on the form, "Mathematics" and "Presentation," sinceI wanted to clarify for the students what they were being judged on. The "Mathematics"section reflected what I believe are the four essential characteristics used in solvingmathematical problems. These were also the components we as a class had discussed andfocused on throughout the year.

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I included the "Presentation" section since I wanted the students to know that justdoing the mathematics was not enough. They would have to communicate theirunderstanding of the mathematics to the judges. Although I did not want to overemphasizethis component (presentation is not more important than mathematical understanding), Idid want it to be a part of the process.

I asked the judges to keep in mind my goal for the final exam, which was tounderstand what my students understood. Also, I reminded them that this was the first timemost students had faced an assessment situation like this and, therefore, the students mightbe nervous. I cautioned judges about using leading questions that might cause students toexplain problems without truly understanding them. Yet, I encouraged judges to use probingquestions to help students, when necessary, get started on their responses. I wanted studentsto be able to say something and feel confident that their preparation for this final benefittedthem. If students were able to provide reasonable explanations during their discussions, and

not just get hints from judges that would make it easy, I could be somewhat assured thatstudents' explanations reflected their understandings.

The Day of the ExamI reserved the library for the entire exam period (which was, under our school policy,

an hour-and-a-half long). I arranged the furniture into four areas so that four groups couldbe taking the exam at the same time. Each group had as much privacy as possible duringthe exam. I circulated around the room in order to handle procedural questions if theyarose.

During the actual assessment, students went before the panel of judges as a group,but only one student presented a problem. The judges picked which problem each studentpresented, therefore requiring every student to be able to discuss all the problems and notjust the one or two they felt most comfortable with. After the judges heard a student discussa problem, they would open the floor to other students in the group who wanted to addanything or refute what they had heard. I set it up this way because I wanted to know whateach student understood but at the same time I did not want the students to feel totallyalone (without the peers they'd studied with) before the judges.

I combined the information I received from the judges and the problems the studentsdid while in my classroom for a final exam score. I used the average of the three judges'scores for two thirds of the total exam score; and the in-class problems counted for onethird. These scores were combined to make up 20 percent of the students' nester grade.

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What I Learned About What My Stu lents LearnedThe results of this assessment gave me plenty of information to digest about my

students, my teaching, and our curriculum. First, I learned that many of my students werestill only superficially learning and understanding the mathematics. In their groups duringthe year I overheard excellent discussions about issues and topics we studied in mathematics,and they were also getting better at writing explanations and justifications. However, whenit came to explaining the mathematics to the judges, they were only able to tell the stepsthey took in solving the problem. They fell short when it came to explaining why theyapproached a particular problems in a certain way. A common response judges heard was,"That's the way we did it in class."

I am still trying to figure out why what we did in class did not translate into betterperformances during the final. I suspect that some of the students froze in the testingsituation despite all I had done to help them relax. Some of the students seemed to havetrouble explaining problems individually. They did not have their partners to provide themwith some connecting ideas that would allow them to give a coherent explanation of theirunderstanding. Finally, I wonder if the students needed more practice throughout the schoolyear with performance assessments since the practice might help them to understand betterwhat they need to do to prepare and carry out a good discussion of a mathematical problem.

In addition, I was surprised that some of the students I expected to do well didn't,and some I didn't expect to do well did! I think some prompting from judges helped thesestudents begin to respond to the questions. While several students did well on their own,others gave good explanations after the judges asked several questions to help them focustheir thinking. I felt especially pleased about this finding. If these students were taking atraditional final examination and got stuck on a problem, they would probably be doomedto be unsuccessful and probably continue not understanding. On a performance assessmentI could tap into what they actually understood. I also could sort out their misunderstandingsfrom simple computational mistakes common on traditional final exams. The performancejudges could help students, through probing, sort out conceptual ideas from mistakes basedon incorrect computations.

Another way this exam differed from traditional finals is that I gained informationfrom three different professionals about each of my students. Each judge helped me piecetogether a picture of my students' understanding that would not have been possible on atypical final. The judges' comments reflected a wide range of observations about whatmathematical understanding my students had. Most judges focused on what sense studentscould make of the mathematics they were doing. Typical of these kinds of comments was

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this example: "The student was accurate mathematically in solving but did not manifest verydeep understanding of what the problem was about." Said another judge, "Started by statingthe sense of the problemthe relationship between pressure and volume."

Often judges commented on how explanations and calculations fit together with theproblem the student was solving and his/her understanding of it. Here are three commentsabout one student:

He calculated the correct equation for the parabola. The only thing he wasunable to do was explain the formula [distance] he used to get his equationfor the parabola. Other than that his explanations were very good.

Did not understand derivation of formula for parabolacould not provideexplanation for why formula workshowever set up problem nicely, clearlyunderstood problem.

[This judge addressed the comments to the student] I hope you continue withyour agility and explanation based on the graphical representation of thisproblem. That's important. Push yourself on why the formula/equationworks.

The judges seemed to agree that this student could perform the calculations correctly.Yet they all pointed to the student's weakness in being able to explain how or why theformula worked. He seemed to be unable to make the connections as to why he would usethe distance formula, though he knew it was necessary to solve the problem. As a teacherI learned that this student knew how and when to use the formula, but could not say whyit workedwhich is what I want my students to be able to do. What I learned about thisstudent I might not have learned on a traditional final exam.

Another set of comments provided another picture:

Explained that she is just doing the problems like the book said. [She didn'tknow why she "logged" things to solve for x]. Did not really explain why shedid things very well. However, she was able to interpret her results andseemed to understand what they meant.

I asked [student] what a log is, and she said "some number to a power" andcould explain nothing more about the concept. Throughout her performanceshe also kept saying she didn't know if she was "right". All of these commentspoint to an emphasis on procedureswhich for the most part were passableuntil #3 where she divided instead of multiplied [even after a judge gave herstrong prompts].

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[Student] has an attitude problem. She thinks that she really understandsmore than the other people in her group and she may be partially right butshe has a long way to go. When questioned she seems to think that it doesn'tmatter if she's wrong if it is her opinion. She doesn't seem to realize that inmath everything isn't wide open, that there are more than opinions. Sheworked on problem four and immediately identified logs. She said she doesn'treally know what a log is or what it means to log both sides. Most peopledon't know. She did most of the problem well and was articulate. I couldn'tjudge her accuracy not having done the problem. One serious error wasfinding that the treatment would take 1,172 days and dividing by 5 to find thenumber of doses. When questioned she didn't revise and simply said shemight be wrong. A judge asked about 10 days and how many doses thatwould be. She seemed to understand that 2 was unreasonable but didn't wantto think about it at this point. She did do a good job explaining why it makessense that it would take longer to get rid of the disease than it would take fordisease to grow.

These comments give me a lot of information about this student. In class she alwaysoffered suggestions and usually could derive a correct answer. Her classroom participationled me to think she understood the concepts very well. However, the judges' commentsallow me to see that this student is very capable of doing the computation without havingthe depth of understanding I had hoped for. On a traditional test she would have made theone mistake with the division for which I would have taken off a few points thinking she hadjust made a simple mistake. I would never have suspected the depth of her misconceptionwould go to the point where, when confronted with it, she would choose to stay with it eventhough she would admit it was unreasonable. If I had this information earlier in the year,I would have been able to address some of these misconceptions to work towards furtherunderstanding.

From this student's comments I learned that she was able to perform themathematics and understand most of the concepts in the problem. However, with some ofthe information she chose to use, she did not understand where it came from. In atraditional paper and pencil exam, with a few lines of computation to illustrate what astudent knows, I might have never known that the student did not really understand thedistance formula.

While learning about my students' substantive mathematical understanding, I alsolearned about their affective mathematical views. I learned that students seemed to enjoythis type of assessment. They felt confident they could do it! Afterwards, several studentstold me that they felt good about the exam and enjoyed taking the final this way instead of

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working problems for one-and-a-half hours. They felt they demonstrated what they reallyknew. These responses gave me information usable in answering another question I havewondered about: How can we help our students feel good about themselves in relationshipto mathematics? During the past two years I have been working with Jan Wilson in tryingto find ways to help our students improve their self-efficacy in mathematics. As stated inEverybody Counts, "In the long run, it is not the memorizations of mathematical skills thatis particularly importantwithout constant use, skills fade rapidlybut the confidence thatone knows how to find and use mathematical tools whenever they become necessary"(National Research Council, 1989, p. 60). When we talk about student self-efficacy, it is thislevel of confidence that we are referring to.

