**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

**Suggested Citation:**"Teaching Issues in the Middle Grades." National Research Council. 2000.

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

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

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.

· 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?

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

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

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

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

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

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

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

used and facilitated for a number of purposes, so it is very important to be clear in your own mind about your purpose (Miller and Kantrov, 19981. It may be that you have multiple goals for using this video case, but you should be clear about what they are. Do you want participants to extract principles from the specifics of the video case? Peda- gogical principles? Mathematical principles? Cognitive principles? In other words, can you abstract bigger ideas about the practice of teaching mathematics? If so, does a facilitator bridge from the concrete and specifics of the video case experience to larger, more abstract ideas? Your decision needs to be informed by what you know about the group's expectations, past experiences, and interests. As in teach- ing, figuring out how to begin with the learner's experiences and bridge to new learning is part of your work as a facilitator in planning and orchestrating video case experiences for learning. THE AUDIENCE (LEARNERS) It is important for the facilitator to learn as much as possible about the audience who wait view the video case session. This wait enable the facilitator to anticipate possible reactions and ways of viewing the video. While people wait attend to the things that they them TEACH~NG ISSUES selves find important, spending time thinking of audiences as a whole can aid in planning. This will help bridge from the au(lience's worI(1 to a new learning that may be outside the experience of the group. It wait also help engage the participants by identifying an entry point (a best guess determined by what is known of the group). Since the Cindy Case is complex and offers a number of possible entry points, it is important both to know the case anti what it offers as well as knowing the participants in order to make an educated guess as to what part of the case is an attractive entry point for the participants. While this may be very (lifferent from the way in which the facilitator connects to this video case, reaching the participants means the facilitator has an understand ing of where the participants will enter the video case. Some entry points into the Cindy video include: student thinking- specifically Lindsey and plus 4; teacher questioning specifically all of the questions Cindy asked in the segment; the role of the table in furthering/ hindering student learning; classroom environment; and the role of mathemati- cal understanding in teacher decision- making around student thinking. It is also important to spend some time thinking about possible audience barriers to analysis. These are things that can get in the way of analyzing the

video segment. It helps to come up with strategies for dealing with these barri- ers. Some barriers might be: . The mathematics. Gaining awareness of the task helps partici- pants to see more in the video than possible without mathematical access. In some cases doing the problem with the squares first helps to highlight the relationship and multiple representa- tions while moving to the triangles prior to viewing the segments. Anywhere from 20 minutes to 45 minutes can be spent on the math- ematics depending on the time available end the audience. Pushing to get the three ways of representing the relationship out has proven helpful in participants' ability to recognize the alternate ways that Nick and Lindsey use to making sense of the relationship. · Lugging Cinc~y as gooc! or bacI. Spending some time up front framing the session as the analysis of one moment in time without enough data to judge Cin(ly as good or ba(1 helps, but it win probably be necessary to seize seemingly judgmental comments and go underneath them asking for specific evidences in the video and why that is important to them. Figur- ing out how to use evaluative state- ments as entries into analysis is something for which strategies are not USING VIDEO OF CLASSROOM PRACTICE yet wall-defined. Comparing this to analyzing student thinking is helpful. The point is not to judge the student as good or bad but to analyze the math- ematical thinking and reasoning. In this way, we are examining teacher work analyzing the mathematical and pedagogical thinking/reasoning. An analogy that is sometimes helpful is an archeological expedition concen- trating on what can be learned from each layer of artifacts, analyzing the relationship between content, stu- dents, and teacher decisions. · "This class cIoes not look like mine!" Cin(ly's class is in an upper- middle class suburban community. The ethnicity of the class is not repre- sentative of the national population of students. It might help to raise this issue up front, acknowle(lging the fact that it is (lifferent from many contexts asking that the video be viewe(1 from what can be stu(lie(1 an learned about mathematics, teaching and learning even though it is situated in a context different from one's own. This issue can emerge when other grade levels view and discuss this video as well. The notion that we can learn from situations like our own as well as those not like our own might be a worthy goal in an initial session. It may also be that this particular vi(leo may not be a good first experience if it presents too much of a barrier to the

