On the Existence of Balanced Chain Rule Task Sets

TL;DR

This paper uses graph decomposition techniques to construct balanced calculus task sets that effectively assess understanding of the Chain Rule in mathematics education.

Contribution

It extends existing graph decomposition results to create balanced, mixed-topic calculus task sets focused on derivative computation and the Chain Rule.

Findings

Constructed balanced task sets for calculus derivative problems

Extended graph decomposition methods to educational task set design

Provided a systematic approach for assessing the Chain Rule

Abstract

In mathematics education research, mathematics task sets involving mixed practice include tasks from many different topics within the same assignment. In this paper, we use graph decompositions to construct mixed practice task sets for Calculus I, focusing on derivative computation tasks, or tasks of the form "Compute $f'(x)$ of the function $f(x)=$ [elementary function]." A decomposition $D$ of a graph $G=(V,E)$ is a collection $\{H_1, H_2, ... , H_t\}$ of nonempty subgraphs such that $H_i=G[E_i]$ for some nonempty subset $E_i$ of $E(G)$, and $\{E_1, E_2, ... , E_t\}$ is a partition of $E(G)$. We extend results on decompositions of the complete directed graph due to Meszka and Skupie\'n to construct balanced task sets that assess the Chain Rule.