Since only some students volunteered their comments, I cannot generalize about allstudents in the class; perhaps there were several students who did not like the exam but whochose not to tell me. However, I have been teaching long enough to know that if studentstruly dislike something, they usually let you know one way or another. Also, by looking atthe expressions on students' faces during and especially after the final was done, I was ableto get a sense of how they felt about it. I did not see the strained and dejected looks Iusually see during and after finals. Rather, I saw students who felt they had accomplishedsomething. They were congratulating each other with "high fives" and commenting on howthey felt they did. They also offered alternative ways of explaining a problem to their peersthat the presenter had not used before the judges.

Several students who normally did not do well on exams were pleased with theirpresentations. One student commented that he was grateful for the judges taking the timeto ask questions since he knew the information but had trouble finding the right words todescribe it. This was similar 'o what the judges said about him. For this student to walkaway feeling good about himself in relationship to a mathematics assessment was worth allmy efforts to plan and organize it. The next day when he found out he got a "B" on theexam he literally jumped two feet off the ground and went down the hall screaming to hisfriends.

What Now?As I enter the 91-92 school year, my task is to use what I learned from this type of

assessment as I plan for instruction that will continue to improve my students' conceptualunderstanding of the mathematics. I now know that good conversations in class do notalways transfer into good understanding down the road. I must look for ways to help

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students transform in-class conversations into meaningful understanding. Transforming what

I learned into realistic changes in my classroom is the hard job that lies ahead.I also have to design a method of doing performance assessment throughout the year.

I cannot wait until the end of the year to gather this information since I can better helpstudents' understanding through ongoing assessments. Having four or five performanceexams during the school year could help me gain an understanding of my students and allowme to help them be reflective of their growth and change. This would provide me withbetter checks on their understanding, and how their sense making does and does not fit withmy instruction.

I am challenged by some hard questions about my instruction, the curriculum, andgeneral conditions of learning high school mathematics. First, how does a teacher come upwith problems that lend themselves to a performance type of assessment? Since theproblems require students to think about something, the problems should reflect somethingworthwhile to wonder about and something real. How does a classroom teacher createproblems that fit these requirements around each issue and theme discussed in thecurriculum. This set of questions surrounds designing problems that invite discussion.

Another set of questions concerns arranging a performance assessment within thetraditional school structures. During a normal school day under normal conditions, I haveto find a way to put together panels of judges I will need several times during the year.Where can I locate people? How can I begin to involve the community outside school?Also, without the benefit of the university personnel who work in our building,' how doesa teacher put together these panels?

I feel very strongly about providing opportunities for performance assessments. The

kind of information I received about each student and the reactions of the students makeit clear to me that this is a much better method of assessing understanding than typicalpaper and pencil tests. If I can assess my students' understanding in a more realisticsituation and at the same time increase their confidence in themselves in relationship tomathematics, how can I simply rely on only traditional tests? '

*The College of Education at Michigan State University has been in a partnership with our school since January 1989.This collaboration is aimed at enhancing the education of practicing professionals (at both institutions), prospective teachers,and high school students. Some of these individuals served as judges are part of our partnership work.

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References

California Mathematics Council. (1989). Assessment alternatives in mathematics. Berkeley,CA: Author.

National Council of Teachers of Mathematics. (1990). Curriculum and evaluation standardsin school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teachingmathematics. Reston, VA: Author.

National Research Council. (1989). Everybody counts. Washington, DC: National AcademyPress.

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Appendix

Algebra 2Discussion Final Guidelines

Please keep in mind that this is a new experience for the students as well asfor us. Give the students plenty of opportunity to explain themselves but ifit is obvious that they are trying to fake it or are unsure of themselves letthem know that it is not what we are after and move on.

Only one student per problem. They have been instructed that they will haveto discuss the problem on their own without help from other members of thegroup. After you feel this student is finished if you want to ask anotherstudent some questions about this problem that is fine.

If you pick a problem that a student seems unprepared for, let them do whatthey can and then come back to that student with a different problem. Pleasemake note of this on the evaluation form.

Please use the following evaluation sheet in assessing the student's discussions.If you find the categories I have outlined unusable or too constraining, feelfree to write comments in the comment section or on the back. In assigningthe final points you need to be as specific in your comments as possible. Alsoremember that I will need these forms to discuss their evaluations for anystudent who wants to check on their performance. If possible, please informthe students of their score. If you can't due to time restrictions andopportunity to confer with the rest of the team, I will be available for themto check grades before school and after on Wednesday and Thursday.

Algebra 2Discussion Final

Mathernatics:

1) Making sense of problem(Understanding concepts)

2) Problem-solving strategies(Methods used)

3) Accuracy of results

Name

4) Interpreting results 1 2 3 4 5(What do the results mean?)

Presentation:

1) Ability to communicate results 1 2 3 4 5(Clarity, use of charts/graphs)

II2) Explanation 1 2 3 4 5(Able to answer questions)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

Comments:

Overall Score

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Craft Paper 93-2

"COULD You SAY MORE ABOUT THAT?"

A CONVERSATION ABOUT THE DEVELOPMENT OF A GROUP'S

INVESTIGATION OF MATHEMATICS TEACHING

Helen Featherstone, Lauren Pfeiffer, Stephen P. Smith, Kathy Beasley, Debi Corbin,Jan Derksen, Lisa Pasek, Carole Shank, and Marian Shears

Published by

National Center for Research on Teacher Learning116 Erickson Hall

Michigan State UniversityEast Lansing, Michigan 48824-1034

September 1993

This work is sponsored in part by the National Center for Research on Teacher Learning, Collegeof Education, Michigan State University. The National Center for Research on Teacher Learning isfunded primarily by the Office of Educational Research and Improvement, United States Depart-ment of Education. The opinions expressed in this paper do not necessarily represent the position,policy, or endorsement of the Office or the Department.

© 1993 by the National Center for Research on Teacher Learning

NATIONAL CENTER FOR RESEARCH ON TEACHER LEARNING

The National Cewerfir Research on Teacher Learning (NCRTL) was founded at Michigan State Universityin 1985 with a grant from the Office of Educational Research and Improvement, United States Departmentof Education.

The NCRTL is committed to research that will contribute to the improvement ofteacher education and teacherlearning. To further its mission, the NCRTL 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 Publication Clerk,National Center for Research on Teacher Learning, 116 Erickson Hall, Michigan State University, EastLansing, Michigan 48824-1034.

Directors: Robert E. RodenG. Williamson McDiarmid

Program Directors: Linda Anderson, Deborah Ball, G. W. McDiarmid, Barbara Neufeld,Kenneth Zeichner, Sharon Feiman-Nemser, Helen Featherstone

Director of Dissemination: Cassandra Book

Communications Specialist: Debra Peterson

Publications Clerk: Tamara D. Hicks

Office Manager: Brenda Sprite

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

Marianne ArnarelDeborah Loewenberg BallJoyce CainSandra CallisBarbara CamilleriAnne ChangDavid K. CohenAda Beth CutlerSharcn Feiman-NemserMary L. GomezSamgeun K. KwonMagdalene LampertParry LanierGlenda LappanSarah Mal artheyJames MeadSusan Melnick

Monica MitchellHarold MorganJames MosenthalGary NatrielloBarbara NeufeldLynn PaineMichelle ParkerRichard PrawatPamela SchramTrish StoddartM. Teresa TattoSandra WilcoxSuzanne WilsonLauren YoungKenneth M. ZeichnerKaren K. Zumwalt

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

Abstract

In this report, the authors describe the first year of an ongoing intervention study and the learningof a group of teachers and researchers who are working together to make changes in the ways theyteach mathematics. When reflecting on their beginning efforts, the researchers found that severalassumptions in their planned interventions were not born out by the group's experience. The subjectmatter content (integers) they selected as an initial focus of study posed major difficulties for severalof the teachers. The teachers' interest in talking about their own practices strongly influenced thegroup's interactions. Collectively, the teachers and researchers created a learning community thatwas grounded in watching videotapes of mathematics teaching in a third grade classroom anddiscussing ideas about teaching and learning in ways that were different from their own experiencesas teachers and students. Here, the researchers pose a set of conjectures that serve as a frameworkfor the continuing collaboration of this group of educators' inquiry into non-traditional approachesin the teaching and learning of mathematics.

