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DEVELOPING PROFICIENCY IN TEACHING MATHEMATICS

In the previous chapter, we examined teaching for mathematical proficiency. We now turn our attention to what it takes to develop proficiency in teaching mathematics. Proficiency in teaching is related to effectiveness: consistently helping students learn worthwhile mathematical content. Proficiency also entails versatility: being able to work effectively with a wide variety of students in different environments and across a range of mathematical content.

What Does It Take to Teach for Mathematical Proficiency?

Teaching in the ways portrayed in chapter 9 is a complex practice that draws on a broad range of resources. Despite the common myth that teaching is little more than common sense or that some people are just born teachers, effective teaching practice can be learned. In this chapter, we consider what teachers need to learn and how they can learn it.

Despite the common myth that teaching is little more than common sense or that some people are just born teachers, effective teaching practice can be learned.

First, what does it take to be proficient at mathematics teaching? If their students are to develop mathematical proficiency, teachers must have a clear vision of the goals of instruction and what proficiency means for the specific mathematical content they are teaching. They need to know the mathematics they teach as well as the horizons of that mathematics—where it can lead and where their students are headed with it. They need to be able to use their knowledge flexibly in practice to appraise and adapt instructional materials, to represent the content in honest and accessible ways, to plan and conduct instruction, and to assess what students are learning. Teachers need to be able to hear and see expressions of students’ mathematical ideas and to design



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Adding + It Up: Helping Children Learn Mathematics 10 DEVELOPING PROFICIENCY IN TEACHING MATHEMATICS In the previous chapter, we examined teaching for mathematical proficiency. We now turn our attention to what it takes to develop proficiency in teaching mathematics. Proficiency in teaching is related to effectiveness: consistently helping students learn worthwhile mathematical content. Proficiency also entails versatility: being able to work effectively with a wide variety of students in different environments and across a range of mathematical content. What Does It Take to Teach for Mathematical Proficiency? Teaching in the ways portrayed in chapter 9 is a complex practice that draws on a broad range of resources. Despite the common myth that teaching is little more than common sense or that some people are just born teachers, effective teaching practice can be learned. In this chapter, we consider what teachers need to learn and how they can learn it. Despite the common myth that teaching is little more than common sense or that some people are just born teachers, effective teaching practice can be learned. First, what does it take to be proficient at mathematics teaching? If their students are to develop mathematical proficiency, teachers must have a clear vision of the goals of instruction and what proficiency means for the specific mathematical content they are teaching. They need to know the mathematics they teach as well as the horizons of that mathematics—where it can lead and where their students are headed with it. They need to be able to use their knowledge flexibly in practice to appraise and adapt instructional materials, to represent the content in honest and accessible ways, to plan and conduct instruction, and to assess what students are learning. Teachers need to be able to hear and see expressions of students’ mathematical ideas and to design

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Adding + It Up: Helping Children Learn Mathematics A Chinese teacher on how a profound understanding of fundamental mathematics is attained One thing is to study whom you are teaching, the other thing is to study the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed…. Believe me, it seems to be simple when I talk about it, but when you really do it, it is very complicated, subtle, and takes a lot of time. It is easy to be an elementary school teacher, but it is difficult to be a good elementary school teacher. SOURCE: Ma, 1999, p. 136. Used by permission from Lawrence Erlbaum Associates. appropriate ways to respond. A teacher must interpret students’ written work, analyze their reasoning, and respond to the different methods they might use in solving a problem. Teaching requires the ability to see the mathematical possibilities in a task, sizing it up and adapting it for a specific group of students. Familiarity with the trajectories along which fundamental mathematical ideas develop is crucial if a teacher is to promote students’ movement along those trajectories. In short, teachers need to muster and deploy a wide range of resources to support the acquisition of mathematical proficiency. In the next two sections, we first discuss the knowledge base needed for teaching mathematics and then offer a framework for looking at proficient teaching of mathematics. In the last two sections, we discuss four programs for developing proficient teaching and then consider how teachers might develop communities of practice. The Knowledge Base for Teaching Mathematics Three kinds of knowledge are crucial for teaching school mathematics: knowledge of mathematics, knowledge of students, and knowledge of instructional practices.1 These can be seen in the instructional triangle (Box 9–1 in chapter 9 and below).2 Mathematics and students are two of the triangle’s vertices, and instructional practices are the interactions portrayed by the arrows.

