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r ~ ~~li- The second day of the workshop focused on two examples from the United States of the use of teaching practice as a medium for professional development. Both sessions dealt with the use of records of practice records of what teachers do as they teach as a way to discuss mathematics teaching. Deborah Ball and Hyman Bass discussed the use of classroom video and addressed the questions · How do observations of what teachers do in the act of teaching enable teachers to learn mathematics? · How do such observations enable teachers to learn how to teach the mathematics they need to teach? Margaret Smith presented two written cases of classroom teaching and addressed the questions How can cases designed to investigate teaching and learning be a site for learning about teaching? What does it mean for teachers to use the study of others' practice to learn mathematics and about teaching mathematics? Professional Development Through Records of Instruction Deborah Loewenherg Ball, Professor, University of Michigan Hyman Bass, Professor, University of Michigan Professional Development Through Written Cases Margaret S. Smith, Assistant Professor, University of Pittsha~rgh
~11 ~~- ~- Deborah Loewenberg Ball, University of Michigan Hyman Bass, University of Michigan Ball: Our focus today wall be on profes- sional development using what we call, "records of practice." Like "lesson study," this is a form of professional development that uses practice to learn about teaching. However, while lesson study engages teachers in examining their own practice and in the practice of colleagues, the work we wall be discussing involves examining records of practice. What is a record of practice? It is a detailed documentation of leaching andlearning. Examples might be videotapes, either segments from lessons or whole lessons; written cases of teaching and learning; students' written work from classrooms; transcripts of classroom discussions; teachers' notes and lesson plans. The point is that these are documents taken directly from teaching and learning of mathematics, without an analysis, which enable teachers to look at practice (Erectly, together with other teachers. WHY USE PROFESSIONAL DEVELOPMENT IN RECORDS OF PRACTICE? Some of the reasons for using records of practice are intended to address a number of problems in our professional (levelopment and teacher education system. For example, recor(ls of practice provide a common context for teachers to work on teaching. Teachers in the United States (lo not usually have opportunities to work (Erectly on teaching and learning with other teachers. When they (lo meet with other teachers, often all they do is tell each other about their work or work on something else like a new technique or a new curriculum or some mathematics. But rarely do they have the sorts of opportunities that we saw teachers in Japan have regularly. Recor(ls of practice provide a common opportunity to study teaching anti learning. A second advantage is that records of practice provide a way for professional development to be grounded in practice so that the problems and issues that teachers work on are directly connected to the work of teaching. Sometimes teachers learn from professional develop- ment experiences but are then unable to use that knowledge in their teaching. Records of practice provide an opportu- nity for teachers to learn knowledge as they would need to use it with students. Therefore, one compelling reason to use records of practice is to ensure that the
knowledge that teachers generate as they work is both useful for practice and usable · ~ In practice. Records of practice allow professional development leaders and teacher educa- tors to select particular problems of mathematics or of the teaching and learning of mathematics. A teacher educator could select a specific challenge of teaching mathematics and then select records that provide an occasion for teachers to consider that challenge. This is different from discussing one another's teaching where the problems that arise are dependent on what teachers happen to bring up. Using records of practice allows a teacher educator to design work around a particular problem of practice. Another benefit of using records of practice is that teacher educators can provide opportunities for teachers to see practices, problems, or issues that they have not seen yet in their own practice. They might see, for example, children discussing mathematics in a way that their own students do not yet know how to do. So it allows teacher educators to expose teachers to issues beyond teachers' own individual classrooms. Working with records of practice can also develop teachers' abilities to learn from their own practice, to learn to look more carefully at student work, to learn to listen more attentively to students' talk, to analyze mathematical tasks in ways they have not done before. One thing that shruck several of us while watching lesson stu(ly was how skilled many Japanese teachers are in the discussion, analysis, and study of practice. Records of practice provide opportunities for teacher educa- tors and teachers to develop some of those skills anti capabilities, not just of teaching, but of the study of teaching. And finally, records of practice can enable teachers to talk safely about problems of teaching and learning, because the teaching and learning that they are looking at is not their own and not their colleagues. There can be more freedom to raise hard questions or to consider problems without the worry of being polite or not hurting someone else's professional pri(le. Many of these advantages apply also to lesson study. One thing we might discuss together later is how using records of practice is similar to and different from lesson stu(ly. POSSIBLE PROBLEMS IN USING RECORDS OF PRACTICE FOR PROFESSIONAL DEVELOPMENT There are also problems in using records of practice. Because the material is not from a teacher's own classroom, it might not be relevant. Teachers might say, "T (lon't have this problem; these students are not like my students; this classroom is not like my classroom." So teacher educators may face problems of making sure that the work seems relevant to teachers. Similarly, it may not seem interesting to teachers when the problems they witness or stu(ly are not their own. There are problems in (leveloping good records of practice. Not every videotape is suitable for stu(ly. Not every lesson is provocative for teachers' learning. Not all examples of chil(lren's work are equally useful in professional (development. Gathering, cataloging, examining, anti becoming familiar with high-quality records of practice is a problem. We have learned that for this work to be profitable for teachers, the tasks that teachers work on with these materials makes a (lifference. This is no (lifferent from the knowledge that the mathematical tasks that chil(lren work on make a PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
