Important work has created approaches to calculus based on crucial quantitative reasoning. For integration, however, the major topic of u-substitution has generally not been fully detailed in these paradigms. This paper presents a study where students were taught u-substitution from a quantitative perspective based on a three-part quantitative structure: differential quantity, integrand quantity, bounds quantity. The students reasoned about the quantitative conversions in flexible ways, and used various quantitative relationship types in their reasoning. However, reasoning about the differential quantity was difficult, and a new type of “collapse” metaphor was identified. By the end, the students had all developed a good quantitative basis for u-sub.
Presenters: Leilani Fonbuena and Steven Jones, Brigham Young University