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

## Article excerpt

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.

Prologue

Saul (1), a 16-year-old member of the team representing Israel in the International Mathematics Olympiad (IMO), was interviewed in a study of gifted and average students' strategic behaviors in mathematical problem solving (2). At the beginning of the interview, the following dialogue took place between Saul and the interviewer:

Interviewer: How do you approach difficult geometry problems?

Saul: Generally speaking, I try to understand what the fuzziest point in the problem is and then I apply my intuition to this point.... If I don't have any idea what to do, I just use different, not nice, methods.

Interviewer: What do you mean?

Saul: I use the special methods that I have learned, like complex numbers in geometry, or trigo ... where there is no choice.

Interviewer: Why do you think that these methods are not nice?

Saul: They could be nice, but ... [sighs, pauses 5 seconds].

Interviewer: Which methods are "nice"?

Saul: Nice is when I have a geometry problem and I solve it by means of classic geometry.

In this dialogue, Saul expressed the opinion that some algebra-laden methods in geometry problem solving may be effective but are not nice. Furthermore, we observed that Saul and some other top-achieving Olympians experience similar mixed feelings when solving problems presumably solvable within Euclidian geometry, using advanced analytical or trigonometric techniques. As researchers, we became intrigued by apparent conflict between the gifted students' conceptions of effectiveness and elegance in problem solving. Hereafter, we will refer to this conflict as the EEC.

In this paper, the EEC is explored as a force potentially guiding problem-solving behaviors of mathematically gifted students. Specifically, we analyzed geometry problem-solving experiences of three top-achieving Olympians and approach the following questions:

1. How is the EEC reflected in the top-achieving Olympians' geometry problem solving during clinical task-based interviews?

2. What are possible didactical and epistemological roots of the EEC?

Theoretical Background

In this section, we briefly discuss the concepts of effectiveness and elegance and review the relevant literature concerning problem solving and intellectual giftedness.

Effectiveness and Elegance in Mathematical Problem Solving

A problem-solving method is often considered effective if it leads to a solution of a given problem without unnecessary effort, and elegant if it is characterized by clarity, simplicity, parsimony, and ingenuity (Baker, 2004; Dreyfus & Eisenberg, 1986; Krutetskii, 1976; Silver & Metzger, 1989). Note that this assertion, produced as an extract from the aforementioned literature sources, has (at best) some communicative value, but cannot serve as a satisfactory definition of the concepts of effectiveness and elegance. Indeed, in the above assertion, effectiveness and elegance appear as different, yet not completely alien concepts. Moreover, the explanatory terms clarity, simplicity, parsimony, and ingenuity are not better institutionalized than the target term elegance (see Baker, 2004, and Sinclair, 2004, for philosophical and educational views on this issue, respectively). …

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