Is the inverse method more effective than the balance method on learning linear equations? A cross-cultural experimental study between Australia and Malaysia

In relation to Figure 1, we note that for the balance method of learning, Line 1 has five elements: a pronumeral (x), two numbers (+6, 3), and two concepts. As described earlier, we consider a concept as an element because it needs learning (Chen et al., 2017). The two concepts are as follows: (i) the “=” sign that shows a quantitative relationship where the left side of the equation is equaled to the right side of the equation and (ii) the number 6 is added to x. Moreover, Line 2 has seven elements: a pronumeral (x), four numbers (+6, –6, 3, –6), and two concepts—namely, cancel +6 with –6 on the left side (i.e., Concept 1), and perform 3 and –6 on the right side to maintain the equality of the equation (i.e., Concept 2). Line 3 has three elements: a pronumeral (x), one number (–3), and one concept—namely, once Line 1 and Line 2 have been processed successfully, x equals to –3 being the solution would be obvious.

4.2 The inverse method of learning

As we mentioned earlier, the balance method and the inverse method differ in the operational line (Line 2). A learner, in this case, would conceptualize +6 as an inverse to –6, as such, he would move +6 from the left side of Line 1 to become –6 on the right side of Line 2 to preserve the equality of the equation. For both the balance method and the inverse method, the learner is required to process multiple interactive elements within and across one operational and two relational lines in order to comprehend the solution procedure of the one-step equation. Differential levels of element interactivity in the operational line would favor the inverse method. The interaction between elements occurs on the right side of the equation for the inverse method (i.e., –6 interacts with 3), but on both sides of the equation for the balance method (i.e., –6 interacts with +6 on the left side and 3 on the right side). Having said this, however, we contend that the inverse method is not better than the balance method for one-step equations, given that the total cognitive load required to process element interactivity would have been low, irrespective of whether it is the balance method or the inverse method (Ngu et al., 2015).

Given that the present study includes both one-step and two-step equations and that the latter has a higher number of relational and operational lines than the former, we expect the inverse method to be better than the balance method. For consideration, then, a question noteworthy for development is as follows: is there credence, as some existing research studies have found (e.g., Ngu et al., 2018a), to advocate for the use of the inverse method over that of the balance method?

4.3 Prior research on the balance method and the inverse method

A majority of secondary school mathematics textbooks use a balance scale to scaffold the “=” sign concept in the context of solving linear equations (Vincent et al., 2011). Aligning the concrete items (e.g., unit block) in a balance scale and the elements (i.e., number, pronumeral) in linear equations (e.g., x + 3 = 10) can, in fact, encourage analogical comparison between the two, leading to the acquisition of the “balance concept” in linear equations (Vlassis, 2002; Warren & Cooper, 2005). Having stated this, however, it is impossible to “take away” a negative pronumeral or a negative number from both sides of a balance scale (Vlassis 2002). As highlighted in a review by Otten et al. (2019), researchers have yet to discover the optimal condition by which the balance model would work effectively to solve linear equations. Despite its limitation, mathematics educators have used the balance scale to scaffold and to develop the balance method for learning to solve linear equations (Vincent et al. 2011). For example, using the balance method, students are taught to cancel identical items (e.g., pronumeral, number) from both sides of the linear equations to preserve equality and then to solve for x (Linchevski & Herscovics, 1996). Interestingly, more capable students were able to generate a shortcut version of the balance method (i.e., the inverse method) to help them solve subsequent linear equations.

In a previous study, we used the balance method to assist eighth-grade students to solve various types of one-step equations, some of which have special features, such as the presence of a negative pronumeral (e.g., 5 – x = 12) (Ngu & Phan, 2017). We found the mean posttest scores to be relatively modest across different types of one-step equations—for example, the mean posttest score was 0.41 for linear equations that involve a negative number (e.g., w/5 = –5). Such results are somewhat similar to those results obtained by Rittle-Johnson and Star (2009) (e.g., posttest, 44.5%), who also used the balance method to facilitate middle school students’ learning of linear equations.

Over the past several years, we have conducted a few empirical studies to compare the balance method and the inverse method for learning to solve linear equations. We noted that the inverse method is better than the balance method for the learning of two-step equations (e.g., 6x + 10 = 22) (Ngu et al., 2015), multistep equations (e.g., 5x – 2 = 3x + 9) (Ngu et al., 2018a), and linear equations with negative pronumerals (e.g., 7 – x = 15) (Ngu & Phan, 2022). To our knowledge, there is no other research study that has used the inverse method to promote effective learning of linear equations. An earlier study by de Lima and Tall (2008), interestingly, found that some students could not appreciate and understand the role of the inverse operation when using the inverse method. For example, when solving a linear equation such as 4x – 7 = 26, some students mechanically pick a term from the equation (e.g., –7) and perform the procedure of “change side, change sign” to solve for x. In other words, the students are not able to comprehend how the inverse operation actually works in the context of linear equations.

