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The sessions on teaching issues in the middle grades focused on the questions . What are the important characteristics of effective teaching in the middle grades? Of effective teaching of mathematics in the middle grades? How can instruction in middle grades classrooms be organized to maximize . learning? How can we tell when learning is happening? What tools and strategies wait make a difference in how middle grades students learn mathematics? USING VIDEO OF CLASSROOM PRACTICE AS A TOOL TO STUDY AND IMPROVE TEACHING Nanette Seago, Project Director, Video Cases for Mathematics Professional Development, Renaissance Project. PANEL REACTIONS TO THE "CINDY VIDEO" TEACHING AND LEARNING MATHEMATICS IN THE MIDDLE GRADES: STUDENT PRESPECTIVES Linda Foreman, President, Teachers Development Group, West Linn, Oregon. PANEL RESPONSE TO FOREMAN STUDENT VIDEO SUMMARY OF SMALL GROUP DISCUSSION ON TEACHING ISSUES IN THE MIDDLE GRADES

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r]d Her [zag far ~ Nanette M. Seago Project Director, Video Cases for Mathematics Professional Development A newly-funded NSF projects which direct, is focused on developing video cases as tools for use in mathematics professional development. Currently, staff are in the process of hypothesizing, testing-out, analyzing, anti revising work- ingtheories around the use of video as a too} to promote teacher learning. There is limited knowledge in the field about how and what teachers' learn from professional development experiences. It is mostly uncharted territory. Not much is known about what teachers learn from profes- sional education (Ball, 1996), especially as it pertains to the effect on teachers' practice. Even less appears to be known about how and why teachers develop and use new un(lerstan(lings in their own contexts. Conversations with colleagues,2 experiences of classroom teachers, and experiences of professional developers responsible for teacher learning are the basis of our current knowledge. Renecting on the practice of teachers anti teachers of teachers, ~ have (level- ope(1 some "working hypotheses" (Ball, 1996) about the use of video to promote teacher learning. These frame the design, development and formative evaluations of my current work: iNational Science Foundation; (ESI #9731339~; Host Institutions: San Diego State University Founda- tion, West Ed 2I would like to acknowledge my colleagues Judy Mumme, Deborah Ball and her research group at University of Michigan, Magdalene Lampert, Deidre LeFevre, Jim Stigler, Joan Akers, Judy Anderson, Cathy Carroll, Gloria Moretti, and Carole Maples for their on-going help in thinking about these ideas. 3Ioan Akers is the formative evaluator of the VideoCases Project; Deidre LeFevre is a graduate student of Magdalene Lampert and Deborah Ball at the University of Michigan and is using the developmental process of our project to research the impact of different forms of video case facilitation on teacher engagement and learning around issues of pedagogy, student learning, and mathematical content.

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teachers' mathematical and pedagogi- cal learning is greatly enhanced by connecting learning opportunities to classroom practice to the actual work of teaching. video cases of teaching can afford teachers' opportunities to learn mathematics as well as pedagogy; video cases of teaching can afford teachers' opportunities to study and learn about the complexities and subtleties of teaching and gain ana- lytic tools; and video case materials cannot stand alone; the facilitation process is as critical to what teachers learn as teaching is to what students learn. This paper is organized around some questions to explore in the attempt to understand how video can be used to promote effective learning: What is the work of teaching? What do teachers need to know and be able to do in order to do the work of teaching well? How can video cases be used effectively to develop teacher learning? Putting forth working hypotheses and questions invites others responsible for teacher education to make explicit the teacher learning theories or working hypoth- eses that guide and frame their own work so that the community can engage in critical professional discourse and inquiry into practice as professional educators responsible for teacher TEACHING ISSU learning. Educational researchers can use this work across multiple and varied contexts to research why and how teachers learn from professional educa- tion experiences whether it is using video or other practice-based materials such as student curriculum, student work, or written cases. Teaching is thinking, intellectually demanding work. Teaching is complex and involves the interactive relation- ship between content, students, and teacher (Figure 11. Typically in profes- sional development we tend to isolate and separate this relationship which can pull apart and oversimplify the work of teaching. What gets left out Figure 1. What Is the Work of Teaching?

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when this happens is the actual role of the teacher in the thinking work of teaching. For example: we spend time doing mathematics because under- standing mathematics is crucial to teaching mathematics for understand- ing and interpreting student thinking. But learning mathematical content in and of itself hasn't helped teachers deal with figuring out the mathematical validity in students' thinking, to recog- nize the validity in partially-formed and inadequately communicated student thinking, or to analyze mathematical misconceptions. But this kind of mathematical analysis is a large part of the mathematical content needed in teaching for understanding. Another example of professional development is the examination of student work and student thinking. This is a critical part of the work of teaching but not sufficient to help teachers deal with the work of figuring out what to do with student thinking once they have it. Are there certain strategies or solutions that the teacher ought to highlight in whole-cIass discus- sion? Does the order in which solutions are shared matter in creating a cohesive mathematical story line? How does one use individual student thinking and alternative approaches to further the collective mathematical learning of the whole class? Does a teacher stop and pursue every chil(l's thinking, always, in USING VIDEO OF CLASSROOM PRACTICE every lesson? How does context factor into the decisions? Other efforts focus on teaching strategies such as use of cooperative groups, manipulatives, or writing in mathematics separate from mathematical learning goals and student learning. Does one always use cooperative groups? Are there advantages or disadvantages in terms of student learning? For what mathematical learning can manipulatives be helpful? Do they ever hinder learn- ing? Are some manipulatives more helpful than others? If so, why? Does how they get used make a difference? These are just some of the recurring dilemmas teachers face without suffi- cient help in acquiring the skins and knowle(lge necessary to make intellectu- ally flexible decisions. The work of teaching involves more than "in-cIass time." It involves acts of professional practice before, during, and after the moments of face-to-face teach ing. Some examples of the kind of work teaching entails are: Before Setting mathematical learning goals Choosing/analyzing tasks anti cur riculum in relation to goals and students Figuring out various student ap proaches and possible misconceptions Learning about students Planning a mathematical story

