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.

“Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students

“…2-4 demonstrate several characteristics of mathematically capable students, such as generalization and abstraction abilities, flexibility in applying solution strategies, creativity and reflection, which are in accord with former research (Dahl 2004;Koichu and Berman 2005;Krutetskii 1976;Sriraman 2003). The solutions demonstrate a high level of generalization and abstraction abilities as well as good foundations of algebraic thinking.…”

Giftedness and Aesthetics

“…Despite the rigorous selection process of students in the study, it became clear not only that mathematical "beauty" was not a consideration that young problem solvers grasped automatically but also that they had not been exposed to such aesthetic appreciations, as defined by expert mathematicians, until much later when serious mathematical work might be involved. Related to the findings by Silver and Metzger (1989) and Koichu and Berman (2005) was the three expert mathematicians' constant reference to geometric reasoning in their explanations of their most preferred approaches for the three problems with respect to simplicity and originality. Nonetheless, such persistent searching for geometric interpretations did not appear greatly in the ways that the 54 students explained their most preferred approaches.…”

“…Using a similar scope of analysis as Silver and Metzger (1989), Koichu and Berman (2005) examined how three members of the Israeli team participating in the International Mathematics Olympiad coped with conflict in their conceptions of effectiveness and elegance. An effective approach led directly to a final result in answering a mathematics problem with minimum memory retrieval of concepts and terms and procedural knowledge.…”

Cognitive Development of Proof

“…They also found that college students had not yet developed the sense of aesthetics of a proof and proposed that such a sense should be encouraged. Koichu and Berman (2005) found that gifted high school students, who were asked to prove the Steiner-Lehmus theorem ('If the bisectors of two angles of a triangle are equal, then it is an isosceles triangle'), could fluently operate in the mode of proving by contradiction. In addition, they manifested the developed aesthetic sense, by their incentive to find the most parsimonious proof.…”