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From page 369... ...
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 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.

From page 370... ...
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 fir Teaching Mathematics Three kinds of knowledge are crucial for teaching school mathematics: knowledge of mathematics, knowledge of students, and knowledge of instructional practices.

From page 371... ...
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

From page 372... ...
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 placevalue 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.

From page 373... ...
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 upperdivision 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.

From page 374... ...
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)

From page 375... ...
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.

From page 376... ...
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.

From page 377... ...
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.~9 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.

From page 378... ...
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.

From page 379... ...
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.

From page 380... ...
In the context of teaching, proficiency requires: conceptual understanding of the core knowledge required in the practice of teaching; fluency in carrying out basic instructional routines; strategic competence in planning effective instruction and solving problems that arise during instruction; adaptive reasoning in justifying and explaining one's instructional practices and in reflecting on those practices so as to improve them; and a productive disposition toward mathematics, teaching, learning, and the .

From page 381... ...
But they also have to know how to use both kinds of knowledge effectively in the context of their work if they are to help their students develop mathematical proficiency. Similarly, many inservice workshops, presentations at professional meetings, publications for teachers, and other opportunities for teacher learning focus almost exclusively on activities or methods of teaching and seldom attempt to help teachers develop their own conceptual understanding of the underlying mathematical ideas, what students understand about those ideas, or how they learn them.

From page 382... ...
lust as students who have acquired procedural fluency can perform calculations with numbers efficiently, accurately, and flexibly with minimal effort, teachers who have acquired a repertoire of instructional routines can readily draw upon them as they interact with students in teaching mathematics. Some routines concern classroom management, such as how to get the class started each day and procedures for correcting and collecting homework.

From page 383... ...
There is never an ideal solution to the more difficult problems of teaching, but teachers can learn to contend with these problems in reasonable ways that take into account the mathematics that students are to learn; what their students understand and how they may best learn it; and representations, activities, and teaching practices that have proven most effective in teaching the mathematics in question or that have been effective in teaching related topics. Teacher education and professional development programs that take into account the strategic decision making in teaching can help prepare teachers to be more effective in solving instructional problems.

From page 384... ...
lust as students must develop a productive disposition toward mathematics such that they believe that mathematics makes sense and that they can figure it out, so too must teachers develop a similar productive disposition. Teachers should think that mathematics, their understanding of children's thinking, and their teaching practices fit together to make sense and that they are capable of learning about mathematics, student mathematical thinking, and their own practice themselves by analyzing what goes on in their classes.

From page 385... ...
We consider below examples of four such program types that represent an array of alternative approaches to developing integrated proficiency in teaching mathematics.39 Focus on Mathematics Some teacher preparation and professional development programs attempt to enhance prospective and practicing teachers' knowledge of mathematics by having them probe more deeply fundamental ideas from elementary school

From page 386... ...
It is familiar content, and although they have not had occasion to divide fractions recently, they feel comfortable, remembering their own experiences in school mathematics and what they learned. But now, what are they being asked?

From page 387... ...
When a second student offers the sesame cracker problem, most nod again, not noticing the difference. The instructor poses a question: How does each problem we heard connect with the original computation?

From page 388... ...
(fine principle behind the instructor's efforts is to engage the prospective teachers in a kind of mathematical work that focuses on developing their proficiency with the mathematical content of the elementary school curriculum. A second principle is to link that work with larger mathematical ideas and structures.

From page 389... ...
Discussions in these programs are conducted in a spirit of supporting the teachers' inquiry. The analysis of children's thinking is not presented as a fixed body of knowledge, and the teachers engage not only in inquiry about how to apply knowledge about students' thinking in planning and implementing instruction but also in inquiry to deepen their understanding of students' thinking.40 The workshop described in Box 102 forms part of a professional development program designed to help teachers develop a deeper understanding of some critical mathematical ideas, including the equality sign.

From page 390... ...
. The teachers work in small groups to construct true and false number sentences they might use to elicit various views of equality.

From page 391... ...
Professional development programs focusing on helping teachers understand both the mathematics of specific content domains and students' mathematical thinking in that domain have consistently been found to contribute to major changes in teachers' instructional practices that have resulted in significant gains in students' achievement.44 For example, in an experimental study of CGI with firstgrade teachers, teachers who had taken a monthlong

From page 392... ...
Although the cases focus on classroom episodes, the discussions the teachers engage in as they reflect on the cases emphasize mathematics content and student thinking. The cases involve instruction in specific mathematical topics, and teachers analyze the cases in terms of the mathematics content being taught and the mathematical thinking reflected in the work the children produce and the interactions they engage in.

From page 393... ...
10 DEVELOPING PROFICIENCY IN TEACHING MATHEMATICS 393 Box 103 Investigating Mathematical Tasks Using Cases from Real Practice A dozen teachers are gathered around a table. They have read a case of a teacher teaching a lesson on functions.

From page 394... ...
It is quite clear that this is no generic skill, for the mathematical sensitivity and knowledge entailed are quite visible throughout. Another teacher notices how the teacher's own mathematical knowledge seems to shape her skilled questioning.

From page 395... ...
Through such lesson study groups, teachers engage in very detailed analyses of mathematics, of students' mathematical thinking and skill, of teaching and learning. Although the process results in a wellcrafted lesson, in the process of developing and refining the lesson, teachers work on analyzing students' responses and learn from and revise their own teaching practices.

From page 396... ...
In this case, a followup 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.

From page 397... ...
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 K12 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 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.

From page 398... ...
For this reason, summer leadership training programs have been used to develop mathematics specialists. corrective 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.

From page 399... ...
was conducted in the late 1980s and early l990s with high school sophomores and juniors. Student achievement data were based on items developed for NAEP.

From page 400... ...
is a professional development program for teachers that focuses on helping them construct explicit models of the development of children's mathematical thinking in welldefined 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.

From page 401... ...
(1990~. The mathematical understandings that prospective teachers bring to teacher education.

From page 402... ...
(1996~. Cognitively Guided Instruction: A knowledge base for reform in primary mathematics instruction.

From page 403... ...
Paper presented at the meeting of the American Educational Research Association, Chicago. (ERIC Document Reproduction Service No.

From page 404... ...
Elementary School Journal, 93, 213228. Rowan, B., Chiang, F

From page 405... ...
~ 1996~. The QUASAR Project: The "revolution of the possible" in mathematics instructional reform in urban middle schools.

From page 406... ...
(~1996~. The QUASAR Project: The "revolution of the possible" in mathematics instructional reform in urban middle schools.

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