## Computer-Generated Representations of Mathematical Reasoning

*Jack Carter*

`jcarter@csuhayward.edu`

Mathematics & Computer Science

California State University

**USA**
*Shigekazu Yanagimoto*

`Yanagim@edu00.f-edu.fukui-u.ac.jp`

Mathematics

Fukui University

**JAPAN**

*Beverly Ferrucci*

`bferrucc@keene.edu`

Mathematics

Keene State College

**USA**

### 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.

© ATCM, Inc. 2002.