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fir I. r~rm.r~_ The National Convocation on Middle Grades Mathematics was followed by three separate Action Conferences, funded by support from the Department of Education and additional support from the American Educational Research Association. The following three papers synthesize the activities and discussion that occurred during the three conferences. ACTION CONFERENCE ON THE NATURE AND IMPACT OF ALGEBRA AT THE MIDDLE GRADES Organized by Hyman Bass, Columbia University This action conference focused on providing school based decision makers with an unclerstancling of the importance of bringing algebra into the micic3le gracles and the issues involved in making this happen. ACTION CONFERENCE ON RESEARCH IN THE TEACHING AND LEARNING OF MATHEMATICS IN THE MIDDLE GRADES Organized by Sandra Wilcox, Department of Teacher Education, Michigan State University The conference was clesignec3 around the question: What are the characteristics of research that would be helpful and informative for teaching mathematics in the micic3le gracles? ACTION CONFERENCE ON THE PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATHEMATICS IN THE MIDDLE GRADES Organized by Deborah Ball, University of Michigan. The participants worked through a frame for considering the design and practice of teacher development at the micic3le gracles, examining the ideas that drive professional development in the light of what is known and unknown about teacher learning.

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r.~.r~ Synthesis by Bradford Findell Program Off cer, National Research Council The Action Conference on the Nature and Teaching of Algebra in the Middle Grades brought together mathematicians, mathematics educa- tors, middle school teachers, math- ematics supervisors, curriculum developers, and others to discuss the role of algebra in the middle grades. The conference was provoked in part by recent events and policy decisions that have focused attention on algebra in the middle grades. Perhaps most prominent among these events was the release in 1997 of results in the Third International Mathematics anti Science Stu(ly MESS, which indicated, among other things, that the U.S. curriculum at eighth grade is about one year behind many other countries, and that U.S. eighth graders perform, as a group, below the international average in mathematics achievement. In the policy arena, several states recently mandated that algebra be a required course for high school gradua tion, and the pressure has expanded into the middle grades as well. In the U.S. Department of E(lucation's White Paper, Mathematics Equals Opportunity, Secretary Riley recommends, on the basis of a strong correlation with college attendance, that students take algebra or courses covering algebraic concepts by the en(1 of eighth gra(le. Despite the growing public attention to algebra in the middle grades, there has been little discussion (and perhaps little public acknowle(lgment) of the fact that there are more than a few possibili- ties for what algebra in the mi(l(lle gra(les might look like. Thus, the agenda for the Action Conference was (lesigne(1 to bring some of these possi- bilities to light through presentations on various views of algebra anti on re- search in the teaching and learning of algebra, together with practical experi- ences of teachers anti (1istricts who have been implementing some version of algebra in the middle grades. Discus

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sion was framed by six questions (See Figure 1) that were presented by the Action Conference Organizer, Hyman Bass, a mathematician from Columbia University and Chair of the National Research Council's Mathematical Sciences Education Board. VIEWS OF ALGEBRA The first presenter was km Fey, a curriculum developer and mathematics education researcher from the Univer- sity of Maryland. He began the after- noon by reminding the participants that a big factor in the debate about the role of algebra is the social and political context. Algebra serves as a gateway to post-secondary study and to scientific and technical careers. And if algebra is good in ninth grade, then it is better in eighth grade, and some think even better in seventh. But there has been little attention to what the content of algebra is or may be. In the conventional view, algebra is primarily about calculation with sym- bols Bob Davis called it a dance of symbols. And algebra is used to solve word problems. But students are not very goo(1 at the wor(1 problems that we teach them, never mind problems that they haven't seen before. The increase(1 availability of calcula- tors and computers provides new ACTION CONFERENCE demands and opportunities for the teaching and learning of algebra. We may concentrate on the design and use of algorithms, on data modeling and predictions, or on analyzing anti project- ing trends. Spreadsheets anti other computing tools allow us to approach such ideas graphically, numerically, anti symbolically. Fey acknowledged that there is a fair amount of debate about what algebra is, but suggeste(1 the question might not be productive. Instead, he suggested that what we want from algebra are some concepts and techniques for reasoning about quantitative conditions and rela- tionships. There are four major aspects of such reasoning: representation, calculation, justification, anti application. Development of student understanding and skill in these areas can begin in the mi(l(lle gra(les. With such an approach, he notes, it will not be sufficient to take high school algebra and move it into the middle grades. Representation is about expressing complex relationships in efficient, condensed, symbolic form. Tradition- ally, the typical algebra question has been ' What is x?" But representations such as data tables, graphs, symbolic rules, and written expressions may be used to record and describe numerical patterns, formulas, patterns that change over time, and cause and effect relation- ships.

