Teaching Proofs to Elementary and Middle School Education Majors

摘要

This paper presents strategies for presenting proofs in undergraduate courses targeted for future elementary and middle school teachers. Several pedagogic theories are cited including those of Van Hiele, Piaget, and Gagne. These theories suggest that teaching proofs at an early undergraduate level poses unique challenges since proof ability is a mathematically later development. This paper's approach utilizes several ideas including emphasis on experientially familiar domains of knowledge and use of specific theories of tutorial and instructional design. This paper's approach is broad: Students may be taught several sophisticated proof methods, including proof by cases, proof by contradiction, proof logic, lemma-theorem approaches, algebraic-geometric duality, motivating proof ideas with computer games and simulations, computer exploration for counterexamples, and computer exploration as a motivation for definitions. This paper illustrates proof methods from a variety of mathematical disciplines including geometry, number theory, and limits.