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 student s and how they formulated abstractions and generalizations, with implications for acceleration and the need for differentiation in the secondary mathematics classroom.
A fascinating aspect of human thought is the ability to generalize from specific experiences and to form new, more abstract concepts. The Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) calls for instructional programs that emphasize problem solving with the goal of helping students develop sophistication with mathematical processes such as representation, mathematical reasoning, abstraction, and generalization. It goes on to proclaim that students should develop increased sophistication with mathematical processes, especially problem solving, representation, and reasoning, and their increased ability to reflect on and monitor their work should lead to greater abstraction and a capability for generalization. Thus, the ability to generalize is the result of certain mathematical experiences and is an important component of mathematical ability. The development of this ability is an objective of mathematics teaching and learning (NCTM).
Psychologists have also been interested in the phenomenon of generalization and have attempted to link the ability to generalize to measures of intelligence (Sternberg, 1979) and to complex problem-solving abilities (Frensch & Steinberg, 1992). Greenes (1981) claimed that mathematically gifted students differed from the general group in their abilities to formulate problems spontaneously, their flexibility in data management, and their ability to abstract and generalize. There is also empirical evidence of differences in generalization in gifted and nongifted learners at the preschool level (Kanevsky 1990). At the secondary level, there are very few studies that document and describe how gifted students approach problem solving, abstract, and generalize mathematical concepts. This leads to the following questions:
1. What are the problem-solving behaviors in which high school students engage?
2. What are the differences in the problem-solving behaviors of gifted and nongifted students?
3. How do gifted students abstract and generalize mathematical concepts?
Problem-solving situation: a situation involving:
a. a conceptual task
b. the nature of which the subject is able to understand by previous learning (Brownell, 1942; Kilpatrick, 1985), by organization of the task (English, 1992), or by originality (Birkhoff, 1969; Ervynck, 1991);
c. the subject knows no direct means of satisfaction;
d. the subject experiences perplexity in the problem situation, but does not experience utter confusion; and
e. an intermediate territory in the continuum that stretches from a puzzle at one extreme to the completely familiar and understandable situation at the other (Kilpatrick, 1985). …