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Figure ~ . Framing Questions SOME IMPORTANT QUESTIONS Attention to subject matter ~ attention to students In teaching algebra in the middle Oracles, what are the tensions between attention to serious ancl challenging mathematical content, on the one hancl, ancl, on the other hancl, sensitivity to clevelop- mental, social, ancl equity issues pertinent to adolescent chilclren? Algebra as the language of mathematics 2. If one thinks of algebra as the language of mathematics, then does the learning of algebra entail some of the same challenges encountered in the learning of reacling, ancl therefore call for a more cleliber- ate ancl focused attention to the task of teaching this formalization of mathematical expression ancl communication? Real world contexts ~ generalization and abstraction 3. Many have argued that in order to motivate student learning of mathematics, it should be presented concretely in terms of real life problems ancl situations. This has been interpreted by some to require that all mathematics learning be embeclclecl in complex empirical investigations ancl measurements. Is this really warrantecl, both in terms of the presumptions about student motivation, ancl as effective peclagogy? Does this shortchange the equally important mathematical processes of generalization ancl abstraction, i.e., the distillation or clecontextualization of mathematical ideas from multiple contexts? Mathematics curriculum: covering mathematics ~ uncovering mathematics A. TIMMS characterizes the U.S. curriculum in mathematics as a mile wide ancl an inch creep. One manifestation of this is the pressure on teachers to race through an overloaded curriculum in both standard ancl accelerated tracks-with little time for student reflection ancl inquiry with new icleas, a practice that flies in the face of what constructivist ideas tell us about the nature of learning. Is this true? Ancl, if so, what can be clone to change this conclition? Situating algebra in the mathematics curriculum? 5. How should algebra be situated in the curriculum? As a traditional focused algebra course, or integrated with other subject areas, such as geometry? As a strand across many Oracles? Mathematical curriculum: materials, design, selection criteria, . . . 6. What are the characteristics of currently available curriculum materials in terms of topic coverage, pedagogical approaches, use of technology, support ancl guidance for teachers, etc. What kinds of tradeoffs must be made in the adoption of one over another of these curricula? How can one measure the impact of curricular choices on issues of equity, teacher preparation, community understanding, program assessment, ancl articulation with elementary ancl high school programs? THE NATURE AND IMPACT OF ALGEBRA

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Traditional approaches to algebra have focused on symbolic calculation, but calculation can be used to con- struct tables anti graphs of relation- ships, to solve equations, and to de- scribe rates of change and optimal values; major ideas in calculus that are often disregarded by calculus students who rely only on symbolic calculation. Calculation can aid in the construction of equivalent representations for quantitative relationships. The reason the representations are equivalent is that they both make sense for model- ing the same situation. In algebra, justification, reasoning, and proof are often considered in conjunction with the properties of the number system. But the properties of the number system are the way they are because the properties make sense. It is not the case that 2 + 3 = 3 + 2 because addition is commutative. Rather, addi- tion is commutative because verification that they both equal 5 can be general- ized to all numbers. Finally, applications can be pervasive from start to finish, providing frequent opportunities to move back and forth between abstraction and the real worI(l. Fey closed by pointing out that these curricular goals should be implemente with consideration of the ways students learn. He suggested that the way students encounter these i(leas must be changed from the "demonstrate, imitate, ACTION CONFERENCE and practice" style that has dominated algebra instruction. Al Cuoco, Director of the Center for Mathematics Education at EDC, pre- sented a view of algebra that was closer to a traditional view, in that symbol manipulation was more prominent anti the problems were more often from the world of mathematics. The emphasis was not on the manipulation, however, but rather on the ways of thinking that emerge from consideration of the historical (levelopment of the subject. Many of his points were illustrated by engaging mathematical problems. Cuoco prefaced his presentation with an acknowledgment that the points made by Fey were important, and then offered a list of possible answers to the question, ' What is algebra?" · Algebra is an area of mathematical research. · Algebra is the language of mathematics. . Algebra is a collection of skills. · Algebra is generalize(1 arithmetic. · Algebra is about "structure." · Algebra is about functions. · Algebra is about graphs. · Algebra is about mo(leling. · Algebra is a tool. Historically, algebra grew out of a long program of mathematical research that looked for ways to solve equations-ways that (li(ln't (lepen(1 upon the particulars of

