RUME 28
2026
Alexandria, Virginia
Two Approaches to Rates and Foundations for Derivatives
Page: 817
We examine how 2 approaches to coordinating quantitative meanings for rates, proportional relationships, and variation can support reasoning about 2 aspects essential for instantaneous rates of change and derivatives. First, rates are single measures and, second, rates of change remain approximately constant as changes in 2 variables tend to 0. The first approach––based on coordinating unit rates, multiple batches, and chunky variation––has dominated research on rates. The second approach––based on coordinating measurement rates, variable parts, and smooth variation––has been largely overlooked. We argue that approaching instantaneous rates of change and derivatives through unit rates is inherently tied to chunky covariational reasoning and that approaching these same topics through measurement rates fits well with smooth covariational reasoning and should be investigated in future empirical research.