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During Figuring out what kids are thinking/ understanding the mathematical validity of student thinking on-the-spot Figuring out what to do with student thinking once you have it seize or not to seize? For what mathematical learning goal? Balancing between individual student and whole class mathematical think- ing and learning analyzing and deciding Building a mathematical story Budgeting time Learning about students Abler Figuring out what to do tomorrow based on today Analyzing student work Learning about students PURPOSES FOR USING VIDEO When most people think of the purpose for using video with teachers, they think of models, exemplars, or illustrators of a point. The question this poses is one worth researching. What is being modeled or exemplified? What is it that teachers learn and use in practice from viewing models or exemplars? Teachers approach new learning in the way in which they were taught by TEACHING ISSUES following the procedure in an example. Using this familiar "way of learning," they watch the video in search of proce- dures to follow or features to copy. This can create barriers to a deep level examination of teaching, for it often keeps the focus on surface and superfi- cial features of classroom practice. Video can be used with a different frame, an analytic frame which focuses on the analysis of teaching practice, gaining the awareness necessary to analyze and interpret the subtleties and complexities involved in the relationship between knowledge of content, stu- dents, learning, and teaching. Analysis involves studying the same video episode through multiple lenses, as well as the comparative study of multiple and varied videos through the same lens. It involves examining the details and specifics of practice in this instance, in this context, under these circumstances, and perceiving the subtle particulars (Schwab, 1978) in classrooms while recognizing how these together form a part of an underlying structure or theory of teaching. Analysis involves supporting and disputing assertions or conjectures with evidence or reasoning. Taking an analytic stance in framing the use of video creates a set of teacher learning issues for teacher educators/ professional developers to consider. Gaining awareness of the subtleties and

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complexities of practice does not hap- pen easily. We exist in a strong culture of quick fixes, definitive answers, and oversimplification of the practice of teaching. It wail take purposeful teach- ing of analytic skills (Ball and Cohen, 1998) in order for teachers to use analytic skills. This means it wait also take the study of how teachers develop and use those skills over time to under- stand how to help teachers acquire them. Developing a culture where teachers can learn to critically examine the practice of others as well as themselves wall be no easy task either, for there currently exists a strong culture of being nice. Teacher struggles and challenges need to be seen as opportu- nities to learn rather than secrets to hide. Teachers need to be active mem bers of an intellectual professional community that values risk-taking, diversity of opinion, critical debate, and collaborative analysis. This has implica- tions for the teachers of the teachers, for they wall be responsible for develop- ing this culture a culture foreign to most U.S. teachers. Video can be used for educative purposes (Lampert an(1 Ball, 1998), as opposed to a more open-ended "what do you think?" approach. Open-ended discussions can be useful in gathering data on what teachers attend to, are USING VIDEO OF CLASSROOM PRACTICE aware or unaware of, or how they think about teaching, learning and math- ematics in practice, but for teaching something such discussions are not usually effective. Often the conversa- tions become unfocused and scattered. When viewing video, there is so much data that sorting through it needs focused guidance and structure. An educative point of view means to plan and focus on specific goals for teacher learning. What can teachers learn from this particular video and how can they best learn it? The mathematical an(1 pe(lagogical terrain of the vi(leo needs to be mapped out in relationship to what each episode offers teachers to learn about mathematics, teaching, and learning. An educative point of view explores the question, how can we go beyond this particular video or set of videos to learn the big ideas of practice? the big ideas of mathematics? HOW DO WE USE VIDEO TO PROMOTE TEACHER LEARNING? What might it mean to plan anti orchestrate professional education that is guided by analytic and educative purposes? What are the implications for the work of teaching teachers? Just as the practice of teaching is complex anti involves the interactive relationship

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between teacher, students and content, the practice of teaching teachers in- volves the interactive relationship between professional educators, teach- ers, and teaching content (Figure 21. While there is no one right way to plan and facilitate a video session, purposeful planning around three key areas is helpful: (~) the content of the video segments, (2) the learning goals of the session and (3) audience (or learners). A discussion of the planning and facilitating these three areas using the Cindy video as a context for analytic and educative purposes in the work of teaching teachers follows. THE CONTENT OF THE VIDEO The Cindy video clip lesson begins with Cindy posing the problem, If you lined up 100 equilateral triangles in a row i_ (shared edges), what would the perimeter be? This is the first part of the larger mathematical problem of the lesson: if you lined up 100 squares, pentagons, hexagons in a row, what would the perimeter be? Can you generalize and find a rule for the perim- eter of any number of regular polygons lined up in a row? Figure 3 shows a graph of Cindy's lesson with an arrow marking the point in time that we drop in with the video clip. Figure 2. Teaching Teachers Content of Video or Other Professionc~l Development Curriculum TEACHING ISSU