difference for their opportunities to learn mathematics. The same is true for teachers' learning; not all tasks are equally useful in professional develop- ment. Quite often in our early work we neglected to frame tasks at all and thought that simply by looking at video- tapes, teachers would learn. It matters what the task is. In the United States we do not have, in general, highly developed norms for the study of teaching. Teachers are not used to discussing, analyzing, or closely probing teaching. Working with recor(ls of practice means that we also have to work on developing a culture and a set of norms for work of that kind. One chal- lenge that we face is to move from a habit of evaluating and judging teaching to analyzingit. Yesterday we saw many examples of Japanese teachers and of ourselves as workshop participants- closely analyzing the teachers' decisions, the nature of a task, a child's contribution. That sort of work, that kind of analysis is different from saying this was a good lesson, this was a smart child. Learning to do this kind of analysis is part of what we have to develop in the U.S. culture of teaching. At the same time, too much analysis can move very far away from teaching. Teachers are not just researchers. They must act with students. In work that analyzes records of practice, it is impor- tant to maintain a balance between analytic work and practice and to strive for the development of knowledge usable for teachers' work. There is the challenge, like the challenge of working with chil- dren, of bringing sessions to closure so that teachers go away with knowledge and ideas that make them feel the work has been useful and they have something to take with them to their own classrooms. DEVELOPING OPPORTUNITIES FOR TEACHERS' LEARNING USING RECORDS OF PRACTICE Interestingly, the work using records of practice has much the same structure as the structure for lesson study: prepara- tion, enactment, and analysis. Preparation Phase: Design. In (leveloping opportunities for teachers' learning, there is a phase of (resign for the teacher educator that inclu(les asking such questions as, What is the goal or purpose for teachers' learning? The preparation phase includes designing a task that teachers would work on together using records of practice and designing the enactment of that task in a session with teachers. It includes selecting resources to support that learning. For example, what sorts of records of practice ~ ~ ~ ~ ~ =~ . . ~ . . ~ WOU1d nelp' what other materials mlgnt be needed? Might teachers need the curriculum materials on which chil(lren were working in order to interpret a videotape? This process looks very much like the design work for teaching math- ematics to children. Enactment Phase: Facilitation. There is the complicate(1 process of enacting the work with teachers. What (foes it take to enact useful, constructive, productive sessions with teachers where analysis of teaching anti learning are the subject? These are some of the tasks involved in this phase of the work: setting purposes; posing the task or problem to be worked on; organizing how time gets spent; attending to teachers' engagement anti learning, to teachers' i(leas anti lifficulties. Anti there is the work of processing the (liscussions, sharing the work from the sessions, and developing ideas that everyone takes from the work. This involves keeping the work groun(le(1 in PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
mathematics, so that teachers have opportunities to develop content knowI- edge. Keeping the work connected deeply to concrete materials of practice means learning to use evidence for statements that are made about teaching and learning. It includes learning how to generalize from studying particular examples and forming more general ideas that can be useful in classrooms other than the one being studied. Analysis Phase: Reflection and Design of Next Steps. And of course there is the phase of analyzing and reflecting on how the sessions with teachers work. EXAMPLES OF USING RECORDS OF PRACTICE IN PROFESSIONAL DEVELOPMENT We will take brief tours of two (lifferent examples of this work. There are many such examples in the United States, and although this is not the main form of professional development at this point, it is also not rare. Deborah Schifter (Schifter et al., 1998), Ed Silver (Stein et al., 1999), and Alice Gill (American Federation of Teachers, 2000), among many others, have all engaged in this kind of work. So together, we will (liscuss some of the work we have been doing to develop approaches to the study of practice. These have some resemblance to lesson study but are also different from it. We have to examine how these are different, how these are similar, anti to learn together ways that we might join some of the special features of each. The example we would like to share draws on work that we have been design- ing over the past ten years with several of our colleagues at the University of Michi- gan. We have been working with a very large collection of records from two classrooms one grade three (8 year olds), and one grade five (~1 year olds) across an entire school year. We collected videotapes every day in these classrooms for a whole school year. We also collecte(1 all of the chil(lren's work, all the tests they took, all the materials they used. This includes a detailed teacher's journal with indications of what the teacher expected over a range of lessons. We have been designing materials and experiences for teachers' learning that draw from this very rich collection of records. What follows is one short exposure to the sort of work we do with teachers using this material. THE PROBLEM OF THE DAY Imagine you are a group of teachers. The problem on which we are going to work is that of (resigning anti enacting mathematical work at the beginning of a school year, actually the fourth day of class. In many schools in the United States, teachers get entirely new groups of students at the beginning of every school year. Those chil(lren have often been in many schools anti have not worke(1 together before. The school where ~ was teaching, for example, was very mobile: Chil(lren move(1 in anti out all the time. My school ha(1 the a(l(litional challenge of serving an international community anti many of my students (li(1 not speak English language anti cultural (liffer- ences increase(1 the complexities of bringing the students together. But, even without cross-cultural anti multilingual considerations, classroom teachers throughout the Unite(1 States must take into account the fact that many chil(lren come from different schools with different PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