"CouLD You SAY MORE ABOUT THAT?"A CONVERSATION ABOUT THE DEVELOPMENT OF A

GROUP'S INVESTIGATION OF MATHEMATICS TEACHING

Helen Featherstone, Lauren Pfeiffer, Stephen P. Smith, Kathy Beasley, Debi Corbin,Jan Derksen, Lisa Pasek, Carole Shank, Marian Shears

Flelen Featherstone, associate professorof teacher education at Michigan StateUniversity, is a senior researcher with theNational Center for Research on TeacherLearning.

Lauren Pfeiffer, a doctoral candidate ineducational psychology at Michigan StateUniversity, is a research assistant with theNational Center for Research on TeacherLearning.

Stephen P. Smith, a doctoral candidate inteacher education at Michigan State Uni-versity, is a research assistant with theNational Center for Research on TeacherLearning.

Kathy Beasley is a third grade teacher atAverill Elementary School in Lansing,Michigan.

Debi Corbin is a third grade co-teacher atAverill Eicmentary School in Lansing,Michigan.

Jan Derksen is a third grade teacher atBath Elementary School in Bath, Michi-gan.

Lisa Pasek is a sixth grade teacher atMayfield Elementary School in Lapeer,Michigan.

Carole Shank is a second grade teacher atAverill Elementary School in Lansing,

ichigan.

Marian Shears is a middle school teacherat Kinawa Middle School in Okemos,M ichigan.

I remember walking timidly into the first ses-sion of the Investigating Mathematics Teach-ing class. I had been intrigued by the descrip-tion of the class which would be looking atteaching math in a third grade classroom. Ihad just started my first teaching job, a parttime, temporary position in a ProfessionalDevelopment School. I would be teachingmath and since I had graduated more than 10years ago and had only a bad experience in amasters level math course, I decided I betterfind out "how to teach" math. I knew I wouldn'tbe satisfied to only use the teacher's guide.Little did I know that a year and a half later Iwould still be meeting with this group of edu-cators.

Jan

During my undergraduate work at MSU, I hadprofound "rebirth" in the area of mathemat-ics teaching. After two years of teaching, Ireturned to MSU to begin graduate studiesand underwent another refocusinga kind of"a-ha" experience. I realized tha, I was miss-ing something: the element of genuine dis-course and deliberation about mathematicsamong students. In spite of this experiencewhich was, at the time, quite shaking, I cameto the IMT group the following fall feelingrefocused and confident that I now had ahandle on what I wanted for my mathematicsstudents and for myself

I was intrigued with the idea of utilizing mul-timedia capabilities organized around a math-ematics classroom. I envisioned independentlymanipulating the HyperCard facilities to.findout "what a teacher could learn" from usingsuch technology.

I had no idea that the technology would serve.first as a springboard and later only as abackdrop to the very personal and collectiveprofessional investigation of mathematicsteaching and learning.

Lisa

Michigan State University, Last Lansing, Michigan 48824-1034 8 5 CI' 93-2 Page I

Jan and Lisa are remembering the feelingsthey had the first time they came to Investigat-ing Mathematics Teaching (IMT), which hadbeen described to them as an experimentalcourse for practicing teachers. Helen, Steve,and Lauren, researchers with the NationalCenter for Research on Teacher Learning, hadorganized this course in order to learn aboutthe ways in which a multimedia collection ofmaterials documenting teaching and learningin Deborah Ball's third grade mathematicsclass might be useful to teachers who wereinterested in thinking about new ways to teachmath.' They advertised a "group independentstudy" for elementary and middle school teach-ers in the masters program in Michigan StateUniversity's College of Edu.-:ation, but alsowelcomed teachers who wanted to participatewithout enrolling for course credit.

As the recollections of Jan and Lisa suggest,the seven teachers who became the IMT groupassembled with different agendas, hopes andfears. They were all, however, committed tothinking hard about the teaching and learningof mathematics. And like Steve, Helen, andLauren, the three university people, they feltsure that they wanted to teach math in waysthat were different from those they had expe-rienced as students in elementary and second-ary school. They were, however, in differentplaces on the journey away from traditionalmathematics teaching and toward somethingelse. And they were in quite different placesthen, on that afternoon in early October, 1991than they are today.

Much of the history of the group is the historyof individuals rethinking their own practiceand their relationship to and attitudes towardmathematics. Some parts of that story we havetold elsewhere (see, for example, Featherstone,Beasley, Corbin, Shank, and Smith, 1993;Featherstone, Pfeiffer, and Smith, in press;and Pfeiffer, Featherstone, and Smith, 1993).Some of it, however, is the story of an ongoingand evolving conversation about the teachingofmathernatics that has lasted through most oftwo academic years In December of 1991, atthe last meeting of the "group independentstudy," several of the teachers expressed aninterest in continuing to meet. Steve. Lauren,and Helen were also eager to continue thework that we all seemed to have begun and so

we set a schedule of biweekly meetings for thefollowing quarter. Over winter break all theother teachers decided that they too would liketo continue the connection. At the end ofwinter quarter, we decided to continue throughspring quarter, and at our last meeting, in June,1992, we agreed to reassemble in late Augustto discuss plans for launching the 1992-93school year. We have been meeting at leastevery other week this year.

By the end of the 1991-92 school year, severalof us had begun to comment on changes in thenature of the conversation that occurred at ourThursday night meetings. Over the summerLauren, Steve, and Helen decided to examin -these changes empirically and to try to under-stand them better. In January 1993, they askedthe teachers to help them to think about thechanges that had occurred over time; severalintriguing conversations followed. The con-jectures that we2 generated in these conversa-tions included the idea that, over time, theteachers had begun to "push each other more."In an effort to develop a clearer understandingof what this meant and how the ecology ofgroup meetings might support or discouragethis "pushing," we began to look at one meet-ing through the lens of "discourse analysis(Tamen, 1989; Coultard, 1992; Shultz, Florio,and Erickson, 1982; Cazden, 1989)" and toplan further study of the evolution of thisphenomenon over time (Pfeiffer, Featherstone,and Smith, 1993).

Discourse analysis provides one tool for look-ing carefully at aspects of the group's conver-sation. It helps us to see, in the actual talk, theways in which the group members participatein a collective study of the teaching and learn-ing of mathematics and, in turn, support oneanother's social, emotional, and intellectualefforts to make changes. We learned w' atsome aspects of the discourse looked andsounded like, and how the group worked to-gether to understand, for example, the math-ematical and pedagogical questions surround-ing the efforts of third graders to grapple withthe relative size of one-half and one-fifth. Wedid not explore, however, the way individualsexperienced our meetings, or the way theythought about the relationship between whatwe did on Thursday evenings and what theydid in their classrooms. Hoping to capture and

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communicate some of this, we returned to ourusual tools: writing and conversation. Jan,Lisa, and Marian volunteered to write aboutsome of their thoughtsabout their own par-ticipation in the group, about the relationshipbetween our joint conversations and theirteaching, about the evolution of the conversa-tion. The nine of us then came together on aSunday afternoon to read and to talk abouttheir texts. Helen then edited the texts and thetrenscribed conversation into this paper.

Lisa and Jan's writing (p. 1) took the groupback to the moment when teachers and re-searchers came together for the first time.During our first meeting we watched a video-tape of a third grade mathematics class dis-cussing a few of the number sentences that thestudents had generated in response to Ball'srequest that they "write number sentencesequal to 10." One student's suggestion that"200 take away 190 equals ten" launches adebate which remains unresolved by the endof the period: When a second student comes tothe board and shows why she thinks that 200-190=190, many of her classmates agree al-though some do not. For the next three mathperiods the students work on and discuss prob-lems that their teacher creates and poses inorder to explore and challenge the concep-tions that underlie the approach which leadsthe third graders to assert that "you can't takenine from zero so you write down the nine."Our conversation begins with Jan and Lisa'swritten recollections of what they said and feltduring these early meetings.