4.4 Preference: Balance method vs. inverse method

Prior studies have demonstrated an advantage of the inverse method over the balance method within the framework of cognitive load (Sweller et al., 2011; Sweller et al., 2019) mainly in Malaysian sociocultural-learning contexts (e.g., Ngu et al., 2018a). It is formally acknowledged that mathematics educators in Western countries (e.g., Australia), in general, promote the use of the balance method for learning linear equations. There is a firm belief, in this sense, that the application of the balance operation (e.g., +3 on both sides) addresses the concept of equality, which is central to the success of solving linear equations. By the same token, many mathematics educators in Western countries are skeptical of the inverse method, as this instructional method emphasizes the notion of “change side, change sign,” which may fall short of addressing the critical concept of equality.

A review by Cai et al. (2005) indicates that a number of Asian countries (e.g., China, Singapore, South Korea) have introduced the inverse method for learning linear equations in primary school education. Teachers in China, for example, provide ample opportunities for their students to learn about inverse operations, using word problems in real-life contexts (Ding, 2016). As an example, “There are four rows of students in a classroom. If the total number of students is 20, then how many students are there in each row?” Students, in this case, are expected to form () × 4 = 20 and then conceptualize “× 4” as “÷ 4” to solve for (). Interestingly, according to Ding (2016), mathematics teachers in China introduce linear equations using the balance method in order to meet formal curriculum requirements. Subsequently, students gain competence in solving linear equations, via means of the inverse method.

In their theoretical review, Rittle-Johnson et al. (2015) indicated that Asian mathematics educators, in general, have the view that “practice makes perfect.” Specifically, providing ample and continuing practice would enable the attainment of procedural fluency, resulting in the development and finetuning of conceptual understanding of mathematical concepts. Therefore, in general, many Asian mathematics educators consider the balance method of learning as cumbersome, error-prone, and ineffective. We acknowledge that there is a need, however, to substantiate such claim (e.g., that the balance method is error-prone) with evidence-based research data. We contend that Asian mathematics educators may prefer the inverse method of learning, which relies on theoretical understanding of the inverse operation to acquire the conceptual knowledge associated with solving linear equations.

8.1 Participants

The materials used were adapted from a previous study (Ngu et al., 2021), which consisted of a pretest that has identical number and type of linear equations as the posttest, an instruction sheet, acquisition problems, and a concept test. The instruction sheet (Appendix) provided the definition of a linear equation, explained the solution procedure of linear equations, scaffolded the balance operation and the inverse operation, and presented four worked examples. For the balance method, a balance scale was used to scaffold the balance operation. Moreover, each worked example had a prompt “–2 on both sides” (Appendix) to assist students to apply the balance operation appropriately in the context of solving linear equations.

For the inverse method, we scaffolded the inverse operation such as 5 + 2 = 7 is the same as 5 = 7 – 2. Building on students’ prior knowledge of primary numeracy education, we expected them to understand the “=” sign concept where the application of an inverse operation (i.e., conceptualize +2 as the inverse operation of –2) would preserve the equality of the equation. Moreover, each worked example included a prompt “+2 becomes –2” as well as an arrow to show how to conceptualize an inverse operation (Appendix). We expected the prompts to help students apply the inverse operation appropriately when solving linear equations. It should be noted that academic success in senior mathematics education (e.g., differentiation and integration in calculus) requires a sound understanding of the inverse operation (Ding, 2016).

There were 16 one-step and two-step linear equations in the pretest (or posttest) (Appendix). Of the 16 linear equations, nine were one-step equations (e.g., 31 – m = 16), and seven were two-step equations (–7 + 3p = 11). We classified “31 – m = 16” as a one-step equation based on a popular mathematics textbook that has been used in Australia (McSeveny et al., 2004, p. 245). One may query that the linear equations with a negative pronumeral (e.g., 31 – m = 16) require the application of two balance operations, and, thus, this involves more than one solution step (Ngu & Phan, 2016): –31 + 31– m = –31 + 16 (–31 on both sides, first step), – m = 15 (divide –1 on both sides, second step). Instead of using the balance method, the inverse method has the flexibility by which one could apply two inverse operations concurrently to both sides of the linear equation: 31 – 16 = m (–m becomes m, 16 becomes −16, first step), 31 – 16 = m (simplify the answer, second step) (Ngu & Phan, 2022).

