Cognitive complexity of geometric proof and geometric calculation: a problem-solving perspective

Abstract

In this study, we explored the relationship between proof and calculation in geometry from a problem-solving perspective. We designed a set of problems comprising two types—geometric proof (GP) and geometric calculation with number (GCN)—to discern factors that appear to affect the cognitive complexity of these tasks. Using GP/GCN problem pairs with identical diagram configurations that could be solved using the same geometric properties, we controlled within-pair differences other than problem type. Additionally, we systematically varied two between-pair factors: diagram complexity (simple vs. complex) and minimum number of reasoning steps (2 steps vs. 3 steps) to reach a solution. Based on data from 915 Taiwanese eighth- and ninth-grade students, performance was better on problems with simple diagrams than on those with complex diagrams. Similarly, performance was superior on two-step problems compared to three-step problems. However, we found no significant difference in performance by problem type at either grade level. Based on the empirical results, we further discuss implications for research on geometric problem solving and proof.