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Adding + It Up: Helping Children Learn Mathematics Mathematical knowledge includes knowledge of mathematical facts, concepts, procedures, and the relationships among them; knowledge of the ways that mathematical ideas can be represented; and knowledge of mathematics as a discipline—in particular, how mathematical knowledge is produced, the nature of discourse in mathematics, and the norms and standards of evidence that guide argument and proof. In our use of the term, knowledge of mathematics includes consideration of the goals of mathematics instruction and provides a basis for discriminating and prioritizing those goals. Knowing mathematics for teaching also entails more than knowing mathematics for oneself. Teachers certainly need to be able to understand concepts correctly and perform procedures accurately, but they also must be able to understand the conceptual foundations of that knowledge. In the course of their work as teachers, they must understand mathematics in ways that allow them to explain and unpack ideas in ways not needed in ordinary adult life. The mathematical sensibilities they hold matter in guiding their decisions and interpretations of students’ mathematical efforts. Knowledge of students and how they learn mathematics includes general knowledge of how various mathematical ideas develop in children over time as well as specific knowledge of how to determine where in a developmental trajectory a child might be. It includes familiarity with the common difficul-

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Adding + It Up: Helping Children Learn Mathematics ties that students have with certain mathematical concepts and procedures, and it encompasses knowledge about learning and about the sorts of experiences, designs, and approaches that influence students’ thinking and learning. Knowledge of instructional practice includes knowledge of curriculum, knowledge of tasks and tools for teaching important mathematical ideas, knowledge of how to design and manage classroom discourse, and knowledge of classroom norms that support the development of mathematical proficiency. Teaching entails more than knowledge, however. Teachers need to do as well as to know. For example, knowledge of what makes a good instructional task is one thing; being able to use a task effectively in class with a group of sixth graders is another. Understanding norms that support productive classroom activity is different from being able to develop and use such norms with a diverse class. Knowledge of Mathematics Because knowledge of the content to be taught is the cornerstone of teaching for proficiency, we begin with it. There is a substantial body of research on teachers’ mathematical knowledge, and teachers’ knowledge of mathematics is prominent in discussions of how to improve mathematics instruction. Improving teachers’ mathematical knowledge and their capacity to use it to do the work of teaching is crucial in developing students’ mathematical proficiency. Many recent studies have revealed that U.S. elementary and middle school teachers possess a limited knowledge of mathematics, including the mathematics they teach. The mathematical education they received, both as K-12 students and in teacher preparation, has not provided them with appropriate or sufficient opportunities to learn mathematics. As a result of that education, teachers may know the facts and procedures that they teach but often have a relatively weak understanding of the conceptual basis for that knowledge. Many have difficulty clarifying mathematical ideas or solving problems that involve more than routine calculations.3 For example, virtually all teachers can multiply multidigit numbers, but several researchers have found that many prospective and practicing elementary school teachers cannot explain the basis for multidigit multiplication using place-value concepts and the underlying properties for adding and multiplying.4 In another study,5 teachers of fourth through sixth graders scored over 90% on items testing common decimal calculations, but fewer than half could find a number between 3.1 and 3.11.