analysis of practice. This raises the important issue of providing opportu- nity to use many video cases to analyze through the same lenses, so learners have a chance to bump up against the big ideas of practice in multiple en c} varier} contexts. · Too many things to pay atten- tion to. There are multiple things to pay attention to when viewing the complexities of classroom practice. While participants can be asker} to focus on one aspect (i.e., student learning, teacher clecision-making, mathematics), a group will pay attention to multiple anc} varier} things what they care about as inclivicluals. It is helpful to recognize this up front anc} to plan for at least two viewings the first to get out on the table the things they will pay attention to anyway en c} the seconc} to focus using a particular lens. Asking that viewers choose incliviclual stu- clents to follow en c} figure out their apparent thinking along with the apparent teacher decisions anc} possible reasoning arounc} that thinking is a helpful task in learning to focus. In planning professional education · · · ~ experiences, time IS a necessary conslc oration. Do you have 2 hours? 4 hours? Multiple sessions? Consider how variables of time, goals, en c} number of TEACHING ISSUES participants interact. What can be clone in advance? Can readings help? What about after the session? If you have multiple sessions, what can participants do between sessions? Think about what can be clone realistically in the time available. Using some time up front to access the mathematics of the task is worthwhile. This time is not intenclec} to mine the task for all of its mathemat- ics but rather to get enough experience to access the video case for analysis. The viewing en c} discussion can offer more opportunities to examine the mathematics with purposeful question- ing by the facilitator. Video of classroom practice offers potentially powerful resources for the professional education curriculum. Video en c! surrounding contextual material constituting a video case study can provide much neeclec} tools for professional clevelopers to use with teachers for observing en c} analyzing complex practice. Increasinglyacces- sible en c! afforciable technologies create exciting possibilities for the use of video in teacher clevelopment. Yet, the most powerful, well-clesignecI, en c! technologi- cally acivancec} tools will clo little to improve teaching in this country by themselves. As gooc! as the curriculum may be, it can't teach. Teaching matters in what students learn; teaching teach- ers matters in what teachers learn.

REFERENCES Ball, D.L (1996~. Developing mathematics reform: What don't we know about teacher learning but would make good working hypotheses. In S. Friel and G. Bright (Eds.), Reflecting on our work: NSF teacher enhance- ment in Kid mathematics (pp. 77-111~. Lanham, MD: University Press of America. Ball, D.L, & Cohen, D.K (1998~. Developing practice, developing practitioners: Toward a practice-based theory of professional educa- tion. In G. Sykes (Ed.), The heart ofthe matter: Teaching as the learningprofession. San Francisco: lossey-Bass. USING VIDEO OF CLASSROOM PRACTICE Eisner, E.W. (1994~. The educational imagina- tion: On the design and evaluation of school programs (3rd ea.) New York: Macmillan. Lampert, M., & Ball, D.L (1998~. Teaching, multimedia, and mathematics: Investigations of real practice. New York: Teachers College Press. McDonald,J.P. (1992~. Teaching: Making sense of an uncertain craft. New York: Teachers College Press. Miller, B., & Kantrov, I. (1998~. A guide to facilitating casesineda~cation. Portsmouth, NH: Heinemann. Schwab, J.J. (1978~. Science, cavrric?`l?`m, and liberal education: Selected essays. In I. Wesbury & N. Wilkof (Eds.~. Chicago: The University of Chicago Press.

Age Hyman Bass, a mathematician from Columbia University in New York, spoke from the perspective of a math- ematician about his observations on mathematics and the teaching of math- ematics after viewing the video. He pointed out that in his experience, the mathematics preparation of teachers comes from a list of topics that have been incorporated into the design of a course. Either the teachers do not learn what is taught at all or they learn the content, but see no connection to what they will (lo in their classrooms. He indicated that the use of a video might be an appropriate too} to help bridge the gap between mathematics as content anti mathematics in practice. From his perspective, although the video was framed as an algebra lesson, he found it hard to think of it that way. The lesson covered functions, data, patterns, graph, but algebra as he thinks about it was not dominant. He noted that the teacher had navigated through measurement and the topic of unit and that some students moved past symbolic patterns to use geometry to solve the problem. There was also some math- ematical tension because using T+2 to find the perimeter seems on the surface to conflict with the fact that the perim- eter increases by increments of 1. What the expressions T+1 and T+2 each mean in the context of this problem, as well as how they map onto each other, are crucial mathematical issues in this lesson, according to Bass. Choices made by the teacher indi- cate(1 that she was on unfamiliar terri- tory in some respects anti approached some of the topics with uncertainty. While some teachers would have been insecure in this role and moved toward something they understood, Cindy was a courageous teacher welling to move into unfamiliar terrain to explore the mathematics. He note(1 that the impor- tant analysis is not just about the math- ematics but also about how that math ematics is begin taught. Deborah Ball, a mathematics e(luca- tor anti researcher, a(l(lresse(1 the vi(leo from the platform of teaching anti frame(1 her observations in the context of conducting a lesson by (liscussion. She pointe(1 out that it is much simpler if the teacher (toes the talking because the teacher then has control. In (liscussion situations, the teachers anti students are