For the pretest and posttest, each correct solution was assigned one point with or without the solution steps provided. We chose to ignore computational errors. However, no point was assigned if a student made a conceptual error—for example, when solving 6 – k = 0, a student wrote +6 on both sides, which is incorrect. The acquisition problems (Appendix) comprised 12 example-problem pairs, and each pair consisted of a worked example paired with a practice problem that shares a similar problem structure (Reed, 1987).

There were eight example-problem pairs in relation to one-step equations, and the remaining were two-step equations. We graded the six practice problems only. Again, in this case, one point was assigned for a correct solution. For the concept test, for example, we assigned one point if a student could accurately judge 2x + 6 = 5 and 2x + 6 – 6 = 5–6 as being equivalent (Appendix). The first question required students to indicate the meaning of the “=” in the linear equation, x + 6 = 11. The answer was graded correctly if students wrote “the same as, equal” (Asquith et al., 2007). We did not analyze the result for the first question because only some students answered this question. The second question assessed students’ understanding of the “=” sign after they had applied the balance operation or inverse operation. Of the eight pairs of equations, five pairs involved the balance operation, and three pairs involved the inverse operation. For example, we assigned one point if students could accurately judge 2x + 6 = 5 and 2x + 6 – 6 = 5 – 6 as equivalent (Appendix). To encourage consistency in terms of grading, only one of the authors who is a mathematics educator and a mathematics teacher from a participating school graded the assessment tasks. The interscorer reliability for 25% of the sample was about .96. We resolved inconsistency through discussion.

8.3 Procedure

We obtained ethics approval from relevant authorities prior to the collection of data. Two of the authors in Australia collected the data in Australia with the assistance of two mathematics teachers. In a similar vein, two local mathematics teachers together with the two co-authors from Malaysia collected data in Malaysia. We used the same selection process (i.e., recruitment of participants) and the same experimental procedure across the two countries. We identified the targeted participants, sought consent for voluntary participation, and then randomly assigned participating students in each class to either the balance group or the inverse group.

For the Australian sample, the balance group had 29 students, and the inverse group had 28 students. For Malaysian students, the balance group had 37 students, and the inverse group had 38 students. We distributed the written task for each phase and collected it when the allocated time to complete the task had elapsed with one exception—the instruction sheet was collected after the acquisition phase. We informed the students to review their work if they finished earlier than the allocated time for the task.

We gave a briefing to all participating students about the experiment. We informed the students that the purpose of the experiment was to help us understand how to assist students, in general, learn linear equations. They were required to complete a number of written tasks individually and not to discuss the tasks with their classmates. Each task had a specific allocated time for them to complete: (i) a pretest (10 min), (ii) an acquisition phase that consisted of an instruction sheet (5 min), acquisition problems (15 min), (iii) a posttest (10 min), and (iv) a concept test (10 min). We encouraged the students to try their best to complete the written tasks and, likewise, informed them that their responses (i.e., correct or incorrect) would not count toward their school results. They were advised to read the instructions carefully, which appeared on the first page of each task before working on the task. They could seek help if they had trouble understanding the materials in the instruction sheet and acquisition problems; but they could not seek assistance to solve the practice problems.

Prior cognitive load research (Carroll, 1994; Chen et al., 2015; Cooper & Sweller, 1987; Kalyuga et al., 2001; Ngu et al., 2018a) has implemented variable intervention lengths to ensure that participants complete various tasks, which could take approximately one hour to complete. On this basis, likewise, we implemented an approximately 1-hour intervention, consisting of various learning tasks to examine the comparative effectiveness of the balance method vs. the inverse method for learning to solve linear equations. The experimental procedure consists of a number of steps in a sequential order: (i) all of the students sat for a pretest, (ii) all of the students studied their respective instruction sheets and then completed their respective acquisition problems (Appendix), (iii) all of the students sat for the posttest, and (iv) all of the students completed the concept test. Identical time duration and learning tasks were allocated to both the balance group and the inverse group. The only difference between the two groups, however, was in the design of the respective worked examples (i.e., balance method vs. inverse method) in the acquisition phase. We speculated that learning would predominately occur for both cohorts when the students completed the acquisition problems (i.e., study a worked example paired with a practice problem across multiple example-problem pairs), though they could also learn from studying the materials in the instruction sheet.

