# Establishing Appropriate Conditions: Students Learning to Apply a Theorem

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## ABSTRACT

During a sequence of tutorials conducted by the first author, it became evident that students were not seeing how to apply the theorem concerning a continuous function on a closed and bounded interval attaining its extreme values in situations in which it is necessary first to construct the closed and bounded interval by reasoning about properties concerning asymptotes. The paper describes 1 tutorial with 10 students and a follow-up revision session with 2 of the students the same day concerning the use of this theorem in which the tutor used the pedagogic strategy of learner-generated examples. The students’ responses and the tutor’s observations are analysed using constructs of *example spaces*, *structure of attention* and *dimensions of possible variation*.

## KEY WORDS

appropriate conditions attention coming-to-mind example construction examples## Preview

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## References

- Abdul-Rahman, S. (2005). Learning with examples and students’ understanding of integration. In A. Rogerson (Ed.),
*Proceedings of the Eighth International Conference of Mathematics Education into the 21st Century Project on “Reform, Revolution and Paradigm Shifts in Mathematics Education”*. Johor Bahru, Malaysia, November 25–December 1, 2005.Google Scholar - Bauersfeld, H. (1988). Interaction, construction, and knowledge–Alternative perspectives for mathematics education. In D. A. Grouws & T. J. Cooney (Eds.),
*Perspectives on research on effective mathematics teaching: Research agenda for mathematics education*(Vol. 1, pp. 27–46). Reston: NCTM/Lawrence Erlbaum.Google Scholar - Benham, C. (1704).
*Benham’s book of quotations, proverbs and household words*. London: Ward, Lock.Google Scholar - Berger, M. (2006). Making mathematical meaning: From preconcepts to pseudoconcepts to concepts.
*Pythagoras, 63*, 14–21.Google Scholar - Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H. Steiner (Ed.),
*Theory of mathematics education, Paper 54*(pp. 110–119). Bielefeld: Institut fur Didaktik der Mathematik der Universitat.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics: Didactiques des mathématiques, 1970–1990,*N. Balacheff, M. Cooper, R. Sutherland, V. Warfield (Trans.), Dordrecht, Netherlands: Kluwer.Google Scholar - Burke, E. (1796).
*Thoughts on the prospect of a regicide peace*. London: Owen.Google Scholar - Burkhardt, H. (1981).
*The real world of mathematics*. Glasgow: Blackie.Google Scholar - Chi, M. & Bassok, M. (1989). Learning from examples via self-explanation. In L. Resnick (Ed.),
*Knowing, learning and instruction: Essays in honour of Robert Glaser*. Hillsdale: Erlbaum.Google Scholar - Chi, M., Bassok, M., Lewis, P., Reiman, P. & Glasser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems.
*Cognitive Science, 13*, 145–182.CrossRefGoogle Scholar - Courant, R. (1981). Reminiscences from Hilbert’s Göttingen.
*Math Intelligencer, 3*(4), 154–164.CrossRefGoogle Scholar - Dahlberg, R. P. & Housman, D. L. (1997). Facilitating learning events through example generation.
*Educational Studies in Mathematics, 33*, 283–299.CrossRefGoogle Scholar - Davis, B. (1996).
*Teaching mathematics: Towards a sound alternative*. New York: Ablex.Google Scholar - Dreyfus, T. & Eisenberg, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann & S. Cunningham (Eds.),
*Visualization in teaching and learning mathematics. MAA Notes No. 19*(pp. 25–37). Washington: Mathematical Association of America.Google Scholar - Fischbein, E. (1993). The theory of figural concepts.
*Educational Studies in Mathematics, 24*(2), 139–162.CrossRefGoogle Scholar - Floyd, A., Burton, L., James, N. & Mason, J. (1981).
*EM235: Developing mathematical thinking*. Milton Keynes: Open University.Google Scholar - Gallagher, W. (2009).
*Rapt: Attention and the focused life*. New York: Penguin.Google Scholar - Halmos, P. (1983).
*Selecta: Expository writing*. In D. Sarasen & L. Gillman (Eds.), New York: Springer.Google Scholar - Halmos, P. (1994). What is teaching?
*The American Mathematical Monthly, 101*(9), 848–854.CrossRefGoogle Scholar - Holt, J. (1964).
*How children fail*. Harmondsworth: Penguin.Google Scholar - James, W. (1890, reprinted 1950).
*Principles of psychology, vol 1*. New York: Dover.Google Scholar - Lave, J. & Wenger, E. (1991).
*Situated learning: Legitimate peripheral participation*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Malara, N., & Navarra, G. (2003).
*ArAl: A project for an early approach to algebraic thinking*. Retrieved January 2007 from http://www.pitagoragroup.it/pited/ArAl.html. - Marton, F. & Booth, S. (1997).
*Learning and awareness*. Mahwah: Lawrence Erlbaum.Google Scholar - Mason, J. (2001). Teaching for flexibility in mathematics: Being aware of the structures of attention and intention.