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the equations. Sometimes the methods for solving these equations worked in situations that had nothing to do with the originalsituation. People started thinking about properties of operations. Algebraic thinking involves reasoning about calculations. When is the aver- age of two averages the average of the whole lot? What is the sum, ~ + 2 + 4 + ~ + ... + 28? Which numbers can be written as sums of a sequence of consecutive whole numbers? In Euclid- ean division, why do the remainders have to keep getting smaller? Algebra involves reasoning about operations. Consider the following two problems: 1. Mary drives from Boston to Washington, a trip of 500 miles. If she travels at an average of 60 MPH on the way clown and 50 MPH on the way back, how many hours floes her trip take? 2. Mary drives from Boston to Washington, and she travels at an average of 60 MPH on the way clown and 50 MPH on the way back. If the total trip takes 18 1/3 hours, how far is Boston from Washington? The first problem can be solved by a sequence of direct computations. The second requires reasoning about opera- tions. Students can solve it using a "function machine" computer program that allows reasoning about operations. Triangular numbers have the recur- sion formula, a (n) = a (n - 1) + n. While THE NATURE AND IMPACT OF ALGEBRA investigating these numbers, one student saw that a (n) = n2 - a (n - 1) . This is surprising enough. Cuoco wants people to see that it is useful: Add the two formulas to get the closed form. Units digit arithmetic provides some opportunities for reasoning about calculations. What is the units digit of 24~3485 _ 75~332? Through questions like this, students can very quickly begin to reason about the system of arithmetic module ten. Some have argued that skins in symbol manipulation are less important today, but symbol manipulation is not just a skill. Knowledge is not a copy of reality. To know an object, to know an event, is not simply to look at it and make a mental copy or image of it. To know an object is to act on it. To know is to mollify, to transform the object, and to unclerstanc3 the process of this transformation, and as a consequence to unclerstanc3 the way an object is constructed. An operation is thus the essence of knowledge; it is an interiorizec3 action, which mollifies the object of knowledge. Cognitive development in children: Development and learning (Piaget, 1964) Symbol manipulation can also support mathematical thinking. An(1 many important mathematical ideas, such as geometric series, the binomial theorem, and the number theory behind Pythagorean triples, require rather sophisticated symbol manipulation.

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During the question/answer period, both Cuoco and Fey agreed on the importance of algorithmic thinking an reasoning about relationships. Fey elaborated that the contexts provide a series of hooks, but it is important to go beyond the individual contexts to find the commonalities among the representations of various contexts. Hyman Bass sug- geste(1 that Fey's an(1 Cuoco's approaches are not in opposition, but emphasize two different aspects of the same thing. RESEARCH ON THE TEACHING AND LEARNING OF ALGEBRA Grit Zaslavsky, a mathematics educa- tion researcher from Israel, began her presentation by noting that in Israel there is no debate about whether to teach algebra. Because complete coverage of the research was not pos- sible, she decided to share an example of her own mathematical learning, which developed into a collaborative study of her graduate student, Hagit Sela, and her colleague, Uri Leron. Through this example she addressed research issues associated with teach- ing and learning algebra. She asked participants first to sketch the graph off = x on axes without scales, and then to sketch the graph of fix) = x on axes where the scales were different. For each case, she posed the following questions: 1. What is the slope of the line you sketched? How clic3 you determine it? 2. Does the line that you sketched clivicle the angle between the two axes into two congruent angles? How c30 you know? 3. Can you calculate tan fad, for the angle a between the line you sketched and the x-axis? If you were able to, how clic3 you calculate it? If not, why not? 4. Describe your considerations, reactions, dilemmas and other thoughts. Typical graphs are shown in Figure 2. For the first graph, most participants Figure 2. Typical graphs ACTION CONFERENCE -2 -3 3 2 y 1- l n ~3 -2 -1 ~ 1 ~ -1 Ad/ 1 Z x 6 Y4 --- - 1 ha - . 2 4