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lust as teachers need to know the content of tasks they pose to their students, teacher educators need to know the content of the material they are using as tools for educating learn- ers. Analyzing the material means to analytically examine the terrain of the video. What opportunity does it offer to learn about mathematics? pedagogy? students? learning? In what ways can it be used to learn new professional habits of mind? How can it be used to develop skills in analysis and develop disposi- tions of inquiry? The Cindy video has been used to: . . [earn mathematics. The math- ematical content allows for teachers to learn about multiple algebraic representations of a geometric rela- tionship in non-simplified forms. It offers the opportunity to examine and learn about recursive and relational generalizations. The role of math- ematical language and definitions are also embedded opportunities for working on mathematics within this video. [earn about student thinking/ reasoning. Focusing on student conversation and responses (Nick, Chris, and Lindsey) provides the opportunity to learn what we can or cannot tell about the student's appar- ent understanding. Nick may have USING VIDEO OF CLASSROOM PRACTICE Figure 3. Cincly Lesson 1 . Overview of whole lesson (50 minutes) 10 min 8 min 10 min 1 5 min 7 min Posing triangle problem Individual/small group Posing of additional polygons problem in small groups the beginnings of making sense of the geometric relationship by viewing the whole of each in(livi(lual polygon's perimeter multiplie(1 by the number of total polygons, anti then subtract- ing out the inside shared edges: (n represents the number of polygons; s represents the number of si(les of each polygon). We drop in

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Nick's Response Total Number of Sides - Shared Sides: p = ns - 2tn - 1 ) ns is the number of polygons multiplied by the number of sicles of each poly- gon. 2 (n - 1) is the number of shared sicles, i.e., sicles of the polygons not contribut- ing to the perimeter. The perimeter is the total perimeters of all polygons minus the perimeter of the shared sicles. (Congruent 's) hi\ hi,.\ clot\ got\ End side Chris sees the relationship like Cindy does tops + bottoms + 2 end sides, where the tops + bottoms are the contri- bution each polygon makes toward the perimeter of the whole: Chris's Response Tops and Bottoms ~ 2 End Sides: p = nils- 2) ~ 2 If top and bottom are consiclerec3 together, each polygon contributes the same amount to the perimeter. Thus, the perimeter equals the number of polygons times the perimeter contrib- utec3 by each polygon plus the two end sicles. TEACHING ISSU Lindsey may be trying to make sense of the relationship by taking out the en polygons, counting the outsi(le e(lges of each and dealing with the middle polygons separately. Lindsey's Response Interior Polygons ~ End Polygons: p = In - Hits - 2) ~ 2ts - 1 ) (n - 2) is the number of interior polygons (s - 2) is the amount contributed to the perimeter by each interior polygon (s - 1) is the contribution of each end polygon to the perimeter The perimeter equals the number of interior polygons times the amount contributed to the perimeter by each interior polygon, plus the contribution toward the perimeter by the two end polygons. Examining students' thinking can provi(le opportunities to work on math- ematics. It also offers the opportunity to examine the teacher-(lecisions around each student's thinking. What appears to be the teacher thinking anti reason- ing for the (recisions around each student's thinking? For what math- ematical learning goals would she seize in(livi(lual student thinking, when would she not? When anti why might she slow things ([own? or spee(1 things up? Veronica introduces a recursive pattern she sees from the table. Examining the

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table and being able to generalize the pattern is an opportunity that can be utilized. . [earn about teacher cIecision- making. While it may seem obvi- ous that teachers make constant 1 . conscious or unconscious cleclslons while planning and orchestrating discourse, this is not typically recog- nized and critically examined by teachers and administrators. In order to analyze teacher decision-making in light of content and students, one first needs to become aware such deci- sions even take place. You can't analyze and interpret what you can't see. It is the work of the facilitator to "lift the veils so that one can see" what isn't normally seen (Eisner, 19911. Using this video over multiple sessions helps to move the learning from awareness of decisions to the level of analysis of decisions in light of content and context. When focusing on teacher decisions, the opportunity exists to push at the current tendency for teachers to be definitive in their claims around teacher decisions. Pushing for alternative possibilities, alternative possible reasoning or conjectures with supporting evidence and arguments over time can create the necessary dissonance for learning new norms and practices. USING VIDEO OF CLASSROOM PRACTICE Learn about the uncertainty in teaching. These video segments offer the opportunity for participants to learn about the uncertainties involved in a complex practice (McDonald, 1992), especially the recognized uncertainties involved in the moments of not knowing if a student's reasoning has mathematical validity Lindsey and Nick). It can highlight the recurring teacher dilemma of figuring out what a stu- dent is thinking and offer the opportu- nity for teachers to gain multiple tools for (leafing with these uncertainties themselves. This learning can create a tension for the facilitator between not wanting teachers to leave with the notion that all of teaching is uncertain and on-the ny nor with the notion that one can always predict what will happen with certainty. The learning focus is to gain understanding of the importance of planning for possible student approaches as well as gaining tools for better (recisions in their own moments of uncertainty. GOALS FOR TEACHER LEARNING It is important to (leci(le on goals for teacher learning when using a video case. Why are you using this case? What outcomes (lo you seek from the video case experience fs)? A case can be

