Discussion groups were an integral part of the Workshop, providing the opportunity to take advantage of the experience and expertise of the participants and to engage all of those present in analyzing and discussing the Workshop activities. The participants were assigned to one often working groups. Each group was asked to prepare a response to one of five questions critical to considering the mathematics elementary teachers have to know and how to help them learn that mathematics. Throughout the Workshop, the groups met three times to consider their task—interspersed with presentations, activities, and informal discussion, using the Workshop experiences as a way to provoke their thinking. The goal was to collectively develop ideas about the mathematics content preparation of teachers. The following papers provide the responses of two groups to each question and illustrate the beginning of some core ideas about the nature of developing teacher content knowledge in preservice programs.

**Question #1**. Often teaching is seen as presenting material to students. But of course teaching includes many more small and large tasks—figuring out what students know, composing good questions, assessing and revising textbook lessons, and so on. What are some of these recurrent tasks of teaching that require the use of mathematics?

**Question #2**. Not everything is a question of knowledge. From what we have done together, what are some mathematical instincts, sensibilities, dispositions that seem crucial to teaching mathematics? What mathematics beyond what is taught in class must a teacher know to do a good job teaching mathematics in that class?

**Question #3**. There is so much to know of mathematics. Creating longer and longer lists of what teachers should know does not seem promising. What are some of the big ideas in mathematics that would seem to have a lot of leverage in practice? How do teachers need to understand these ideas—for example, what does it mean to be able to “unpack” ideas, as well as to connect them?

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OCR for page 129

Knowing and Learning Mathematics for Teaching
DISCUSSION GROUP REPORTS
Discussion groups were an integral part of the Workshop, providing the opportunity to take advantage of the experience and expertise of the participants and to engage all of those present in analyzing and discussing the Workshop activities. The participants were assigned to one often working groups. Each group was asked to prepare a response to one of five questions critical to considering the mathematics elementary teachers have to know and how to help them learn that mathematics. Throughout the Workshop, the groups met three times to consider their task—interspersed with presentations, activities, and informal discussion, using the Workshop experiences as a way to provoke their thinking. The goal was to collectively develop ideas about the mathematics content preparation of teachers. The following papers provide the responses of two groups to each question and illustrate the beginning of some core ideas about the nature of developing teacher content knowledge in preservice programs.
Question #1. Often teaching is seen as presenting material to students. But of course teaching includes many more small and large tasks—figuring out what students know, composing good questions, assessing and revising textbook lessons, and so on. What are some of these recurrent tasks of teaching that require the use of mathematics?
Question #2. Not everything is a question of knowledge. From what we have done together, what are some mathematical instincts, sensibilities, dispositions that seem crucial to teaching mathematics? What mathematics beyond what is taught in class must a teacher know to do a good job teaching mathematics in that class?
Question #3. There is so much to know of mathematics. Creating longer and longer lists of what teachers should know does not seem promising. What are some of the big ideas in mathematics that would seem to have a lot of leverage in practice? How do teachers need to understand these ideas—for example, what does it mean to be able to “unpack” ideas, as well as to connect them?

OCR for page 129

Knowing and Learning Mathematics for Teaching
Question #4. What are some promising ways to help teachers learn mathematics that also help them develop mathematically? What are the key features of what makes an approach promising?
Question #5. What are some promising ways to help teachers not only develop mathematical understanding but learn to use mathematical insight and knowledge in the context of practice? What are the key features of what makes an approach promising? Are there ways to engage preservice teachers in learning mathematics through the tasks they will actually do in practice?