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Appreciating mathematical structure for all

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Abstract

We takemathematical structure to mean the identification of general properties which are instantiated in particular situations as relationships between elements or subsets of elements of a set. Because we take the view that appreciating structure is powerfully productive, attention to structure should be an essential part of mathematical teaching and learning. This is not to be confused with teaching children mathematical structure. We observe that children from quite early ages are able to appreciate structure to a greater extent than some authors have imagined. Initiating students to appreciate structure implies, of course, that their appreciation of it needs to be cultivated in order to deepen and to become more mature. We first consider some recent research that supports this view and then go on to argue that unless students are encouraged to attend to structure and to engage in structural thinking they will be blocked from thinking productively and deeply about mathematics. We provide several illustrative cases in which structural thinking helps to bridge the mythical chasm between conceptual and procedural approaches to teaching and learning mathematics. Finally we place our proposals in the context of how several writers in the past have attempted to explore the importance of structure in mathematics teaching and learning.

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References

  • Abelson, H., & diSessa, A. (1980).Turtle geometry: The computer as a medium for exploring mathematics. Cambridge, MA: MIT Press.

    Google Scholar 

  • Balacheff, N. (1987). Processus de preuve et situations de validation.Educational Studies in Mathematics, 18, 147–176.

    Article  Google Scholar 

  • Balacheff, N. (1988).Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.),Mathematics, teachers and children, (pp. 216–235). London: Hodder and Stoughton.

    Google Scholar 

  • Beaulieu, L. (1990). Proofs in expository writing: Some examples from Bourbaki’s early drafts.Interchange, 21(92), 35–45.

    Article  Google Scholar 

  • Biggs, J. (1999).Teaching for quality learning at university. Buckingham: SRHE and Open University Press.

    Google Scholar 

  • Biggs, J., & Collis K. (1982).Evaluating the quality of learning: The SOLO taxonomy. New York: Academic Press.

    Google Scholar 

  • Birkoff, G., & Maclane, S. (1958).A survey of modern algebra. New York: Macmillan.

    Google Scholar 

  • Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.),The proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra, (Vol. 1, pp. 155–162). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Cohn, P. (1965).Universal algebra. New York: Harper & Row.

    Google Scholar 

  • Cooper, T., & Warren, E. (2008). The effect of different representations on Years 3 to 5 students’ ability to generalise.Zentralblatt für Didaktik der Mathematik, 40, 23–37.

    Article  Google Scholar 

  • Davis, R. (1984).Learning mathematics: The cognitive science approach to mathematics education. Norwood, NJ, USA: Ablex.

    Google Scholar 

  • Euler, L. (1810). J. Hewlitt (Trans.)Elements of algebra (Translated from the French, with the additions of La Grange, and the notes of the French translator, to which is added an appendix). London: Johnson.

  • Fischbein, E., & Muzicant, B. (2002). Richard Skemp and his conception of relational and instrumental understanding: Open sentences and open phrases. In D. Tall á M. Thomas (Eds.),Intelligence, learning and understanding in mathematics: A tribute to Richard Skemp (pp. 49–77). Flaxton, Queensland: Post Pressed Publishers.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2001). Fostering understanding of algebraic generalisation through numerical expressions: The role of the quasi-variables. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.),The proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra, (Vol. 1, pp. 258–264). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. Greenes & R. Rubenstein (Eds.)Algebra and algebraic thinking in school mathematics: Seventieth Yearbook, pp. 127–140. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Gattegno, C. (1987).The science of education part I: Theoretical considerations. New York: Educational Solutions.

    Google Scholar 

  • Halford, G. (1999). The development of intelligence includes capacity to process relations of greater complexity. In M. Anderson (Ed.)The development of intelligence, pp.193–213. Hove: Psychology Press,.

    Google Scholar 

  • Harel, G. & Confrey, J. (Eds.) (1995).The development of multiplicative reasoning in the learning of mathematics. Albany, NY: SUNY Press.

    Google Scholar 

  • Hewitt, D. (1991).Working mathematically on symbols in KS3. PM647 Videotape. Milton Keynes: Open University.

    Google Scholar 

  • Hiebert, J. (1986).Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Hilbert, D., & Cohn-Vossen, S. (1952).Geometry and the imagination. New York: Chelsea.

    Google Scholar 

  • Irwin, K., & Britt, M. (2005). The algebraic nature of students’ numerical manipulation in the New Zealand Numeracy Project.Educational Studies in Mathematics, 58, 169–188.

    Article  Google Scholar 

  • Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Developing children’s algebraic reasoning.Journal for Research in Mathematics Education, 38, 258–288.

    Google Scholar 

  • Kieran, C. (1981). Concepts associated with the equality symbol.Educational Studies in Mathematics, 12, 317–326.

    Article  Google Scholar 

  • Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001).Adding it up: Helping children learn mathematics. Mathematics Learning Study Committee. Washington DC: National Academy Press.

    Google Scholar 

  • Lakoff, G., & Nunez, R. (2000).Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

    Google Scholar 

  • Latimer, J., & Smith, T. (1937).A course in geometry. London: George Harrap.

    Google Scholar 

  • Lins, R., & Kaput, J. (2004). The early development of algebraic reasoning: The current state of the field. In K. Stacey, H. Chick, & M. Kendal (Eds.),The Future of the teaching and learning of algebra: The 12th ICMI Study, pp. 47–70. Boston: Kluwer Academic Publishers.

