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Association: Name: North American Chapter of the International Group for the Psychology of Mathematics Education URL: http://www.pmena.org
Citation: URL: http://www.allacademic.com/meta/p117693_index.html Direct Link: HTML Code:

Bloom, Irene. "Mathematics for Teaching: Facilitating Knowledge Construction in Prospective High School Mathematics Teachers" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada, Oct 21, 2004 <Not Available>. 2009-05-26 <http://www.allacademic.com/meta/p117693_index.html>

Bloom, I. , 2004-10-21 "Mathematics for Teaching: Facilitating Knowledge Construction in Prospective High School Mathematics Teachers" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF>. 2009-05-26 from http://www.allacademic.com/meta/p117693_index.html

Publication Type: Conference Paper/Unpublished Manuscript Review Method: Peer Reviewed Abstract: This paper reports results of an ongoing design-based research project that seeks to both promote and characterize the kind of deep, well-connected and flexible conceptual understanding of mathematics that is advocated for teachers (e.g. Ball & McDiarmid, 1990; CBMS, 2000; Fennema & Franke, 1992; Shulman, 1986). In particular, this paper investigates how students access relevant mathematical information in the context of mathematical problem solving. Background A subject’s mathematical background is an important component of problem solving abilities (Schoenfeld, 1985); yet adequate mathematical preparation does not guarantee success. Studies show that undergraduates (Schoenfeld, 1985; 1992), graduate mathematics students (Carlson, 1999), and even some professional mathematicians (DeFranco, 1996) struggle to access the appropriate mathematics needed to solve a particular problem. In their work with mathematicians, Carlson and Bloom (2003) suggest that a well-developed and well-connected conceptual understanding of mathematics facilitated the mathematicians’ problem solving success. Additionally, the mathematicians reported that content knowledge was more useful when solving difficult problems than other factors such as heuristics. Research into the mathematical understandings of teachers indicate that even when teachers (both preservice and in-service) appear to be adequately prepared, their mathematical knowledge base is shallow and compartmentalized (Ball, 1990; Bloom, 2001; Bryan, 1999; Post, Harel, Behr, & Lesh, 1991; Tirosh & Graeber, 1990). A rich, well-connected knowledge base allows teachers to conduct the kind of inquiry and discourse recommended by NCTM (Ball, 1991; Ball & McDiarmid, 1990; Koency & Swanson, 2000; Ma, 1999; McDiarmid, Ball, & Anderson, 1989; NCTM, 2000) and inadequate mathematical understandings tend to inhibit efforts to implement “reform” curricula (Behr, Khoury, Harel, Post, & Leah, 1997; Koency & Swanson, 2000; Mathematical Sciences Education Board, 2001; Post et al., 1991). Theoretical Framework The Multi-Dimensional Problem Solving Framework developed by Carlson and Bloom (2004) provides the means to scrutinize the resources accessed or overlooked in the process of solving mathematical problems. By attending to both the repetitive cycles of orienting, planning, executing and verifying, and the aspects of resources, affect monitoring and heuristics that influence each of the phases, one can tease out the nature of the subject’s knowledge base and its influence during the solution process. The Study This study uses Design-based research methods -- the iterative refinement of instructional innovations employing thought revealing activities and artifacts in conjunction with inquiry into theoretical considerations regarding the nature of the development of mathematical knowledge (Kelly & Lesh, 2002; Lesh, 2002). The subjects are preservice high school mathematics teachers who have completed most or all of their required mathematics courses and will student teach within the next year. The study is set in an upper division mathematics education course called Mathematics in the Secondary School. The curriculum that has been developed over several iterations of the experiment provides students with opportunities to revisit the major concepts of school mathematics while at the same time, enriching their knowledge base and improving their problem solving abilities. For instance, challenging problems are selected for their capacity to reveal student thinking and understanding, and to stimulate mathematical discourse. Students also take typical high school mathematics problems and then they generalize them and analyze them for mathematical structure. Students share their work and their reasoning, orally and in written form, with their peers on a regular basis. Data sources include clinical and task-based interviews, video recordings of classroom sessions, homework, classwork, reflective journals and pre/posttests consisting of NAEP (grade-12) items. Findings Careful analyses of the data show that students enter the class exhibiting a compartmentalized, proceduralized knowledge base. When stripped of examples and cues, these students struggle to solve problems from high school materials. When asked to list the concepts used to solve a problem, students tend to be general and vague. For example, when asked find the rate of change at a moment in time, students tended to find the average rate of change over an arbitrary period of time, and list “algebra” as the main concept