approach the tasks and how they would want elementary teachers to approach them. They were also asked to reflect on the mathematical knowledge, skill, and even sensibilities that their approach would require of a teacher.

Participants were also asked to read material from Chapter 1 and Chapter 4 of Liping Ma's book Knowing and Teaching Elementary Mathematics. Ma used the two tasks above to interview Chinese elementary teachers and then compared the Chinese teachers' responses to those obtained by the Michigan State researchers who had interviewed U.S. teachers. Participants were given the following questions concerning the Ma excerpts.

1. Ma raises the issue of vocabulary and appropriate word choice in teachers' mathematical talk with students. How does this play out in both the teachers' grasp of mathematics and how the students came to understand the mathematical concept?

2. Ma quotes Jerome Bruner (1977) on the notion that the more fundamental the concept, the greater the applications. How is this idea reflected in the different approaches used by the teachers? What does this indicate about the mathematics teachers must know in order to be effective teachers?

3. Ma describes two ways of thinking about knowing a mathematical concept: as a sequence of steps leading to the concept or as a package of knowledge whose elements contribute in different ways and at different points to the knowing. With which of these ways are you most comfortable? Select another topic and try to analyze it from both perspectives.

1. Is there any indication of how attitude towards mathematics affected the teachers' approaches both to the problem and to how they responded to the question?

2. Read the responses of the teachers carefully. Can you categorize the responses according to the level of the teachers' mathematical understanding?

3. Ma writes that only teachers with mathematical inquiry themselves can foster this in their students. Do you agree? How do the teachers interviewed display mathematical inquiry?

Ball, D. L. ( 1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, East Lansing.

Bruner, J. ( 1977). The process of education. Cambridge, MA: Harvard University Press.

Kennedy, M. M., Ball, D. L., & McDiarmid, G. W. ( 1993). A study package for examining and tracking changes in teachers' knowledge (NCRTL Technical Series 93-1). East Lansing, MI: The National Center for Research on Teacher Education.

Ma, L. ( 1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum.

Front Matter (R1-R16)

Introduction (1-2)

Workshop Overview: Knowing and Learning Mathematics for Teaching (3-7)

Pre-Workshop Tasks (8-10)

Teachers' Understanding of Fundamental Mathematics (11-22)

Reconsidering the Mathematics That Teachers Need to Know (23-24)

Investigating Teaching Practice: Setting the Stage (25-38)

Investigating Teaching Practice: What Mathematical Knowledge, Skills, and Sensibilities Does It Take? (39-64)

What Kinds of Mathematical Knowledge Matter in Teaching? (65-74)

How Can Teachers Develop Such Mathematical Knowledge? (75-76)

Investigating Alternative Approaches to Helping Teachers Learn Mathematics (77-104)

Promising Approaches for Helping Prospective Elementary Teachers Learn Mathematics for Teaching (105-126)

Concluding Remarks (127-128)

Discussion Group Reports (129-130)

Question #1 (131-136)

Question #2 (137-143)

Question #3 (144-148)

Question #4 (149-154)

Question #5 (155-162)

Appendix A: Pre-Workshop Reading (163-165)

Appendix B: Workshop Agenda (166-168)

Appendix C: Homework Problems (169-169)

Appendix D: Workshop Participant List (170-173)

Appendix E: Transcript of Ball Videos (174-182)

Appendix F: Explanation of the Unit on Weight (183-192)

Appendix G: Excerpts from Investigations (193-200)

Appendix H: Mathematics Case Methods Project (201-209)

Appendix I: Biographical Information (210-217)