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Adding + It Up: Helping Children Learn Mathematics Teachers frequently regard mathematics as a fixed body of facts and procedures that are learned by memorization, and that view carries over into their instruction. Many have little appreciation of the ways in which mathematical knowledge is generated or justified. Preservice teachers, for example, have repeatedly been shown to be quite willing to accept a series of instances as proving a mathematical generalization.6 Nowhere in their education have they had opportunities to study and experience the nature and role of justification in mathematics, a notion central to developing mathematical knowledge. Although teachers may understand the mathematics they teach in only a superficial way, simply taking more of the standard college mathematics courses does not appear to help matters. The evidence on this score has been consistent, although the reasons have not been adequately explored. For example, a study of prospective secondary mathematics teachers at three major institutions showed that, although they had completed the upper-division college mathematics courses required for the mathematics major, they had only a cursory understanding of the concepts underlying elementary mathematics.7 The mathematics of the elementary and middle school curriculum is not trivial, and the underlying concepts and structures are worthy of serious, sustained study by teachers. To develop prospective teachers’ understanding of the mathematics they will teach, careful attention must be given to identifying the mathematics that teachers need in order to teach effectively, articulating the ways in which they must use it in practice and what that implies for their opportunities to learn mathematics. This sort of attention to teachers’ mathematical knowledge and its central role in practice is crucial to ensure that their study of mathematics provides teachers with mathematical knowledge useful to teaching well. Teachers’ mathematical knowledge and student achievement. Conventional wisdom asserts that student achievement must be related to teachers’ knowledge of their subject. That wisdom is contained in adages such as “You cannot teach what you don’t know.” For the better part of a century, researchers have attempted to find a positive relation between teacher content knowledge and student achievement. For the most part, the results have been disappointing: Most studies have failed to find a strong relationship between the two. Many studies, however, have relied on crude measures of these variables. The measure of teacher knowledge, for example, has often been the number of mathematics courses taken or other easily documented data from college

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Adding + It Up: Helping Children Learn Mathematics transcripts. Such measures do not provide an accurate index of the specific mathematics that teachers know or of how they hold that knowledge. Teachers may have completed their courses successfully without achieving mathematical proficiency. Or they may have learned the mathematics but not know how to use it in their teaching to help students learn. They may have learned mathematics that is not well connected to what they teach or may not know how to connect it. Similarly, many of the measures of student achievement used in research on teacher knowledge have been standardized tests that focus primarily on students’ procedural skills. Some evidence suggests that there is a positive relationship between teachers’ mathematical knowledge and their students’ learning of advanced mathematical concepts.8 There seems to be no association, however, between how many advanced mathematics courses a teacher takes and how well that teacher’s students achieve overall in mathematics.9 In general, empirical evidence regarding the effects of teachers’ knowledge of mathematics content on student learning is still rather sparse. In the National Longitudinal Study of Mathematical Abilities (NLSMA), conducted during the 1960s and still today the largest study of its kind, there was essentially no association between students’ achievement and the number of credits a teacher had in mathematics at the level of calculus or beyond.10 Commenting on the findings from NLSMA and a number of other studies of teacher knowledge, the director of NLSMA later said, It is widely believed that the more a teacher knows about his subject matter, the more effective he will be as a teacher. The empirical literature suggests that this belief needs drastic modification and in fact suggests that once a teacher reaches a certain level of understanding of the subject matter, then further understanding contributes nothing to student achievement.11 The notion that there is a threshold of necessary content knowledge for teaching is supported by the findings of another study in 1994 that used data from the Longitudinal Study of American Youth (LSAY).12 There was a notable increase in student performance for each additional mathematics course their teachers had taken, yet after the fifth course there was little additional benefit.13 Data from the 1996 NAEP on teachers’ college major rather than the number of courses they had taken provide a contrast to the general trend of this line of research. The NAEP data revealed that eighth graders taught by teachers who majored in mathematics outperformed those whose teachers

