RUME 27
2025
Alexandria, Virginia
Integration by Substitution: A Justification Based on Multiplication, Measurement, and Inverse Proportions
Page: 918
Researchers have argued that interpreting definite integrals and their notation as summations of small amounts of a target quantity (Adding Up Pieces) is especially productive for applied problems. Jones and Fonbuena (2024) extended research on Adding Up Pieces by providing an analysis of integration by substitution grounded in quantitative reasoning. The present theoretical paper develops a complementary justification that makes three contributions. First, whereas past research on integration has rarely foregrounded explicit, quantitative meanings for multiplication, the analysis presented here is based on connections between multiplication and coordinated measurement with two units. Second, the analysis presented here uses coordinated measurement to explain how, across diverse situations, small amounts of a target quantity can be understood as products. This supports reasoning about Riemann sums and areas as approximations for definite integrals. Finally, the analysis provides a new justification for integration by substitution that relies on rectangular areas and inversely proportional relationships.