employed, rather than calculating the derivative at a point. As the semester progresses, they demonstrate an increasing facility with both doing and discussing mathematics. Mathematical discussions and written work become richer and clearer. Students appear to be more comfortable explaining their work, more confident of their work, and can produce multiple solutions. When asked to determine the mathematical ideas involved in a problem situation, students are able to determine both the “big ideas” as well as unpack them to consider the skills and routines that might also come into play. At the end of the semester, problem-solving sessions are markedly different. Students are able to articulate the mathematics needed and verbalize a solution path. In addition, they report a better understanding of how topics fit together and relate as well as greater confidence in problem solving in general, and the content of high school mathematics in particular. Improved scores on posttest support this evidence. Finally, students report that they feel they are more effective in their tutoring or internships because they don’t have to “see how the book did it” first, and are able to explain concepts in multiple ways. This ongoing project provides some insights into a process by which teacher educators can successfully enrich the content knowledge base of prospective mathematics teachers. References Ball, D. L. (1990). Prospective Elementary and Secondary Teachers' Understanding of Division. Journal for Research in Mathematics Education, 21(2), 132-144. Ball, D. L. (1991). Research on Teaching: Making Subject Matter Knowledge Part of the Equation. In J. E. Brophy (Ed.), Advances in Research on Teaching (Vol. 2, pp. 1-48). Greenwich, CT: JAI. Ball, D. L., & McDiarmid, W. (1990). The Subject-Matter Preparation of Teachers. In W. R. Houston (Ed.), Handbook for Research on Teacher Education. New York: Macmillan. Behr, M. J., Khoury, H. A., Harel, G., Post, T., & Leah, R. (1997). Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-operator Task. Journal for Research in Mathematics Education, 28(1), 48-69. Bloom, I. (2001). Curriculum Reforms That Increase the Mathematical Understanding of Prospective Elementary Teachers. Paper presented at the Twenty-Third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Snowbird Utah. Bryan, T. J. (1999). The Conceptual Knowledge of Preservice Secondary Mathematics Teachers: How Well Do They Know The Subject Matter They Will Teach? Issues in the Undergraduate Mathematics Preparation of School Teachers: The Journal, 1, 1-12. Carlson, M., & Bloom, I. (2004). A Multidimensional Framework for Analyzing Problem Solving Behavior. Educational Studies in Mathematics, accepted. Carlson, M. P. (1999). The mathematical behavior of six successful mathematics graduate students: Influences leading to mathematical success. Educational Studies in Mathematics, 40, 237-258. Conference Board of Mathematical Sciences. (2000). The Mathematical Education of Teachers. Issues in Mathematics Education. Providence: The American Mathematical Society. DeFranco, T. C. (1996). A perspective on mathematical problem-solving expertise based on the performances of male Ph.D. mathematicians. In Research in Collegiate Mathematics II (Vol. 6, pp. 195-213). Providence, RI: American Mathematical Association. Fennema, E., & Franke, M. L. (1992). Teachers' Knowledge and its Impact. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 147-164). New York, NY: MacMillan Publishing Company. Kelly, E., & Lesh, R. (2002). Understanding and Explicating the Design Experiment Methodology. Journal of the ESRC Teaching and Learning Research Programme Research Capacity Building Network(3), 1-3. Koency, G., & Swanson, J. (2000). The Special Case of Mathematics: Insufficient Content Knowledge a Major Obstacle to Reform. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA. Lesh, R. (2002). Research Design in Mathematics Education: Focusing on Design Experiments. In L. D. English (Ed.), Handbook of International Research in Mathematics Education (pp. 27-49). Mahwah, NJ: Lawrence Erlbaum Associates. Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates. Mathematical Sciences Education Board. (2001). Knowing and Learning Mathematics for Teaching. Washington, DC: National Research Council. McDiarmid, G. W., Ball, D. L., & Anderson, C. (1989). Why Staying One Chapter Ahead Doesn't Really Work: Subject-Specific Pedagogy. In M. C. Reynolds (Ed.), Knowledge Base for the Beginning Teacher (pp. 193-205). Elmsford, NY: Pergamon Press. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston: National Council of Teachers of Mathematics,. Post, T. R., Harel, G., Behr, M. J., & Lesh, R. (1991). Intermediate Teachers' Knowledge of Rational Number Concepts. In E. Fennema, T. P. Carpenter & S. J. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 177-198). Albany, NY: University of New York Press. Schoenfeld, A. (1985). Mathematical Problem Solving. Orlando Florida: Academic Press. Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition and sense-making in mathematics. In D. A. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan Publishing Company. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Tirosh, D., & Graeber, A. O. (1990). Evoking Cognitive Conflict to Explore Preservice Teachers' Thinking About Division. Journal for Research in Mathematics Education, 21(2), 98-108.

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