EXPLOITING UNEXPECTED SITUATIONS IN THE MATHEMATICS CLASSROOM

Abstract

The professional development of mathematics teachers needs to support teachers in orchestrating the mathematics classroom in ways that enable them to respond flexibly and productively to the unexpected. When a situation arises in the classroom which is not connected in an obvious way to the mathematical learning intentions of the lesson, it can be challenging for the teacher to improvise so as to craft this situation into an opportunity for doing and learning mathematics. In this study, as teacher-researcher I maintained a record of unexpected situations as they arose in my own secondary mathematics classroom. Details are given of four unexpected situations which I found ways to exploit mathematically, and these are analysed to highlight factors which may enhance a mathematics teacher’s preparedness for dealing with the unexpected. The results of this study indicate that deviating from the intended lesson to exploit an unexpected situation in which students have shown some interest can lead them into enjoyable and worthwhile mathematical engagement.

References

  1. Armstrong, P. (2003). Teaching as stand-up comedy: The metaphor of scripted and improvised performance of teaching. In I. Davidson, D. Murphy & B. Piette (Eds.) Speaking in tongues: Languages of lifelong learning, Proceedings of the 33rd Annual Conference on University Teaching and Research in the Education of Adults. Bangor: University of Wales Bangor/SCUTREA. Available at www.leeds.ac.uk/educol/documents/00003086.htm.

  2. Atkinson, T. & Claxton, G. (2003). The intuitive practitioner: On the value of not always knowing what one is doing. Maidenhead: The Open University.

    Google Scholar 

  3. Berliner, D. C. (2001). Learning about and learning from expert teachers. International Journal of Educational Research, 35(5), 463–482.

    Article  Google Scholar 

  4. Brookfield, S. D. (2006). The skillful teacher: On technique, trust, and responsiveness in the classroom (2nd ed.). San Francisco: Wiley.

    Google Scholar 

  5. Bryant, J. & Sangwin, C. (2008). How round is your circle?: Where engineering and mathematics meet. Oxford: Princeton University Press.

    Google Scholar 

  6. Chick, H. & Stacey, K. (2013). Teachers of mathematics as problem-solving applied mathematicians. Canadian Journal of Science, Mathematics, and Technology Education, 13(2), 121–136.

    Article  Google Scholar 

  7. Claxton, G. (2003). The anatomy of intuition. In T. Atkinson & G. Claxton (Eds.), The intuitive practitioner: On the value of not always knowing what one is doing (pp. 32–52). Maidenhead: The Open University.

    Google Scholar 

  8. Coker, J. (1964). Improvising jazz. Englewood Cliffs: Prentice-Hall.

    Google Scholar 

  9. Cryer, B. (2009). Butterfly brain. London: Orion Books Ltd.

    Google Scholar 

  10. Csikszentmihalyi, M. (1988). The flow experience and its significance for human psychology. In M. Csikszentmihalyi & I. S. Csikszentmihalyi (Eds.), Optimal experience: Psychological studies of flow in consciousness (pp. 15–35). Cambridge: CUP.

    Google Scholar 

  11. Csikszentmihalyi, M. (2002). Flow: The classic work on how to achieve happiness. London: Rider.

    Google Scholar 

  12. Csikszentmihalyi, M. & Csikszentmihalyi, I. S. (1988). Optimal experience: Psychological studies of flow in consciousness. Cambridge: CUP.

    Google Scholar 

  13. Davis, B. (2009). Inventions of teaching: A genealogy. London: Routledge.

    Google Scholar 

  14. Davis, B. & Simmt, E. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education, 34(2), 137–167.

    Article  Google Scholar 

  15. Davis, B. & Sumara, D. (2004). Becoming more curious about learning. Journal of Curriculum and Pedagogy, 1(1), 26–30.

    Article  Google Scholar 

  16. Davis, B. & Sumara, D. (2005). Complexity science and educational action research: Toward a pragmatics of transformation. Educational Action Research, 13(3), 453–466.

    Article  Google Scholar 

  17. Davis, B., Sumara, D. & Luce-Kapler, R. (2008). Engaging minds: Changing teaching in complex times (2nd ed.). London: Routledge.

    Google Scholar 

  18. Fernandez, C. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. London: Lawrence Erlbaum.

    Google Scholar 

  19. Foster, C. (2007). Mathematical behaviour. Mathematics Teaching, 202, 12–13.

    Google Scholar 

  20. Foster, C. (2013a). Resisting reductionism in mathematics pedagogy. Curriculum Journal, 24(4), 563–585.

    Google Scholar 

  21. Foster, C. (2013b). Mathematical études: Embedding opportunities for developing procedural fluency within rich mathematical contexts. International Journal of Mathematical Education in Science and Technology, 44(5), 765–774.

    Google Scholar 

  22. Geertz, C. (1994). Thick description: Toward an interpretive theory of culture. In M. Martin & L. C. McIntyre (Eds.), Readings in the philosophy of social science (pp. 213–231). London: MIT Press.

