Proof Frames of Preservice Elementary Teachers

Abstract

This study asked 101 preservice elementary teachers enrolled in a sophomore-level mathematics course to judge mathematical correctness of inductive and deductive verifications of either a familiar or an unfamiliar statement. For each statement, more than half students accepted an inductive argument as a valid mathematical proof. More than 60% accepted a correct deductive argument as a valid mathematical proof; 38% and 52% accepted an incorrect deductive argument as being mathematically correct for familiar and unfamiliar statements, respectively. Over a third of students simultaneously accepted an inductive and a correct deductive argument as being mathematically valid. The concept of proof is of great importance in study of mathematics. Smith and Henderson (1959) stated, for example, that the idea of proof is one of pivotal ideas in mathematics. It enables us to test implication of ideas, thus establishing relationship of ideas and leading to discovery of new knowledge (p. 178). Research has explored this important topic with elementary and high school students (Bell, 1976, 1979; Galbraith, 1981; Lester, 1975; Williams, 1980). Two studies are most pertinent to our research. Fischbein and Kedem (1982) investigated whether high school students understand that mathematical proof requires no further empirical verification. They verified empirically their assumption that students, after finding or learning a correct proof for a certain mathematical statement, will continue to consider that surprises are still possible, that further checks are desirable in order to render respective statement more trustworthy (p. 128). Vinner (1983) focused on question, What makes a given sequence of correct mathematical arguments a mathematical proof in eyes of high school students? He asked students to give their preference for proving a particular case of a previously proved statement. He found they preferred using a particularization of deductive proof rather than general result. The general proof was viewed as a method to examine and to verify a particular case. Vinner further observed that students judged a mathematical proof on its appearance, relying on ritualistic aspects of proof. In this study, we investigated views of proof of a different population, preservice elementary school teachers. Further, we investigated a different aspect of proof, one related to inductive and deductive reasoning. The views of proof held by preservice elementary school teachers are important. Because proof receives very limited attention in elementary school curriculum, main source of children's experience with verification and proof is classroom teacher. Classroom teachers' understanding of what constitutes mathematical proof is important, even though they do not directly teach that topic. If teachers lead their students to believe that a few well-chosen examples constitute proof,