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assumed that the scales were the same. Zaslavsky pointed out that most people think offal) = x as bisecting the right angle of the coordinate system. None of the four questions were problematic for the first graph. For the second graph, however, there was disagreement, as some participants focused on the scales of the axes and others focused on the angle visually made by the line in the first quadrant. Zaslavsky pointed out that with the advent of graphing technologies and the possibilities for scale change, there was a sense that everything was invariant under scale changes. But she thought that there were some things that were varying the angle in particular. This example raised many questions: What is the slope of a linear function? What is the relation between the slope and angle? Is the slope a characteristic of a Linear) function that is independent of its representation? Or is it a charac- teristic of its graphical representation? (Similar questions may be asked about derivative.) Does it make sense to talk about a function without reference to its representation? As part of a research project, Zaslavsky and her colleagues were investigating approaches to this set of tasks in various populations. Students, mathematics teachers, mathematics educators, and mathematicians all shared a sense of confusion and to THE NATURE AND IMPACT OF ALGEBRA some extent, inconsistency or disequi- librium. All felt a need to re-think, re- define, or re-construct meaning for what they hall thought of as fun(lamen- tal and rather elementary concepts: slope, scale, anti angle. In their research (as in the Action Conference) there were several qualita- tively different approaches to tackling the problem. Individual people brought their own perspectives. For some, slope is a geometric concept. When the scale is 1:3, the line does not bisect the angle. For others, slope is an analytic concept, and the "fact" that the line bisects the angle is a clear result of the analytic calculation. Still others questione(1 the meaning of y = x if the units of x are ifferent from the units of y. What (toes all this have to (lo with learning algebra in the middle grades? Learning is about constructing mean- ing. This meaning can change over time, across learners (even experts) and across contexts. The teacher shoul provide a rich context for building different perspectives and meanings. In algebra, even the notion of variable has several meanings: unknown, vary- ing, generalization, etc. Meaningful learning takes place when the learner (leals with a "real problem" in the sense that the problem is real to him or her. Research supports a contextual ap- proach in which students engage with problems to which they can relate.

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of trajectories for teachers' professional growth (what is appropriate to learn at different points in a teacher's career), how to support communities of inquiry for continuing professional growth. 5. We need to better understand what the system needs to be like, in order to take the best prepared novice and support her/him. 6. We need to better understand what makes rich teaming environments for teachers (including standards-based curriculum materials), what they learn in these environments, and how to scale up this activity. This includes understanding productive connections between re- search and professional development. 7. We need to develop research tools and methodologies for examining and describing complex settings like classrooms. 8. We nee(1 to enlist teachers as colleagues in developing knowledge about these issues. Student learning 1. We know a lot about student thinking in some domains of the middle grades curriculum. We need to expand this knowledge to other content areas as well as across content domains 2. Much of what we know about student thinking has unstated premises (e.g., social anti instructional conditions are assumed) and we need to be ready to question what we know. ACTION CONFERENCE 3. We need to broaden the "we" in the research community. We may learn that there is more shared knowledge than there is share(1 theoretical commitment. 4. We nee(1 to combine mo(le} buil(l- ing with mode} testing. 5. We need to better understand whether and in what ways reform curricula support the development of mathematical power for all students, particularly poor students, students of color, and students with special needs. 6. We need to look at good teaching as a context for student learning. 7. We nee(1 to learn more about distributed learning communities, and about the relation of individual thinking to shared social practice. Communicating with various constituencies Here we raised a number of issues. 1. Who are we doing research for? 2. What does it mean to do system- atic research? What is the role of collaboration between teachers and researchers? 3. Does our research address problems that are widely recognized as significant? Do we know the concerns that parents have and how concerns change? 4. How do we package what we believe is useful and compelling for others (e.g., school boards, parents, commu- nity)? How do we present date thatis credible and acceptable to the public?