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JOEL RECORDING IMPORTANT IDEAS As we work on different problems, we come across "big ideas" that seem to keep coming up, even in seemingly unrelated topics.... This is where our journals come in. They allow us to record our thoughts and processes so that we may look back later and work through our thought processes again. Each student's journal is different, as a journal is a place for records of per sonal struggles, discoveries, and in- sights that help illustrate what we have Figure 1. Journal excerpts ~ and 2 ~ _. 1 ~ 't7'-~.111111111~___ . ~ ~ ~ A ~ n ~ ~ ~ j ~ ; ~ e ~ ~ ~- ~ j By; ~ ~ ~ ~ ~ r ~ ; ~ ~ ~ e -_ ~ ~ t ~ t ~ ~ ~ ~ ~ ~ ~.~..~nd~-~ ~ .~~ phi i~:~' ~tttLn~6.~l~1 .. r ci461g4:JF~~w-I ~ 5 .^ ~ ~ - ~ O44t~ e it; I | t ~ Of ~ ~ ~ ~ v 0. ~; F ~ ~ ~ ~ J A 4 t-9 ~ ~ ~ F ~ 1 ~ I I 1 ~ ~ ~4 8& ~ ~ inch ~ ~9. t .~ ~ 1 ,~ ~ L A _ -. F ~ ~ ~ .s ~., ~ ~ rib ~ a~ * ~ ~- ~ ~ ~ e ~ ~ ~ *|~fv/~ ~ __ ~ _ _ jr ' ~ r --a in N Ilk i IS - MA 1 ~ --A 4 ~j ~ ~ V 1 t t~ ~ ~ + - ~ ~- ~- \ ~ ~ ~ ^~-Limpid _ ~ li~ e~ it~ r ~ ~r /~ ~ ~ ~ r ~ ~ ~ ~ ~ i ~ ~-_ r I , ~ i t l ~ ~ ~ or _ - - v ~ . - ~ ~ ~ ~ ~ 1 _ . _ ta L ~ ~ ~ 1 ~ r ~ ~ 1 ~ F ~ ~ ah o ~ -~ ~ ~ ~ ~ ~ : ~ -~- fit-~ ~ -~ 1 l! - ~ Amp. ~I=~y. f A ~ ~-~;~ ~IL^~ A 4 ~ HI Rae 1. ~ ' ., t. ~ I ~ ~ r ~ ;. MA ~ . A ~ ~ . ~ ~ _ _ L ~ ~ 4 __ _ 1 ~ _, ~ Am; 4~J t 4 ~t r 1 ~ ~ ,.~.. :~ so*-'' ~ ~ ~ . ~ ~_ _ ~' ~ ~ _ A ~ ~ A, ~ _ ~ ~ _. ~ ~ A ~ -0 ~ _; ~ ~ - : 7 ~ ~ ~ I: : t - d - . ~ F ~ - ~ r t t e F ~ !~ ~ ~~ ~1 ~_-~;L if. uib~ I 6wt. F 4 I t A $ _ ~ ~-r ~$ 11~ ~ 0~ ~ _ t~4 -~_0 ~.~v-;l-,~411~-~4^,~S.~. ~ TEACHING ISSUES been working on in class.... I have chosen a few excerpts from my journal to help demonstrate how we record important ideas and look back later if we are stuck. It is by coming across examples such as this that we have learned that it is very important to record the important mathematics ideas we come across, and to try applying methods we came up with earlier, even if at first look it seems as though the methods have nothing to do with the idea we're examining. In this way, new i(leas make sense and are easier to un(lerstan(l.... o~e . ----- 81~ ~-_ ~ ~ ~ F ~ ~ t 4 t ~t ~ j ~ ~ Hi ~ ~ ~ Hi i~ ~ e I ~ ~ 4/0 ~ ~i t ~ F 441-~ *if--if i - '^I - ~.W~-~--00 $~-~ kl,Ci i- -~ #-' ~ ~-~ ~ ~ it ~ ~ ~ ~ ~ ~ _ _ -.~ . ~ ~ H ~ ~ ~ _ _ ~ ~ Fin ~ Ian ~ r ^~ n t {tti}t ~ i ~~ - ^ 1~ - an ~ - . be- ~ ~ ~ s ~ ~ . i ~ ~ rem En. . t - ! i - . ;- h ~ ~ it ~- ~-~~+r~ , tail jig ~-J~1~4t~ A-I -' y --ihid4L~ hi, ---r-l~ ~ j t~, _ i - - ~ ~ _ - , ~ ^~ A' t. ~ . ~ .. t d. ~ 4 p I ~F ~ ~ ~ ~67~ ~ ~7hj(5 >_ _~L4 ~_0j~t ~ _, _ - - ~ ., _ L _ _ . J _ . ~ _ ~ _ _ _ . f _ _ . ~ . ;- r. . 4 ~ r ~ , ~ I, 6~, . 5 I~e - ~ ~ - t I t ~ 4 i ;3~ ~ ''rick - . ~~ . ~ ~ ~ ~ i~ Or - -A ~ ~ r ~ .~ ~t ~ t. . ~ -I i_- t tail ~ -i t-~ ~4~--~ ~ ~r ~ ~c ~ 4 1 t ~L ~-- ~J~3~` ~] ~ r _~ ~ ~ -~ ~ _ ~ ~ ~ ~ ~ ~ ~ ~ ~ L ~ t I. ~ 4 ~ I t 4 ~ d t _ ~_ _ F e ~ i_

