Students all over the world learn linear equations, which is a universal topic in junior secondary mathematics curriculum. In relation to learning linear equations, the balance method is usually preferred in Western countries. Apparently, the balance method highlights the concept of "balance" on both sides ·of a linear equation, which is critical in understanding the equal (i.e.,- '=') sign concept in equation solving. In contrast, the inverse method is popular in many Asian countries. The inverse method conceptualizes, for example, addition as an inverse operation to subtraction in equation solving. Asian mathematics teachers tend L view the balance method as complicated, error prone, and inefficient for effective learning. Western mathematics teachers, in contrast, regard the inverse method, which emphasizes the importance of procedural manipulation (change sign, change side), a being limited in addressing the equal sign concept. The main difference between the inverse method and the balance method lie in the critical procedural step (e.g., + 2 on both side versus- 2 becomes + 2). For the balance operation (e.g., + 2 on both side. ), the interaction between elements occurs on both sides of the equation. ln contrast, for the inverse operation (- 2 becomes+ 2), interaction between elements occurs on one side of the equation only. Research has indicated that the balance method imposes higher cognitive load than the inverse method, and therefore is inferior in learning linear equations. The present study, cross-cultural in nature, intends to shed light on the ongoing debate between Asian countries and Western countries in regard to the effectiveness of instructional practices (i.e., balance method versus inverse method) for learning linear equations . .Drawing from our previous research inquiries, we implemented an intervention design by which secondary school students in Australia and Malaysia (N = 147) who had no prior knowledge of linear equations were randomly assigned to the balance method or the inverse method lo learn how to solve one-step equations (e.g., x - 3 = - 7). Both the balance group and the inverse group completed a pre-test, studied an instruction sheet completed multiple example-equation pairs, and a post-test. Each example-equation pair comprised of a worked example paired with an equation. For both Australian and Malaysian students, the inverse group outperformed the balance groups for the practice equations not but the post-test. Malaysian students outperformed Australian students on practice problems irrespective of the balance group or the inverse group, whereas Malaysian student outperformed Australian students on post-test for the inverse group only. The present study, in line with the scope of this edited book, is significant for its inquiries into comparative instructional approaches for effective mathematics learning from the perspective of cognitive load imposition. Om examination of an in-class intervention has clarified the myth concerning cross-cultural differences in perceptions, appreciation and understanding of different instructional approaches ( i.e., balance method versus inverse method). The findings have indicated an advantage of the inverse method over the balance method in facilitating learning of one-step equations irrespective of cultural context.

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).

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.