A case for teaching about parameters in calculus courses

TL;DR

The paper argues for explicit instruction on parameterized functions in single-variable calculus, illustrating that constants such as $a,b,c$ in $y=ax^2+bx+c$ can vary across settings. It discusses conceptual challenges and promotes Calc 1–2 as natural sites to teach parsing notation, end behavior via parameters (e.g., the logistic model limit as $t\to\infty$), and reasoning about entire families of functions. It presents two implementation outlines: a Calc 1 plan emphasizing dynamic graphing, variable-parameter identification, and problems like $Q=\frac{\alpha M}{P}$; and a Calc 2 plan focusing on convergence and generalization through $p$-integrals and related series. Preliminary results suggest improved student confidence and understanding, motivating broader adoption and assessment across courses.

Abstract

Working with letters that represent unknown constants, i.e., parameters, has been historically challenging for students. This is an important skill for their success in many future quantitative settings, and yet it appears this topic is rarely included explicitly in math curricula. We argue that we should be explicitly teaching our students how to work with parameters, and that single variable calculus courses are a natural place to do so. We offer justification for this as well as examples and sample outlines for incorporating parameters into these classes.