i' ointly authoring the text and if the teacher has a goal in mind, it can be difficult to reach the goal with this joint authoring. She raised four points, and in each, indicated that there were two ways to approach the situation and there were costs in choosing either. Her first point was the mix of Cindy's reactions to student responses. Some were open and elicited student ideas, "What do you think?" and some closed off discussion, "That's right." When teachers elicit responses, they leave what they hear to chance, not knowing what kind of input they wait get. When they close off discussion, they don't have access to student thinking but can be sure they direct the lesson in ways to finish within the class period. Ball indicated it would be interesting to analyze what Cindy's choices in this respect meant for the learning that went on in the class. In handling student errors teachers make the same choices. In one case, Cindy moved the student past an error to the right vocabulary and in another instance asked the class what they thought of the statement, again either closing off any discussion or opening it up and taking a chance on hearing something unexpected. Ball's second point was to examine who was being called on to respond. Teachers make choices based on a variety of considerations knowing what a student is likely to say, who is likely to PANEL REACTIONS TO THE "CINDY VIDEOU be having problems, and so on. These are complicated decisions and the way teachers make them contribute in significant ways to creating lessons out of (liscussions. Her third point was that the lesson containe(1 very little praise. Japanese teachers use praise and reinforcement in ways much different from teachers in the United States. Praise can be moti vating and a clue to students that they are on the right track, but too much reinforcement increases student reli- ance on the teacher and makes them less self directed. The clues about whether the class was (loin" well in the video seemed to be embedded in the content. The students seemed to fee} they were thinking appropriately al- though there were very few verbal clues. Ball's final point was to emphasize the relation between the ability to ask questions anti the teacher's un(lerstan(l- ing of the mathematics in the problem. She commented that it strikes her as particularly unhelpful in professional development to talk about good ques- tioning techniques absent a content base. The teacher has to have the ability to hear students and to keep an eye on where she is going with the (liscussion. She has to have a sense of the mathematics anti of the students she is teaching to make the right decisions about the questions she will ask.

The final remarks came from Sam Chattin, a middle grades science teacher. Chattin began with some comments on the TIMSS video of an eighth grade Japanese classroom that he had viewed during lunch. He observed that the Japanese classroom had students in rows, no decorations, the map on the wall was in only two colors, there were no interruptions, and there were two teachers for 36 students. This is a significant cultural difference compared with middle grades classes in the United States. Chattin drew a parallel between this and some other observations about Cindy's teaching behavior. If students were to answer any questions, they had to do it very fast because she moved on. When the students spoke, you TEACHING ISSU could not hear which is typical be- cause in U.S. classrooms it is often not required that everyone hear student responses. Jenny gave a wrong an- swer, but it is important to know that you can have wrong answers and still (lo well. The U.S. culture uses some prompt such as "excuse me" to spee(1 up and move on. The question "Does everyone agree?" elicits no response or just one everyone knew they did not have to pay attention because she was moving on. Remarks such as 'think about this," are immediately followed by "Okay, no..." with no time to think. Teachers can (lo all of the changes they are asked to make as teachers but don't have the time to internalize them so they will be useful in the actual process of teaching.

