Visualization techniques for proofs: Implications for enhancing conceptualization and understanding in mathematical analysis

Abstract

Visual images are frequently utilized to elucidate concepts in general mathematics and geometry; however, their application in mathematical analysis remains uncommon. This paper demonstrates how visual imagery can enhance the proof of certain theorems in mathematical analysis. It emphasizes the importance of visualization in the learning and understanding of mathematical concepts, particularly within mathematical analysis, where diagrams are seldom employed. The paper focuses on the reasoning processes used by mathematicians in proving selected fundamental theorems of mathematical analysis. It provides illustrative examples where visual images are instrumental in performing specific subtasks within proof development and in completing the proofs. The proofs discussed include the sum of the first n natural numbers, the sum rule of integration, the mean value theorem for derivatives, the mean value theorem for integrals, and Young’s Inequality. This paper underscores that visual images serve not only as persuasive tools but also as bridges between symbolic representations and real-world understanding.

Keywords

• definite integrals
• imagery
• mathematical analysis
• mean value theorem
• visualisation