    Google Scholar 

  23. George, M. (2012). How mathematics teaching can be like improv theater. MathAMATYC Educator, 3(2), 21–23.

    Google Scholar 

  24. Griffiths, J. (2007). Improvisando. Mathematics Teaching Incorporating Micromath, 205, 31.

    Google Scholar 

  25. Hoftstadter, D. (2007). Thoughts on geometrical thinking. Mathematics in School, 36(4), 27.

    Google Scholar 

  26. LeCompte, M. D., Preissle, J. & Tesch, R. (1993). Ethnography and qualitative design in educational research. San Diego: Academic Press.

    Google Scholar 

  27. Leinhardt, G. (1990). Capturing craft knowledge in teaching. Educational Researcher, 19(2), 18–25.

    Article  Google Scholar 

  28. Mason, J. (2002). Researching your own practice: The discipline of noticing. London: RoutledgeFalmer.

    Google Scholar 

  29. Mason, J. & Davis, B. (2013). The importance of teachers’ mathematical awareness for in-the-moment pedagogy. Canadian Journal of Science, Mathematics, and Technology Education, 13(2), 182–197.

    Article  Google Scholar 

  30. Mason, J. & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 38, 135–161.

    Article  Google Scholar 

  31. McMahon, A. (2003). The development of professional intuition. In T. Atkinson & G. Claxton (Eds.), The intuitive practitioner: On the value of not always knowing what one is doing (pp. 137–148). Maidenhead: The Open University.

    Google Scholar 

  32. Middleton, J. A. & Spanias, P. A. (1999). Motivation for achievement in mathematics: Findings, generalizations, and criticisms of the research. Journal for Research in Mathematics Education, 30(1), 65–88.

    Article  Google Scholar 

  33. Pelletier, L. G. & Sharp, E. (2008). Persuasive communication and proenvironmental behaviours: How message tailoring and message framing can improve the integration of behaviours through self-determined motivation. Canadian Psychology, 49(3), 210–217.

    Article  Google Scholar 

  34. Polanyi, M. (2009). The tacit dimension. Chicago: University of Chicago Press.

    Google Scholar 

  35. Remillard, J. T. (1997). Mathematics teaching as improvisation: A problem for policy implementation. Chicago: Paper presented at the annual meeting of the American Educational Research Association.

    Google Scholar 

  36. Rothenberg, J. J. (1994). Memories of schooling. Teaching and Teacher Education, 10(4), 369–379.

    Article  Google Scholar 

  37. Rowland, T. & Zazkis, R. (2013). Contingency in the mathematics classroom: Opportunities taken and opportunities missed. Canadian Journal of Science, Mathematics, and Technology Education, 13(2), 137–153.

    Article  Google Scholar 

  38. Rowland, T., Huckstep, P. & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255–281.

    Article  Google Scholar 

  39. Sawyer, R. K. (2004). Creative teaching: Collaborative discussion as disciplined improvisation. Educational researcher, 33(2), 12–20.

    Article  Google Scholar 

  40. Sedig, K. (2007). Toward operationalization of ‘flow’ in mathematics learnware. Computers in Human Behavior, 23(4), 2064–2092.

    Article  Google Scholar 

  41. Skovsmose, O. (2011). An invitation to critical mathematics education. Rotterdam: Sense.

    Google Scholar 

  42. Tanner, H., Jones, S., Beauchamp, G. & Kennewell, S. (2010). Interactive whiteboards and all that jazz: Analysing classroom activity with interactive technologies. In L. Sparrow, B. Kissane & C. Hurst (Eds.), Shaping the future of mathematics education. Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia, Freemantle (Vol. 2, pp. 547–554). Adelaide: MERGA.

  43. Towers, J. & Davis, B. (2002). Structuring occasions. Educational Studies in Mathematics, 49(3), 313–340.

    Article  Google Scholar 

  44. von Glasersfeld, E. (1995). Radical constructivism: A way of learning (studies in mathematics education). London: RoutledgeFalmer.

    Google Scholar 

  45. Walls, R. T., Sperling, R. A. & Weber, K. D. (2001). Autobiographical memory of school. The Journal of Educational Research, 95(2), 116–127.

    Article  Google Scholar 

  46. Ward-Penny, R. (2010). Context or con? How might we better represent the “real-world” in the classroom? Mathematics in School, 39(1), 10–12.

    Google Scholar 

  47. Williams, G. (2002). Associations between mathematically insightful collaborative behaviour and positive affect. In A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 402–409). Norwich: PME.

    Google Scholar 

  48. Zodik, I. & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom. Educational Studies in Mathematics, 69(2), 165–182.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Colin Foster.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Reprints and Permissions

About this article

Cite this article

Foster, C. EXPLOITING UNEXPECTED SITUATIONS IN THE MATHEMATICS CLASSROOM. Int J of Sci and Math Educ 13, 1065–1088 (2015). https://doi.org/10.1007/s10763-014-9515-3

Download citation

Key words

  • contingency
  • discipline of noticing
  • flow
  • mathematics teaching
  • orchestrating
  • researching own practice
  • unexpected situations