[a. r~r.r I Linda Foreman President, Teachers Development Group, West Linn, Oregon In the spring of 1994, ~ spent six weeks teaching mathematics to a heterogeneous class of 4th and 5th graders so that ~ could field test several activities ~ was developing for the Mathematics Alive! curriculum.) Work- ing with these students informed my writing in remarkable ways, and since my assignment was to create a complete four-year curriculum, this six-week project extended to four years. During this time, a few students moved away and a few were added, but most of this group remained, receiving their only mathematics instruction for three or four years in my classroom. In the spring of 1997, ~ read several excerpts from the TIMSS report to my class. The students respon(le(1 strongly to descriptions of "typical" American and Japanese classrooms and suggested they would like an opportunity to share their ideas about what is possible. Hence, they applied and were invited to present at the 1998 annual meetings of the National Council of Teachers of Mathematics anti the National Council of Supervisors of Mathematics in Washington, DC. Each of the students identified a "big idea" related to teach- ing and learning mathematics, wrote a two-page paper about the i(lea, anti based a 3-4 minute presentation on that paper. Inclu(le(1 below are excerpts from five students' papers/presentations. Although students receive(1 input anti ~ Mathematics Alive! is a comprehensive mathematics curriculum for grades 5-8, written by Linda Cooper Foreman and Albert B. Bennett, Ir., and developed with support from the National Science Foundation.

support from one another as they read each other's drafts and practiced their presentations, each student's ideas and work are original. Please note that ~ did not begin this project intending to do research or expecting the students to share stories as they have. Rather, ~ chose a school that was convenient to get to and from so ~ could test ideas and maximize my writing time. ~ chose the class because the teacher had been thoughtful about assuring a heterogeneous mix of stu- dents (interestingly, the students' achievement levels were very diverse in the beginning but grew to be very similar, e.g., by the end of their Sth gra(le year, they average(1 in the 92n(1 percentile on the statewide standardized test). With this in mind, ~ encourage you to set aside the concerns you have about ways these students may differ from yours or others you know, and simply consider the issues and possibili- ties prompted by these young mathema- ticians' comments. As you read their papers, you might speculate about the nature of the cIass- room instruction anti culture that enable the students' views about learning, teaching, and mathematics. While these students are explicit in their recommenda- tions, are there implicit messages about teaching and learning in the middle grades? What can you "read between the lines" about their mathematics experi TEACH~NG ISSUES ence? In particular, what thoughts (lo the students' comments and work samples prompt about the following: · What are the important characteris- tics of effective teaching in the middle gra(les? Of effective teaching of mathematics in the middle grades? · How can instruction in mi(l(lle gra(les 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? . ERICA QUESTIONING Ben: "Ms. Martin, I am stuck on this problem. How c30 I figure out the area of a triangle? Can you help me get started?" Teacher: "Sure, Ben. First, here's what you need to c30. Think about what a triangle looks like. Now where wouIc3 you finc3 the formula in your book for cleter- mining the area?" Ben: "Okay. I've got it. It's 2 bin." Teacher: "Good job. Now what is the base measurement? What is the height measurement? Do you see how to plug in those measurements for base and height to get the right answer?" Ben: "Thanks! I really unclerstanc3 now!" No way!! ~ (lon't think Ben really un(lerstan(ls. This (lialogue is an

example of what ~ call ineffective ques tioning by a well-intentioned teacher. This type of questioning leads a student to a certain way of thinking instead of having them figure it out on their own. ... Also, using questions like this only generates one method, focuses on the answer not the process, and stops the student from thinking very much.... Ineffective questions do not allow the students to explain anything that they have done or thought about. If students don't have the chance to explain their thoughts, then the teacher can't know if they really understand the problem. Students may have the answer, but that doesn't help them in the long run. ... ~ know how powerful it is to come up with an answer all by myself. My teacher might help me get started, but prefer to solve a problem myself. Once ~ find an answer ~ am intrigued to continue thinking of other possible methods. Also, when ~ find the answer, am more likely to remember how to do that problem, or ones like it, than if am told the solution by a teacher or a book.... Some examples of effective ques- tions to ask a student or a group of students are: · ' VVhat do you think?" · ' VVhat if ?" · "Is there another way to think about that problem?" S T U D E N T P E R S P E C T I V E S · 'VVhat are some observations you can make?" "Can you explain your thinking?" · "Can you predict what might happen next?" ...Being asked questions that (li(ln't lead me right to the answer was hard to adjust to at first. ~ wasn't used to finding more than the answer. Eventually the whole class got used to being asked these open-ended questions. Then we asked the same type of questions that the teacher had asked us, but to each other.... In the beginning dialogue, Ben was led to his answer by the teacher and he wasn't able to fee} the joy of finding a solution on his own. Sooner or later he may realize that he really doesn't under- stand the mathematics concept; he just figured out the answer by plugging into a formula without thinking. Ben would have had a greater understanding if the questions his teacher had asked had been something like this instead.... Teacher: "First, tell me what you unclerstanc3 about area." Ben: (explains his thinking) Teacher: "Can you tell me what you already know about the triangle from the picture?" Ben: (describes what he "sees") Teacher: 'What c30 you think you need to know in order to finc3 the area of a triangle?"

