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Class #5: Wednesday, September 26. 5. Skemp and instrumental/relational understanding. Skemp, R. R. (1971/1987). The psychology of learning mathematics.

McCrory

1 CEP913 Psychology and Pedagogy of Mathematics Fall 2007 Raven McCrory, Instructor Schedule of readings and assignments

Class #1: Wednesday, August 29 1. Introduction Erlwanger, S. H. (1973). Benny's conception of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1(2), 7-26. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics. Assignment 1: Statement of Interest due; see Assigment 1 on Angel. Assignment 2 is Benny, and nothing written is due. Class #2: Wednesday, September 5 2. Thorndike and mathematical knowing as the formation of bonds Thorndike, E.L. (1922). Psychology of arithmetic. New York: Macmillan. Required: pp. xi – xvi, 51 – 168, 227 – 265. Optional: Rest of book. Weekly Analytic Summaries (WAS): Every week, you are asked to write an analytic summary of the readings. Assignment 3, found on Angel, explains what I expect you to do for upir Weekly Analytic Summaries. Class #3: Wednesday, September 12 3. Brownell and mathematical knowing as meaning making Brownell, W., & Chazal, C. (1935). The effects of premature drill in third-grade arithmetic. Journal of Educational Research, 29(1). Brownell, W. (1938). Two kinds of learning in arithmetic. Journal of Educational Research, 31(9). Brownell, W. (1947/2004). The place of meaning in the teaching of arithmetic. The Elementary School Journal, 47: 156-65. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics with a perspective from Mary Lindquist. Brownell, W. (1948). Learning theory and educational practice. Journal of Educational Research, 41(7). McConnell, T. (1934). Discovery vs. authoritative identification in the learning of children. University of Iowa Studies in Education, 9(5), 11-62. Assignment 4: Research question and justification due, along with IRB certification. See Assignment 4 on Angel.

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Class #4: Wednesday, September 19 4. Wertheimer and productive thinking Wertheimer, M. (1945/1959). Productive thinking. New York: Harper & Brothers. Required: pp. 1 – 123; Optional: Rest of book. Class #5: Wednesday, September 26 5. Skemp and instrumental/relational understanding Skemp, R. R. (1971/1987). The psychology of learning mathematics. Hillsdale, NJ: Lawrence Erlbaum. Required: pp. 9 – 65, 101 – 127, 152 – 176; Optional: Rest of book. Class #6: Wednesday, October 3 6. Resnick and the psychology of math for instruction Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale, NJ: Lawrence Erlbaum. Required: pp. 3 – 37, 97 – 236; Optional: Rest of book. Assignment 5: Research proposal due. See Assignment 5 on Angel. Class #7: Wednesday, October 10 7. Conceptions of mathematical knowledge Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up : helping children learn mathematics. Washington, DC: National Research Council, Mathematics Learning Study Committee, National Academy Press. Chapters 1,2,4, and 5. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum Associates. Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 181-198). Hillsdale, NJ: Lawrence Erlbaum Associates. Carpenter, T. (1986). Conceptual knowledge as a foundation for procedural knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113-132). Hillsdale, NJ: Lawrence Erlbaum Associates. Optional: Another conception of understanding Pirie, S. E. B., & Kieren, T. E. (1989). A recursive theory of mathematical understanding. For the Learning of Mathematics, 9(3), 7-11. Pirie, S. E. B., & Kieren, T. E. (1992). Watching Sandy’s understanding grow. Journal of Mathematical Behavior, 11, 243-257. Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26(2-3), 165-190.

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Pirie, S. E. B., & Martin, L. (2000). The role of collecting in the growth of mathematical understanding. Mathematics Education Research Journal, 12(2), 127-146. Pirie, S. E. B., & Kieren, T. E. (1994). Beyond metaphor: Formalising in mathematical understanding within constructivist environments. For the Learning of Mathematics, 14(1), 39-43. Class #8: Wednesday, October 17 Midterm Exam Class #9: Wednesday, October 24 9. Recent views on conceptual and procedural knowledge Baroody, A. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1-33). Mahwah, NJ: Lawrence Erlbaum. Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge: Does one lead to the other? Journal of Educational Psychology, 91(1), 1-16. Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-110). East Sussex, UK: Psychology Press. Schneider, M., & Stern, E. (2005). Conceptual and procedural knowledge of a mathematics problem: Their measurement and their causal interrelations. Proceedings of the twenty-seventh annual conference of the Cognitive Science Society. Mahwah, NJ: Lawrence Erlbaum Optional: Additional readings on conceptual and procedural knowledge: VanLehn, K. (1986). Arithmetic procedures are induced from examples. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 133179). Hillsdale, NJ: Lawrence Erlbaum Associates. Davis, R. B. (1983). Complex mathematical cognition. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 253-290). New York: Academic Press. Greeno, J. (1978). Understanding and procedural knowledge in mathematics instruction. Educational Psychologist, 12(3), 262-283. Assignment 6: Research protocols due – interview, observation, survey or whatever instrument(s) you plan to use. See Angel for assignment 6 details. Class #10: Wednesday, October 31 10. Teacher’s conceptions of student’s thinking Nathan, M. J. (2000). Teachers' and researchers' beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31(2), 168-190.