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REFERENCES Fennema, E., Sowder, J., & Carpenter, T. (in press). Creating classrooms that promote understanding. In T. Romberg and E. Fennema, (Eds.), Teaching and learning mathematics with understanding Hillsdale, NI: Erlbaum. Huntley, M.A., Rasmussen, C.L, Villarubi, R.S., Sangtong, J., & Fey, J.T. (in press). Effects of Standards-based mathematics education: A study of the Core-Plus Mathematics Project Algebra/Functions strand. {oavrnalfor Research in Mathematics Education. Gravemeijer, K (1998~. Developmental research as a research method. In A. Sierpinska and I. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (An ICMI Study Publication, Book 2 (pp. 277-295~. Dordrecht, The Netherlands: Kluwer Aca- demic Publishers. Greenwald, A., Pratkanis, A., Lieppe, M., & Baumgardner,M. (1986~. Under what conditions does theory obstruct research progress? Psychological Review, 93, 21~229. Henningsen, M., & Stein, M.K (1997~. Math- ematical tasks and student cognition: Class- room-based factors that support and inhibit high-level mathematical thinking and reason- ing. {oavrnalfor Research in Mathematics Education, 28~5), 524-549. Hiebert, I. (1998~. Aiming research toward understanding: lessons we can learn from children. In A. Sierpinska & I. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity. Dordrecht, The Netherlands: Kluwer Academic Publishers. Hiebert, I. (1999~. Relationships between research and the NCTM Standards. {oavrnal for Research in Mathematics Education. Hoover, M., Zawojewski, J., & Ridgway, J. (1997~. Effects of the Connected Mathematics Project on student attainment. www.mth.msu.edu/ cmp/effect.html. Lambdin, D., & Lappan, G. (1997, April). Dilemmas and issues in curriculum reform: Reflections from the Connected Mathematics project. Paper presented at the Annual Meeting of the American Educational Re- search Association in Chicago, IL. National Center for Educational Statistics. (1996~. P?`rsaving excellence: A study of U.S. eighth- grade mathematics and science achievement in international context. Washington, DC: Author. National Center for Educational Statistics. (1997~. P?`rsaving excellence: A Study of U.S. foavrth- grade mathematics and science achievement in international context. Washington, DC: Author. Reese, C.M., Miller, KE., Mazzeo, I., & Dossey, I.A. (1997) NAEP 1996 mathematics report cardfor the nation and the states: Findingsirom the NationalAssessment of Educational Progress. Washington, DC: National Center for Education Statistics. Sowder, I.T., Philipp, RA., Armstrong, B.E., & Schappelle, B. (1998~. Middle grades teachers' mathematical knowledge and its relationship to instruction. Albany,NY: SUNY Press. Sowder, J.T., & Philipp, R (in press) The role of the teacher in promoting learning in middle school mathematics. In T. Romberg & E. Fennema (Eds.), Teaching and learning mathematics with understanding Hillsdale, NI: Erlbaum. Stein, M.K, Grover, B.W., & Henningsen, M. (1996~. Building student capacity for math- ematical thinking and reasoning: An analysis of mathematical tasks used in reform class- rooms. American Educational Research {oavrnal, 33~2), 455-488. Stein, M.K, & Lane, S. (1996~. Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2~1), 50-80. Stigler, I.W., & Hiebert, I. (1999~. The teaching gap. New York: Free Press. Thompson, P. W. (1995~. Notation, convention, and quantity in elementary mathematics. In Sowder, J.T., & Schappelle, B.P. (Eds.~. Providing a foundation for teaching mathemat- ics in the middle grades (pp. 199-219~. Albany, NY: SUNY Press. RESEARCH IN THE TEACHING AND LEARNING OF MATHEMATICS

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Synthesis by Megan Loef Franke, University of California, Los Angeles and Deborah Ball, University of Michigan National attention to teaching, higher student achievement, and the need for more and better-qualified teachers is on the rise. Professional development and teacher education are of increasing interest and concern. All the concern for professional development is occurring at a time when views on teacher learning con- tinue to evolve. The change that teach- ers are being asked to make in order to enact standards-based reforms are ambitious and complex (Little, 1993; Cohen and Hill, 19981. Little points out that the current reforms require teach- ers "to (liscover anti (1evelop practices that embody central values and prin- ciples." Here teachers are seen as learners, "teaching and learning are inter(lepen(lent, not separate functions... [teachers] are problem posers and problem-solvers; they are researchers, and they are intellectuals engaged in unraveling the process both for them- seives anti for [their stu(lents]" (Lieberman & Miller, 19901. The Action Conference on Profes- sional Development was (lesigne(1 to afford an opportunity to examine promising approaches to professional levelopment. The premise was that extant knowI- edge about professional development is underdeveloped. Ideology and belief all too often dominate practice and policy. The Conference was intended to create an analytic and practical conversation about the sorts of opportunities in professional development most likely to lea(1 to teachers' learning anti improve- ments in their practice. With a focus on mathematics at the mi(l(lle gra(les, the structure of the Action Conference was groun(le(1 in analysis of the practice of teaching middle grades mathematics, considering the major tasks teachers

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face and what knowledge and skill it takes to perform those tasks. This analysis of teaching practice was used to take a fresh look at the kinds of opportu- nities for learning that teachers need. What does teaching cIemanc! of teachers arc! what cIoes this imply for teacher learning? Deborah Ball, as conference leader, framed the workshop discussion with the diagram (Figure 11. Ball pointed out that the focus of the discussion through- out the workshop would be from the vantage point of teacher educators, and would consider sites through which teachers might most profitably learn mathematics content needed in teach ing, based on tasks which teachers regularly do as part of their teaching. Participants observed that the diagram helps make clear that the content to learn becomes more complex at each level. Schoolchildren are learning mathematics. Teachers are learning about mathematics, but also about children's learning of that mathematics, and about the teaching of that math- ematics. The diagram does not incorpo rate all elements of what would be needed in a national infrastructure for systemic change. It depicts relations among learning mathematics, learning to teach mathematics, learning to teach about the teaching of mathematics. Ball in(licate(1 that mathematicians, mathematics education researchers and teacher educators have all created lists specifying knowle(lge for teachers. Although these lists enable profession- als in the field to discuss content in professional development, these lists are not grounded in an analysis of the work of teaching. Such lists tend not to consider questions about the math- ematical content knowledge that is necessary in the context of teaching or the knowledge of mathematics required when teaching extends beyond how to a(l(1 fractions or identify geometric patterns. It includes being able to frame a mathematically strategic question, come up with the right example, con- struct an equivalent problem, or under- stan(1 a chil(l's non-stan(lar(1 solution. The kind of mathematical knowledge it Figure 1. Teaching, Learning, and Learning to Teach Mathematics PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS \

