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A five-phase model for mathematical problem solving: Identifying synergies in pre-service-teachers’ metacognitive and cognitive actions

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Abstract

Based on empirical data from a study of pre-service teachers engaged in non-routine mathematics problem solving, a five-phase model is proposed to describe the range of cognitive and metacognitive approaches used. The five phases are engagement, transformation-formulation, implementation, evaluation and internalization, with each phase being described in terms of sub-categories. The model caters for a variety of pathways that can be adopted during any problem-solving process by recognizing that the path between these five phases is neither linear nor unidirectional.

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References

  • Artzt, A. F., & Armour-Thomas, E. (1992). Development of a cognitive–metacognitive framework for protocol analysis of mathematical problem solving in small groups. Cognition and Instruction, 9, 137–175.

    Article  Google Scholar 

  • Artzt, A. F., & Armour-Thomas, E. (1997). Mathematical problem-solving in small groups: Exploring the interplay of students’ metacognitive behaviours, perceptions, and ability levels. Journal of Mathematical Behavior, 16, 63–74.

    Article  Google Scholar 

  • Berardi-Coleta, B., Dominowski, R. L., Buyer, L. S., & Rellinger, E. R. (1995). Metacognition and problem solving: A process-oriented approach. Journal of Experimental Psychology, 21, 205–223.

    Google Scholar 

  • Carlson, M. P. (2000). A study of the mathematical behaviour of mathematicians: The role of metacognition and mathematical intimacy in solving problems. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 137–144). Hiroshima, Japan: PME.

  • Carlson, M., & Bloom, I. (2005). The cyclic nature of problem-solving: An emergent multi-dimensional problem-solving framework. Educational Studies in Mathematics, 58, 45–75.

    Article  Google Scholar 

  • Casey, D. P. (1978). Failing students: A strategy of error analysis. In P. Costello (Ed.), Aspects of motivation (pp. 295–306). Melbourne: Mathematical Association of Victoria.

    Google Scholar 

  • Chicola, N. A. (1992). The effect of metacognitive strategy instruction on sixth-graders’ mathematics problem-solving ability. Unpublished doctoral dissertation, University of Colorado.

  • Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190.

    Article  Google Scholar 

  • Crawford, P. (1998). Fostering reflective thinking in first-semester calculus students. Unpublished doctoral dissertation, Western Michigan University, Kalamazoo.

  • Davidson, J. E., Deuser, R., & Sternberg, R. J. (1994). The role of metacognition in problem solving. In J. Metcalf & A. P. Shimamura (Eds.), Metacognition (pp. 207–226). Boston, MA: The MIT Press.

    Google Scholar 

  • DeBellis, V. A. (1998). Mathematical intimacy: Local affect in powerful problem solvers. In S. Berenson, et al. (Eds.), Proceedings of the 20th annual meeting of PME-NA (Vol. 2, pp. 435–440). NC: Raleigh.

  • DeFranco, T. C., & Hilton, P. (1999). Distinguishing features of mechanical and human problem solving. Journal of Mathematical Behavior, 18, 79–84.

    Article  Google Scholar 

  • Ellerton, N. F. (2003). Metacognitive benefits accruing from teacher education students’ reflections on problem solving. In H. S. Dhindsa, L. S. Bee, P. Achleitner, & M. A. Clements (Eds.), Studies in science, mathematics and technical education (pp. 202–211). Gadong, Brunei Darussalam: Universiti Brunei Darussalam.

    Google Scholar 

  • Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B. Resnick (Ed.), The nature of intelligence. Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Flavell, J. H. (1987). Speculations about the nature and development of metacognition. In F. E. Weinert & R. H. Kluwe (Eds.), Metacognition, motivation, and understanding (pp. 21–29). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Garofalo, J. (1989). Beliefs and their influence on mathematical performance. Mathematics Teacher, 82, 502–505.

    Google Scholar 

  • Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16, 163–176.