past experiences and may never have done the sort of work in which the teacher aims to engage them. Teachers must find out what their students know how to do, and they must begin to teach them the curriculum. They must also teach them the ways of working that the teacher aims to use during that school year. Before watching the classroom video, there are two kinds of questions we want you to consider. CONSIDERATIONS FOR THE DESIGN AND ENACTMENT OF MATHEMATICAL WORK AT THE BEGINNING OF THE SCHOOL YEAR Bass: Place yourself in the position of having to do this work which faces the teacher at the beginning of the school year. What are some of the consider- ations in the design of such a lesson, the enactment of such a lesson? What are the things you want the students to be doing? With what things should the teacher be concerned? What would the teacher want to find out in these early lessons? What are the problems that should be on the teacher's mind at this stage? U.S. Participant: At the beginning of the school year, my most important goal is creating a mathematical culture, to get the students asking questions, making conjectures. ~ don't worry so much about particular content, starting the textbook or anything like that. So ~ have sort of favorite activities that ~ know are very engaging, that bring up a lot of ideas, and that bring up a lot of questions. ~ start the year, to set the stage for how they are to act for the rest of the year. U.S. Participant: ~ think with the student population that Deborah Ball described, the language issue is one that is especially critical for the teacher to both be under- stood and to be sensitive to understanding what the children are saying. So sensitivity to language is important. U.S. Participant: For me, it's developing a culture of people being respectful to each other in these conversations. The content is one thing, but the social dynam- ics of listening, appreciating each other's ideas is very important to develop in the beginning of the school year. U.S. Participant: ~ think one of the things that ~ like to do at the beginning of the year is to give students an opportunity to let me understand some of what they know and how they are used to working. Are they used to responding to a question, or are they simply looking for something the teacher says to say back? CONSIDERATIONS OF THE MATHEMATICAL TASK AND ITS USE Bass: Let us move now to another aspect and consider the mathematical work itself. You were given a homework assignment (Appendix G) that includes the math- ematical task you will see worked on in this lesson. There are questions about the capacity of this task to support the considerations you brought up about doing serious mathematical work, estab- lishing norms for communication with each other, learning and showing respect for other students. What are the kinds of questions you would pose in the enact- ment of the three-coin problem (Box I) ? How would you pose the questions for the work on this task? How and when would you do so? What kinds of student response might you anticipate? What kinds of responses would you want to PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
r.~.e ~ ~ i-1 ! 1 [~ I ~ 1 I have pennies (1 ¢), nickels (5¢), and Jimes (10¢) in my pocket. If I pull three coins out, what amount of money could I haven SOURCE: NCTM (1989). elicit? And what elaboration of the task might you want to be prepared to do in case it turns out to be too difficult or too easy? How can you be ready to incorpo- rate responses from the students in the development of the lesson? This task does involve some serious mathematical work. How do you recon- cile that with the fact that this is a very diverse class? There are large differences in background and in background knowI- edge, in the students' sense of how to work with each other. What is the suit- ability of this kind of a task, given that kind of diversity, and so many unknowns? U.S. Participant: It's a good task in that it has many entry levels. For example, the task could be set up in such a way that you could find one answer or many answers. And the ultimate part of the task, ~ would imagine for third grade, is considering how do you know you have them all? You might start by asking students in the whole group to think of one way, one amount of money they could have, and start to collect the ways. And then allow the students to go off, perhaps in pairs or by themselves, giving them choices about how they want to work initially. This woul(1 allow you to observe who works in(livi(lually, who works with a partner, who works with a small group, who knows who, who doesn't all of those social issues. Anti who has confi(lence as they begin to solve the problem? Who takes it further? U.S. Participant In a class where children have many languages, ~ might take this problem and indicate that the pennies are worth one; the nickels, five; and the dimes, ten. ~ would not be certain that a multilingual class woul(1 un(lerstan the value of those coins. But ~ think it's a good task to get them to begin to think about things. U.S. Participant ~ guess I'm thinking just a little differently about it. Actually, approach this at the beginning of the course in methods of teaching elementary school mathematics. ~ use a problem that ~ have airea(ly tried out, perhaps many times, and have (levelope(1 a (letaile(1 lesson plan. So I'm familiar with the kinds of responses that ~ woul(1 likely get from the students and also how ~ woul(1 (teal with those responses. U.S. Participant: ~ woul(1 also have jars of money with the coins, for those stu- (lents who felt they wanted them. U.S. Participant ~ (lon't teach elemen- tary school, so I'm not quite sure if my PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
