Mathematical Giftedness, Problem Solving, and the Ability to Formulate Generalizations: The Problem-Solving Experiences of Four Gifted Students

Abstract

Complex mathematical tasks such as problem solving are an ideal way to provide students opportunities to develop higher order mathematical processes such as representation, abstraction, and generalization. In this study, 9 freshmen in a ninth-grade accelerated algebra class were asked to solve five nonroutine combinatorial problems in their journals. The problems were assigned over the course of 3 months at increasing levels of complexity. The generality that characterized the solutions of the 5 problems was the pigeonhole (Dirichlet) principle. The 4 mathematically gifted students were successful in discovering and verbalizing the generality that characterized the solutions of the 5 problems, whereas the 5 nongifted students were unable to discover the hidden generality. This validates the hypothesis that there exists a relationship between mathematical giftedness, problem-solving ability, and the ability to generalize. This paper describes the problem-solving experiences of the mathematically gifted students and how they formulated abstractions and generalizations, with implications for acceleration and the need for differentiation in the secondary mathematics classroom.