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Adding + It Up: Helping Children Learn Mathematics majored in education or some other field. Fourth graders taught by teachers who majored in mathematics education or in education tended to outperform those whose teachers majored in a field other than education.14 Although studies of teachers’ mathematical knowledge have not demonstrated a strong relationship between teachers’ mathematical knowledge and their students’ achievement, teachers’ knowledge is still likely a significant factor in students’ achievement. That crude measures of teacher knowledge, such as the number of mathematics courses taken, do not correlate positively with student performance data, supports the need to study more closely the nature of the mathematical knowledge needed to teach and to measure it more sensitively. The persistent failure of the many efforts to show strong, definitive relations between teachers’ mathematical knowledge and their effectiveness does not imply that mathematical knowledge makes no difference in teaching. The research, however, does suggest that proposals to improve mathematics instruction by simply increasing the number of mathematics courses required of teachers are not likely to be successful. As we discuss in the sections that follow, courses that reflect a serious examination of the nature of the mathematics that teachers use in the practice of teaching do have some promise of improving student performance. Teachers need to know mathematics in ways that enable them to help students learn. The specialized knowledge of mathematics that they need is different from the mathematical content contained in most college mathematics courses, which are principally designed for those whose professional uses of mathematics will be in mathematics, science, and other technical fields. Why does this difference matter in considering the mathematical education of teachers? First, the topics taught in upper-level mathematics courses are often remote from the core content of the K-12 curriculum. Although the abstract mathematical ideas are connected, of course, basic algebraic concepts or elementary geometry are not what prospective teachers study in a course in advanced calculus or linear algebra. Second, college mathematics courses do not provide students with opportunities to learn either multiple representations of mathematical ideas or the ways in which different representations relate to one another. Advanced courses do not emphasize the conceptual underpinnings of ideas needed by teachers whose uses of mathematics are to help others learn mathematics.15 Instead, the study of college mathematics involves the increasing compression of elementary ideas into the more and more powerful and abstract forms needed by those whose professional uses of mathematics will be in scientific domains. Third, advanced mathematical

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Adding + It Up: Helping Children Learn Mathematics study entails using elementary concepts and procedures without much conscious attention to their meanings or implications, thus reinforcing the making of prior learning routine in the service of more advanced work. While this approach is important for the education of mathematicians and scientists, it is at odds with the kind of mathematical study needed by teachers. Consider the proficiency teachers need with algorithms. The power of computational algorithms is that they allow learners to calculate without having to think deeply about the steps in the calculation or why the calculations work. That frees up the learners’ thinking so that they can concentrate on the problem they are trying to use the calculation to solve rather than having to worry about the details of the calculation. Over time, people tend to forget the reasons a procedure works or what is entailed in understanding or justifying a particular algorithm. Because the algorithm has become so automatic, it is difficult to step back and consider what is needed to explain it to someone who does not understand. Consequently, appreciating children’s difficulties in learning an algorithm can be very difficult for adults who are fluent with that algorithm. The necessary compression of ideas in the course of mathematical study also shortchanges teachers’ mathematical needs. Most advanced mathematics classes engage students in taking ideas they have already learned and using them to construct increasingly powerful and abstract concepts and methods. Once theorems have been proved, they can be used to prove other theorems. It is not necessary to go back to foundational concepts to learn more advanced ideas. Teaching, however, entails reversing the direction followed in learning advanced mathematics. In helping students learn, teachers must take abstract ideas and unpack them in ways that make the basic underlying concepts visible.16 For example, most adults have lost sight of the fact that there are different interpretations of division. For adults, division is an operation on numbers. Division, however, is rooted in quite different physical situations, and distinctions among those situations are important for understanding children’s thinking, developing their understanding of the meaning of division, and helping them apply that understanding to solve problems.17 For example, although both of the following problems can be represented as dividing 24 by 6, young children think about them in very different ways and use quite different strategies to solve them:18 Jane has 24 cookies. She wants to put 6 cookies on each plate. How many plates will she need?

