PROCEEDINGS OF THE 21ST ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2018
San Diego, California
Scaling-Continuous Variation:A Productive Foundation for Calculus Reasoning
Page: 1180
This paper introduces a new mode of variational and covariational reasoning, called scaling-continuous reasoning. Scaling-continuous reasoning builds on Leibniz’ ideas of increments and infinitesimals and does not rely on images of motion. Instead, it entails (a) imagining a variable taking on all values on the continuum at any scale, (b) understanding that there is no scale at which the continuum becomes discrete, and (c) re-scaling to any arbitrarily small increment for x and coordinating that scaling with associated values for y. We present one clarifying example of this type of reasoning and argue that scaling-continuous reasoning can support a robust understanding of foundational ideas for calculus, including rates of change, differentiation, and the definite integral.