concern would be valid, but listening to what Deborah Ball said about the kinds of students that she worked with, one of the concerns that ~ would have with this particular problem is the fact that there are many answers. And many children might not be used to this idea of a math problem having several solutions. ~ think that's something we want children to understand, but is that too big of a prob- lem to bring up on the very first three or four days of the school year? ~ might consider actually softening that fact by maybe putting this into a game context or something that they see players playing, and whoever gets the most wins. Some- thing where there are many solutions but where that doesn't become an initial focus. Japanese Participant: It's important to know what students have learned in the previous years. What is the level of knowledge of those children in your classroom? ~ think that kind of context is a major concern for Japanese teachers. Japanese Participant ~ think the three- coin problem is too difficult for the elementary school student. How do you position this problem in the context of classroom teaching? U.S. Participant: ~ did teach third grade most of my career, and ~ do think it is suitable for third graders. The problem would give me as a teacher a sense of whether or not these students were able to approach a problem in an organized manner. This is important to me as a teacher because if they can't, then ~ need to do some things to help them organize their thinking. Bass: Your questions indicate where you want to speculate a little bit. When you watch the enactment in the class, think about how to reevaluate that concern. One thing that is often emphasized is the importance of the teacher doing the math of the lesson prior to the lesson. The math is typically elementary, but the insoles of it often involve intricacies and complications that vitally affect the instruction. That was part of the intention, in fact, of having you do the homework: It would allow you to be inside the territory where the children are working. EXAMINATION OF THE RECORDS OF PRACTICE Ball: A lesson plan (Appendix G) starts with the problem and provides an explana- tion of what the purposes are for the class. There were three purposes that ~ had as the teacher. The first was to develop students' habits of searching out multiple solutions and establishing whether all solutions to a problem have been found. This inclu(le(1 (1eveloping students' ability to produce a mathematical explanation. In this problem, an explanation for a solution must establish two things. First, that three coins were used from among these three types, anti second, that the amount of money produced is correct in total. The second purpose is to communicate to the students what (loin" mathematics wall mean this year in this class. For example, students will learn that math- ematical work will inclu(le producing explanations for one's work to the teacher anti to other students; they will learn to listen, to critique, anti to use other stu- dents' ideas; and they wall learn to be accountable for their own mathematical ideas. The third purpose was for the teacher to begin to learn about the student to learn, for example, about the students' PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
addition and multiplication skills, their openness to multiple answers and solu- tions, the strategies they use for finding solutions, how they keep track of their work and of the solutions, how skeptical they are that they are finished, and how they go about determining whether they are done with the problem. How do they work with concrete materials, in this case coins? What is their disposition to confer with other students and to consider others' ideas? The rest of the lesson plan outlines the steps. On the second page is a list of strategies that ~ knew that students of this age were likely to use and different approaches to recording their work that anticipated them using. This was a problem that ~ had used many times before with this age and also with both younger and older students. The video includes a segment from the beginning of the class. The problem was posed, and there was some discussion about what the problem meant, including an example of a solution. Then the chil(lren worked alone. After about ten minutes of work they came back into large group discussion of solutions to the problem. (Transcript anc! class description in Appenclix G} See the vicleo clip: RecorcIs of Instruction: Reasoning About Three Coins at Thirc! Gracle ANALYSIS OF THE LESSON Ball: The next phase of our work is to return to the problem that we're consi(ler- ing in this session, which has to (lo with organizing mathematical work in the beginning of the year in light of a whole host of considerations that bear on how one begins a school year. In this lesson segment we saw how the teacher's considerations about the beginning of the year were handled. What came up? How did the segment correspond to the teacher's goals and anticipations? What seemed unexpected either to the teacher or to you? How did this problem work out so far? Obviously, we haven't seen the whole completion of the work. How did this problem work out so far, given the teacher's goals? What surprised you? Was this what you had expected? Having had an opportunity to examine one example of a teacher's work in this problem (lomain of (resigning anti enact- ing mathematical work at the beginning of the year, what comments woul(1 you like to make? U.S. Participant: The first question the teacher aske(1 after bringing the class back together was to predict how many different solutions there were. Then the students worked on the problems a little bit more on their own. But when the class reconvened as a whole class, the (liscus- sion was on what amounts can you get. was wondering what purpose that initial question served? U.S. Participant ~ thought the lesson just came to something of an abrupt end. was anticipating some request of the students to think in a more structure way about the (lifferent solutions they were generating. U.S. Participant The thir(1 state(1 purpose for the goal inclu(le(1 a list of the many things the teacher wants to learn. Unless the teacher vi(leotape(1 a lesson anti sat (town afterwards, ~ saw no evi- (lence or any specific way the teacher recor(le(lthatinformation. In other PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
words, it is nice to ask, 'what strategies did the children use?," but unless there is a systematic way to record that, especially the first week of school, it's very difficult to keep track of that sort of information. So I'm not sure how the teacher assessed that objective. Japanese Participant: Three coins are taken out, and students asked what amount could be made. That was the situation. However, what was the motiva- tion in this class? In this situation, the Japanese teacher would take three coins out of his pocket and say, for example, '~his would buy you an apple." Then the children would think about wanting to purchase something and would get more interested in working on the problem. In the video, the children did not think about why they needed to solve the problem. It was just because the teacher said so. This content would be in the sixth grade in Japan where they would consider how many cases, not how much the value could be. The students would solve it structurally, by drawing a diagram or counting each case. The mathematical content would have a focus on analyzing the number of distinct combinations. Bass: One of the advantages of having the very complete record of what hap- pened over the entire year is that when we see something from a small window of a lesson, we