Our conversation did not, however, move fromthese early memories into a linear history ofthc group or of our memories of it. Rather,these texts--some of which are reproducedhere in italicslaunched us into some reflec-tions on schools, teachers, and administrators,and on the reasons that the teachers havefound the gT oup useful and even necessary.

The development of this Sunday afternoonconversation mirrors that of the IMT group.Like the group, it starts with some texts: Thetexts here are the writings of Lisa, Jan, andMarian. The texts for the IMT group were, tobegin with, videotapes and other materialsdocumenting teaching and learning in DeborahBall's third grade mathematics class. In recent

months we have sometimes started watchingvideotapes of our own math classes. Like theconversation in the group's regular Thursdaynight meetings, this conversation moves backand between these texts and other matters.And like the conversation in the regular IMTmeetings, this conversation ends up diggingdeeply into questions that come up because ofthe texts but are not always directly related tothem.

Kathy:

Carole:

GETTING TOGETHERWhen I read what Jan wrote, I just startedthinking about the first time I walked in.And "timid" is a good word. I had thisgreat fear that everyone here was going tobe really good at math. I knew I wasn't,and I kept thinking, "What are they goingto think when they find out that this per-son is in this math group that doesn'tknow anything about math?"

I was really confused about why I wasthere. I knew I waited to do somethingabout math, and Kathy and other peopletalked about the NCTM Standards.] didn'tknow a thing about the Standards, plus Ididn't know much about math.

FIRST IMPRESSIONS: WATCHING TAPEWhen I first watched Deborah Ball end hermath class "in the middle of a problem" Ipanicked. How could she do this to thesechildren? Would they be able to sleep at nightnot having heard the answer? Is one problema day enough? How would I ever get throughall my "material" if I taught like that? Whereare the manipulatives? How can you teachnegative numbers to third graders? Why wouldyou?

In some of our early session as I recall theothers talking about "teaching this way," the"Standards," and "discourse in the class-room." "This way of teaching" was even com-pared to the "Whole Language" method. Thiswas all new to me and I was afraid to speak upin the beginning. I spent a lot of time listeningand thinking. I often heard doubt in some ofthe other voices. Some of this had to do with a

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lack of support in their district but I alsosensed that it had to do with the fact that thiswas a new way of looking at teaching math.Maybe we were all "walking on thin ice."

Jan

Mv first response to Deborah spending 3 dayson 200-190=? was to turn to one trustedfriend in the group and say, "I can 't believeshe spent 3 days on those problems! As thegroup discussed the lesson I commented thatadministrators in my district would have avery hard time with this idea.

Late in the school year, I talked of my ownperspective on the 200-190 problem as beingan important reference point in the changes inmy own thinking. Seven months after thatrestless, urgent interjection (that my adminis-trators wouldn't understand spending 3 dayson one problem), I found myself acknowledg-ing that regardless of opposition or resis-tance, real or perceived, that I had been expe-riencing in my teaching, it was ME who hadhad so much apprehension and conflict withthis kind of genuine discourse as a primarypedagogical tool.

Now, I can't believe how much I've changed.I've come to see what really happens in aclassroom where students use discourse toconstruct their own understanding. To a largeextent, I had used my district administrationas a scapegoat for why I couldn't "do" thatkind of discussion in my mathematics class-room. By this later point, a trust has devel-oped in the group that has enabled me to putout on the table that I had been unwilling tosay "I have a problem with this. . . ."

But it wasn 't just a matter of emotional sup-port, but also ofproviding an environment forintellectual exploration.

Lisa

As a beginning teacher I was in essence start-ing with a clean slate. I was able to make myown curriculum decisions and I had no onewatching over me. This was actually a fright-ening experience. It seemed that the othermembers in the group had either been teach-ing for several years and "needed" to makechanges in their teaching for survival or theyhad "grown up" with this way of looking atteaching. I wasn 't sure what I thought or

Jan:

believed. I had no foundational experiencesto compare this to. I didn 't have a store ofnegative experiences nor of positive teachingexperiences. A lot of teachers at my schoolwere looking at math in different ways but 1can still remember being adv.sed to "just usethe teachers guide your first year."

I remember those early experiments with "test-ing the ice." I bought math journals and,borrowing from Deborah Ball, 1 asked mythird graders to "write number sentences equalto 10." 1 was amazed by their responses andby what I learned about their understandings.I looked forward to sharing this with the restof the group.

Jan

I agree so much with what Lisa wrote.Because I think she is saying that so muchis self-imposed, and I feel that way, too.I actually get support from myprincipal,but not from my colleagues.When I say, "Have you tried this or that?"it's as if I don't exist because I don't havecredibility. I think I had some feelingsreaffirmed here. And so I began thinkingthat maybe I am making some right deci-sions, decisions that are best for kids,regardless of the "What chapter are youon?" questions. and I've decided, eventhough it's frustrating and I hope thatmaybe someday we'll be able to engage,that right now I have to sort of do mything, and talk to the one or two peoplewho want to talk.

Kathy: Do you think it's partly because you are afirst year teacher?

Jan:

Marian:

8

Probably. They've been around. . . .

Lisa used to talk a lot about the feelingthat no one would listen. No one.

CI' 93-2 l'agc 4 co 1993 by thc National Center for Research on Teacher Learning

Lisa: I think that they listen, but. . . . A teacherin my district is really struggling to makechanges, and she made the comment, "Fortwenty-five years I've tried to do every-thing that they've asked me to do and it'snever been what they've wanted." It wasterrible hearing the pain in that statement.

So often teachers will look and they'llsay, "Here's a teacher who's enthusias-tic, they're dedicated to their own profes-sional development; they're inexperi-enced, obviously. They don't know whatteaching is all about yet." That's why it isso revitalizing to come here and see thatthere are teachers here who have been inthe classroom for twenty-five years, theyknow the ropes and they're still fighting.

Kathy: It's true. I remember when I first startedout, if I'd make a comment that was dis-couraging, or not optimistic, people wouldsay, "Now you sound like a real teacher!"It wa'; like they were saying, "Now peoplewill listen to you."

Jan: That's what makes this group unique.Teachers, we beat each other up. We aren'tsupportive of one another, that's my ex-perience. It's as though there's a compe-tition: Who's going to have the cutestbulletin board display in the hall? or what-ever.

The IMT group is a place in which we beganto share our frustrations and insecurities. Wecame to accept that teaching is strugglingofan honorable, honest sort. We also continuedto examine interactions and issues in the vid-eotapes and connect them to our own teach-ing, our own experience and ideas. We beganto question episodes of mathematics teachingin the videotapes and echoes of those ques-tions reverberated in our own heads, pushingus to ask such questions of ourselves and ofeach other.

The group has provided me with a safe envi-ronment in which I can discuss my own teach-ing, listen to others, and force myself to thinkabout and question what I am doing. But mostimportant, this nurturing and challenging at-mosphere has encouraged me to take risks.

Lisa:

Lisa

I have a time every week where about halfof the kids go out of the room to band. Theones who are loft seem to be the ones whohave more trouble with math, so I talkedto them last week about having a mathsupport time. I had them break into groupsin which some people would be givingsupport and others would be receivingsupport.

I was watching two students who wereworking together and one was pushingthe person who was having trouble, actu-ally pushing on them. It was very vivid.And I said, "Stop!" I held up my pen andI told them that there's a difference be-tween pushing a pen and sending it fly-ing, and just supporting it. I asked them tothink about the difference. And that de-veloped into thinking with the kids aboutwhat you do when you help someone ridea bicycle: You hold that back bar and yourun with them, and you don't know whatto say that's going to help, you just en-courage them, saying, "You can do this."You're giving the support you can, butyou can't "teach" them: The other personreally has to get it for themselves.

Anyway, at the end of the period I hadthem spend the last five minutes writingabout what they learned, or what theywere thinking. And this one boy wrotc, "Ilearned that getting support is much harderthan giving support." And I thought hehad unlocked the mystery of the universe.