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takes to teach a problem goes beyond knowing the content that the students are learning, and includes the capacity to know and use mathematics in the course of teaching. Participants considered the implica- tions of this perspective on mathemati- calknowledge. In the practice of teaching, the teacher's mathematical knowledge is called upon in many different ways in figuring out what a student means, in listening to what students say, in choosing and adapting mathematical tasks, and in knowing whether the students are understand- ing. How do teachers use their knowI- edge of the content, their knowledge of pedagogy anti other knowle(lge in an interaction with their students? How do they decide which students' ideas to pursue? How do they decide which paths will further the particular content in which they are engaged? How do teachers decide what a student means by a question or an answer, or judge whether students actually understand the concept in question? It is in these interactions that knowledge of the content is critical. Ball, together with Joan Ferrini- Mun(ly, Center for Science, Mathemat- ics and Engineering Education at the National Research Council, led an activity designed to engage participants in considering mathematics knowledge as it is used in one task of teaching. Ball ACTION CONFERENCE presented a mathematics problem and asked participants to solve the problem and to describe the mathematical territories into which it might head. I have some pennies, nickels, and climes in my pocket. I put three of the coins in my hand. How much money conic I have? (NCI~M, 1989) Having analyze(1 the mathematical territory of the problem, participants were then asked to make a "downsized or unsized version" that was mathemati- cally similar in structure. The partici- pants considered what students might say or do. Was making the problem more complicated numerically a means of "upsizing"? What (li(1 it mean to make a "similar" problem, anti what sorts of mathematical knowle(lge anti reasoning did it take to do this? Participants discussed different versions of the problem and considered the mathemat- ics they used to create and evaluate them. What cIo we know about professional cIeve~opment, teacher learning, arc! the improvement of practice? Although there is a lot of professional development in the U.S., much of it is ineffective. Approaches to professional development in the U.S. are often fragmented and incoherent, with little basis in the curriculum teachers will

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have to teach. Much professional development centers around features such as manipulatives or cooperative groups rather than on the substance of improving students' mathematics learning. Generally, evidence about the effects of professional development on student achievement is scattered and thin. Too often, professional develop- ment is defined by belief and propa- gated by enthusiasm. Some profes- sional development works, but there is a large gap even between effective profes- sional development and changes in practice. Often teachers view the focus of mathematics professional development as engaging in process: learning how to be a mathematics teacher or figuring out what the process of teaching should look like. Yet as teachers identify process goals, the substance of the process is left unchallenged. Teachers need opportunities to think about how the processes in which they engage serve the students' [earning as well as their own, how the processes relate to the mathematical ideas, student think- ing, and mathematical discourse. As one example, Ball cited the Cohen and Hill (1998) study in California that found that professional development that was grounded in the curriculum that teachers had to teach that is, provided teachers with opportunities to learn about the content, to learn how children think about that same content, and to learn ways to represent that content in teaching made a significant difference in student scores. Their findings indicated that professional development that made a difference provided opportunities to learn that · grounded content in the student curriculum; · were about students' thinking about that content; were about ways to connect students and content; · and were situated in context, materi- als, and sites of practice. . OPPORTUNITY TO LEARN SITUATED IN SITES OF PRACTICE The Action Conference explored the promise and potential problems of situating opportunities for teachers' learning in practice. Much teacher professional development offers teachers the opportunity to learn a new idea or activity and transfer that idea to their classroom. Professional development focused on helping teachers develop expertise within the context of their practice emphasizes the interrelationship of ideas and practice. Some professional developers engage teachers with cases of teachers engaged in teaching mathemat- ics and discuss what the teacher did and PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS

