The Effect of Prior Knowledge on Learning to Solve Linear Equations with a Negative Pronumeral: The Balance Method versus Inverse Method

Abstract

<p>Within the framework of cognitive load theory, this study hypothesized that the inverse method will be better than the balance method for learning to solve linear equations with a negative pronumeral (e.g., 6 – 2x = 13) rather than a positive pronumeral (4x + 2 = 15). The critical design feature that distinguishes the balance method and inverse method lies in the application of a mathematical operation (e.g., + 3 on both sides vs. – 3 become +3). Higher level of element interactivity was associated with the balance operation in which the interaction between elements occurs on both sides of the equation rather than on one side of the equation as in the case of the inverse operation. Another advantage of using the inverse operation over the balance operation is the flexibility of applying two inverse operations concurrently to solve linear equations with a negative pronumeral. We invited 55 Year 9 Australian students aged 15 to participate in the study. They were randomly assigned to either the balance group or inverse group. They completed a pre-test, an acquisition phase, a post-test and a concept test. We used 2 (group: balance vs. inverse) × 2 (type of equation: positive pronumeral vs. negative pronumeral) to examine the effect of the balance method and inverse method upon learning to solve two types of linear equations (positive pronumeral vs. negative pronumeral).</p> <p>The balance group and inverse group did not differ on the pre-test that comprised the positive and negative pronumeral equations. However, both groups scored significantly better on the positive pronumeral rather than the negative pronumeral equations for the pre-test, confirming our prediction that the negative pronumeral equations posed a challenge to students. Furthermore, for the practice equations, both groups also performed better on the positive pronumeral equations than the negative pronumeral equations. Contrary to our expectation, for the post-test, the balance group outperformed the inverse group for solving the negative pronumeral equations. For the concept test, the inverse group performed better on the inverse operation than the balance operation. However, the inverse group was inferior to the balance group for the balance operation. In regard to the subjective rating of difficulty that reflects cognitive load imposition, contrary to expectation, the inverse group did not score lower than the balance group for the negative pronumeral equations. As hypothesized, both groups indicated higher mean scores on the subjective rating of difficulty for the negative rather than positive pronumeral equations. Overall, some of the results contradicted previous findings. We attributed such findings to students’ prior experience of using the balance method to solve linear equations prior to participating in this study.</p>