Cascading cognitive failures in geometric proof construction: A qualitative study of prospective mathematics teachers

Abstract

Constructing a valid geometric proof requires prospective mathematics teachers to coordinate premises, deductive warrants, proof goals, and representations. Although empirical studies have documented multiple proof-related difficulties, less is known about how a breakdown at one point constrains subsequent reasoning. This qualitative descriptive-exploratory study examined four direct proof tasks and task-based semi-structured interviews with six prospective mathematics teachers from an initial sample of 20. Data were analyzed using an adapted Explorative Mathematical Argumentation (EMA) framework composed of initiation, tools, goals, and modalities. A response was classified as a failure when both written and interview data indicated that the participant possessed relevant knowledge but did not use or apply it appropriately. Failures were present in all components of the EMA framework, and were most prevalent in the application of theorems and in the use of deductive logic. Within-participant analysis indicated possible pathways, as the under-integration of a premise limited access to an appropriate warrant; the use of a theorem without support resulted in invalid intermediate assertions, and disrupted the continuity of the argument; and the use of a theorem along with a drawing resulted in a lack of checking the theorem against the conditions. These pathways were not rigid but indicated the interplay of different conceptual, strategic, representational, and communicative demands. The results of the study provide support for a stage-differentiated and functionally interrelated description of proof, and for the construction of pathways of failure as a process-oriented description of gaps across different classifications of failure.