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Figure 2. Journal excerpt 3 ~. , . ~ -~ t (~ ~ ~ W~j ~,~]-L_ ....1. r-(J~ti~ Amid i 6~ ii t W t ~4 -.~- ~54i-~ ~ `+ '-~' ~ i: i l~ &7 jam ~l t_~ ~ ~ ~ t t t ~65 ~. ~ If: ~ ~ ~ .. r-ll~$~ aim- ~ ~ phi ~ |=~ art *I iJh in ^~;=a-A; ~ =14 ~ - AWL* r ~. t t t __ r ~ ~ ~ _ . . __ ~4 ~_ 1 ~ ~4 ~[ ~ ~ _ ~ ~ -~5 ~j . f J. ~_~, ,,~ ~ _ _,~_ it; ~I'w.p fat 1! :n +. ~ ~ Jim ~ ~ v ~ i~-\ i ~ ~13 .~ Hi-~ it F '-I 1 \.~./ 13 =- \~tt~ <. ~~ I ; ~ r ~ Sew I. e . ~f ~,,,~ 1 4 ~; . I $. ~t ~.-` ~ .' \~. ~jj' r . _, ~ ~ t: : r .~. ~fffl. ~. .. ~ 3 ~S_~ . ~-~L. ~ 7~..,~ ~ (~ 1t,}~- ;OJCi~r~2-~t ~ _ A ~ ~ ~ ~ p ~ . , . ~ ' t: . t ~ t...,\ =~..s~ ^-1~-~'' ~ ~ r ~ ~;, ji. _4, j~ I1~ ~ ~ ~ ~ A ''_ _ .:t... ~', ,~, J,,~.. - ,.~ - l~. ~%,1~$ 2~2 $ - ~-~_-' ~,~-,._,~2- L. ~ 4:: ~ ~ ~ ~ t' ;~ ~_ ~ 4 - t ~t-:~^~ ~ ~f~ .h- ~5~s - i~ r-~_~ 40~r~ ~ ~f~ t ~ . , I ~J ~s.- ,- ~. r KYLE MATHEMATICAL TRUST Learning mathematics is a journey. ... Our teacher trusts us as capable Answer a question without letting the stu(lents get involve(l, Not allow students to invent their own mathematical thinkers who can find our procedures, own way.... That is, she believes there is a mathematician w~thin each of us. Therefore, she does not lead, show, or ~ gu~de us ~n our Journey.... A teacher that does NOT have math- ematical trust in her students may: Show a best way to solve a problem, Ask a leading question that looks for an expected answer, S T U D E N T P E R S P E C T I V E S Not allow students to fee} disequilibrium. The above actions discourage the use of new ideas and different approaches. They also take away the stu(lent's opportunity to fee} the joy of learning and doing mathematics. ~ have noticed that when someone telIs me how to solve a problem, my thinking stops. On the other han(l, when someone allows