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Adding + It Up: Helping Children Learn Mathematics Jeremy has 24 cookies. He wants to put all the cookies on 6 plates. If he puts the same number of cookies on each plate, how many cookies will he put on each plate? These two problems correspond to the measurement and sharing models of division, respectively, that were discussed in chapter 3. Young children using counters solve the first problem by putting 24 counters in piles of 6 counters each. They solve the second by partitioning the 24 counters into 6 groups. In the first case the answer is the number of groups; in the second, it is the number in each group. Until the children are much older, they are not aware that, abstractly, the two solutions are equivalent. Teachers need to see that equivalence so that they can understand and anticipate the difficulties children may have with division. To understand the sense that children are making of arithmetic problems, teachers must understand the distinctions children are making among those problems and how the distinctions might be reflected in how the children think about the problems. The different semantic contexts for each of the operations of arithmetic is not a common topic in college mathematics courses, yet it is essential for teachers to know those contexts and be able to use their knowledge in instruction. The division example illustrates a different way of thinking about the content of courses for teachers—a way that can make those courses more relevant to the teaching of school mathematics. A recent study indicates that teachers’ performance on mathematical tasks that have been set in the context of teaching practice is positively related to student achievement.19 In the study, teachers’ ability to interpret four student responses to a ratio problem and to determine which were correct was strongly related to their students’ mathematics achievement. Teachers’ mathematical knowledge and their teaching practice. Conventional wisdom holds that a teacher’s knowledge of mathematics is linked to how the teacher teaches. Teachers are unlikely to be able to provide an adequate explanation of concepts they do not understand, and they can hardly engage their students in productive conversations about multiple ways to solve a problem if they themselves can only solve it in a single way. In the last 15 years, researchers have investigated how teachers’ mathematical knowledge shapes the way they teach. Most of the investigations have been case studies, almost all involving fewer than 10 teachers, and most only one to three teachers. In general, the researchers found that teachers

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Adding + It Up: Helping Children Learn Mathematics with a relatively weak conceptual knowledge of mathematics tended to demonstrate a procedure and then give students opportunities to practice it. Not surprisingly, these teachers gave the students little assistance in developing an understanding of what they were doing.20 When the teachers did try to provide a clear explanation and justification, they were not able to do so.21 In some cases, their inadequate conceptual knowledge resulted in their presenting incorrect procedures.22 Some of the same studies contrasted the teaching practices of teachers with low levels of mathematical knowledge with the teaching practices of teachers who had a better command of mathematics. These studies indicate that a strong grasp of mathematics made it possible for teachers to understand and use constructively students’ mathematical solutions, explanations, and questions.23 Several researchers found, however, that some teachers with strong conceptual knowledge did not necessarily use that knowledge to understand their students’ mathematical explanations, preferring instead to impose their own explanations.24 Knowledge of Students Knowledge of students includes both knowledge of the particular students being taught and knowledge of students’ learning in general. Knowing one’s own students includes knowing who they are, what they know, and how they view learning, mathematics, and themselves. The teacher needs to know something of each student’s personal and educational background, especially the mathematical skills, abilities, and dispositions that the student brings to the lesson. The teacher also needs to be sensitive to the unique ways of learning, thinking about, and doing mathematics that the student has developed. Each student can be seen as located on a path through school mathematics, equipped with strengths and weaknesses, having developed his or her own approaches to mathematical tasks, and capable of contributing to and profiting from each lesson in a distinctive way. Teachers also need a general knowledge of how students think—the approaches that are typical for students of a given age and background, their common conceptions and misconceptions, and the likely sources of those ideas. Over the last decade, researchers have produced an impressive body of evidence about how children’s thinking about various mathematical concepts progresses over time. We have described some of those progressions in chapters 6 through 8. Using that body of evidence, researchers have also

