A comparison of the element interactivity effect associated with balance and inverse methods in equation solving

Abstract

Understanding mathematical concepts entails simultaneous assimilation of interaction elements because each element cannot be separately manipulated from others. According to cognitive load theory (Sweller & van Merrienboer, 1998; van Merrienboer & Sweller, 2005), the learners need to invest cognitive load to understand the interacting elements. The more the learners need to process the number of the interacting elements, the higher the cognitive load. Due to the limitation of working memory, manipulating a large number of interacting elements imposes high cognitive load which interferes with understanding. This study compared the relative merit of balance and inverse methods in equation solving from the element interactivity perspective. To solve an equation such as, x - 5 = 10, the learner needs to understand that the '=' sign describes a relation where the left side equals to the right side. For the balance method, the learner is required to cancel -5 with +5 in the left side, and the same +5 must be done in the right side to balance the equation. In contrast, the inverse method requires the learner to move -5 from the left side to interact with 10 in the right side to balance the equation. The interaction between elements occurs on both sides of the equation in the balance method but on one side only in the inverse method. For this one-step equation, both the inverse and balance methods may be equally effective because the degree of element interactivity may lie within the working memory capacity of the learners. However, as the complexity of the equations increases, we predicted that the inverse method would outperform the balance method. We tested this hypothesis with 41 Middle school students (mean age=14) in a regular mathematics classroom. Students were randomly assigned to inverse or balance group to complete a pre-test, an acquisition phase, and a post-test. During the acquisition phase, students were required to study an instruction sheet that explained how to solve equations through the use of two worked examples. For each of the 12 pairs of acquisition problems, students studied a worked example and solved an equivalent equation. As hypothesized, both methods were equally effective for low element interactivity equations. However, the inverse group outperformed the balance group for high element interactivity equations. The inverse method allowed mental processes to involve fewer interacting elements leading to better learning outcomes.