Rebecca Walker Grand Valley State Univ. Natasha Speer Michigan State Univ.

1. Rebecca Walker Grand Valley State Univ. Natasha Speer Michigan State Univ. Mathematics capstone courses for prospective teachers: Reflections from the perspective of instructors and researchers

2. Overview of session • Why are we and others offering capstone courses for pre-service teachers? • An activity from the course • Discussion about the activity • General discussion about the course and associated issues

3. Some things we have learned from educational research • Content knowledge is not all that matters • more courses in content are not strongly correlated with higher student achievement (Begle, 1979; Monk, 1994)‏ “The conclusions of the few studies in this area are especially provocative because they undermine the certainty often expressed about the strong link between college study of a subject matter and teacher quality.” (Wilson, Floden, & Ferrini-Mundy, 2002, p. 191)‏ • Then what should pre-service mathematics teachers study in college?

4. Some more things we have learned from educational research • Pedagogical content knowledge matters • One example: • knowledge of student thinking shapes teachers’ practices (Carpenter et al, 1988, 1989) and their students’ learning (Fennema et al., 1996)‏ • Developing PCK entails connecting mathematical content knowledge to learning/teaching knowledge • What content knowledge is needed and how do those connections get made?

5. Even more things we know from research • Mathematical knowledge for teaching matters • used to do the “mathematical work” of teaching • to follow and understand students’ mathematical thinking • to evaluate the validity of student-generated strategies • to make sense of a range of solution paths • shown to play a role in teachers’ practices and correlate with students’ learning (Ball & Bass, 2000; Hill et al 2004, 2005; Ma, 1999)‏ • But specifically what do teachers need to know to do this work and how can that knowledge be developed?

6. The mathematics education community’s response? • Mathematics capstone courses for pre-service teachers! • Recommendations from various sources, including The Mathematical Education of Teachers (MET) report (CBMS, 2001)‏ • Resources developed, such as text Mathematics for High School Teachers, An Advanced Perspective (Usiskin et al)‏ • Many such courses offered around the country

7. But: There is a lot of diversity in the foci for such courses • Examine advanced topics that are related to school math content • Examine mathematical connections between school math content and advanced topics • Examine how to use knowledge of advanced topics to inform decisions about the teaching of school math content • Develop capacity to use knowledge of advanced mathematics to understand students’ thinking about school math content

8. But: There is a lot of diversity in the foci for such courses • Examine advanced topics that are related to school math content • Examine mathematical connections between school math content and advanced topics • Examine how to use knowledge of advanced topics to inform decisions about the teaching of school math content • Develop capacity to use knowledge of advanced mathematics to understand students’ thinking about school math content

9. A Window into One Capstone Course • MTH 495 • Grand Valley State University • Instructors: Ed Aboufadel and Rebecca Walker

10. Who were our students? • Mathematics majors • math – no teaching emphasis (3)‏ • secondary education emphasis (12)‏ • elementary education emphasis (3)‏ • Courses taken • Two semesters of Calculus - all • Proof writing course - all • Modern Algebra - all • Linear Algebra – all • Discrete Mathematics – most • Education emphasis students have also taken 2-3 mathematics education courses where they have • considered the school curriculum • explored reasoning behind the mathematics taught in schools • explored how students might reason about the topics

11. Major Mathematical Topics • Real and complex number systems • Cardinality and Infinity • Quadrature • Solving equations and inequalities over different fields • Polynomials over R[X] and C[X] • History of π

12. Solving Equations Activity • In groups complete the designated problem. • Please write your solutions on the paper provided and be prepared to share a bit about how you arrived at your solution.

13. In what ways does this activity: • Examine advanced topics that are related to school math content? • Explore mathematical connections between school math content and advanced topics? • Help develop the ability to use knowledge of advanced topics to inform decisions about the teaching of school math content? • Help develop ability to understand students’ thinking about school math content?

14. Other comments or questions?

15. Acknowledgements • Others involved in the course • Ed Aboufadel • Sharon Senk • Dick Hill • Bruce Sagan • Kirk Weller • This work supported in part by NSF grant DUE-0536231

16. An advertisement • Jan. ‘09 JMM • Washington, DC, January 5-8 • Mini-course on capstone courses • More activities, more information about the courses, more about collaboration, more about research

17. References cited • Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics. Westport, CT: Ablex. • Begle, E. G. (1979). Critical variables in mathematics education: Findings from a survey of the empirical literature. Washington, DC: Mathematical Association of American and National Council of Teachers of Mathematics. • Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’ pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19(5), 385–401. • Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.-P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531. • CBMS. (2001). The Mathematical Education of Teachers. Providence, Rhode Island: Mathematical Association of America, in cooperation with the American Mathematical Society. • Fennema, E., Carpenter, T., Franke, M., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children's thinking in mathematics Instruction. Journal for Research in Mathematics Education, 27(4), 403-434. • Hill, H., Rowan, B., & Ball, D. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406. • Hill, H., Schilling, S., & Ball, D. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11-30. • Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Assoc. • Monk, D. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125-145. • Wilson, S. M., Floden, R. E., & Ferrini-Mundy, J. (2002). Teacher preparation research: An insider’s view from the outside. Journal of Teacher Education, 53, 190-204.