Abstract

Recent literature in Education and Philosophy has emphasized the importance of explanatory proofs for both mathematical practice and pedagogy. However, it remains unclear when, exactly, a proof qualifies as an explanation for a reader. This thesis investigates one potential factor in a reader's satisfaction with a proof as an explanation: the agreement between the reader's conceptual metaphors and the proofs metaphorical language. A conceptual metaphor is a cognitive mechanism, first posited in Cognitive Linguistics, used to understand abstract concepts. When reading a proof involving continuity, do a reader's conceptual metaphors for continuity influence their satisfaction with the proof as an explanation? To answer this question, we conducted a case study of four students in an undergraduate course in Analysis. Using two semi-structured clinical interviews - conducted before and after the classroom lecture on continuity - we determined the subjects' conceptual metaphors for continuity and their satisfaction with three proofs as explanations. Before instruction, every subject appeared to understand continuity using the CONTINUITY IS GAPLESSNFSS metaphor. After instruction, a new metaphor was apparent: CONTINUITY IS PRESERVATION OF CLOSENESS. The post-instruction interview presented three proofs of the same theorem, which - while mathematically equivalent - differed in their metaphorical language. The subjects were more satisfied with a proof as an explanation when it employed metaphorical language that reflected their own conceptual metaphors. Thus, the results support the conjecture that a reader's conceptual metaphors playa role in their satisfaction with a proof as an explanation. We discuss implications for teaching