Planning and Teaching Mathematics Capstone Courses for Pre-service Secondary School Teachers

1. Planning and Teaching Mathematics Capstone Courses for Pre-service Secondary School Teachers Joint Mathematics Meetings Washington, DC January, 2009

2. Presenters Grand Valley State University Edward Aboufadel and Rebecca Walker Michigan State University Richard Hill and Sharon Senk Bruce Sagan, on leave at the NSF, 2008-2010 University of Maine Natasha Speer Ferris State University Kirk Weller

3. Day 1 Introduction and rationale for offering capstone courses for pre-service teachers Approximations of p and the development of knowledge over time Definitions of functions and the role of definitions Structure of capstone courses at MSU and GVSU Day 2 Role of writing in capstone courses Metric spaces and convergent sequences in high school and college The use of technology Team-teaching: lessons learned Minicourse Evaluation Minicourse Outline

4. Request for feedback during the course Now: On an index card, answer the question: “What do you hope to learn in this course?” Today: On the index cards, identify issues that you want to hear about or discuss during Day 2. [Including your name is optional.]

5. Part 1Rationale for Offering Capstone Courses for Pre-service TeachersPresenter:Natasha Speer

6. 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?

7. Things learned from educational research (cont.) 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?

8. Additional lessons from research Mathematical knowledge for teaching matters - It is used to do the “mathematical work” of teaching to: • follow and understand students’ mathematical thinking • evaluate the validity of student-generated strategies • make sense of a range of solution paths • It is shown to play a role in teachers’ practices and correlate with elementary students’ learning (Ball & Bass, 2000; Hill et al 2004, 2005; Ma, 1999)

9. As stated in the Mathematics Education of Teachers (MET) Report: “It appears that instead of (or perhaps in addition to) acquiring knowledge of advanced mathematics, what effective teachers need is mathematical knowledge that is organized for teaching.” (CBMS, 2001, p. 121)

10. Some unanswered questions Specifically what do teachers need to know to do this work? How does that knowledge develop? What can we do to promote that development? What does this mean for secondary school teachers?

11. The mathematics education community’s response? Develop resources, e.g., Mathematics for High School Teachers, An Advanced Perspective (Usiskin et al., 2003) Offer mathematics capstone courses for pre-service teachers MSU first offered such a course for pre-service secondary teachers in Fall 2003 Many others exist now.

12. What a capstone course should look like Mathematics departments should "support the design, development, and offering of a capstone course sequence for teachers in which conceptual difficulties, fundamental ideas, and techniques of high school mathematics are examined from an advanced standpoint.” (CBMS, 2001, p. 39)

13. Diversity in the foci for such courses Examineadvanced topicsthat 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

14. 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

15. Who should teach a capstone course? “Such a capstone sequence would be most effectively taught through a collaboration of faculty with primary expertise in mathematics and faculty with primary expertise in mathematics education and experience in high school teaching." (CBMS, 2001, p. 39)

16. Goals of NSF Award (DUE 0536231) • Support team-teaching of capstone courses by mathematicians and mathematics educators at MSU and GVSU; • Develop additional instructional materials for capstone courses; • Examine the different contributions that mathematics educators and mathematicians make to the project.

17. Time Line Course offered at MSU Course offered at MSU Course offered at GVSU Fall ’03 Fall ’04 Sp.’06 Fall ’06 Sp.’07 Sp.’08Sp.’09 CCLI grant awarded

18. Part 2 Approximations of p and the Development of Knowledge Over Time Presenters: Ed Aboufadel and Rebecca Walker

19. Approximating p • Archimedes’ method – Around 225 BC • Viète’s method – Around 1600 AD • Newton’s method – Around 1665 AD

20. Archimedes’ method for approximating p Read through the whole activity and work with one or two other people on the parts you find interesting. As you work, think and take some notes about: • Mathematical content addressed • Mathematics for teaching content addressed • Why do this with preservice teachers?

21. More advanced methods for approximating p • Viète’s Method - Infinite product that converges to 2/p • Newton’s Method - Develop power series for • Find the area of a slice of a circle by integrating that series - The area can be computed exactly using geometry - Create a truncated series to approximate 

22. Reflection on this sequence of activities Students: - Engaged with a variety of mathematical ideas; - Developed some perspective on historical development of mathematics, and; - Connected this development to the development of their future students’ mathematical abilities and understanding.

23. Questions and Comments

24. Part 3 Definitions of Functions and the Role of Definitions Presenters: Richard Hill and Sharon Senk

25. Definitions of Function Read and answer Questions 1-3 as if you were a student. As faculty members, discuss in small groups what is being brought out by the different examples of Question 3. Are there other examples you would use, and why? Look over Questions 4-6. Discuss possible purposes of these questions.

26. Issues about functions that many upper-class mathematics majors find difficult • Functions whose domain and range are sets other than subsets of the reals • Functions as objects vs. functions as rules

27. Other topics studiedthat involve alternate definitions • Parallelism • Angle • Similarity • Sin ø, cos ø, and tan ø

28. Questions and Comments

29. Part 4 Structure of our Capstone Courses Presenters: Ed Aboufadel & Rebecca Walker GVSU Richard Hill & Sharon Senk MSU

30. Grand Valley State University Capstone Course – MTH 495 Fall 2006 Dr. Ed Aboufadel Dr. Rebecca Walker

31. Capstone students at GVSU • 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 (emphasis on rings and fields)- all - Linear Algebra -all - Discrete Mathematics - most

32. GVSU students (cont.) • Education emphasis students have also taken two or three mathematics education courses where they: - consider the school curriculum - explore reasoning behind the mathematics taught in schools - explore how students might reason about the topics

33. Major mathematical topics at GVSU • 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 p

34. Instructors of Capstone Courses at MSU, 2003 - 2008

35. Capstone students at MSU • Senior mathematics majors • Math courses completed by all - Three semesters of calculus - Real analysis - Linear algebra - Abstract algebra • Most had completed several other higher level math courses.

36. MSU students (cont.) • All had been admitted to teacher preparation program for Grades 7-12, in which GPA ≥ 2.5 overall and in major is required for student teaching. • All were simultaneously taking a year-long mathematics education course.

37. Evolution of major mathematical topics at MSU

38. Questions and Comments

39. Wrapping up Day 1 Before you leave… On a note card, write questions, concerns, or comments that you would like addressed. Leave cards on thetable or hand to one of the presenters. [Including your name is optional.]

40. 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.

41. 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.