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why. Some professional developers engage teachers in the curriculum that they are going to teach and use that as the practice context, while others engage teachers with video examples of cIass- room mathematics practice, or some with student work. Each of these situates opportunities for teacher learning in practice. In each case the teacher is pushed to consider how the mathemati- cal ideas play out in the context of teaching mathematics, and they are competed to consider the details related to the students' thinking, the nature of the mathematics and its relation to their instruction. Each of these approaches affords opportunities forIearning. Each holds pitfalls that might impede such learning. Critical questions then are: How might the potential affordances be exploited? How might the potential pitfalls be managed? To learn more about what people are trying and how it is working, anti to sharpen the ability to examine their own designs and practices as professional developers, participants engage(1 in several examples of profes- sional development projects. WRITTEN CASES Margaret Smith, Pennsylvania State University, engage(1 the participants in a case study of student work The Case of Ed Taylor; Smith, Henningsen, Silver, & ACTION CONFERENCE Stein, 1998) and worked through the S Pattern Task (Figure 21. Analyzing student responses le(1 the participants to discuss the importance of the mathemati- cal knowledge of the teacher in analyzing the mathematics students do. The partici- pants offered the foLow~ng potential affordances for professional developers using cases as a medium for learning. Cases offer the possibilities of metacognition about the mathematics, provi(le an in-(lepth look at the mathemat- ics because the student thinking in the case is visible; provide the opportunity to study the particular mathematics involved in the case in context, provide an opportu- nity to read mathematics, can be used in many ways with different groups including administrators, and create opportunities to learn mathematics in teaching. Potential pitfalls included the challenge of asking questions that focus on the mathematics, focusing more on pedagogy than the mathematics, making the best use of lime, hard to read mathematics, and the fact the teachers may not know the mathematics. C U RRIC U LUM-MATE RIALS Karen Economopoulos, TERC, en gaged participants in an experience designed to show how curriculum materials might serve as a site of practice where teachers might learn mathematics in their work. Economopoulos posed two

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Figure 2. The S Paltern Task ·~ ·~.·~. ·~. ·~.·~. ·. ·. ·~.·~. · ·. ·~.·~. ·. ·~. ·~.·~. a. Sketch the next two figures in the S Pattern. b. Make observations about the pattern that help describe larger figure numbers in the pattern. c. Sketch ancl describe two figure numbers in the pattern that were larger than 20. cl. Describe a method for finding the total number of tiles in the larger figure numbers. e. Write a generalized clescription/formula to find the total number of tiles in any figure in the pattern. questions for rejection and discussion: How might curriculum materials such as these offer professional development opportunities for leachers? How might these materials influence or support a teacher's daily decisions? Potential affordances to exploit suggested by the participants included: The material speaks directly to the teacher; the mathematics is very near the surface, new materials generate thinking and learning. It is possible for administrators to see more depth in the mathematics. Materials can provide the opportunity to see a broader view of mathematics. Materials can move the mathematics preferred by the teacher forward. Conti- nuity is built into the lessons; the com- mon material promotes communication among the teachers. The pitfalls to be managed included the forgoing: It is easy to turn the search for the mathemat- ics into a make-an (l-take situ ation (levoi of thinking. If teachers (lo not un(ler- stand the content, using the materials to focus on the mathematics might increase frustrationIevels. Re-learning familiar materials might result in resistance. In some tasks, it might be easy to avoid the mathematics. VIDEOTAPES OF TEACHING AND LEARNING Nanette Seago, Mathematics Renais- sance Project, engage(1 participants in examining a videotape of a professional (levelopment workshop, "Facilitating the Cindy Lesson Tape." The teachers had previously viewe(1 a videotape of an eighth gra(le math lesson. The tape PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS