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me to wrestle with an idea, ~ find myself inventing more strategies and ~ get a stronger grasp of the idea. Solving a problem myself helps me clarify my understanding of the mathematics and leads to important conjectures. Last year our class invented the strategy "completing the square" to solve quadratic equations. This fall, when the idea came up again, we felt certain we could solve any quadratic equation. So, our teacher sai(l, "Just for fun, if ax2 + be + c = 0, what is x?" After Figure 3. Journal entries - " ~ 1 It ~ ~ 1 "=9 ~ ~ ~ ., , . .. , at ~-~ For rat . : ~ the first day of exploration our teacher asked us if we wanted clues. We pro tested and ended that day and the next (lay of class in (lisequilibrium. Finally, after three (lays of buil(ling mo(lels, cutting and rearranging pieces, debat- ing and discussing, and no clues, we found the value of x. The journal entry on the left below shows how ~ solved a specific quadratic equation. The journal entry on the right shows how ~ use(1 that i(lea to generalize about any quadratic equation. l:)ate ~ ~ I .i~ 7 ~ _ . -or -3?~--~- ~ i~ ~ F ~ ~3 +.h~-~ ~ 4, i t , ~ , r , ~' ; t ; ~ ~.; ' -a - ' i ' ~---act ~ ~ -r I ~ ~ ~ ~ ~ ,~ ~-F ~ r- r -; ~; ~ iI ~ ~- ~ *a ~ t ~ ~- ~- ~ ~ - -~= ~ ~ _ ~ 6A . ~ _ _ _ ~ ~ ~ ~ ~ _ i ~t t , I'm -~-~-~l~_ ~r ~ ~5;~ C'-h .~-1 ~-~~1-~r ~t ~t - ^ ~ ~ t ~ 6 ~ ~ t ~ ~ t ~ ~ ~t ~ ~ ~ i ~ ~ | ~ ~ ~ ~ ~ ~ t ~ t ~ i t ~e~ ~ Jo; ~J,L ~ ~ ~- - h Is ~ - 4~-~ . . v.~ 1 u ~ t _ _ ~ ~ ~ ~ the ~ ~ ~ . : ~ 1 lit In . l: t : ~ ~t , , , , ~ ~ I f t ~ ~ ~ .. ~. . .. 23. , In. Ply ;~$ ~ G.^ ~ 34-r~ki-~ -qv<---- -- ~ ,l;bt4iLK~, , rim 7-~ i 1 ~-~$~5 ~ i ~-~-- ~i,.j i-,- .~ l i, , 11~. l~l;3~..l ---- t of. ~ ~ ~ .~^ ` - ~ . t t ~ . ~ ~ ~ ~ 'am, ql,"~t ~ AL I ~1ET" -$~:~h-~t-- L~1~- ~ I= . ~In--L,hi~ ~1~- ~1 ~\r~--r ~!_ A; ~t~l~o~l & t Byte ~ ~ Ott . :-~ ~ 4: - A: ~ ' ~ ~ ~ ~ ~ r. -e ~ r J~ _ _ _ J ~ ~ ~ ~ ; ~ ~ _ ~W 1i Fit _ ~-3 ~- ; If; ~ r At ~ it_ ~ ~ t ~ ~ t f ~ ~ ~ ~ Vie Ed ~ \~ ~ S t ~ ~ 3 ~ ~e t I: ~ ~ ~ ~ CALM Amp p ~= ~ ~ ~ ~ ~ ~ ~~~ t~~ Am, ~ ~ is ~ ' ' '' ~ ~ ~ ~ ~ --~ j Em ~ +~ ~ to laid 1~ ~ r ~ r A= r ~ ~ ~ ~ ~ hat ~ ~ I I ]-5 ~ ~ ~-t ~ 3~ ., Ad ~ ~ it ~ ~'i~t t- 11---.---1----;--l - -- -in ~..a~ ~_~4~ t ~ . ~O~~ 84r~~ A ~1-~~ ~~~~~l ~ - ~ ~ i: ~ t ~q_~hits~ ~ . . . ~ ~ 3 : .,~ *~,~bJ~r~,, 3 ~5.~v-Ad, }- ~ - ~ ~ ~ - ~ ~ a ~ - ~ 1~_ ~ _ ~ ,L _ ~ ~ ~ ~ - ~ ~^ ~ ~ ~ ~ . ~ . ~ ~ . . ~ . . . it: . . ~ ~ :- . ~ ~. ~ ~ ~ . ~ ~ t ~ I: ~ ~ I ~ ~ ~ ~ i , A~.3 hit ~- ~- -- --I --- --- - - -- 3--- -3 - -- 3--- - - ~ -- 3 ~ 4'--~ ~I I I t I: ~ f ~f ~L _ _ ~ t ~- ~ ^ arm ; . ~ t --urn Am ma --i ~ ~~~-~3rXt-~- ~t -a _ _ _ ~ _ _ ~ ~ I' ~ ~ ~ - ~ ~ - ~ K ~ { ~ ~ =: - . . ~- . . ~ ~ ~ . . _ _ t - ~ _ ~ J: ~ A ~ . ~ _ _ ~ ~ _ _ _ i . _ . . ~ _ ~ . ~ . _ _ . . ~ ~ _- ~$ ~ r--- ,~ ~ ~I; ~- ~ Al 56 8~ ~-~ ~~ ~ ~ ~-( I 1 4. ~_~ _ ~ i, ~_ ., i ~ -0 ^- =^ -~ ~ ~ ' ~ ~~ ' i: : ~>_ _ _ _ Lam _ ~ _ tam ~ ~ ~- ~ ~ ~ ~ ~ ' - ~ ~I Fatal _~J ~,nauat_S~E-~-._ TEACHING ISSUES

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Because our teacher trusted us and we trusted ourselves, we invented an algorithm for solving any quadratic equation dater we found out that other mathematicians had also invented that aIgorithm). Learning is a journey. Mathematical trust keeps us going and allows us to travel in new directions without worry- ing about getting lost or taking the routes that others do. CONCLUSIONS Working with these students for four years has stirred my thinking about learning and teaching and enabled my growth as a writer. More importantly perhaps, it has left me with food for thought about teaching mathematics. Following are a few ideas on my mind, prompted by the students' comments and work and by the Convocation panelists and participants. Perhaps they wait provide thought or discussion starters for you, an(l/or perhaps you have other ideas to add to the list. It is possible to form a remarkable mathematics community and cIass- room culture when youngsters and their teacher stay together over time. How can this mode} be adapted to work in the mainstream mi(l(lle school setting? S T U D E N T P E R S P E C T I V E S We teachers and curriculum develop- ers can impose artificial limits on students by the questions we don't invite or pose and by our own concep- tions of learning, teaching, anti mathematics. How do we learn to recognize this in our actions and work? While it is the case that I~in(lsay, Joel, and Kyle each went on to explore applications of the area of a trapezoid, center of rotation, and quadratic formula, their papers suggest it was the mathematical ideas themselves that were engaging. What motivates students to engage in thinking about mathematical ideas? What makes a problem "real" for students? What is meaningful context? If one agrees that these students provide evidence that it is possible to cultivate interest in serious math- ematical content, what are the instruc- tional practices that are most influen- tial in cultivating such interest? What is important and relevant mathematics for mi(l(lle gra(les? What is worthwhile mathematical activity? Note: (leriving the quadratic formula was not a part of my original lesson plan; however, as Kyle pointe(1 out, the class spent 3 (lays wrestling with the challenge. What may have been gained or lost by taking this math- ematical excursion? It seems to fit Julie's criteria for worthwhile activi