*Quaestiones Mathematicae, 24*(Suppl 1), 1–15.Google Scholar - Mason, J. (2002). Generalisation and algebra: Exploiting children’s powers. In L. Haggerty (Ed.),
*Aspects of teaching secondary mathematics: Perspectives on practice*(pp. 105–120). London: Routledge Falmer.Google Scholar - Mason, J. (2003). On the structure of attention in the learning of mathematics.
*Australian Mathematics Teacher, 59*(4), 17–25.Google Scholar - Mason, J. & Davis, J. (1989). The inner teacher, the didactic tension, and shifts of attention. In G. Vergnaud, J. Rogalski, & M. Artigue, (Eds.).
*Proceedings of PME XIII*, Paris, vol 2, pp. 274–281.Google Scholar - Mason, J. Drury, H. & Bills, E. (2007). Explorations in the zone of proximal awareness. In J. Watson & K. Beswick (Eds.)
*Mathematics: Essential research, essential practice: Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia*. Adelaide, SA, Australia: MERGA, vol. 1, pp. 42–58.Google Scholar - Mason, J. Burton L. & Stacey K. (1982/2010).
*Thinking Mathematically*, London, UK: Addison Wesley.Google Scholar - Mason, J. & Johnston-Wilder, S. (2004).
*Fundamental Constructs in Mathematics Education*. London: Routledge Falmer.Google Scholar - Mason, J. & Johnston-Wilder, S. (2006).
*Designing and Using Mathematical Tasks*(2nd ed.). St. Albans: Tarquin.Google Scholar - Mason, J. & Pimm, D. (1984). Generic Examples: Seeing the General in the Particular.
*Educational Studies in Mathematics, 15*(3), 277–290.CrossRefGoogle Scholar - Mason, J. & Watson, A. (2001). Stimulating Students to Construct Boundary Examples.
*Quaestiones Mathematicae, 24*(Suppl 1), 123–132.Google Scholar - Nardi E. (1998). Mapping Out the First-year Mathematics Undergraduate’s Difficulties with Abstraction in Advanced Mathematical Thinking: Didactical observations related to the teaching of advanced mathematics from a cross-topical study, in
*Proceedings of the International Conference on the Teaching of Mathematics*, Pythagorion, Greece. p224–226. New York, NY, USA: Wiley.Google Scholar - Oxford (private communication). Internal mock exam set for mathematics undergraduates at Oxford University.Google Scholar
- Pirie, S. & Kieren, T. (1989). A Recursive Theory of Mathematical Understanding.
*For the Learning of Mathematics, 9*(4), 7–11.Google Scholar - Pirie, S. & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it?
*Educational Studies in Mathematics, 26*(2–3), 165–190.CrossRefGoogle Scholar - Pólya, G. (1962).
*Mathematical Discovery: on understanding, learning, and teaching problem solving*. New York: Wiley.Google Scholar - Pólya, G. (1965).
*Let Us Teach Guessing, (film)*. Washington: Mathematical Association of America.Google Scholar - Popper, K. (1965).
*Conjectures and Refutations: the growth of scientific knowledge*. London: Routledge & Kegan Paul.Google Scholar - Schoenfeld, A. (1985).
*Mathematical Problem Solving*. New York: Academic.Google Scholar - Schön, D. (1983).
*The Reflective Practitioner: how professionals think in action*. London: Temple Smith.Google Scholar - Sierpinska, A. (1994).
*Understanding in Mathematics*. London: Falmer Press.Google Scholar - Sweller, J. & Cooper, G. (1985). The use of worked examples as a substitute for problem solving in learning algebra.
*Cognition and Instruction, 2*, 58–89.CrossRefGoogle Scholar - Tall, D. & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity.
*Educational Studies in Mathematics, 12*(2), 151–169.CrossRefGoogle Scholar - Usiskin, Z. (1982).
*Van Hiele levels and achievement in secondary school geometry*. Chicago: University of Chicago.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.Google Scholar - Watson, A. & Mason, J. (2002). Student-Generated Examples in the Learning of Mathematics,
*Canadian Journal of Science, Mathematics and Technology Education, 2*(2), 237–249.Google Scholar - Watson, A. & Mason, J. (2005).
*Mathematics as a Constructive Activity: students generating examples*. Mahwah: Erlbaum.Google Scholar - Whitehead, A. (1911). (reset 1948).
*An Introduction to Mathematics*. Oxford University Press, London.Google Scholar - Wittgenstein, L. (1922).
*Tractatus Logico-Philosophicus*. London: Routledge & Kegan Paul.Google Scholar - Zaslavsky, O. & Lavie, O. (2005).
*Teachers’ Use of Instructional Examples*. Paper presented at the 15th ICMI study conference: The Professional Education and Development of Teachers of Mathematics. Águas de Lindóia, Brazil.Google Scholar - Zaslavsky, O. & Peled, I. (1996). Inhibiting Factors in Generating Examples by Mathematics Teachers and Student-Teachers: The Case of Binary Operation.
*Journal for Research in Mathematics Education, 27*(1), 67–78.CrossRefGoogle Scholar - Zazkis, R. & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool.
*For the Learning of Mathematics, 27*(2), 15–21.Google Scholar