Integration is a core concept of calculus. As such, significant work has been done on understanding how students come to reason about integrals, including both the definite integral and the accumulation function. A path towards understanding the accumulation function first, then the definite integral as a single point on the accumulation function has been presented in the literature. However, there seems to be an accessible path that begins first with understanding the definite integral through an Adding Up Pieces (AUP) perspective and extending that understanding to the accumulation function. This study provides a viable hypothetical learning trajectory (HLT) for beginning instruction with an AUP perspective of the definite integral and extending this understanding to accumulation functions. This HLT was implemented in a small-scale teaching experiment that provides empirical data for the type of student reasoning that can occur through the various learning activities. The HLT also appears to be a promising springboard into developing the Fundamental Theorem of Calculus. Additionally, this study offers a systematic framework for understanding the process- and object-level thinking that occurs at different layers of integration.

In much of calculus teaching there is an overemphasis on procedures and manipulation of symbols and insufficient emphasis on conceptual understanding of calculus topics. As such students to struggle to understand and use calculus ideas in applied settings. Research shows that learning calculus topics from a quantitative reasoning-perspective results in more powerful and flexible conceptions of calculus topics like integration. However, topics beyond introducing integrals and the Fundamental Theorem of Calculus, like u-substitution, have yet to be explored from a quantity-based perspective. In this study, I conducted a set of two clinical interviews where we discussed quantitative meanings of integrals, derivatives, and differentials and used those meanings to quantitatively develop u-substitution. This study suggests that given the scaffolding of the quantity-based tasks students can develop the u-substitution structure (substitution of the bounds, the function, and the differential) by applying quantitative reasoning. It also suggests that two-quantity quantitative relationships are critical to students' productive thinking about substitution. Finally, this study offers a theoretical and quantitatively grounded framework for understanding u-substitution.