Michigan State University, East Lansing, Michigan 48824-1034 CP 93-2 Page 5

And I sort of thought that the bike anal-ogy kind of connects to what we do here.Even though we are not world class cy-clers, we can still help each other, balanceeach other out. Even though we are notexperts.

Kathy: It's harder to be on thelearn, than to be running

Helen: And it makes you thinkand what's involved inport.

bike, trying toalongside.

about yourselfaccepting sup-

THE DEVELOPMENT OF DISCOURSEOur group began as a collection of individu-als who were all interested in examining theteaching of mathematics. Some of us kneweach other (three even taught at the sameschool), and some of us met for the .first timeat our first session. We all thought that wewere simply participating in a ten-week uni-versity class. But what happened in that classover time changed our perceptions and ourgoals.

A (first our meetings centered around watch-ing and discussing videotapes of DeborahBall 's third grade mathematics class. Theinstructors selected the tape each week, andwe discussed what was happening in thatclass and how it pertained to our own class-rooms. Through these weekly discussions apattern began to emerge. At first we focussedon what was happening in Ball's classroom:what representations the teacher used, howspecific childrei. made .sense of problems, etcetera. Over time, however, our focus beganto shift. Instead of looking ;list at what washappening, we also began to look at why thatmight be happening. We, collectively, took astep back to look at the bigger classroompicture.

As a part of this new way of looking at thetapes, we began to examine the patterns ofclassroom talk: the kinds of questions theteacher asked and the kinds of responses kidsgave to her and lo each other. Specifically,our group started to jocus on the classroomculture and the part that discourse played inft.

By discourse we meant more than just class-room talk. We meant that students were in-volved in explaining their ideas to each other,that they had, and shared, reasons for theirideas, and they listened to each other. Gooddiscoursc requires an environment where stu-dents don't look to the teacher for "answers,"but look to each other (and within them-selves). They don 'tjust accept answers blindlyeither; they ask for (and offer) logical rea-sons.

As our group examined discourse in DeborahBall 's classroom, we began to change in subtleways. We found ourselves asking each otherthe same questions that Ball asked her stu-dentsquestions like "Why do you thinkthat?" What do others think?" or "Could yousay more about that?" We began to push eachother's thinking in ways that Deborah pushedher students' thinking. In other words, whilestudying the discourse on the tapes, we cre-ated a classroom culture of discourse withinour group.

This development ofdiscourse seemed to havea dual relationship with trust within our group.First, we were able to develop it because wewere beginning to trust each other. But ourtrust also grew as a result of participating indiscourse, perhaps because respect foranother 's ideas is inherent in good classroomdiscourse. Interestingly, no one in our groupeve ,. talked about the pr:rallels that were de-veloping between our group's talk and thetalk in Deborah Ball 's third grade classroom.We weren't consciously aware of what washappening, but we were modeling somethingthat grew out of our shared experience.

We did, however, openly discuss how we couldcultivate discourse in our own classrooms.We wanted our students to question each other,and to give and expect reasons for theirthoughts. As teachers, we tried to model dis-ourse for our students. We also taught it

directly, saying things like, "It 's importantthat we listen to each other."

Perhaps we were better able to teach andmodel good discourse only after we had expe-rienced it ourselves.

It is difficult to trace this experience and itvaries for different members of the group.But, there is a shared sense that we movedfrom a set of concerned individual teachers toa collective that seeks and supports a criticalbut trusting atmosphere in which we purse an

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emerging shared vision ofmathematics teach-ing. This vision includes fostering a class-room culture of discourse much like the onewe have experienced.

Marian

Lisa: When I looked at this description of theway we developed discourse in here, acouple ofthings struck me. One was whereshe says "Good discourse requires an en-vironment where students don't look tothe teacher for `answers' but look to eachother (and within themselves). They don'tjust accept answers blindly either; theyask for (and offer) logical reasons." I

crossed out "students" there and put"teachers," because that begins to de-scribe the discourse within our group.

Then, further down here, it says, "Per-haps we were better able to teach andmodel good discourse only after we hadexperienced it ourselves." I felt verystrongly about that and thought a lot aboutcomments that we've made about howthe support that we've felt, the support wecontinue to feel, supports the develop-ment of what we do in our classrooms.

Debi: I was struck by that same thing, becausethe thing that I realized was that I reallydidn't know what discourse was and Iwas trying to create it in my classroom.And here we have done it ourselves,slowly. It hadn't really occurred to methat that was what we were doing in here,but we have kind of taught ourselves, orat least I have been taught on my ownlevel, so maybe I can take that and trans-fer it more easily to what I do with mystudents.

PUSHING EACH OTHER ANDDEVELOPING TRUST

At our first meeting of our second year, I wassharing my own ideas about whether to startthe year with a set of lessons which wouldengage students in a very "safe" introduction

to discoursefocusing on setting classroomnorms. The students would be encouraged toagree or disagree with each other's puzzlemodels given very cut and dry criteria.unexpectedly, Kathy asked a question thatcame at me from across the table like a bullet:"But, isn 't the discourse in the task? " THUMP.And I think to myself.. . she's right. If my taskis well constructed, won't students engage ina more genuine discourse? But it is the begin-ning of the year, so maybe we can start off ina safer way, introduce norms and vocabularyin a set of lessons which would require lessintellectual risk-taking for these new sixthgraders who have never been asked to thinkabout whether they agree or disagree with amathematical idea.

Already we were challenging each other andourselves to not take anything for granted, butto really dig into what decisions 1,e weremaking and why we were making them. Thisintellectual pursuit of teaching could onlytake place in a setting of trust. In this context,a hard hitting question is not meant to embar-rass or demeanIt is a supportive pushOften these questions are ones that are hardto push ourselves on because we may tooquickly settle for our own perspective.

Lisa

Kathy: I don't remember even asking that ques-tion.

Lisa:

Helen:

Kathy:

Helen:

You don't?

[to Kathy] I thought Lisa had asked itbecause the next week you told us thatyou had gone home aftcr the meeting andsaid to yourself, "Oh, okay, now I under-stand: The discourse is in the task."

That makes sense to rne, Helen.

But, actually, it turns out that you askedthc question.

Michigan State University, East Lansing, Michigan 48824-1034 CP 93-2 Page 7

Kathy: I think that's an example of trust. Lisawas telling a story, almost in a way likeshe's kind of telling us how to do dis-course or get ready for discourse. And itjust made us all think about it. I don't Lisa:know. I don't think you ask questions likethat to people if you don't think that it'sgoing to be okay with them.

Like, if I ask a question in my staff meet-ing, people generally assume that I'masking it not because I'm seeking infor-mation, but I'm sort of maybe trying toput their idea down, or play out a problemwith it. And I think here, when we askquestions people understand that what wewant is information.

Marian: You just said it: "Assume." We don'tassume the motive. If something doesn'tstrike us right, we say, "What do youmean by that?" or "Are yousaying that . . ?" So instead of assumingnegative motives, we try to clarify.

Jan:

Debi:

Do any of the rest of you find yourselves,when you're talking with people, saying,"I'm not trying to be argumentative?" Ifind myself saying that a lot, because I doask questions. I want to understandpeople's rea,sons.

GETTING TO TRUSTI want to know how we got to this trust.Because it's really hard for me to get to apoint where I feel free to talk. I wastalking to another instructor about why Idon't talk in that class, versus why I dotalk in here. She was wondering why, andI'm not sure why except that we've beentogether longer and I was allowed to bequiet for as long as I needed to bc, until Ifelt safe enough to start sharing things.

Marian: I think too, that the shared experience ofwatching and discussing thc tapes ofDeborah Ball's class was a big part of it.We were building a common frame of

reference: We could always say, "Like inDeborah's class. . . ."

And maybe that common frame of refer-ence was a safely net: We were talkingabout things that we were really thinkingthrough in our own classrooms when wewere talking about what we saw in theother class. We could say, "Look at howdirective she was," without saying,"You're being directive," "I'm being di-rective." We could say, "This person onthis tape that isn't here is being direc-tive," and we could discuss whether ornot that's okay without turning it intosomething personal.

Marian: So part of that trust is because we couldtake those risks with someone else first.Deborah took those risks for us.