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provided an opportunity to consider the challenges of using videotape as an instructional medium in professional development. Watching this video focused on the task of "facilitation" of a teacher group discussion and raised explicit issues about the challenges of keeping the professional development work centered on specific learning goals teachers' learning of mathemat- ics, for example. There was considerable interest expressed in the potential of videotape as a medium and discussion about its role in helping teachers think about their practice. Seago encouraged the viewers to think back to the framing diagram of the conference and the use of video in the context of teachers as learners. Affordances cited included the opportunity to look at a shared instance of teaching, to see mathematics in use by teachers and students. Some of the pitfalls included the problem of keeping teachers focused on particular content, questions about the quality of the teach- ing portrayed, anti the (difficulty of concentrating on the mathematics in the midst of the many other things to watch. REFLECTIONS ON THE IMPROVEMENT OF PROFESSIONAL DEVELOPMENT A panel, moderated by Mark Saul, from BronxviDe Public Schools, consisting of ACTION CONFERENCE Iris Weiss, Horizon Research; Stephanie Williamson, Louisiana Systemic Initiative; and John Moyer, from Marquette Univer- sity, reacted to the issues raised at the Conference through the lens of their experiences. Weiss, from the perspective of large scale research about implementa- tion, argued for the need to help teachers develop some way to filter and make decisions. She also raised a concern about the issue of scale; how to move from a teacher and a classroom to the system, and in the process, the need for a central vision an(1 coordination an(1 for some form of quality control. Weiss in(licate(1 her belief that the issue is a design problem. Professional develop- ment mo(lels (lesigne(1 for a best case or for small numbers (lo not work when scaled up. Moyer, resecting on profes- sional development done with urban, large city mi(l(lle gra(les teachers, in(li- cated that scaling up had to be done creatively and in small steps. Leaders who had been developed from earlier small projects became facilitators in the larger one but also remained as part of the ca(lre of learners. He observe(1 that one of the most successful efforts was to observe teachers as they taught, with the observer writing down what the teacher sai(l. When the observer later aske(1 why the teacher ha(1 use(1 those wor(ls, the teachers began the process of resection on their practice that le(1 to some lasting changes. Williamson (lescribe(1 the work

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done in the state of Louisiana through the National Science Foundation funded ment leaders to engage in the study of professional development. As a LASIP systemic initiative and the focus on consequence, professional developers leacher consent knowledge. Williamson noted that the guidelines for the program require collaboration among school systems and universities and stimulated partnerships among K-12 and higher education. An outgrowth of the programs has been the development of a "Core of Essential Mathematics for Grades K-~" that focuses on growth of important ideas across grade levels and is now guiding decisions for both professional develop ment and preservice programs about what teachers need to know and be able to do to teach these concepts in depth. At the close of the panel, Nora Sabelli from the National Science Foundation discussed the new opportunities for research on middle grades mathematics in the Department of Education/ National Science Foundation Research Initiative. She indicated that the focus of the initiative will be on long-term agendas, strategic plans for implementa tion, and ensuring that there is the human, methodological and institutional capacity for converting schools into learning communities. SUMMARY Participants discussed the lack of must use their work with teachers as sites for ongoing learning and re- search as they create opportunities for teachers to learn working to ensure that those opportunities impact teach- ers' knowledge and practice as well as student achievement. Professional developers are expected to work with large numbers of teachers and do so quickly. Under these conditions, professional developers have the opportunities to see themselves as learners. The Action Conference did, however, take seriously professional development as a field and attempted to create a frame for thinking about theoretical, research, and practice- based learning. Most professional development providers are convince(1 that the approaches they take enable teachers to learn and students to benefit; other- wise, they would not pursue the ap- proaches. However, little is known about what various approaches afford or do not afford, especially in relation to classroom practice and student achievement. Little is known about the details of the various approaches to professional (levelopment. The mes- sage as the field considers the issues raised at the Conference is to reflect on opportunities for professional develop- the circle of learners and on their PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS

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relation to middle grades mathematics and to each other, to design research studies around professional develop- ment approaches, and to think deeply about the mathematics middle grades teachers need to know to teach well and how they can come to know that mathematics for themselves. REFERENCES Cohen, D.K, & Hill, H. (1998, January). State policy and classroom performance: Mathemat- ics reform in California. CPRE Policy Briefs. Philadelphia, PA: University of Pennsylvania. Lieberman, A, & Miller, L. (1990~. Teacher development in professional practice schools. Teachers College Record, 92~1), 105-122. Little, I.W. (1993~. Teachers' professional development in a climate of educational reform. Educational Evaluation and Policy Analysis, 1562}, 129-151. ACTION CONFERENCE National Council of Teachers of Mathematics. (1989~. Cavrric?`l?`m and evaluation standards for school mathematics. Reston, VA: Author. Smith, M., Henningsen, M., Silver, E.A, & Stein, M.K (1998) The case of Ed Taylor. Unpub- lished document, University of Pittsburgh. PARTICIPANT BACKGROUND MATERIAL Ball, D.L (1997~. Developing mathematics reform: What don't we know about teacher learning but would make good working hypotheses. In Friel, S., & Bright, G. (Eds.), Reflecting on our work. NSF Teacher Enhance- ment in K-6 Mathematics (pp. 79-111~. Lanham, MD: University Press of America. Good Teaching Matters (1998~. Thinking K-16. Vol 3, Issue 2. Education Trust. Rhine, S. (1998, June-July). The role of research and teachers' knowledge base in professional development. Research News and Comment (pp. 27-31~.