    Article  Google Scholar 

  • Geiger, V., & Galbraith, P. (1998). Developing a diagnostic framework for evaluating student approaches to applied mathematics. International Journal of Mathematics, Education, Science, and Technology, 29, 533–559.

    Article  Google Scholar 

  • Goldin, G. A. (1988). Affective representation and mathematical problem solving. In M. J. Behr, C. B. Lacampagne, & M. M. Wheeler (Eds.), Proceedings of the 10th annual meeting of PME-NA (pp. 1–7). DeKalb, IL: University of Northern Illinois.

  • Goos, M. (2002). Understanding metacognitive failure. Journal of Mathematical Behavior, 21, 283–302.

    Article  Google Scholar 

  • Goos, M., & Galbraith, P. (1996). Do it this way! Metacognitive strategies in collaborative mathematical problem solving. Educational Studies in Mathematics, 30, 229–260.

    Article  Google Scholar 

  • Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49, 193–223.

    Article  Google Scholar 

  • Kramarski, B., Mevarech, Z. R., & Arami, M. (2002). The effects of metacognitive instruction on solving mathematical authentic tasks. Educational Studies in Mathematics, 49, 225–250.

    Article  Google Scholar 

  • Krathwohl, D. R., Bloom, B. S., & Masia, B. B. (1964). Taxonomy of educational objectives: Handbook II: Affective domain. New York: David McKay Company Inc.

    Google Scholar 

  • Kuhn, D. (2000). Metacognitive development. Current Directions in Psychological Science, 9, 178–181.

    Article  Google Scholar 

  • Lester, F. K. (1985). Methodological considerations in research on mathematical problem solving instruction. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 41–69). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Lester, F. K. (1994). Musings about mathematical problem-solving research: 1970–1994. Journal for Research in Mathematics Education, 25, 660–675.

    Article  Google Scholar 

  • Lin, X., Schwartz, D. L., & Hatano, G. (2005). Toward teachers’ adaptive metacognition. Educational Psychologist, 40, 245–255.

    Article  Google Scholar 

  • Marcus, A. (2007). Coding strategic behavior in mathematical problem solving. In D. Kuechemann (Ed.), Proceedings of the British Society for research into learning mathematics (Vol. 27, pp. 54–59).

  • Marge, J. J. (2001). The effect of metacognitive strategy scaffolding on student achievement in solving complex math word problems. Unpublished doctoral dissertation, University of California, Riverside.

  • Martinez, M. E. (2006). What is metacognition? Phi Delta Kappan, pp. 696–699.

  • Maykut, P., & Morehouse, R. (1994). Beginning qualitative research: A philosophic and practical guide. London: The Falmer Press.

    Google Scholar 

  • McLeod, D. B. (1989). The role of affect in mathematical problem solving. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving: A new perspective (pp. 20–36). New York: Springer.

    Google Scholar 

  • McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualisation. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575–596). New York: Macmillan.

    Google Scholar 

  • McLeod, D. B., Craviotto, C., & Ortega, M. (1990). Students’ affective responses to non-routine mathematical problems: An empirical study. In G. Booker, P. Cobb, & T. N. de Menicuti (Eds.), Proceedings of the fourteenth international conference on the psychology of mathematics education (Vol. 1, pp. 159–166). Mexico: PME.

  • Mevarech, Z. R., & Fridkin, S. (2006). The effects of IMPROVE on mathematical knowledge, mathematical reasoning and meta-cognition. Metacognition and Learning, 1, 85–97.

    Article  Google Scholar 

  • Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A multidimensional method for teaching mathematics in heterogeneous classrooms. American Educational Research Journal, 34, 365–395.

    Google Scholar 

  • Mildren, J., Ellerton, N., & Stephens, M. (1990). Children’s mathematical writing: A window into metacognition. In K. Clements (Ed.), Whither mathematics (pp. 356–363). Melbourne: Mathematics Association of Victoria.