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ties. What are other criteria to add to Julie's list? Many teachers were educated in a system that promoted the notion that only certain people can do mathemat- ics. Can students come to recognize and nurture their "inner mathemati- cians" if the teacher does not believe every student has a capable math- ematical mind? What professional development experiences are neces- sary for teachers to develop a sense of "mathematical trust" in their students? Contrary to many common (lescrip- tors, doing mathematics is an emo- tional experience, and those emotions can be positive ones, e.g., empower- ment and passion for the subject matter, pride in discovery and inven- tion, respect for disequilibrium, and joy over solving challenging prob- lems. What teacher actions best facilitate the development of such feelings about mathematics? When students are called upon to communicate mathematically, they learn the language of mathematics as they learn to speak any language- simultaneously using invented lan- guage (e.g., smooching) and formal language (e.g., disequilibrium, trans- formations). It can be uncomfortable for teachers as they strike a balance between accepting students' invented language anti teaching formal math- ematical language. TEACHING ISSU As Erica pointed out, teaching that focuses on how students think about mathematics has a powerful influence on students' learning as well as their views of themselves as mathemati cians. It also provides the teacher rich information about the extent to which students un(lerstan(1 anti are able to integrate mathematical ideas and processes into their own way of thinking. For example, in Lindsay's and Toel's explanations of their thinking, their use of transformations provides evidence of their sense of geometry as a process; anti we see evidence of I~in(lsay's anti Kyle's sense about the integrated nature of algebra and geometry by their use of algebraic symbols to represent the geometric and algebraic relationships they could "see" in models. Because teaching that centers on how stu- dents' think is so different from the mathematics instruction most teach- ers and parents have experienced, it is particularly challenging to shift away from practice that emphasizes telling students what to think. Long term professional (levelopment anti a curriculum that emphasizes students' mathematical thinking are essential for teachers to make this shift. The examples given in this paper provide a glimpse of what is possible when students are immerse(1 over time

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in a Standards-based curriculum; however, it is important to keep in mind the fact that this project lasted for four years. lust as implementing reform is a challenge and requires long-term support for teachers, the benefits of reform-based teaching are not immediately apparent in students. Had these students written their papers even a year earlier, some students would have expressed doubts about how and what they were learning (they kept close tabs on activity in their peers' more traditional cIassrooms S T U D E N T P E R S P E C T I V E S the media, and even some mathematics teachers, told them they would never learn what they needed to know); there would have been more disequilibrium about certain mathematical i(leas that are described with confidence here; their parents may have expressed doubt due to the uproar about math- ematics reform in the local news; and their teacher would have been a little less secure in her conviction to main tain high expectations, trusting everyone's (lisequilibrium was a sign of new learning about to occur.

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=.1.= 1~01~1 ~. rig Three panelists were invited to comment: Hyman Bass, a university mathematician, Sam Chattin, a career middle grades teacher, and Deborah Ball, a researcher on teaching and teacher educator. Sam Chattin, who does not teach math, reacted to the video thinking about middle grades students in the context of social groups. He observed that the students in the video had formed an effective social group and seemed to have made a conscious choice to stay in the group, probably because of the affirmation they received about their ability to do math- ematics and be successful. The teacher had clearly done some modeling about learning mathematics and about group behaviors. He noted that a strong commitment to the social group was evident in the students' willingness to raise money to travel to Washington, DC. The body language of the students making their presentations indicated they felt very secure, and it was clear that a(lults hall ma(le them that way, free to make mistakes with no censure. They use(1 their own wor(ls (e.g., "smooched"), an indication they felt free to translate what they knew to their own world. Probably the most significant observation for the audience was his remark that the audience laughed when the students were the most serious. Chattin pointed out that it is hard for adults to recognize just how serious middle grades students are. Hyman Bass described himself as a university mathematician "infected" with observing elementary teaching and thinking about how students learn mathematics. He shared Sam's impres- sions anti saw in the students a renec- tionoftheteacher'spractice: attitude towards content, classroom culture, philosophy, anti principles of teaching. Bass felt the students saw themselves as mathematicians with the pri(le of (liscov- ery when they realize(1 they were part of history. The mathematical topics covere(1 algebra anti geometry, which in his view have been excessively sepa- rate(1 in the stan(lar(1 curriculum. Particularly nice was the use of transfor- mational geometry, cutting anti pasting to find the area of the trapezoid (which preserves area), and in one of the presentations the use of rotations and