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Nathan, M. J., & Koedinger, K. R. (2000). An investigation of teachers' beliefs of students' algebra development. Cognition and Instruction, 18(2), 209-237. Nathan, M. J., & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Educational Research Journal, 40(4), 905-928. Optional readings: If you are interested in student thinking and how teachers’ make sense of it: Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3-20. Lampert, M. (2001). Teaching problems: A study of practice from inside the classroom. New Haven, CT: Yale University Press.

Class #11: Wednesday, November 7 11. Teachers’ beliefs and conceptions of mathematics Pajares, M. F. (1992). Teachers' beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307-332. Speer, N. (2005). Issues of methods and theory in the study of mathematics teachers' professed and attributed beliefs. Educational Studies in Mathematics, 58(3), 361-391. Schoenfeld, A. H. (1983) Beyond the purely cognitive: belief systems, social cognitions, and metacognitions as driving forces in intellectual performance. Cognitive Science 7 (1983): 329–63. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics with a perspective from Frank K. Lester, Jr. Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics 15 (1984): 105–27. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics with a perspective from Douglas B. McLeod and Denise S. Mewborn. Optional Readings (if you are interested in studying teachers’ beliefs): Schoenfeld, A. H. (1989). Explorations of students' mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338-355. Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. New York: Macmillan. Vacc, N. N. (1999). Elementary preservice teachers' changing beliefs and instructional use of children's mathematical thinking. Journal for Research in Mathematics Educ ation, 30(1), 89-110.

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Class #12: Wednesday, November 14 12. Mathematical knowledge and the Math Wars Pesek, D. & Kirschner, D. (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31(5), 524-540. Brown, S., Seidelmann, A., & Zimmermann, G. (n.d.). In the trenches: Three teachers' perspectives on moving beyond the math wars. mathematicallysane.com. Wu, H. H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23, 14-19, 5052. Star, J.R. (in press). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education. Optional readings: If you are interested in the “Math Wars” Askey, R. (1999). Knowing and teaching elementary mathematics. American Educator, Fall. Howe, R. (1999). Knowing and teaching elementary mathematics. Notices of the AMS, 46(8), 881-887. Wilson, S. M. (2002). California dreaming: Rerforming mathematics education. New Haven, CT: Yale University Press. No class on November 21: Happy Thanksgiving! We will make up the class during our exam period. Class #13, November 28 13. Other perspectives. Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., Loef, M. (1989) Using Knowledge of Children’s Mathematics Thinking inClassroom Teaching: An Experimental Study American Educational Research Journal 26: 499–531. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics with a perspective from Victoria R. Jacobs. Cobb, P., Yackel, E. Constructivist, Emergent, and Sociocultural Perspectives in the Context of Developmental Research. Educational Psychologist 31 (1996): 175–90. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics with a perspective from Janet Bowers. D’Ambrosio, U (1985). Ethnomathematics and Its Place in the History and Pedagogy of Mathematics” For the Learning of Mathematics 5: 44–48. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National

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Council of Teachers of Mathematics with a perspective from Beatriz D’Ambrosio. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36. Tall, D., Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity Educational Studies in Mathematics 12: 151–69. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics with a perspective from Guershon Harel. Optional readings: If you are interested in Anna Sfard’s work: Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44-55. Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26(2-3), 191-228. Sfard, A. (2001). There is more to discourse than meets the ears: looking at thinking as communicating to learn more about mathematical learning. Educational Studies in Mathematics, 46(1/3), 13-57. Sfard, A. (downloaded August 2007). Argumentation on learning-throughargumentation: Commentary on Baker and Schwarz & Asterhan.Unpublished manuscript. Class #14: Wednesday, December 5 14. Knowledge for teaching mathematics Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up : helping children learn mathematics. Washington, DC: National Research Council, Mathematics Learning Study Committee, National Academy Press. Chapters 9, 10. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal 27 (1990): 29–63. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.). (2004). Classics in mathematics education research. Reston, VA: National Council of Teachers of Mathematics with a perspective from Miriam Gamoran Sherin. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum. Selections to be announced. Hill, H., Ball, D., & Rowan, B. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406. Assignment 7: Research project paper due. See Angel for Assignment 7 details. Class #15: Exam period, Tuesday December 11, 3-6 pm, Raven’s house, followed by dinner. Details to be announced. During this class, you will present your research project, AERA-style (10 minute presentation followed by 5 minutes for questions).

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List of assignments All written assignments are due by class time, in your individual drop box. I greatly appreciate assignments turned in early, especially the weekly summaries, for it gives me time to read them before class. NAME (WAS = Weekly Analytic Summary) Assignment 1, 2 WAS 1 Assignment 3 (explanation of WAS) WAS 2 Assignment 4 WAS 3 WAS 4 WAS5 Assignment 5 WAS 6 Midterm Exam Class discussion WAS 7 Assignment 6 WAS 8 WAS 9 WAS 10 WAS 11 Assignment 7 Final Presentation

Description Statement of interest, Benny reading

CLASS 8/29/07 9/5/07

Research question & justification; IRB certification

Research proposal Prepare and lead, with partner Research protocols

Research project report

9/12/07 9/12/07 9/19/07 9/26/07 10/3/07 10/3/07 10/10/07 10/17/07 varies 10/24/07 10/24/07 10/31/07 11/7/07 11/14/07 12/5/07 12/5/07 12/11/07