The respective Cronbach's alpha values for the pretest were .86 for Australian students and .84 for Malaysian students. Using the univariate ANOVA test, differences were not found between the balance method and inverse method for Australian students, F(1, 123) = 1.46, p = .23, partial η² = 0.01, and Malaysian students, F(1, 123) = 0.33, p = .57, partial η² = 0.00. Furthermore, the Australian students and Malaysian students did not differ with respect to the balance method, F(1, 123) = 0.02, p = .88, partial η² = 0.00, and the inverse method, F(1, 123) = 0.36, p = .55, partial η² = 0.00. Thus, there was no statistically significant difference between the balance group and inverse group with respect to individual countries and between two countries prior to intervention.

We hypothesized that prior exposure to the balance method (i.e., for Australian students) and the inverse method (i.e., for Malaysian students) would impact on comprehension and learning of linear equations. As such, we analyzed the strategies (e.g., the balance method) used in the pretest for both Australian and Malaysian students. For the Australian cohort, about 48% of the students used the balance method to solve the linear equations, whereas the remaining provided answers but did not show the solution steps. For the Malaysian cohort, in contrast, 50% used the inverse method, 19% used the balance method, and the remaining provided answers without showing the solution steps.

9.2 Practice problems

The respective Cronbach's alpha values for the balance method and the inverse method were .87 and .78 for Australian students and .84 and .51 for Malaysian students. The rather low Cronbach's alpha value for the inverse method could have been caused by our removal of item 5, which had zero variance. The main effect of the country was statistically significant, F(1, 123) = 9.50, p < .001, partial η² = 0.07. The main effect of the method was statistically nonsignificant, F(1, 123) = 0.30, p = .59, partial η² = 0.00. The country × method interaction effect was statistically significant, F (1, 123) = 9.28, p < .001, partial η² = 0.07. Within individual countries, we hypothesized that differential performance would favor the inverse group due to the lower level of element interactivity of the inverse operation, regardless of whether they were Australian students or Malaysian students. As revealed in Figure 2a, contrary to the hypothesis, the balance group outperformed the inverse group (p = .02) for Australian students. As hypothesized, for Malaysia students, the inverse group marginally outperformed the balance group (p = .06). Viewing the two countries together, contrary to the hypothesis, both the Australian students and the Malaysian students did not differ with respect to the balance method (p > .05); but the Malaysian students outperformed the Australian students with respect to the inverse method (p < .001), supporting our hypothesis (Figure 2a).

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Figure 2.

2 (country) × 2 (method) analysis of variance (ANOVA) on (a) practice problems and (b) posttest.

Cronbach's alpha values for the posttest were .84 and .78 for Australian students and Malaysian students, respectively. Statistically nonsignificant differences were found for the main effect of the country, F(1, 123) = 0.18, p = .67, partial η² = 0.00; main effect of the method, F(1, 123) = 2.51, p = .12, partial η² = 0.02; and the country × method interaction effect, F(1, 125) = 2.51, p = .12, partial η² = 0.02. Consider each country separately, contrary to the hypothesis, for Australian students, the balance group outperformed the inverse group (p = .04), and the balance group and the inverse group did not differ statistically for the Malaysian students (p > .05) (Figure 2b). Examining two countries together, the Australian students and the Malaysian students neither differed with respect to the balance method (p > .05) nor with respect to the inverse method (p > .05). Such results do not support our hypothesis.

9.4 Concept test

The Cronbach's alpha value for the concept test was .64 for Australian students and .69 for Malaysian students. A statistically significant main effect for the country was observed, F(1, 123) = 9.21, p < .001, partial η² = 0.07, and a statistically significant main effect for the method was found, F(1, 123) = 14.47, p < .001, partial η² = 0.11. However, the country × method interaction effect was statistically nonsignificant, F(1, 125) = 1.54, p = .22, partial η² = 0.01. Viewing individual countries separately, contrary to the hypothesis, which stated that differential performance would favor the inverse group owing to the lower level of element interactivity of the inverse method, the balance group outperformed the inverse group for both Australian students (p < .001) and Malaysian students (p = .05) (Figure 3a). Examining the two countries together, as hypothesized, the Malaysian students outperformed Australian students with respect to the inverse method (p < .001). But contrary to our hypothesis, no statistically significant difference between the Australian students and Malaysian students was observed with respect to the balance method (p > .05).

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Figure 3.

2 (country) × 2 (method) analysis of variance (ANOVA) on (a) concept test, (b) balance operation, and (c) inverse operation.

The main effect of the country was statistically significant, F(1, 123) = 5.17, p = .03, partial η² = 0.04. The main effect of the method was statistically significant, F(1, 123) = 19.81, p < .001, partial η² = 0.14. The country × method interaction effect was statistically nonsignificant, F(1, 123) = 0.83, p = .37, partial η² = 0.01. Considering each country separately, the balance group outperformed the inverse group for both Australian students (p < .001) and Malaysian students (p = .01) (Figure 3b), supporting our hypothesis. Viewing the two countries concurrently, no statistically significant difference between Australian students and Malaysian students was observed with respect to the balance method (p > .05). Such results do not support the hypothesis.