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Adding + It Up: Helping Children Learn Mathematics studied how teachers’ knowledge of students’ mathematical thinking is related to how they teach and to how well their students achieve. From the many examples of misconceptions to which teachers need to be sensitive, we have chosen one: An important mathematical notion that poses a major stumbling block when students are moving from arithmetic to algebra is the role played by “=,” the sign for equality.25 As we discussed in chapter 8, many if not most elementary school children have the misconception that the equality sign is a signal to do something, to carry out the calculation that precedes it.26 The number immediately after the equal sign is seen as the answer to the calculation. For example, in the number sentence 8+4=□+5, many students would put 12 in the box. Children can develop this impression because that is how the notation is often described in the elementary school curriculum and most of their practice exercises fit that pattern. Few teachers realize the degree of their students’ misunderstanding of such sentences.27 Moreover, although most teachers have some idea that equality is a relation between two numbers, few realize how important it is that students understand equality as a relation, and few consider this need for understanding when they use the equals sign. Knowledge of Classroom Practice Knowing classroom practice means knowing what is to be taught and how to plan, conduct, and assess effective lessons on that mathematical content. It includes a knowledge of learning goals as expressed in the curriculum and a knowledge of the resources at one’s disposal for helping students reach those goals. It also includes skill in organizing one’s class to create a community of learners and in managing classroom discourse and learning activities so that everyone is engaged in substantive mathematical work. We have discussed these matters in chapter 9. This type of knowledge is gained through experience in classrooms and through analyzing and reflecting on one’s own practice and that of others. In the sections that follow, we consider how to develop an integrated corpus of knowledge of the types discussed in this section. First, however, we need to clarify our stance on the relation between knowledge and practice. We have discussed the kinds of knowledge teachers need if they are to teach for mathematical proficiency. Although we have used the term knowledge throughout, we do not mean it exclusively in the sense of knowing about. Teachers must also know how to use their knowledge in practice. Teachers’ knowledge is of value only if they can apply it to their teaching; it cannot be

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Adding + It Up: Helping Children Learn Mathematics Box 10–4 The Japanese Lesson Study Small groups of teachers form within the school around areas of common teaching interests or responsibilities (e.g., grade-level groups in mathematics or in science). Each group begins by formulating a goal for the year. Sometimes the goal is adapted from national-level recommendations (e.g., improve students’ problem-solving skills) and is translated into a more specific goal (e.g., improve students’ understanding of problems involving ratios). The more specific goal might focus on a curriculum topic that has been problematic for students in their classrooms. A few lessons then are identified that ordinarily deal with that topic, and the group begins its yearlong task to improve those lessons. Lesson study groups meet regularly, often once a week after school (e.g., 3:00 to 5:00 pm), to develop, test, and refine the improved lessons. Some groups divide their work into three major phases, each taking about one third of the school year. During the first phase, teachers do research on the topic, reading and sharing relevant research reports and collecting information from other teachers on effective approaches for teaching the topic. During the second phase, teachers design the targeted lessons (often just one, two, or perhaps three lessons). Important parts of the design include (a) the problems that will be presented to students, (b) the teachers’ predictions about how students will solve the problems, and (c) how these different solution methods are to be integrated into a productive class discussion. During the third phase, the lessons are tested and refined. The first test often involves one of the group members teaching a lesson to his or her class while the other group members observe and take notes. After the group refines the lesson, it might be tested with another class in front of all the teachers in the school. In this case, a follow-up session is scheduled, and the lesson study group engages their colleagues in a discussion about the lesson, receiving feedback about its effectiveness. The final task for the group is to prepare a report of the year’s work, including a rationale for the approach used and a detailed plan of the lesson, complete with descriptions of the different solution methods students are likely to present and the ways in which these can be orchestrated into a constructive discussion.

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Adding + It Up: Helping Children Learn Mathematics is teacher development. Working directly on improving teaching is their means of becoming better teachers. Communities of Practice Learning in ways that continue to be generative over time is best done in a community of fellow practitioners and learners, as illustrated by the Japanese lesson study groups. The foregoing discussion of teacher proficiency focused on individual teachers’ knowledge, but teaching proficiency does not easily develop and is not generally sustained in isolation. Studies of school reform efforts suggest that professional development is most effective when it extends beyond the individual teacher.48 Collaboration among teachers provides support for them to engage in the kinds of inquiry that are needed to develop teaching proficiency. Professional development can create contexts for teacher collaboration, provide a focus for the collaboration, and provide a common frame for interacting with other teachers around common problems. When teachers have opportunities to continue to participate in communities of practice that support their inquiry, instructional practices that foster the development of mathematical proficiency can more easily be sustained. Professional development can create contexts for teacher collaboration, provide a focus for the collaboration, and provide a common frame for interacting with other teachers around common problems. The focus of teacher groups matters for what teachers learn from their interactions with others. When sustained work is focused on mathematics, on students’ thinking about specific mathematical topics, or on the detailed work of designing and enacting instruction, the resources generated for teachers’ own practice are greater than when there is less concrete focus. For example, general sharing, or discussion of approaches, ungrounded in the particulars of classroom artifacts, while possibly enjoyable, less often produces usable knowledge that can make a difference for teachers’ work. Mathematics Specialists Because of the specialized knowledge required to teach mathematics, there has been increased discussion recently of the use of mathematics specialists, particularly in the upper elementary and middle school grades. The Learning First Alliance, comprising 12 major education groups, recommends that mathematics teachers from grades 5 through 9 have “a solid grounding in the coursework of grades K-12 and the teaching of middle grades mathematics.”49 The Conference Board of the Mathematical Sciences recommends in its draft report that mathematics in middle grades should be taught by mathematics specialists, starting at least in the fifth grade.50 They further recommend that teachers of middle school mathematics have taken 21 semester