can ask questions about how the students became acclimated to the meaning of the coins or how the children worked with the coins, and we can trace what happened in the earlier lesson where the teacher worked on the two-coin problem. And very much what you were suggesting a Japanese teacher might do, happened in that lesson. There was time spent moving around the room, showing the students the coins, asking for their understanding of what the coins were worth, and things of that kind. So that preparation in fact was in place before the enactment of this problem. Japanese Participant In Japan, the same problem is also handled in the first year of senior high school. The focus is how to solve this problem and how to find those strategies. There needs to be time for students to think about the problem. Japanese Participant If students understand the structure of the two-coin problem, then they can utilize it in prob- lem one (see the Homework handout in Appendix G) and also in problem three. If they understand the structure of problem two, even if they (lo not write out every- thing, they can use the results of problem two for problem four. By drawing tables they can grasp the structure of the problem. If they learn that in elementary school, and if they go up to the high school, they can always think about the meaning of the problem that is given. U.S. Participant: What we have here is what happens with many rich problems. This problem is a different problem at the third-grade level than it is at the sixth- gra(le level, than it woul(1 be at the first year of senior high school. You look for (lifferent reactions from the students. The only concern ~ had about this problem at the thir(l-gra(le level was that the entire time was spent in un(lerstan(ling the problem. At the en(1 of the 30 minutes, the students had no better way to solve the problem than they had at the begin- ning, which was trial and error. But maybe at the third-grade level, that's all we want. That is, perhaps it is enough to simply introduce a problem of this kind, to introduce the idea of trial and error, and to solve it. At later grades of course, we look PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
for systematic ways. And at the higher grades, we look for the underlying math- ematics of combinations and so on to solve it. Ball: The nature of these materials is important to highlight. Because there is a record of every day, the comments that people are making right now in a profes- sional development context would be converted by the teacher educator into questions for further inquiry by the teacher e.g., Did students at the third- grade level develop a more general structure for understanding this class of problems rather than simply discussing whether children can or cannot do that? When someone wonders about these kinds of things, they would be invited to turn back to the record, to look at the next day's work, to look at students' written work perhaps, or to read the teacher journal. And this enables the kind of developmental tracing of children's learning not possible by simply visiting a school or by reading one example. This is a special feature of this kind of year-Ion" record of practice. After all, teaching and learning occur across a school year, across time. One kind of work that this kind of record permits is the opportunity to look across days to see what happened at the beginning of the next class, or what sort of structure third graders ended up developing? And how did the children differ within the third-grade class? U.S. Participant: ~ got the feeling that after you looked past the mathematics, one of the major purposes of this class was creating a community of learners. Especially from the way the teacher at the end remarked about the way students were listening to one another, giving each other enough time, promoting the think- ing of students without interfering others. A teacher can get a lot of information from the students about whether they were using multiplicative methods or additive methods in fin(ling out the amount of money from the coins. U.S. Participant: The lesson needs to be seen in the context of an ongoing activity, and the record we see is only a fragment of a particular lesson. ~ think it did very well in establishing certain rules of behavior and operation. Listening to each other, giving explanations, taking turns, getting input from many students, all these features were being established very carefully. The full story is certainly not here. But notice that Mick (li(1 come up with nine solutions. In fact, he came up with ten, which is also quite interest- ing. So ~ thought as a recor(l, it woul(1 be very interesting to see whether from the recor(1 of the practice, the teacher was able to observe what the students (li(l. Japanese Participant As teachers make a lesson plan for the lesson, it's very important to think about the multiple solutions that might come up and about motivation as a very important factor. Using the setting to naturally urge the students to come up with the questions, then come up with the solutions is impor- tant. We (lo our best to urge the students to come up with the solutions, to interest them. Take the coins out of your pocket. Then say, "Now ~ have three coins in my han(l. How much it coul(1 it be?" This is one way to check whether the chil(lren un(lerstan(1 this problem. Some people woul(1 say three cents or fifteen cents, just haphazardly. But that's a good chance to motivate them. Here you have a right answer. ~ woul(1 probably repeat this two or three times to interest the children. Then maybe ~ woul(1 ask what is the possible minimum amount of money? PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
What is the possible maximum amount of money? And ~ would say, is it possible to come up with four cents? Impossible. What about six cents? What about thirteen? Maybe it's possible. Here we have three coins. Some amounts they can make; some amounts they cannot. Okay, now let us think about the cases, impos- sible cases and possible cases, and let us come up with all possible cases. The question of how many solutions are there is a mathematically interesting question. But for children, is it very interesting? Maybe they will wonder why is it impor- tant to have knowledge about how many solutions there could be? Maybe the more important thing is four cents is impossible, five cents is okay. What about six, about seven? And do they have all of the possible solutions? ~ think this kind of approach is more important. If they can make the complete table of those cases, with three coins, they will list those impossible cases. What about with four coins? ~ think that could be the next step Japanese Participant: The important thing is to urge the children to think about all of those