9.6 Inverse operation

A statistically significant main effect of the country was observed, F(1, 123) = 8.02, p = .01, partial η² = 0.06. Neither the main effect of the method, F(1, 123) = 0.36, p = .55, partial η² = 0.00, nor the country × method interaction effect, F(1, 123) = 1.42, p = .24, partial η² = 0.01, was statistically significant. With reference to individual countries, the inverse group and the balance group neither differed for Malaysian students (p > .05) nor for Australian students (p > .05). Such results do not support our hypothesis, which stated that the inverse group would outperform the balance group for the inverse operation. Viewing the two countries together, in line with our hypothesis, the Malaysian students outperformed the Australian students with respect to the inverse method (p = .01) (Figure 3c).

10.1 Theoretical and research consideration

From a cognitive load perspective (Sweller et al., 2011; Sweller et al., 2019), the present study has emphasized the importance of the concept of element interactivity (Chen et al., 2017), which may help to discern the complexity of linear equations. Element interactivity, in this sense, enables us to classify linear equations hierarchically by using the number of relational and operational lines. Such discourse, significantly, may offer useful insights for mathematics educators to identify challenges that are associated with the learning of one-step and two-step linear equations. Furthermore, as existing research has shown, the notion of “level of element interactivity” can also distinguish the relative effectiveness of different instructional methods for usage (e.g., the balance method vs. the inverse method). As noted earlier in our discussion, Australian students tend to struggle with the solving of linear equations, as evidenced by the generation of multiple errors (Steinle et al., 2022). Given that the level of element interactivity favors the inverse method of learning, mathematics educators may wish to explore the use of this instructional method to assist students to overcome challenges that are associated with learning to solve linear equations.

We advocate for the advancement of our research inquiry, which considers the importance of the comparative effectiveness of different instructional methods. Central to this advocation, we contend, is the inclusion of cognitive load theory (Sweller et al., 2011; Sweller et al., 2019), which may help to provide a theoretical framework for the design of different instructional methods for usage. Over the past several years, researchers have situated cognitive load theory (Sweller et al., 2011; Sweller et al., 2019) within other theoretical concepts, inquiries, and contexts—for example, the importance of cognitive load theory and “achievement of optimal best” (Phan et al., 2017), “the negative impact of cognitive entrenchment” (Phan & Ngu, 2021a), and “the relevance of perceived optimal efficiency” (Phan & Ngu, 2021b). As such, researchers may wish to use cognitive load theory as a basis to conceptualize other instructional methods for future inquiries—for example, how does one's cognizance of a need to maximize optimal efficiency (Phan & Ngu, 2021b) influence his design of appropriate instructional methods for usage, which take into account the relevance of cognitive load theory?

10.2 Practical implication

How can mathematics educators use our findings to inform their classroom practices? By the end of the day, regardless of whether one chooses to use the balance method or the inverse method for learning, it is important to consider the notions of efficiency and effectiveness. Maybe a complementary use of both the balance method and the inverse method would showcase and demonstrate effective learning (e.g., better comprehension). From our point of view, introducing the use of the relational line and the operational line to describe the solution procedure of linear equations could help students to distinguish the interplay between both conceptual knowledge and procedural knowledge in the solving of linear equations. Exposing students to the balance method and the inverse method (Figure 1), via means of learning by comparison (Rittle-Johnson et al., 2017), likewise, may assist them to gain an in-depth understanding of the role of the “=” sign concept with respect to the relational line and operational line.

To reiterate, it is not our intention to specifically confirm that the balance method is superior to the inverse method or vice versa. To a certain degree, we contend, personal preference for a particular instructional method (e.g., the balance method) may arise from several factors—for example, a student's educational history (e.g., the student was taught using a particular instructional method from an early age), a teacher's personal teacher training, which emphasizes the importance of a particular instructional method. As mathematics education researchers, we are interested to seek new educational and research frontiers for development (e.g., establishing a new theoretical tenet for consideration). As a possibility, for example, we advocate for a formal reform of different mathematics curricula (e.g., Australian Years 7-10), which could consider the use of both relational line and operational line in order to improve middle school students’ acquisition of linear equations. By the same token, we recommend exposing pre-service teachers to a myriad of comparative instructional methods of teaching linear equations (e.g., the balance method vs. the inverse method) in higher education institutions.