An Investigation of the Relationships between Conceptualization of Limits and Proof Comprehension
Herrera, Christine A.
This study investigates eighteen Real Analysis students’ informal and formal understanding of the mathematical concept of limits and the relationship on their comprehension of limit proofs. This study utilizes Tall and Vinner’s (1981) notion of concept image and concept definition. The framework examined students’ conceptualization of limit by eliciting their mental images, associated properties, processes, and example space (Watson & Mason, 2005). The study analyzed surveys on the conceptualizations of limits of sequences and limits of functions, class observations, and task-based interviews to explore the different varieties of conceptual understanding held by Real Analysis students. From the data emerged two cognitive categorization of participants’ thinking. Those whose concept image held serious potential conflict factors with the formal definitions of limits and those who had resolved their serious potential conflict factors by the end of the semester, called the cognitive conflict and cognitive resolution groups, respectively. The students were given an end-of-semester proof comprehension assessment that was designed based on Mejia-Ramos, Fuller, Weber, Rhoads, and Samkoff’s (2012) model for assessments for advanced mathematics’ proof comprehension. The analysis showed that Real Analysis students with cognitive resolution had a better local understanding of limit proofs. However, both cognitive groups had difficulty generating examples that illustrated the main ideas of the limit proofs.
Limits, Limits of sequences, Limits of functions, Real analysis, Undergraduate mathematics, Proof, Proof comprehension, Concept image, Concept definition, Example space
Herrera, C. A. (2016). <i>An investigation of the relationships between conceptualization of limits and proof comprehension</i> (Unpublished dissertation). Texas State University, San Marcos, Texas.