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The Fundamental Cycle of Concept Construction Underlying Various Theoretical Frameworks

Part of the Advances in Mathematics Education book series (AME)

Abstract

In this paper, the development of mathematical concepts over time is considered. Particular reference is given to the shifting of attention from step-by-step procedures that are performed in time, to symbolism that can be manipulated as mental entities on paper and in the mind. The development is analysed using different theoretical perspectives, including the SOLO model of John Biggs and Kevin Collis and various theories of concept construction to reveal a fundamental cycle underlying the building of concepts that features widely in different ways of thinking that occurs throughout.

Keywords

  • Mathematical Thinking
  • Fundamental Cycle
  • Free Vector
  • Global Framework
  • Concept Construction

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  • Biggs, J., & Collis, K. (1982). Evaluating the Quality of Learning: The SOLO Taxonomy. New York: Academic Press.

    Google Scholar 

  • Biggs, J., & Collis, K. (1991). Multimodal learning and the quality of intelligent behaviour. In H. Rowe (Ed.), Intelligence, Reconceptualization and Measurement (pp. 57–76). New Jersey: Laurence Erlbaum Assoc.

    Google Scholar 

  • Bruner, J. S. (1966). Towards a Theory of Instruction. New York: Norton.

    Google Scholar 

  • Case, R. (1992). The Mind’s Staircase: Exploring the Conceptual Underpinnings of Children’s Thought and Knowledge. Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Collis, K. (1975). A Study of Concrete and Formal Operations in School Mathematics: A Piagetian Viewpoint. Melbourne: Australian Council for Educational Research.

    Google Scholar 

  • Crick, F. (1994). The Astonishing Hypothesis. London: Simon & Schuster.

    Google Scholar 

  • Czarnocha, B., Dubinsky, E., Prabhu, V., & Vidakovic, D. (1999). One theoretical perspective in undergraduate mathematics education research. In O. Zaslavsky (Ed.), Proceedings of the 23 rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 95–110). Haifa, Israel.

    Google Scholar 

  • Davis, R. B. (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, NJ: Ablex.

    Google Scholar 

  • Dienes, Z. P. (1960). Building Up Mathematics. London: Hutchinson.

    Google Scholar 

  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced Mathematical Thinking (pp. 95–123). Dordrecht: Kluwer.

    Google Scholar 

  • Edelman, G. M., & Tononi, G. (2000). Consciousness: How Matter Becomes Imagination. New York: Basic Books.

    Google Scholar 

  • Fischer, K. W., & Knight, C. C. (1990). Cognitive development in real children: Levels and variations. In B. Presseisen (Ed.), Learning and Thinking Styles: Classroom Interaction. Washington: National Education Association.

    Google Scholar 

  • Gray, E. M., & Tall, D. O. (1991). Duality, ambiguity and flexibility in successful mathematical thinking. In F. Furinghetti (Ed.), Proceedings of the 13 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 72–79). Assisi, Italy.

    Google Scholar 

  • Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.

    Google Scholar 

  • Gray, E. M., & Tall, D. O. (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 65–72). Utrecht, The Netherlands.

    Google Scholar 

  • Gray, E. M., Pitta, D., Pinto, M. M. F., & Tall, D. O. (1999). Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics, 38, 1–3, 111–133.

    CrossRef  Google Scholar 

  • Greeno, J. (1983). Conceptual entities. In D. Gentner & A. L. Stevens (Eds.), Mental Models (pp. 227–252). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Gruber, H. E., & Voneche, J. J. (1977). The Essential Piaget. New York: Basic Books, Inc.

    Google Scholar 

  • Halford, G. S. (1993). Children’s Understanding: The Development of Mental Models. Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Lakoff, G., & Nunez, R. (2000). Where Mathematics Comes From. New York: Basic Books.

    Google Scholar 

  • Lave, J., & Wenger, E. (1990). Situated Learning: Legitimate Peripheral Participation. Cambridge: CUP.

    Google Scholar 

  • Panizzon, D., & Pegg, J. (2008). Assessment practices: Empowering mathematics and science teachers in rural secondary schools to enhance student learning. International Journal of Science and Mathematics Education, 6, 417–436.

