Computer-Generated Representations of Mathematical Reasoning

Abstract

The representations described in this paper were developed to encourage secondary school students to set goals, to plan to reach goals, and to use observation and discussion as fundamental techniques for proving results by deduction. Since an important aim of instruction in mathematics is to insure that students experience the deductive development of facts, particularly geometry teachers face a dilemma in trying to meet the needs of students whose preparedness for deductive reasoning varies considerably. The representations presented here were designed to enable teachers to overcome this dilemma by creating a classroom environment in which students can both prove deductions individually and combine their individual deductions with those of other students to construct and prove new results. Recent research has emphasized the benefits of cooperative activities in learning deductive reasoning and with these computer-generated representations, students collectively use diagrams and charts to represent the components of reasoning and then build deductive statements by transforming these representations. The classroom use of these representations is illustrated in a case study of a teacher's work to enable students to prove theorems about the segment connecting midpoints of sides in a triangle. The graphic representations from the teacher's enabling efforts are then transformed into a comparable set of language-based representations, and both these computer-generated representations are then contrasted with those corresponding to conventional textbook proofs. The paper concludes with observations of common features in classrooms where these representations have been used to improve students' deductive reasoning.