WHAT ARE QUESTIONS FOR?Lauren: I wanted to pick up on Kathy's comment

that when questions were asked in thegroup it was to find out more information.I wanted everybody to talk more aboutthat, because that surprised me: I wouldhave thought questions also served otherfunctions besides getting information.

Helen: I think what she was saying is that they'renot a backhanded way of criticizing. Thatwe assume a wholesome motive, if youwill.

Debi: And that if you want to know something,it's safe to ask.

Marian: What I think you're saying is that we alsouse questions in anothcr way: Maybe weask questions to get each other to thinkabout things from a different perspective.

Lauren: I guess that's what I'm curious about. Isthat truc? And how do you think aboutthat?

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Marian: I was going to say that I think they'rerelated. That we're pushing each other,but it's with the assumption that it's re-ally in there. We're not saying, "This iswhat you don't know," or "I don't thinkyou've thought about this." We're say-ing, "Have you thought about all theseangles?" or maybe we're pushing it alittle, because we realize from our expe-rience that pushing does help us clarifyfor ourselves. And so we're helping eachother to clarify.

Lisa:

QUESTIONS AND THECOCONSTRUCTION OF AN IDEA

And we are helping to develop an idea inprogress. I think it's really fascinatingthat Kathy didn't remember asking thequestion rIsn't the discourse in thetask?"] that drove my journal entries forweeks. I had pages and pages that I wroteabout this question; it really pushed me tothink about my rationale.

to move beyond where they are, I'm justtrying to understand it. Sometimes I askthem question in order to make themthink harder about this or to move themwith their reasoning. Probably there areother reasons.

Marian: And that would hold here in the grouptoo?

Kathy: I think so.

I think about the conversation we had thenight when Steve was taking his verystrong stand on the multiplication tables.I'm thinking about the questions I wasasking him that night: Some of the time Iwanted to know what he was thinking,and some of the time, my questioning wasto say to him, "Stop and think about this,Steve."

Lauren: So that's not just understanding; that'sYou asked that question and later on youwere saying, "I understand now," as if itwere somebody else's idea. It was kind ofthis in-progress thinking that you threw Kathy:out, "the discourse in the task." And thenCarole built on it and it became this idearight there on the table and we reallylooked at it and it still is really with us, Lisa:pushing us. And then it kind of came backand you left with your own new versionof what the question was. It was kind oflike what Helen talked about when shesaid that our ideas are not just celebrated,but people grab onto a half a conjectureand run with it. Especially in this in-stance, where you threw out somethingthat just seemed so profound.

Kathy: I think questioning is several differentthings. I want to revise. Jan made methink about the questions I ask in myclassroom, and in this group as well.Sometimes I ask students questions be-cause I want to understand what they'rethinking. I don't want them to changewhat they're thinking. I don't want them

trying to push him to think hardcr?

Yeah. It almost feels like there's a thirdone, likc . . .

One is "Keep going with that idea, saymore," and other is "Stop and look backon it."

Lauren: Another word that as come up in thewriting and the conversation is "chal-lenging," and I've been trying to think ifpushing and challenging are the samething.

Kathy: I think they're different.

Lauren: How would you define the difference?

Michigan State University, East Lansing, Michigan 48824-1034 CI' 93-2 Page 9

Carole: Kathy, in your class when you say, "Iwant you to look at a new idea that's onthe board. I want you to think about it," ina sense aren't you challenging them?

Kathy:

Kathy:

Helen:

Steve:

Kathy:

Jan:

Kathy:

Lisa:

That's more like pushing.

Challenging is almost . . .

Confrontational?

It's a way of disagreeing?

But it's a strong way of disagreeing. Ithink you have to have a sense that peopleare going to stay with you and not takeoffense and get angry before you go tochallenge them.

Are you sort of thinking of challengingand pushing an idea?

I think it's gentler.

Lisa: I didn't feel that it was supportive, so Ibacked out of it. But other people werestill engaging it so they may have thoughtit was supportive.

Kathy: I thought it was a great discussion and Ijust loved it. But now I'm wondering,"What did I say? How did I say it?"

Lisa: It could have just been where I was at thatparticular day: I was feeling urgency aboutfiguring out today what I was going to dofor this marking period, and it was a verytheoretical discussion. That was prob-ably frustrating for me, wanting to haveanswers or some feedback about whatmight I do this week when I'm calculat-ing grades. Really, you know, it was moreabout questioning answers and not aboutanswering questions. But if we say thegroup is about questioning answers andnot answering questions, that's what itwas.

Kathy: I guess I left feeling that it was unre-solved. I didn't think people had sanc-tioned anything.

There are supportive questions, and then Marian:there is pushing with support. But wehave had some discussions where there issome challenging going on, and, for me,it was uncomfortable. That assessmentdiscussion that we had was very uncom-fortable for me. I think I probably saidtwo words that whole night. I didn't wantto be confrontational, but I was frustrated Steve:about how to ask questions without beingconfrontational. There was a lot of con-frontation going on and it felt like it wasbackhanded so I kind of retreated.

Lauren: So that wasn't a time that you had a sensethat thc interaction was supportive push-ing? Lisa:

I never felt like we had to come to ananswer. So I guess what I'm saying is thatthat's why I didn't view it as a confronta-tional meeting, I wasn't looking for ananswer. Maybe Lisa was looking for ananswer.

Maybe what made it seem confrontationalis that it didn't seem like there was asmuch open thinking going on as usual.On other occasions when people are push-ing each other, there's more listening andwondering. I didn't seem like there wasany change taking place.

It felt like we were talking at each otherand not with each other.

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Jan:

Jan:

Helen:

Jan:

Lisa:

Carole:

I was listening more than talking, be-cause I'm trying to resolve a lot of thesethings in my own mind. When I grade, Isort of hide and don't let anyone elseknow how I do it. I got my school to goaway from A, B, C, D. So I just liked tohear the rationale. I have a parent whosays, at every report card, "I don't agreewith this form of grading, I want to know Jan:how she has improved." So I wrote up anexplanation that I thought would showhim that, but he wants to know if she hasgone to a 98 from a 95!

Kathy: I guess I want to know what people think:Do you think confrontation is bad, then,not a good thing for our group to do? I feelthat reaching a point where we can actu-ally confront each other and be challeng-ing is a good place to be. It implies trust.

Is CHALLENGING BAD?Are we assuming that we all think thatchallenging is negative? Because it seemslike we are using the word that way. I Jan:wonder if some of the negative feelingsthat we might have about this are be-cause, often when you are challengingstudents in your classroom you're tryingto control their thinking. I mean, you'rehoping they'll go in a certain direction.

It's our ideas that are being challengedand not our being. But we're so used, asteachers, to feeling that when our ideasare being challenged our very being isbeing torn apart.

Kathy: We can't separate ourselves from our

You don't want them to conclude thatdivision is commutative.

Well, if you're challenging their thinkingabout multiplication, your goal is to try toget them to come to some understandingabout what multiplication is. There issome control because you 're asking thequestions, and they're not just questionsthat are out there somewhere and aremeaningless. So, I wonder whether chal-lenge implies control.

practice.

But maybe we can do that here, right?

Kathy: But I think conflict deepens a relation-ship. lt might be kind of uncomfortablewhen you're in it. . . .

Debi: Then there are attack questions, and thoseyou don't want. I don't think anybodyfeels safc in that sort of conflict. Butconfrontation, to me, is okay.

Kathy: See, I come from a family where peopleare pretty confrontational and do a lot ofattacking, but we've always kind of en-joyed that. But a lot of people don't, andI'm alwiys kind of surprised by thatwell, I'm not surprised anymore, but Iforget.

With wrestlers, you have a prize winnerand you have a challenger who is hopingto take control over the prize. The word"challenge" has the connotation that there Lisa:is going to be a winner.

I drew a mountain: I think of a challengeas going bcyond, it's not getting at justthe basics, it's going beyond.

9 5

If anything, I feel bad that I didn't noticethat Lisa bailed out.

Debi is saying that she was allowed tostay quiet until she was ready. I wasallowed to stay quiet because I didn't feelcomfortable engaging at the level thatother people were engaging. But I wasstill engaged. I was still very much corn-

Michigan State University, East Lansing, Michigan 48824-1034 CP 93-2 Page 11

Steve:

Debi:

pelled by the discussion, but I had justretreated to a safe place where I could gcta handle on what was going on because Ididn't feel safe.