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Adding + It Up: Helping Children Learn Mathematics hours of mathematics, 12 of which are on fundamental ideas of school mathematics appropriate for middle school teachers. Implicit in the recommendations for mathematics specialists is the notion of the mathematics specialist in a departmental arrangement. In such arrangements, teachers with a strong background in mathematics teach mathematics and sometimes another subject, depending on the student population, while other teachers in the building teach other subject areas. Departmentalization is most often found in the upper elementary grades (4 to 6). Other models of mathematics specialists are used, particularly in elementary schools, which rarely are departmentalized. Rather than a specialist for all mathematics instruction, a single school-level mathematics specialist is sometimes used. This person, who has a deep knowledge of mathematics and how students learn it, acts as a resource for other teachers in the school. The specialist may consult with other teachers about specific issues, teach demonstration lessons, observe and offer suggestions, or provide special training sessions during the year. School-level mathematics specialists can also take the lead in establishing communities of practice, as discussed in the previous section. Because many districts do not have enough teachers with strong backgrounds in mathematics to provide at least one specialist in every school, districts instead identify district-level mathematics coaches who are responsible for several schools. Whereas a school-level specialist usually has a regular or reduced teaching assignment, district-level specialists often have no classroom teaching assignment during their tenure as a district coach. The constraint on all of the models for mathematics specialists is the limited number of teachers, especially at the elementary level, with strong backgrounds in mathematics. For this reason, summer leadership training programs have been used to develop mathematics specialists. Effective Professional Development Perhaps the central goal of all the teacher preparation and professional development programs is in helping teachers understand the mathematics they teach, how their students learn that mathematics, and how to facilitate that learning. Many of the innovative programs described in this chapter make serious efforts to help teachers connect these strands of knowledge so that they can be applied in practice. Teachers are expected to explain and justify their ideas and conclusions. Teachers’ ideas are respected, and they are encouraged to engage in inquiry. They have opportunities to develop a productive disposition toward their own learning about teaching that contrib-

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Adding + It Up: Helping Children Learn Mathematics utes to their learning becoming generative. Teachers are not given readymade solutions to teaching problems or prescriptions for practice. Instead, they adapt what they are learning and engage in problem solving to deal with the situations that arise when they attempt to use what they learn. Professional development beyond initial preparation is critical for developing proficiency in teaching mathematics. However, such professional development requires the marshalling of substantial resources. One of the critical resources is time. If teachers are going engage in inquiry, they need repeated opportunities to try out ideas and approaches with their students and continuing opportunities to discuss their experiences with specialists in mathematics, staff developers, and other teachers. These opportunities should not be limited to a period of a few weeks or months; instead, they should be part of the ongoing culture of professional practice. Through inquiry into teaching, teacher learning can become generative, and teachers can continue to learn and grow as professionals. Notes 1.   Shulman, 1987. 2.   Cohen and Ball, 1999, 2000. 3.   Ball, 1991; Ma, 1999; Post, Harel, Behr, and Lesh, 1991; Tirosh, Fischbein, Graeber, and Wilson, 1999. 4.   Ball, 1991; Ma, 1999. 5.   Post, Harel, Behr, and Lesh, 1991. 6.   Ball, 1988; Martin and Harel, 1989; Simon and Blume, 1996. 7.   Ball, 1990, 1991. 8.   Mullens, Murnane, and Willet, 1996; but see Begle, 1972. 9.   Monk, 1994. 10.   Begle, 1979. 11.   Begle, 1979, p. 51. 12.   The Longitudinal Study of American Youth (LSAY) was conducted in the late 1980s and early 1990s with high school sophomores and juniors. Student achievement data were based on items developed for NAEP. 13.   Monk, 1994, p. 130. 14.   Hawkins, Stancavage, and Dossey, 1998. 15.   In fact, it appears that sometimes content knowledge by itself may be detrimental to good teaching. In one study, more knowledgeable teachers sometimes overestimated the accessibility of symbol-based representations and procedures (Nathan and Koedinger, 2000). 16.   Ball and Bass, 2000; Ma, 1999. 17.   Carpenter, Fennema, and Franke, 1996; Carpenter, Fennema, Franke, Empson, and Levi, 1999; Greer, 1992.