    Google Scholar 

  • National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

  • Newman, M. A. (1977). An analysis of sixth-grade pupils’ errors on written mathematical tasks. Victorian Institute for Educational Research Bulletin, 39, 31–43.

    Google Scholar 

  • Newman, M. A. (1983). Strategies for diagnosis and remediation. Sydney, Australia: Harcourt, Brace Jovanovich.

    Google Scholar 

  • Polya, G. (1957). How to solve it. Princeton, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Pugalee, D. K. (2001). Writing, mathematics, and metacognition: Looking for connections through students’ work in mathematical problem solving. School Science & Mathematics, 101, 236–245.

    Article  Google Scholar 

  • Robert, S., & Tayeh, C. (2007). It’s the thought that counts: Reflecting on problem solving. Mathematics Teaching in the Middle School, 12, 232–237.

    Google Scholar 

  • Schoenfeld, A. H. (1985a). Making sense of “out loud” problem solving protocols. The Journal of Mathematical Behavior, 4, 171–191.

    Google Scholar 

  • Schoenfeld, A. H. (1985b). Mathematical problem solving. New York: Academic Press.

    Google Scholar 

  • Schoenfeld, A. H. (1987). What’s all the fuss about metacognition? In Cognitive science and mathematics education (pp. 189–215). Hillsdale, NJ: Lawrence Erlbaum.

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grows (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.

    Google Scholar 

  • Schoenfeld, A. H. (2006). Problem solving from cradle to grave. Annales de Didactique et de Sciences Cognitives, 11, 41–73.

    Google Scholar 

  • Schoenfeld, A. H. (2007). Problem solving, teaching, and more: Toward a theory of goal-oriented behavior. In: Proceedings CIEAM (Vol. 59, pp. 48–52). Debogoko-Hungary.

  • Silver, E. A. (1987). Foundations of cognitive theory and research for mathematics problem-solving instruction. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 33–60). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Simon, H. A. (1973). The structure of ill-structured problems. Artificial Intelligence, 4, 181–202.

    Article  Google Scholar 

  • Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem solving experiences of four gifted students. The Journal of Secondary Gifted Education, 14, 151–165.

    Google Scholar 

  • Stillman, G. A., & Galbraith, P. L. (1998). Applying mathematics with real-world connections: Metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36, 157–195.

    Article  Google Scholar 

  • Veenman, M. V. J., Van Hout-Wolters, B. H. A. M., & Afflerbach, P. (2006). Metacognition and learning: Conceptual and methodological considerations. Metacognition and Learning, 1, 2–14.

    Article  Google Scholar 

  • Willburne, J. M. (1997). The effect of teaching metacognition strategies to preservice elementary school teachers on their mathematical problem-solving achievement and attitude. Unpublished doctoral dissertation, Temple University, PA.

  • Yimer, A. (2004). Metacognitive and cognitive functioning of college students during mathematical problem solving. Unpublished doctoral dissertation, Illinois State University, IL.

  • Yimer, A., & Ellerton, N. F. (2006). Cognitive and metacognitive aspects of mathematical problem solving: An emerging model. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures, and learning spaces (pp. 575–582). Adelaide, Australia: Mathematics Education Research Group of Australasia.

    Google Scholar 

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Authors and Affiliations

  1. Chicago State University, Chicago, IL, USA

    Asmamaw Yimer

  2. Illinois State University, Normal, IL, USA

    Nerida F. Ellerton

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  1. Asmamaw Yimer
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  2. Nerida F. Ellerton
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Correspondence to Nerida F. Ellerton.

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Yimer, A., Ellerton, N.F. A five-phase model for mathematical problem solving: Identifying synergies in pre-service-teachers’ metacognitive and cognitive actions. ZDM Mathematics Education 42, 245–261 (2010). https://doi.org/10.1007/s11858-009-0223-3

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