Tuning of Search of the Problem Space for Geometry Proofs

Abstract

In plannm~ a proof. a student searches .through a space of mf erences leadmg forward from the givens of the problem and backward from the to-be-proven statement. One dimension of growth of expertise IS that students become more tuned in the search of th1s problem space. This can be shown to result from the applicallon of various learning operators to production embodiments of the inference rules. Rules are evaluated after the solution of a problem according to whether they led to or led away from the solution. Rules that contributed to a solution are strengthened and an attempt IS made to formulate general versions of these rules that will apply in other situations. Rules that led away from the solution are weakened and a discrimination process is evoked to try to add features to the rules that will try to restrict them to the correct Circumstances of application. Composition is a learning process that collapses successful sequences of rule operations mto smgle macro-rule productions. There IS also a process that converts the backward reasoning rules formed by composition into forward reasoning rules. The effect of these learning processes is to put into producllon conditions tests for problem features that are beunsllcally predicllve of the rule's success.