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the center of rotations as a way to approach the problem. He observed that some of the methods of teaching were obvious in the student presenta- tions, such as keeping a journal of mathematical ideas. When one of the students was given a problem about using perpendicular bisectors of chords to find the center of a circle, although he had the answer, he returned to his records in the journal and made the connections with the mathematics he had recorded to learn why his answer worked. According to Bass, the tape revealed products of enlightened teach- ing where the students were able to communicate about mathematics and had a passion for the subject. Deborah Ball noted the challenge of commenting on teaching when teaching was not visible on the video segment. However, she said, several key observa- tions were possible about what the teacher must have done for students to be able to do the things displayed on the video. She framed her remarks around the conjecture that this teacher had PANEL RESONSE TO FOREMAN STUDENT VIDEO actively held and communicated high expectations for students. She said that teachers can move children if they hold expectations that students can learn and (lo not take refuge in "Most (or my) kills can't (lo this." Ball's first point was that the teacher in this case hall to teach her students how to use her questions as a way to learn; they had to learn to make sense of the way she teaches. Second, the teacher had to cultivate interest in the mathematics. Thinking students are not interested is the static view; the teacher had done something to make these students interested and involved. A thir(1 point was that the teacher hall to cultivate a language within which the class could work. She had taught them some formal wor(ls that were not part of their vocabulary (e.g., "(lisequilibrium") but she also accepted their words ("smooshed"~. And finally, she must have created some incentives for stu- dents to learn to work this way. The students hall been given high incentives to engage in mathematically soun work.

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or Using the videos and pane! observa tions as a backdrop, the discussion groups addressed the following questions: What are the important characteristics of effective teaching in the middle grades? Of effective teaching of mathematics in the middle grades? How can instruction in middle graces classrooms be organized to maximize learning? How can we tell when learning is happening? And, what tools and strategies will make a difference in how middle grades students learn mathematics? What are the important characteristics of effective teaching in the middle graces? Ofeffective teaching of mathematics in the middle grades? The answers to the first question clearly reflected the middle school philosophy, with an emphasis on a safe learning environment where students work in a social caring classroom and learning mathematics is treated as a social activity. There was strong sup- port for student-centered classrooms and reinforcement of student ideas and work. Some caution was voiced about using praise to reward less-than I ~Cw.2 Ark adequate performance. As the groups struggled to identify effective math- ematics teaching, many mentioned that "quiet discomfort" signals new learning and that a sense of disequilibrium is essential to learning, resecting the message from one of the videos. A clearly identified characteristic of effective middle grades mathematics teaching was the need for strong con- tent knowledge on the part of the teacher. This was particularly signifi- cant when mathematics was viewe(1 from the perspective of a challenging mi(l(lle gra(les curriculum that goes beyond computational skills. The groups i(lentifie(1 the following charac- teristics of effective mi(l(lle gra(les mathematics teachers. Effective mi(l(lle gra(les mathematics teachers: have high expectations for their students have students who are involve(1 in active learning situations anti en- gage(1 in communicating mathematics (resign their lessons with well (lefine(1

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goals and a coherent message making connections within the course and the curriculum . are able to build understanding from concrete models, knowing how and when to bring the ideas to closure understand the importance of asking the right questions in ways that promote thinking and allowing sufficient time for students to respond listen to their students' responses so they know where those students are in their mathematical understanding and use this knowledge to develop student learning are flexible yet have created a familiar structure and routine for their classes provide models for student learning by the way they teach. How can instruction in middle grades classrooms be organized to maximize learning? The groups stressed the critical role of the school administration in creating a structure and support for student learning, from setting the school environment to ensuring that class disruptions were minimized. The groups also consistently mentioned the need for time for teachers to work together (leveloping lessons anti think- ing through the curriculum, for flexible blocks of time (not necessarily block scheduling), for manageable class sizes, and for clear articulation between gra(les. Over a fourth of the groups SUMMARY OF SMALL GROUP DISCUSSION supported "looping" having a teacher remain with a class of students over several years. There was strong sup- port for using teams as a way to create a community of teachers. How can teachers tell when learning is taking place? Students provided such evidence when they were engaged and able to explain the mathematics they were learning to others. Students who un(lerstan(1 can apply mathematics to solve problems and have the ability to revise their thinking based on their investigations. What tools anti strategies will make a difference in how middle grades students learn mathematics? Manipulatives, calculators including graphing calculators, and computers all were referenced by the discussants as important tools to help students learn mathematics. Assessment as a too} to enhance learning was suggested, as well as engaging students in writing and projects. Strategies for helping students learn mathematics inclu(le creating a warm and open environ ment, where there were clear and consistent policies among the team members. Parents should be informed anti involve(l. Mentoring anti buil(ling a community of teachers were reoccur- ring themes. Teachers should be working with other teachers on les- sons, visiting classes, anti (1esigning professional (levelopment activities

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around the context of the content teachers were teaching. Teachers should be engaged in posing questions and debating answers to stimulate student thinking. Attention should be paid to the developmental levels of students, although questions were raised about what the term develop mentally appropriate meant and how teachers would understand this in terms of their students. The comment was made that ' five are not taught to teach, only about teach- ing." Video as a way to initiate a discus- sion of teaching was perceived as both positive and negative. The initial ten TEACHING ISSU dency to criticize can overtake the discussion. There was concern that viewers might not recognize good teaching. The risk of going public as a teacher and standing for inspection was too great to make this a useful medium. Those who found viewing actual in- stances of teaching useful, appreciated the different thinking that can result from talking about actual practice. A video allows a situation to be viewed repeatedly from different lenses. The groups did agree, however, that the viewer should have a well (lefine(1 focus in order to make the viewing and ensuing discussion useful.