Investigation into the development of a quantitatively based summation conception of the definite integral

Abstract

Research has shown the majority of students who have completed a university calculus course reason about the definite integral primarily in terms of prototypical imagery or in purely algorithmic and non-quantitative ways. This dissertation draws on the framework of Emergent Quantitative Models to identify how calculus students might develop a quantitative understanding of definite integrals which supports them in modeling activity when the differential from is not a multiplicative product between a rate of change and a differential quantity (e.g. gravitational force). The Emergent Quantitative Models framework describes three quantitative schemes basic, local, and global models which students draw on when reasoning about definite integral tasks. Basic models are quantitative relationships that apply to a contextual situation if the quantities involved are constant values, a local model is a localized version of the basic model applied to a subregion of the overall context, and a global model is derived from an accumulation process applied to the local model, whose underlying quantitative reasoning is encoded in the differential form.

To characterize the mental activity which supports the productive development of a quantitative understanding of definite integrals I outlined a conceptual analysis, designed a hypothetical learning trajectory, and conducted an eight-week teaching experiment with five freshman calculus students to test and refine my hypotheses. The results of this dissertation research provide insight into foundational constructs which position students' development of primitive basic, local, and global models and outlines the mental activity which engenders the progressive coevolution of those models towards a quantitative understanding of the definite integral. I offer two constructs to the Emergent Quantitative Models framework which were identified as productive in such a development: a gross basic model and a generalized local model. Additionally, I provide a refined hypothetical learning trajectory and pedagogical implications towards the development of curriculum that supports a quantitative understanding of definite integrals.