possibilities and set up the steps. Usually the Japanese teachers try to think about how they can best set up several stages of thinking for students. But ~ know that this lesson is at the beginning of the school year, so the mathematical approach and the commu- nity making are also the priorities. ~ know that my comments should not be always applied to that kind of situation, but ~ think the important thing is motivation. Can we ask the proper questions so that the children can follow the steps in a mathematical way? Bass: One important feature of the work on this task that was not explicitly men- tioned was beginning to teach the children how to reason mathematically. So the mathematical task provided a context for that, but the very detailed, fastidious explanations of why certain amounts added up to certain amounts were not only elementary exercises in addition but they were the first steps in learning what it means to give a reasoned, careful explanation for a claim. And this, as one woul(1 see in the later recor(ls of this class, became a very important theme in developing children's capacity to reason mathematically. CONCLUSION Thank you very much for those very interesting comments. ~ think we have used up our time. ~ just want to make one very brief remark. Some of the themes or (Erections that you propose(1 coul(1 be investigated in the actual record to see what was either in the lessons before and after, anti also in the teacher's journal. An others of these, that are not necessarily enacted in the lesson but simply math- ematical elaborations of the task, can become potential material for learning mathematics in the context of practice and using these for professional development. PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
F~.le I- - ~-~el - ~ :~] Aims Margaret S. Smith, University of Pittsburgh Over the past few years, my colleagues en c! ~ at the University of Pittsburgh have been exploring the potential of cases- written accounts of teaching as sites for investigating and analyzing mathematics teaching and learning. The cases that we have createc! are based on ciata that was coDectec! Mom QUASAR, a national project aimed at improving mathematics inshruc- tion for students attending micicIle schools in economically clisacivantagec! communi- ties. Each case is based on actual events that occurrec! as teachers enactec! reform- orientec! instruction in urban micicIle school classrooms. The cases are not meant to represent best practice. Rather, they are intenclec! to represent actual practice what really happenec! when teachers set about to teach mathematics in new ways. The cases provide sites for teachers to · · ~ · · · ~ engage in critique, inquiry, ant 1nvesUga- tion into the practice of teaching. Each case has been conshructec! around a cognitively challenging mathematical task. Prior to discussing a case, we engage teachers in an opening activity intenclec! to give them an opportunity to explore the mathematical icleas that are cenbral to the case. This provides teachers with a personal experience in working through the mathematics on which to ciraw as they interpret en c! analyze the work of the teacher ant! her students cluring the class porbrayec! in the case. The opening activity also provides an opportunity to explore the mathematics in the task in more clepth. The remainder of this discussion focuses on a specific case entitlec! '~he Pattern Trains: The Case of Catherine Evans en c! David Young" (see Appendix for a copy of the case). This is one of a set of cases that was clevelopec! uncler the auspices of COMET (Cases of Mathematics Inshruction to Enhance Teaching), project funclec! by the National Science Founcia- tion that is creating materials for teacher professional clevelopment in mathematics. This case is one of four that explores icleas relater! to algebra as the stucly of patterns and functions. The opening activity in the case of Catherine and David, as shown in Figure I, provides an opportunity for teachers to look for the unclerlying mathematical shructure of a pattern, to use that shructure to continue the pattern, en c! to clevelop a rule that can be used to describe and build larger figures. The task provides an interesting context for discussing what algebra is en c! how algebraic reasoning can be clevelopecI.
FIGURE ~ The opening activity from the pattern: The case of Catherine Evans and David Young. ; ~ if= ~ train 1 train 2 train 3 train 4 Solve For the pattern shown, compute the perimeter for the first four trains, determine the perimeter for the tenth train without constructing it, ancl then write a description that could be used to compute the perimeter of any train in the pattern. (Use the edge length of any pattern block as your unit of measure.} The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is aclclecl. The first four trains in the pattern are shown. Consider Fincl as many different ways as you can to compute (ancl justify} the perimeter. The hexagon pattern task, featured in this opening activity, can be solved in several different ways. Consider, for example, the responses produced by five practicing middle school teachers who participated in a workshop during the summer of 1999. Linda's solution (Figure 2) involves a recursive approach. Linda recognized the general pattern of adding 4 to find the perimeter of each successive train, but her strategy required knowing the perimeter of one train in order to find the perimeter of the next train. Barbara's solution (Figure 3) by con- trast resulted in a generalization that can be applied to any train. She determined that each hexagon added four sides to the perimeter of a train and that the first and last hexagons also each contribute one a(l(litional si(le. Kevin's solution (Figure 4) also involves adding four sides, but differs slightly Tom the one proposed by Barbara. PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES In this approach, Kevin explained that each hexagon a(l(le(1 four si(les to the perimeter of the train for each of the hexagons in the middle of the train and that each of the hexagons on the en(ls of the train contribute(1 five si(les to the perimeter. Michael's approach (Figure 5) involved first counting all six sides of each hexagon. For each hexagon he then subtracte(1 the vertical si(les (two per hexagon), anti then a(l(le(1 on the two vertical sides on the ends of the train. Chris's solution (Figure 6) is a bit more unusual. Chris thought about the hexagon train as having a bottom anti a si(le anti a top anti a si(le Marked by bol(1 lines). She noticed that the perimeter of the bottom anti a si(le was the same as the perimeter of the top anti a si(le. Chris note(1 that the bottom or the top was two times the train number so that ~ nee(le(1 to be added to the bottom and to the top.
