When Do Gifted High School Students Use Geometry to Solve Geometry Problems?

Abstract

This article describes the following phenomenon: Gifted high school students trained in solving Olympiad-style mathematics problems experienced conflict between their conceptions of effectiveness and elegance (the EEC). This phenomenon was observed while analyzing clinical task-based interviews that were conducted with three members of the Israeli team participating in the International Mathematics Olympiad. We illustrate how the conflict between the students’ conceptions of effectiveness and elegance is reflected in their geometrical problem solving, and analyze didactical and epistemological roots of the phenomenon.

Get full access to this article

View all access options for this article.

References

Baker

(2004). Simplicity. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, retrieved November 18, 2005, from http://plato.stanford.edu/archives/win2004/entries/simplicity

Berzsenyi

(1999). Providing opportunities through competitions. In Sheffield

L. J.

(Ed.), Developing mathematically promising students (pp. 185–190). Reston, VA: NCTM.

Clement

(2000). Analysis of clinical interviews: Foundation and model viability. In Kelly

A. E.

Lesh

(Eds.), Handbook of research design in mathematics and science education (pp. 547–589). Mahwah, NJ: Lawrence Erlbaum Associates.

Coxeter

Greitzer

(1967). Geometry revisited. Washington, DC: Mathematical Association of America.

Davis

(1984). Learning mathematics: The cognitive science approach to mathematics education. Beckenham, England: Croom Helm.

Dreyfus

Eisenberg

(1986). On the aesthetics of mathematical thoughts. For the Learning of Mathematics, 6(1), 2–10.

Dreyfus

Eisenberg

(1996). On different facets of mathematical thinking. In Sternberg

R. J.

Ben-Zeev

(Eds.), The nature of mathematical thinking (pp. 253–284). Mahwah, NJ: Lawrence Erlbaum Associates.

Erickson

Simon

(1993). Protocol analysis: Verbal reports as data (Rev. ed.). Cambridge, MA: MIT Press.

Goldin

G. A.

(2002) Affect, meta-affect, and mathematical belief structures. In Leder

G. C.

Penkonen

Torner

(Eds.), Beliefs: A hidden variable in mathematics education? (pp. 59–72). Dordrecht, Netherlands: Kluwer.

10.

Gorodetsky

Klavir

(2003). What can we learn from how gifted/average pupils describe their process of problem solving? [Electronic version]. Learning and Instruction, 13, 305–325.

11.

Grassl

Mingus

(1999). Nurturing the growth of young mathematicians through mathematics contests. In Sheffield

L. J.

(Ed.), Developing mathematically promising students (pp. 291–296). Reston, VA: NCTM.

12.

Hadamard

(1945). The mathematician's mind: The physiology of invention in the mathematical field. Princeton, NJ: Princeton University Press.

13.

Harel

(1998). The dual assertions: The first on learning and the second on teaching (or vice versa). American Mathematical Monthly, 105, 497–507.

14.

Koichu

(2003). Junior high school students' heuristic behaviors in mathematical problem solving. Unpublished doctoral dissertation, Technion, Israel.

15.

Koichu

Berman

Moore

(2004). Promotion of heuristic literacy in a regular mathematics classroom. For the Learning of Mathematics, 24(1), 33–39.

16.

Krutetskii

(1976). The psychology of mathematical abilities in schoolchildren ( Teller

, Trans.). Chicago: University of Chicago Press. (Original work published 1968)

17.

McGregor

G. D.

Jr. (2001). Gifted and talented youth as philosophers of the new millennium. Journal of Secondary Gifted Education, 12, 157–161.

18.

Nokelainen

Tirri

Campbell

J. R.

(2004). Cross-cultural predictors of mathematical talent and academic productivity. High Ability Study, 15, 229–242.

19.

Poincaré

(1946). The foundation of science ( Halsed

G. B.

, Trans.). Lancaster, PA: Science Press. (Original work published 1913)

20.

Prasolov

V. V.

(1995). Zadachi po planimetrii [Problems in plane geometry]. Moscow: NCNMO.

21.

Schoenfeld

A. H.

(1985). Mathematical problem solving. New York: Academic Press.

22.

Silver

Metzger

(1989). Aesthetic influences on expert problem solving. In McLeod

D. B.

Adams

V. M.

(Eds.), Affect and mathematical problem solving (pp. 59–74). New York: Springer-Verlag.

23.

Sinclair

(2001). The aesthetics is relevant. For the Learning of Mathematics, 21(1), 25–33.

24.

Sinclair

(2004). The role of the aesthetics in mathematical inquiry. Mathematical Thinking and Learning, 6, 261–284.

25.

Sternberg

R. J.

Davidson

J. E.

(1983). Insight in the gifted. Educational Psychologist, 18, 51–57.

26.

Von Glasersfeld

Steffe

L. P.

(1991). Conceptual models in educational research and practice. Journal of Education Thought, 25, 91–103.