A study of the relationship between introductory calculus students' understanding of function and their understanding of limit

Abstract

Introductory calculus students are often successful in doing procedural tasks in calculus even when their understanding of the underlying concepts is lacking, and these conceptual difficulties extend to the limit concept. Since the concept of limit in introductory calculus usually concerns a process applied to a single function, it seems reasonable to believe that a robust understanding of function is beneficial to and perhaps necessary for a meaningful understanding of limit. Therefore, the main goal of this dissertation is to quantitatively correlate students' understanding of function and their understanding of limit. In particular, the correlation between the two is examined in the context of an introductory calculus course for future scientists and engineers at a public, land grant research university in the west. In order to measure the strength of the correlation between understanding of function and understanding of limit, two tests-the Precalculus Concept Assessment (PCA) to measure function understanding and the Limit Understanding Assessment (LUA) to measure limit understanding-were administered to students in all sections of the aforementioned introductory calculus course in the fall of 2008. A linear regression which included appropriate covariates was utilized in which students' scores on the PCA were correlated with their scores on the LUA. Nonparametric bivariate correlations between students' PCA scores and students' scores on particular subcategories of limit understanding measured by the LUA were also calculated. Moreover, a descriptive profile of students' understanding of limit was created which included possible explanations as to why students responded to LUA items the way they did. There was a strong positive linear correlation between PCA and LUA scores, and this correlation was highly significant (p<0.001). Furthermore, the nonparametric correlations between PCA scores and LUA subcategory scores were all statistically significant (p<0.001). The descriptive profile of what the typical student understands about limit in each LUA subcategory supplied valuable context in which to interpret the quantitative results. Based on these results, it is concluded that understanding of function is a significant predictor of future understanding of limit. Recommendations for practicing mathematics educators and indications for future research are provided.