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Overcoming the Algebra Barrier: Being Particular About the General, and Generally Looking Beyond the Particular, in Homage to Mary Boole

  • John MasonEmail author
Chapter

Abstract

Consistent with a phenomenographic approach valuing lived experience as the basis for future actions, a collection of pedagogic strategies for introducing and developing algebraic thinking are exemplified and described. They are drawn from experience over many years working with students of all ages, teachers and other colleagues, and reading algebra texts from the fifteenth century to the present. Attention in this chapter is mainly focused on invoking learners’ powers to express generality, to instantiate generalities in particular cases, and to treat all generalities as conjectures which need to be justified. Learning to manipulate algebra is actually straightforward once you have begun to appreciate where algebraic expressions come from.

Keywords

Expressing generality Pedagogic strategies Tracking arithmetic Watch What You Do Say What You See Reasoning without numbers Same and different Invariance in the midst of change 

References

  1. Bednarz, N., Kieran, C., & Lee, L. (Eds.). (1996). Approaches to Algebra: Perspectives for research and teaching. Dordrecht: Kluwer.Google Scholar
  2. Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex and setting. Buckingham: Open University Press.Google Scholar
  3. Bruner, J. (1966). Towards a theory of instruction. Cambridge: Harvard University Press.Google Scholar
  4. Cai, J., & Knuth, E. (2011). Early algebraization: A global dialogue from multiple perspectives. Heidelberg: Springer.CrossRefGoogle Scholar
  5. Chevallard, Y. (1985). La Transposition Didactique. Grenoble: La Pensée Sauvage.Google Scholar
  6. Chick, H., Stacey, K., Vincent, J., & Vincent, J. (Eds.). (2001). The future of the teaching and learning of algebra. Proceedings of the 12th ICMI Study Conference, University of Melbourne, Melbourne.Google Scholar
  7. Conway, J., & Guy, R. (1996). The book of numbers. New York: Copernicus.CrossRefGoogle Scholar
  8. Courant, R. (1981). Reminiscences from Hilbert’s Gottingen. Mathematical Intelligencer, 3(4), 154–164.CrossRefGoogle Scholar
  9. Davis, B. (1996). Teaching mathematics: Towards a sound alternative. New York: Ablex.Google Scholar
  10. Davydov, V. (1990). Types of generalisation in instruction (Soviet studies in mathematics education, Vol. 2). Reston: NCTM.Google Scholar
  11. Dougherty, B. (2008). Algebra in the early grades. Mahwah: Lawrence Erlbaum.Google Scholar
  12. Gattegno, C. (1988). The mind teaches the brain (2nd ed.). New York: Educational Solutions.Google Scholar
  13. Giménez, J., Lins, R., & Gómez, B. (Eds.). (1996). Arithmetics and algebra education: Searching for the future. Barcelona: Universitat Rovira i Virgili.Google Scholar
  14. Halmos, P. (1975). The problem of learning to teach. American Mathematical Monthly, 82(5), 466–476.CrossRefGoogle Scholar
  15. Hewitt, D. (1998). Approaching arithmetic algebraically. Mathematics Teaching, 163, 19–29.Google Scholar
  16. Kaput, J., Carraher, D., & Blanton, M. (2008). Algebra in the early grades. Mahwah: Lawrence Erlbaum.Google Scholar
  17. MacTutor Website. Retrieved from http://www-history.mcs.st-and.ac.uk/index.html
  18. MAphorisms. Retrieved October, 2015, from www.math.ku.dk/~olsson/links/maforisms.html
  19. Marton, F. (2015). Necessary conditions for learning. Abingdon: Routledge.Google Scholar
  20. Marton, F., & Booth, S. (1997). Learning and awareness. Hillsdale, MI: Lawrence Erlbaum.Google Scholar
  21. Mason, J. (2001). Teaching for flexibility in mathematics: Being aware of the structures of attention and intention. Questiones Mathematicae, 24(Suppl 1), 1–15.Google Scholar
  22. Mason, J. (2002a). Researching your own practice: The discipline of noticing. London: Routledge-Falmer.Google Scholar
  23. Mason, J. (2002b). Mathematics teaching practice: A guidebook for university and college lecturers. Chichester: Horwood.CrossRefGoogle Scholar
  24. Mason, J. (2010). Attention and intention in learning about teaching through teaching. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice (pp. 23–47). New York: Springer.CrossRefGoogle Scholar
  25. Mason, J. (2014). Uniqueness of patterns generated by repetition. Mathematical Gazette, 98(541), 1–7.CrossRefGoogle Scholar
  26. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London: Addison Wesley.Google Scholar
  27. Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985). Routes to, roots of Algebra. Milton Keynes: The Open University.Google Scholar
  28. Mason, J., & Johnston-Wilder, S. (2004). Designing and using mathematical tasks. Milton Keynes: Open University.Google Scholar
  29. Mason, J., Oliveira, H., & Boavida, A. M. (2012). Reasoning reasonably in mathematics. Quadrante, XXI(2), 165–195.Google Scholar
  30. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–290.CrossRefGoogle Scholar
  31. Mason, J., & Sutherland, R. (2002). Key aspects of teaching algebra in schools. London: QCA.Google Scholar
  32. Moessner, A. (1952). Ein Bemerkung über die Potenzen der natürlichen Zahlen. S.–B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29 (MR 14 p353b).Google Scholar
  33. Newton, I. (1683) in D. Whiteside (Ed.). (1964). The mathematical papers of Isaac Newton (Vol. V). Cambridge: Cambridge University Press.Google Scholar
  34. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford: Blackwell.Google Scholar
  35. Nunes, T., Bryant, P., & Watson, A. (2008). Key understandings in mathematics learning. Retrieved October, 2015, from www.nuffieldfoundation.org/key-understandings-mathematics-learning
  36. Open University. (1982). EM235: Developing mathematical thinking. A distance learning course. Milton Keynes: Open University.Google Scholar
  37. Pólya, G. (1954). Mathematics and plausible reasoning (Induction and analogy in mathematics, Vol. 1). Princeton: Princeton University Press.Google Scholar
  38. Pólya, G. (1965). Let us teach guessing (film). Washington: Mathematical Association of America.Google Scholar
  39. Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19(1), 19–43.CrossRefGoogle Scholar
  40. Tahta, D. (1972). A Boolean anthology: Selected writings of Mary Boole on mathematics education. Derby: Association of Teachers of Mathematics.Google Scholar
  41. Ward, J. (1706). The young mathematicians guide, being a plain and easy Introduction to the Mathematicks in Five Parts. Thomas Horne: London.Google Scholar
  42. Watson, A. (2000). Going across the grain: Mathematical generalisation in a group of low attainers. Nordisk Matematikk Didaktikk (Nordic Studies in Mathematics Education), 8(1), 7–22.Google Scholar
  43. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah: Erlbaum.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.OxfordUK

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