    Google Scholar 

  • Love, E., & Mason, J. (1992).Teaching mathematics: Action and awareness. Milton Keynes: Open University.

    Google Scholar 

  • Marton, F., & Booth, S. (1997).Learning and awareness. Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Marton, F., & Trigwell, K. (2000).Variatio est mater studiorum. Higher Education Research and Development, 19(3), 381–395.

    Article  Google Scholar 

  • Marton, F., & Tsui, A. (Eds.) (2004).Classroom discourse and the space for learning. Marwah, NJ: Erlbaum.

    Google Scholar 

  • Mashaal, M. (2006).Bourbaki: A secret society of mathematicians. Washington, DC: American Mathematical Society.

    Google Scholar 

  • Mason, J. (2003). Structure of attention in the learning of mathematics. In J. Novotná (Ed.)Proceedings, International Symposium on Elementary Mathematics Teaching, pp. 9–16. Prague: Charles University.

    Google Scholar 

  • Mason, J., & Johnston-Wilder, S. (2004).Fundamental constructs in mathematics education. London: Routledge Falmer.

    Google Scholar 

  • Mason, J., Burton, L., & Stacey, K. (1982).Thinking mathematically. London: Addison Wesley.

    Google Scholar 

  • Molina, M. (2007).Desarrollo de pensiamento relacional y comprehensón del signo igual por alumnos de tercero de educación primaria. Unpublished Ph.D. Thesis. Granada: University of Granada.

    Google Scholar 

  • Papert, S. (1980).Mind storms, New York: Basic Books.

    Google Scholar 

  • Papic, M. (2007).Mathematical patterning in early childhood: An intervention study. Unpublished Ph.D. thesis. Sydney: Macquarie University.

    Google Scholar 

  • Piaget, J. (1970). Genetic epistemology. New York: Norton.

    Google Scholar 

  • Pólya, G. (1945).How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Pólya, G. (1962).Mathematical discovery: On understanding, learning, and teaching problem solving (Combined edition). New York: Wiley.

    Google Scholar 

  • Pólya, G. (1965).Let us teach guessing, (film). Washington, DC: Mathematical Association of America.

    Google Scholar 

  • Rivera, F. (2008). On the pitfalls of abduction: Complicities and complexities in patterning activity.For the Learning of Mathematics, 28(91), 17–25.

    Google Scholar 

  • Seeley Brown, J., Collins A., & Duguid, P. (1989). Situated cognition and the culture of learning,Educational Researcher, 18(1), 32–42.

    Google Scholar 

  • Simon, M. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals.Mathematical Thinking and Learning, 8(4), 359–371.

    Article  Google Scholar 

  • Skemp, R. (1976). Relational understanding and instrumental understanding.Mathematics Teacher, 77, 20–26.

    Google Scholar 

  • Stephens, M. (2006). Describing and exploring the power of relational thinking. In P. Grootenboer, R. Zevenbergen & M. Chinnappan (Eds.),Identities, cultures and learning spaces (Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia, pp. 479–486). Canberra: MERGA.

    Google Scholar 

  • Stephens, M. (2008). Some key junctures in relational thinking. In M. Goos, R. Brown & K. Makar (Eds.),Navigating current and charting directions (Proceedings of the 31st annual conference of the Mathematics Education Group of Australasia, pp. 491–498). Brisbane: MERGA.

    Google Scholar 

  • Stephens, M., Wang, X., & Al-Murani, T. (2008).Some key junctures in relational thinking. Unpublished seminar presentation at East China Normal University, Shanghai PRC, 28 April.

  • Stephens, M., & Wang, X. (2008). Some key junctures in relational thinking.Journal of Mathematics Education (in Chinese), 17(5), 36–40.

    Google Scholar 

  • van der Veer, R., & Valsiner, J. (1991). Understanding Vygotsky. London: Blackwell.

    Google Scholar 

  • van Hiele, P. (1986).Structure and insight: A theory of mathematics education, (Developmental Psychology Series). London: Academic Press.

    Google Scholar 

  • Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.)Acquisition of mathematics concepts and processes, pp. 127–124. New York: Academic Press.

    Google Scholar 

  • Warren, E., & Cooper, T. (2008). Patterns that support early algebraic thinking in the elementary school. In C. Greenes & R. Rubenstein (Eds.)Algebra and algebraic thinking in school mathematics: Seventieth Yearbook, pp. 113–126. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Watson, A., & Mason, J. (2005).Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Wertheimer, M. (1945, 1961 enlarged edition).Productive thinking. London: Tavistock.

    Google Scholar 

  • Whitehead, A. (1911, reprinted 1948).An introduction to mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Zazkis, R. (2001). From arithmetic to algebra via big numbers. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds).The proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra, (Vol. 2, pp. 676–681). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The Open University, MK7 6AA, Milton Keynes, Buckinghamshire, UK

    John Mason

  2. Science and Mathematics Education, University of Melbourne, 3010, Melbourne, Australia

    Max Stephens

  3. Department of Education, Oxford University, OX2 6PY, UK

    Anne Watson

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  1. John Mason
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  3. Anne Watson
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Mason, J., Stephens, M. & Watson, A. Appreciating mathematical structure for all. Math Ed Res J 21, 10–32 (2009). https://doi.org/10.1007/BF03217543

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