FIGURE 2 Linda's solution to the hexagon pattern task. ~ / ~ train 1 train 2 train 3 6 10 14 +4 +4 ~4 train 4 18 You just keep aclcling ~ each time. So the perimeter of the first train is 6, the perimeter of the second train is 10, the perimeter of the third train is 1 A, and the perimeter of the fourth train is 18. FIGURE 3 Barbara's solution to the hexagon pattern task. /\~ ~ ~ train 1 train 2 train 3 There are four sides for each hexagon plus two on the ends. So fortrain 4: 4 ~ 4 + 2 = 18 P = 4x + 2 train 4 FIGURE 4 Kevin's solution to the hexagon pattern task. train 1 train 2 train 3 train 4 Each hexagon on the inside of the train acids four sides to the perimeter. The first ancl last hexagons in the train acid five sides each to the perimeter. ~ · 2+ 10 PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
FIGURE 5 Michael's solution to the hexagon pattern task. train 1 train 2 train 3 train 4 You find the total number of sides for all the hexagons and then subtract two sides for each hexagon for the insicles. Then you need to acid two back on for the encls. So for train A: 6 · ~ - 2 · ~ ~ 2 = 18 P= 6 x - 2x +2 FIGURE 6 Chris's solution to the hexagon pattern task. ~1~1 train 1 train 2 train 3 train 4 For each train, the perimeter of one side and the bottom is the same as the perimeter of one side and the top. So the perimeter of one side and top or bottom is 2x ~ 1, so you have 2 of these. So for train A: 2~2 · ~ ~ 1 } or 2~9} = 1 8 P= 22x+1} As you can see from these sample solutions, teachers have many different yet interesting ways of connecting the diagram with a symbolic representation. After teachers have solved the task, we have found it helpful to explore the mathematics in more depth before moving on to a discussion of the case. One possibility would be to make a list of all the symbolic representations generated by the teachers, and ask them if the PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES representations are equivalent and to explain the rationale for their (recision. Alternatively, you may want to explore the mathematical content and processes embedded in the hexagon pattern task. This can lead to a (1iscussion of math- ematical ideas such as generalization, the or(ler of operations, the (listributive property, equivalence, and perimeter. There are many other questions that could be asked depending on your goals
for teacher learning, the context within which you and the teachers are working, and the teachers' prior knowledge and experience. Once teachers have had a mathematics experience related to the case, they are ready for the case discussion. Since the case is not self-enacting, you must create a professional learning task for teachers which serves to focus their investigation and analysis of the case. One task that my colleagues and ~ have found helpful in analyzing the case of Catherine and David is as follows: Indicate the ways in which you think Catherine's and David's classes are the same and the ways in which you think they are different. Be sure to cite line numbers from the case to support your claims. [At this point participants are given time to work in small groups to generate charts that made salient the similarities and differences between - Catherine and David's classes.] This small group work was followed by a whole group discussion of similarities and (lifferences. Table ~ contains a record of the responses pro(luce(1 by workshop participants during the group discussion. This task of finding the similarities and differences between Catherine and David's classes requires comparing an event that occurred in one class with an event that occurred in the other class, analyzing the two events in order to determine whether they have anything in common, and noting what is the same or what is (lifferent about the events. This activity generally brings to light many key issues related to mathematics teaching and learning that can be further explore(l. Another example of a similarities/(liffer- ences list generated by practicing mi(l(lle school teachers is shown in Box ~ at the end of the (locument. TABLE 1 Chart Generatec! by Practicing Mic~clIe School Teachers in Response to the Similarities anc! Differences Task Similarities Differences . Willingness to change · Same task · Positive attitudes · Encouraged student involvement Commitment to new program Same school and same Oracle level Both teachers were part of a community · Catherine focused on doing proceclures; Davicl focused on understanding relationships (between number of blocks and perimeter). · Catherine was more concerned with success and directed student thinking so they would be more successful; David was more concerned with student unclerstancling. · Types of questions: Catherine's had one right answer; Davicl's required explanation. · Catherine made tasks easier; Davicl Respect students solve the original tasks. · Davicl's students form generalizations "approaching symbolic"; Catherine's could apply rule to large numbers but not any number. PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
The COMET project shares Shulman's (1986) view that the strength of cases is that they can be used to straddle the space between gener- alizations and particularities, between the kinds of abstract, formal, codified knowl- edge that can be taught in the absence of context and the kinds of knowledge that are experientially derived, often informal, and perhaps lacking in precision. Hence a case allows you the ability to go from the very particular things that happen within a specific classroom to seeing those instances as examples of some larger class of phenomena that we consider to be important in teaching and learning. For example, the case of Catherine and David was designed to provide specific instantiations of key mathematical ideas such as generaliza- tions identifying patterns, perimeter, intuitive notions of variable, and connec- tions among representations. Within the case, there are opportunities to look at particular examples of each of these ideas. In addition, if you look across the set of cases related to algebra as the study of pattern and functions, you would see the same set of ideas woven throughout the cases. This provides an opportunity to explore ideas in more than one context and from more than one perspective. Each of the cases is also designed to make salient"pedagogical moves" that support or inhibit student learning. Moves that support student learning include teachers pressing for explanation and meaning, modeling high-level perfor- mance, allowing students sufficient time to explore and think, drawing conceptual connections, and building on prior knowI- edge. Pedagogical moves that inhibit student learning include shifting the focus to following rehearsed procedures; removing problematic aspects of the class, and allowing insufficient time for students to explore and think. So again, in each case you can see specific events that connect to these more general ideas about math- ematics teaching and learning. These ideas can be explored over a set of cases as well as in a teacher's own practice. In facilitating the discussion of the case, we generally start by making a record of the similarities and differences that teachers