But it's one thing to be putting somethingon the table and saying, "Do you thinkthat this is true?" and talking about thatidea. And it's another thing to have "Thisis what I think," "This is what I think,"and having a butting of ideas instead of ameshing of ideas. That was what wasfrustrating: I was hoping that there wouldbe a collaboration, a wondering aboutwhat are some ways to handle this. I wasnot looking for an answer like, "That'swhat Teri does, so that's what I'm goingto do." I was looking for intellectual ex-ploration.

It occurred to me that when Kathy wastalking about all the ways we questionand about the teachers in the staff roommisinterpreting questions, I wonder howkids interpret questions.

Because kids say to me, "Why do youalways ask me why?"

I'm wondering if we think about whatassumptions they're making about whatwe're saying. I kind of wonder how they'refeeling. They need that silent time. I putthem on the spot.

CIRCLING BACKKathy: I want to go back to what Marian wrote

about the development of our discourseand say quickly about something I dis-agreed with: I don't think we developeddiscourse because we saw it in action on Helen:Deborah's tape. I think we developeddiscourse because of our task. I think thediscourse is in the task, and I think that'swhy our group has discourse, not becauseit's something we learned from watchingDeborah's tape.

Marian:

Kathy:

Marian:

Kathy:

Marian:

Kathy:

Helen:

It can't be both?

It could be. But that part isn't in here. Itmakes it seem like we were pretty pas-sive, that somebody taught us how to dodiscourse, and then they did discourse. Ithink we were much more active in that.Because of our task, which was under-standing our own teaching, understand-ing our own mathematics.

I would agree, I would say it's both.

Because I'm thinking about my own class-room. They don't watch how to do dis-course and then learn how to do dis-course. If 1 give them a good task, they doit.

The discourse is in the task?

Now it's not a question, it's a declarativesentence. So that's what I'm disagreeingwith.

Well, do we need a group simply becausewe're trying to do something that's hardand different, or is it also because we aretrying to create conversation, and we needto engage in conversations in order tocreate them?

Kathy: Lucy Caulkens says that teachers can'treally understand how to teach writingunless they write themselves. It doesn'tquite fit for me here, but I guess it must betrue.

I wouldn't agree that it must be true.Maybe it's too pat.

Kathy: I mean, it seems sensible: You can't teachwriting well unless you write, you can'thave good discourse unless you . . .

CP 93-2 Page 12 (c.^, 1993 by the National Center for Research on Teacher Learning

Lauren: But I think that the function of the groupis broader than that. We are learning morethan just about how to create conversa-tions in classrooms, and maybe that'swhy it's too pat to make it that simple. Itmakes sense. But it sort of discounts thepower that we're learning about, the powerthat the group provides. But then whenyou come back to say, well, could youlearn to create conversations in the class-room without a group, then that's clear.

My other question was, is the need for agroup specific to mathematics? Is it re-lated to the fact that we all had bad expe-riences learning math?

Kathy: I don't know, I mean, I want to changehew I'm teaching reading, and I've beenthinking about it a lot this week, but Ithought, "This is going to be really hardto do all alone."

I think it's all the same thing.

Notes'For a detailed description of the agenda of the researchers

and the early history of the group, see Featherstone, Pfeiffer, andSmith, in press.

'The nine authors listed here, plus one other group memberwho has been unable to participate in the writing of this paper.

ReferencesCazden, C. (1988). Classroom discourse, the language of teach-

ing and learning. Portsmouth, NH: Heinemann.

Coultard, M. (1992). An introduction to discourse analysis.New York: Longman.

Featherstone, H., Pfeiffer, L., & Smith, S. P. (1993). Learningin good company (Research Report 93-2). East Lansing:Michigan State University, National Center for Researchon Teacher Learning.

Featherstone, H., Smith, S. P., D nsley, K., Corbin, D., & Shank,C. (in press). Expanding the equation: Learning mathemat-ics through teaching in new ways. East Lansing: MichiganState University, National Center for Research on TeacherLearning.

Pfeiffer, L., Featherstone, H., & Smith, S. P. (in press). "Do youreally mean all when you say all?' A close look at theecology of pushing in talk about mathematics teaching.East Lansing: Michigan State University, National Centerfor Research on Teacher Learning.

Shultz, J., Florio, S., & Erickson, F. (1982). Where is the floor?Aspects of the cultural organization of social relationshipsin communication at home and in school. In Children inand out of school. Washington, DC: Center for AppliedLinguistics.

Tannen, D. (1989). Talking voices: Repetition, dialogue, andimagery in conversational discourse. New York: Cam-bridge University Press.

97Michigan State University, East Lansing, Michigan 48824-1034 CP 93-2 Page 13

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1 6

CHANGING PRACTICE:TEACHING MATHEMATICS

FOR UNDERSTANDING

A PROFESSIONAL DEVELOPMENT GUIDE

Developed by

Steven A. Kirsner, John S. Zeuli,Linda Alford, and Michael J. Michell

The Teaching Mathematics for UnderstandingVideo Project

National Center for Research on Teacher LearningMichigan State UniversityEast Lansing, Michigan

1107

INTRODUCTIONTO THE VIDEO

A. Students' Role as Learners ofMathematics

B. Mathematics Content

1. Mathematics asProblem-Solving

2. Mathematics as Reasoning3. Mathematics as Communication4. Mathematical Connections

C. The Teacher's Role

1. Worthwhile tasks2. Classroom discourse3. Learning environment4. Systematic analysis

ORIENTING QUESTIONSFOR VIEWING THE VIDEO

Questions about the Students' Role

1. When you observe the students inthe video,

a. What are students saying thatyou find particularly interestingor considerably different fromwhat students typically say inconventional mathematicsclassrooms?

b. What are students doing thatyou think is different fromwhat one typically sees duringmathematics instruction?

109

ORIENTING QUESTIONSFOR VIEWING THE VIDEO

Questions about MathematicalContent

1. The NCTM Standards suggestthat students should engage inworthwhile activities. How wouldyou define the content thesestudents are thinking about andhow does this content differ fromthe content offered in yourtextbooks?

no

0 Questions about Mathematical Content(cont.)

2. The NCTM Curriculum andEvaluation Standards recommendsthat mathematics curricula at allgrade levels focus ona. mathematics as problem-

solving;b. mathematics as reasoning;c. mathematics as communication;

andd. mathematical connections.

What examples of each curriculumfocus are shown on the video?

3. How do teachers and othereducators in the video think aboutthe role and relevance of basicskills and procedures?

1U

ORIENTING QUESTIONSFOR VIEWING THE VIDEO

Questions about the Teacher's Role

1. How do the teachers in the videodescribe what students in theirclassrooms are saying or doingdifferently as they learnmathematics?

2_ What kinds of questions doteachers pose and how do they getall students involved in thinkingabout these questions?

3. How do teachers guide students'responses and how do theyrespond to students' unexpectedideas?

Questions about the Teacher's Role (cont.)

4. Teachers in the video use anddescribe different approaches toteaching mathematics (e.g., usingmanipulatives, alternative assess-ments, student writing, groupwork).

a. Could teachers use thesedifferent approaches duringmathematics instruction, yetremain tied to traditionalmathematics teaching? Forexample, could a teacher usegroup work without teachingmathematics for understanding,or use manipulatives withoutany focus on helping themreason about mathematics?

113

0 Questions about the Teacher's Role (cont.)

b. If so, what makes thesedifferent approaches consistentwith teaching mathematics forunderstanding?

5. What difficulties do teachers andother educators describe in learn-ing to teach mathematics forunderstanding?

III

a. What do they describe as thebenefits of trying to overcomethese difficulties?

114

Questions about the Teacher's Role (cont.)

b. What obstacles might youconfront if you want yourstudents to engage in theseactivities? How might youmanage these obstacles in orderto help students focus onunderstanding mathematics?

MS. BALL'S 3RD GRADECLASSROOM

Discussion questions:

1. According to the NCTMCurriculum and EvaluationStandards ,

"Students need to experience genuineproblems regularly. A genuine prob-lem is a situation in which, for theindividual or group concerned, one ormore appropriate solutions have yet tobe developed. The situation should becomplex enough to offer challenge, butnot so complex as to be insoluble...Learning should be guided by thesearch to answer questionsfirst at anintuitive, empirical level; then bygeneralizing; and finally by justifying(proving). " (p. 10)

116

MS. BALL'S CLASSROOM (cont.)

a. Do you think that the studentsin this class N,vere trying tosolve a "genuine problem"?What features of the problemmake it genuine or notgenuine?

b. How did Ms. Ball use thisproblem to engage students andelicit their mathematicalreasoning and communication?

c. What could be a genuineproblem for students at yourgrade level?

117

MS. BALL'S CLASSROOM (cont.)

2. During the whole class discussion,the teacher rarely gave studentsany hint about whether shethought they were right or wrong,or even if they were on the righttrack. Why do you think shebehaved this way? What effectdid her behavior have on thestudents' reasoning about math?

3. One of the purposes for thislesson was to diagnose whatstudents knew about fractions asthey began a unit on fractions.

a. Why might it be important fora teacher to learn what herstudents know about a topicthey had not yet studied?

118

MS. BALL'S CLASSROOM (cont.)

b. What did you find out aboutwhat students knew? Didanything surprise you?

c. Contrast this teacher'sdiagnosis of her students'knowledge of fractions with amore traditional pre-testapproach.

4. Ms. Ball accepted a student'sdefinition of the unit as "cuttedbread." Would you do this? Isthis good practice? What is yourreasoning?

1!9

MS. BALL'S CLASSROOM (cont.)

5. According to the ProfessionalTeaching Standards, "The teacher ofmathematics should orchestrate discourse by:

posing questions and tasks that elicit,engage, and challenge each student'sthinking;

*listening carefully to students' ideas;asking students to clarify and justify their

ideas orally and in writing;deciding what to pursue in depth from

among the ideas that students bring upduring a discussion;

deciding when and how to attachmathematical notation and language tostudents' ideas;

deciding when to provide information, whento clarify an issue, when to model, whento lead, and when to let a student strugglewith a difficulty;

*monitoring students' participation indiscussions and deciding when and how toencourage each student to participate. "(p. 35)

0 MS. BALL'S CLASSROOM (cont.)

Orchestrating discourse depends"on teachers' understandings ofmathematics and of their studentson judgments about the things thatstudents can figure out on their ownor collectively and those for whichthey will need input." (p. 36)

a. In what ways did Ms. Ballattend to these suggestions fororchestrating discourse?

b. Do you see places where ateacher might have madedifferent decisions from thoseMs. Ball made? What do youthink her decisions were basedupon? On what are yourdecisions based?

MS. BALL'S CLASSROOM (cont.)

6. The students in this class seemedcomfortable reasoning aboutmathematics in a whole groupsetting, including at timesdisagreeing with their classmates.How might a teacher establishthese norms of communication inher class?

122

MS. JONES'S7TH GRADE CLASSROOM

Discussion questions:

1. The video segment shows studentsrepresenting numbers with baseten blocks. According to theCurriculum and EvaluationStandards,"Students need to experience genuineproblems regularly. A genuineproblem is a situation in which, for theindividual or group concerned, one ormore appropriate solutions have yet tobe developed. The situation should becomplex enough to offer challenge, butnot so complex as to be insoluble...Learning should be guided by thesearch to answer questionsfirst at anintuitive, empirical level; then bygeneralizing; and finally by justifying(proving)." (p. 10)

123

MS. JONES'S CLASSROOM (cont.)

a. Do you think that the studentsin this class were trying tosolve a "genuine problem"?What features of the problemmake it genuine or notgenuine?

b. How could the problem be re-cast to make it more genuine orless genuine?

c. How does this method oflearning decimals compare withyour textbook's treatment ofdecimals?

.1.?4

0 MS. JONES'S CLASSROOM (cont.)

2. The students in this segmentworked in groups, as they doregularly in Ms. Jones's class(and as is encouraged by theStandards).

a. What is the rationale for groupwork? What benefits do youthink students derive fromworking in groups?

b. This class was filmed in themiddle of the year. Whatnorms of classroom discourseand behavior do you think Ms.Jones had to cultivate through-out the school year for studentsto be able to work together as

0 they did in this segment?

MS. JONES'S CLASSROOM (cont.)

c. During the class discussion Ms.Jones consistently pushedstudents to elaborate when theyresponded to her questions.Why do you think she did this?Compare and contrast how sheasked questions with the moretraditional questioning thatseeks a right or wrong answerfrom students.

d. What other questions ordifferent prompts can you thinkof to help students developdecimal number sense?

MS. JONES'S CLASSROOM (cont.)

3. The Professional TeachingStandards ask teachers to attend tostandards in four areas: selectingworthwhile tasks; creating alearning environment;orchestrating discourse; andengaging in thoughtful analysis.

How did this teacher appear to beattending to any of these standardsduring the lesson?

127

fbMS. JONES'S CLASSROOM (cont.)

4. Four standards in the Curriculumand Evaluation Standards arecommon to all grade levels:mathematics as problem-solving;mathematics as communication;mathematics as reasoning; andmathematical connections.

How did these students appear tobe attending to any of thesestandards during the lesson?

128

0

0

MR. SHERBECK'S CLASSROOM

Discussion questions:

1. The students in this segmentworked in groups.

a. What is the rationale for groupwork? What benefits do youthink students derive from work-ing in groups? What are thedrawbacks?

b. This class was taped in themiddle of the year. What normsof classroom discourse andbehavior do you think Mr. Sher-beck had to cultivate throughoutthe school year for students to beable to work together as they didin this segment?

MR. SHERBECK'S CLASSROOM (cont.)

c. If you wanted your students totalk to each other like thestudents in Mr. Sherbeck's class,how would you model this kindof conversation? How wouldyou give feedback to studentsthat would encourageconstructive comments anddiscourage off-task comments?

2. Why might it be important forstudents to write about theirmathematical thinking?

130

MR. SHERBECK'S CLASSROOM (cont.)0

3. A teacher could ask students to writeabout mathematics withoutnecessarily contributing to theirproblem solving, reasoning, orcommunication skills. How canteachers use students' writing inways that promote meaningfulmathematical learning?

4. The Professional Teaching Standardsask teachers to attend to standards infour areas: selecting worthwhiletasks; creating a learning environ-ment; orchestrating discourse; andengaging in thoughtful analysis.

0

How did this teacher appear to beattending to any of these standardsduring the lesson?

131

MR. SHERBECK'S CLASSROOM (cont.)

5. Four standards in the Curriculumand Evaluation Standards arecommon to all grade levels:mathematics as problem-solving;mathematics as communication;mathematics as reasoning; andmathematical connections.

How did these students appear to beattending to any of these standardsduring the lesson?

132

MR. LEHMAN'SALGEBRA ASSESSMENT

Discussion questions:

1. What are the advantages anddisadvantages of this type ofassessment, where studentsindividually demonstrate theirknowledge and understanding byorally defending their solutions toproblems?

Contrast what students must knowand be able to do for this type ofassessment with what they mustknow and be able to do for moretraditional pencil and paper tests.

133

0 MR. LEHMAN'S ASSESSMENT (cont.)

2. Although the students preparetheir solutions while working ingroups, they were required toexplain their solutionsindividually.

What might be some advantages,or disadvantages, of allowinggroups to demonstrate theirsolutions collaboratively in thistype of assessment activity?

134

0 MR. LEHMAN'S ASSESSMENT (cont.)

3. If classroom instruction is notconsistent with the type of sense-making, conceptual questions thatpanelists ask students, thenstudents cannot be expected to dowell on the assessment.

If you knew your students wouldbe assessed by a panel like this,how would you prepare them?

Where would you find problemsfor them to work on?

How would you organizeclassroom discussions to modelthis kind of reasoning aboutmathematics?

0

*

0

MR. LEHMAN'S ASSESSMENT (cont.)

4. What logistical or technicalproblems might be associated witharranging this type of assessment?

What alternatives might be usedby teachers who might havedifficulty gathering a largenumber of panelists, but whowould like to give studentsopportunities to demonstrate orallytheir mathematical sense-makingand problem solving abilities?

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