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Adding + It Up: Helping Children Learn Mathematics 18.   Carpenter, Fennema, Franke, Empson, and Levi, 1999. 19.   Rowan, Chiang, and Miller, 1997. 20.   Ball, 1991; Leinhardt and Smith, 1985. 21.   Borko, Eisenhart, Brown, Underhill, Jones, Agard, 1992. 22.   Leinhardt and Smith, 1985; Putnam, Heaton, Prawat, and Remillard, 1992. 23.   Ball, 1991; Fernandez, 1997. 24.   Lubinski, Otto, Rich, and Jaberg, 1998; Thompson and Thompson, 1994, 1996. 25.   Kieran, 1981; Matz, 1982. 26.   Behr, Erlwanger, and Nichols,1976, 1980; Erlwanger and Berlanger, 1983; Kieran, 1981; Saenz-Ludlow and Walgamuth, 1998. 27.   Falkner, Levi, and Carpenter, 1999. 28.   Ball and Bass, 2000; Putnam and Borko, 2000. 29.   Leinhardt and Smith, 1985; Schoenfeld, 1998. 30.   Carpenter, 1988. 31.   Clark and Peterson, 1986. 32.   Schon, 1987. 33.   Brown, Collins, and Duguid, 1989; Lewis and Ball, 2000; Schon, 1987. 34.   Franke, Carpenter, Fennema, Ansell, and Behrent, 1998; Franke, Carpenter, Levi, and Fennema, in press. 35.   For an example of how such study might be conducted, see Ma, 1999. 36.   National Research Council, 2000. 37.   Franke, Carpenter, Fennema, Ansell, and Behrend, 1998; Franke, Carpenter, Levi, and Fennema, in press; Little, 1993; Sarason, 1990, 1996. 38.   Franke, Carpenter, Levi, and Fennema, in press. 39.   These programs share the idea that professional development should be based upon the mathematical work of teaching. For more examples, see National Research Council, 2001. A comprehensive guide for designing professional development programs can be found in Loucks-Horsley, Hewson, Love, Stiles, 1998. 40.   Franke, Carpenter, Levi, and Fennema, in press. 41.   Cognitively Guided Instruction (CGI) is a professional development program for teachers that focuses on helping them construct explicit models of the development of children’s mathematical thinking in well-defined content domains. No instructional materials or specifications for practice are provided in CGI; teachers develop their own instructional materials and practices from watching and listening to their students solve problems. Although the program focuses on children’s mathematical thinking, teachers acquire a knowledge of mathematics as they are learning about children’s thinking by analyzing structural features of the problems children solve and the mathematical principles underlying their solutions. A major thesis of CGI is that children bring to school informal or intuitive knowledge of mathematics that can serve as the basis for developing much of the formal mathematics of the primary school mathematics curriculum. The development of children’s mathematical thinking is portrayed as the progressive abstraction and formalization of children’s informal attempts to solve problems by constructing models of problem situations. 42.   Carpenter and Levi, 1999.

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