Ben: "Can you give me a clue?" Teacher: "Sure. VVhatif you builcithe triangle on a geoboarc3 to help you see it in a different way?" Ben: (explores and invents a formula for the area of the triangle) Teacher: "So now you have a formula that works for the area of that triangle. Do you think it will work with a right tri- angle? ...an equilateral triangle? ...any triangle? Why don't you investigate that?" JULIE WORTHWHILE ACTIVITIES In order for a mathematics class to be worthwhile, ~ believe that the teacher needs to provide the students with worthwhile activities. For example, when we explain our thinking out loud, make posters of our work, draw dia- grams, work in groups, anti use manipulatives, we are more productive and therefore learn more.... To deter- mine if an activity is worthwhile, some helpful things to ask are: · Will it make students stretch their thinking? · Will it branch off to other topics? · Is there more than one way to solve the problem? · Will it help students' un(lerstan(ling of the idea? · Will it cause some disequilibrium? TEACHING ISSUES ... The best activities to do are ones for which the teacher can say "yes" to all of the above questions. Teachers can't expect the students to work well in groups if they give them a problem like, "Find the sum of twenty-eight anti seventeen." Better questions to ask would be, "Can you fin(1 more than one way to solve the problem twenty-eight plus seventeen?" or "Can you invent an algorithm for adding any two digit numbers?" . . . As you read on, look for evidence in I~in(lsay's, Joel's, anti Kyle's papers that effective questioning and worthwhile activities are/are not elements of their instruction. What questions and/or activities might have prompted the mathematical thinking that I~in(lsay, Joel, anti Kyle (remonstrate in their examples? LINDSAY VISUAL MODELS Working with models for mathematics i(leas helps me have a better un(ler- stan(ling of mathematics concepts which lea(ls to confidence anti success in mathematics. Visualmodelsdon'tjust help me fin(1 an answer, they help me understand why the answer works. When ~ use visual models, ~ have more confidence in my thinking, and ~ have a stronger grasp of the mathematics ideas. When ~ can find an answer by

myself, and see why it works, it makes me fee} confident and that makes mathematics more fun for me. Models help guide me to the inven- tion of formulas. Visual models also help mereinventideas. If~forgeta formula or concept, all ~ have to do is go back to the mode} and ~ can reinvent the proof. For example, if ~ forget how to find the area of a trapezoid, different formulas just pop right out to me in the model. Before ~ was able to invent the following formulas, ~ invented formulas for the areas of triangles, rectangles, and parallelograms. In diagram i, ~ divided a trapezoid into 2 triangles. ~ found the area of both triangles and added them together. That gave me the area of the trapezoid. Diagram ~ ~ 6, T /--- ~. h ~ ~ Area ~ (b ) ~ (A a h) In diagram 2, ~ took 2 trapezoids and "smooched" them together to form a parallelogram. ~ found the area of the parallelogram and then divided it by 2. In diagram 3, ~ divided the trapezoid into 2 triangles and a rectangle. ~ S T U D E N T P E R S P E C T I V E S "smooched" together the 2 triangles, forming a larger triangle. Then ~ found the area of that triangle and added it to the area of the rectangle. Diagram 2 hrca ~ (b' High Diagram 3 T / ~ / 1/L . Mob, ~ . ~ . 1- in_ v- ~_~ ~ ] Area. (b,h)4 hi ...There are many advantages to using visual models. They help me produce work more efficiently, and give me a better understanding of mathemat- ics concepts because ~ can actually "see" the why behind a concept. Most impor- tantly, these mo(lels enable me to be confident in myself as a mathematics student and know that ~ really (lo have an inner mathematician.

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_

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

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

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

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

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.

=.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

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.

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

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

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.