    CrossRef  Google Scholar 

  • Panizzon, D., Callingham, R., Wright, T., & Pegg, J. (2007). Shifting sands: Using SOLO to promote assessment for learning with secondary mathematics and science teachers. Refereed paper presented at the Australasian Association for Research in Education (AARE) conference in Fremantle, Western Australia, 25–29th November 2007. CD ISSN 1324-9320.

    Google Scholar 

  • Pegg, J. (1992). Assessing students’ understanding at the primary and secondary level in the mathematical sciences. In J. Izard & M. Stephens (Eds.), Reshaping Assessment Practice: Assessment in the Mathematical Sciences under Challenge (pp. 368–385). Melbourne: Australian Council of Educational Research.

    Google Scholar 

  • Pegg, J. (2003). Assessment in mathematics: A developmental approach. In J. M. Royer (Ed.), Advances in Cognition and Instruction (pp. 227–259). New York: Information Age Publishing Inc.

    Google Scholar 

  • Pegg, J., & Davey, G. (1998). A synthesis of two models: Interpreting student understanding in geometry. In R. Lehrer & C. Chazan (Eds.), Designing Learning Environments for Developing Understanding of Geometry and Space (pp. 109–135). New Jersey: Lawrence Erlbaum.

    Google Scholar 

  • Pegg, J., & Panizzon, D. (2003–2005). Australian Research Council. Assessing practices: Empowering mathematics and science teachers in rural areas to improve student learning and curriculum implementation.

    Google Scholar 

  • Pegg, J., & Panizzon, D. (2007). Addressing changing assessment agendas: Impact of professional development on secondary mathematics teachers in NSW. Mathematics Teacher Education & Development, 9, 66–79.

    Google Scholar 

  • Pegg, J., & Tall, D. O. (2005). The fundamental cycle of concept construction uncderlying various theoretical frameworks. International Reviews on Mathematical Education (ZDM), 37(6), 468–475.

    Google Scholar 

  • Pegg, J., Baxter, D., Callingham, R., Panizzon, D., Bruniges, M., & Brock, P. (2004–2008). Australian Research Council. Impact of Developmentally-based Qualitative Assessment Practices in English, Mathematics, and Science on School Policies, Classroom Instruction, and Teacher Knowledge.

    Google Scholar 

  • Pegg, J., Baxter, D., Callingham, R., & Panizzon, D. (under preparation). Enhancing Teaching and Learning through Quality Assessment. Post Pressed, Teneriffe, Queensland, Australia.

    Google Scholar 

  • 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, 1–36.

    CrossRef  Google Scholar 

  • Poynter, A. (2004). Effect as a pivot between actions and symbols: the case of vector. Unpublished PhD thesis, University of Warwick.

    Google Scholar 

  • Tall, D. O. (2002). Natural and formal infinities. Educational Studies in Mathematics, 48(2&3), 199–238.

    Google Scholar 

  • Tall, D. O. (2004). Thinking through three worlds of mathematics. In Proceedings of the 28 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 158–161). Bergen, Norway.

    Google Scholar 

  • Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24.

    Google Scholar 

  • Tall, D. O., Gray, E., Ali, M., Crowley, L., DeMarois, P., McGowen, M., Pitta, D., Pinto, M., Thomas, M., & Yusof, Y. (2000). Symbols and the bifurcation between procedural and conceptual thinking. The Canadian Journal of Science, Mathematics and Technology Education, 1, 80–104.

    Google Scholar 

  • Van Hiele, P. M. (1986). Structure and Insight: A Theory of Mathematics Education. New York: Academic Press.

    Google Scholar 

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Correspondence to John Pegg .

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Dedicated to the memory of Kevin F. Collis 1930–2008.

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Pegg, J., Tall, D. (2010). The Fundamental Cycle of Concept Construction Underlying Various Theoretical Frameworks. In: Sriraman, B., English, L. (eds) Theories of Mathematics Education. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00742-2_19

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  • DOI: https://doi.org/10.1007/978-3-642-00742-2_19

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