identified, resulting in the creation of a chart similar to the one shown in Table 1. Our goal is to then move from the specific things that happen in the case (as represented in the chart) to more general ideas. So the chart becomes not an end in itself but rather a starting point for additional discussion. This discussion might begin by focusing on a specific difference that was noticed. For example, in further discussing the types of questions pose(1 by Catherine anti David (see Table I, third bullet in the differences column), the facilitator might want to press teachers to analyze the learning opportunities that were or were not affor(le(1 by each teachers' question- ing strategies, thereby explicitly connect- ing teaching and learning. The facilitator might also want teachers to look at the list of differences and to begin to look for commonalties across events in the list in order to see specific instances as a subset of a larger class of phenomena. For example, fostering students' thinking might serve as a bigger idea around which to organize a number of different classroom events such as helping students un(lerstan(1 relationships anti requiring students to provide explanations. The ultimate goal of a case (liscussion is to create generalities that teachers will be able to (lraw upon in situations outside the case. The point is for teachers to take something away from the analysis of the case that can be use(1 to think about their own teaching. For example, a (liscussion PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES
Similarities Between Catherine ant! Davic! · Both teachers were struggling to change their practice. · The classes were working on the same task. Each student in the classes just presents one idea. (In a Japanese class, each individual student is encouraged to present multiple ideas.) Both teachers asked students to share their thinking and asked for different solutions. · Both tried to capitalize on student solutions. The problem about the 1 00th train is from the students (lines 490-4951. · Both Jo things that they hadn't planned on and so monitor and adjust their teaching as they are going. Both of them give homework directly from the class. · Both seemed to realize it was important to work toward generalizations of what they were teaching. Both were going to get to big mathematical ideas in the end. There was no activity to create formulas for expressions in either of these classes. There is no evidence that the teachers were selecting students with a sequence of ideas in mind, or that the teachers knew what the students had Jone in small groups or individually. There was no evidence that they were calling on students in a particu- lar order. Differences Between Catherine ant! Davic! · One difference is that Catherine is in the first year of teaching this curriculum, and David is in his second year, which points out that teachers themselves need to learn how to Jeal with reform and what it might look like. · Catherine had a set amount of time in mind for the lesson and when that time was over, she said, "We have to move on. It's time for another topic." Whereas, David felt more comfortable continuing his lesson into another day when the content was not covered to his satisfaction. · Catherine used a square to measure perimeter versus David who used a segment to mark off and measure perimeter. This might have implications for students' unJer- stanJing about what perimeter is. · There was a difference in the level of support. Catherine was going through a change of practice with colleagues who were at the same place in learning how to do this. David was coming into an established community that had gone through this change and was trying to catch up. It wasn't clear whether he had the same opportu- nities to look at videos of his class and discuss it with colleagues. He Jid not seem to have the same opportunities to reflect as Catherine JiJ. PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
· Catherine seems to get a little bit more impatient, and when a student doesn't seem to get an idea right away, she is there helping. For example, she seems to be literally moving the hands of the boy who was showing the perimeter on the overhead. David tends to be more willing to take time, ask more provoking questions, and wait for the students to make sense of things. David seems to push more for the multiple strategies, having one student explain, and then ask whether anybody else had Jone it a different way. He gave five different explanations for the formula, whereas Catherine got one from a student and pre- sented another. They introduce the topic in a different way. Catherine starts the lesson by asking the students to make generalizations about the patterns and only brings up the word "perimeter" when a student mentions it. David's initial introduction to the lesson is to find the perimeter for the four trains. · Catherine seemed to be narrow in how she asked questions, with answers that she wanted from the students, rather than being open to the answers that the students gave. · David had questions for example, about noticing a relationship, giving a bit of direction to the student in terms of what are the kinds of things you might look for. Catherine asks questions such as, "How many on the enJ2" (line 164) How many will there be altogether (line 1661. They are very specific one-answer questions. David says "How are these two numbers related2/' (line 569) in his effort to help students find a connection between the train number and the perimeter. He is giving questions with several possible answers. · It seems as though Catherine was validating the students' answers, which would introduce something that the students would then seek, versus David who was encour- aging open discussions and not necessarily commenting on correctness. The relationship between questions and evidence of student learning or understand- ing possibly came from the relationship between the questions Catherine was asking and what it was she thought she was getting (lines 205 and 2451. Catherine seems to be focused on asking questions with a numerical answer. What's another perimeters Whereas David seemed to be assessing student understanding based on their ability to explain how they got their answer and communicate an understanding that way. . . PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES
regarding Catherine's concern about student success may provide teachers with a new lens for considering what it means for students to be successful and for considering whether "imitation" indicates understanding of mathematics. The hope is that this "lens" would sensi- tize teachers to similar decision points in their own practice. What can be gained by using materials like this? My colleagues and ~ contend: In order to grab hold of classroom events, to learn from examples, and to transfer what has been learned in one event to learning in similar events, teachers must learn to recognize events as instances of something larger and more generalizable. Only then can knowledge accumulate. Only then will lessons learned in one setting suggest appropriate avenues of action in another (Stein et al., 2000, p. 341. PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE