**6. Balance method, inverse method, and element interactivity**

Based on the Framework of Achievement Bests [4], it is plausible to postulate that appropriate instructional designs and pedagogical practices could serve to optimize students' learning experiences in mathematics. This postulation reflects, in part, our previous research undertakings that involved secondary school students in Australia and Malaysia. We contend that pedagogical practices (i.e., instructional designs) used by teachers are comparative, resulting in perceived differences in terms of effectiveness. Cognitive load imposition [5, 43], as explained, may assist and/or determine the effectiveness of a particular pedagogical approach. In mathematics learning, the two popular methods that facilitate the acquisition of equation-solving skills are the *balance* and *inverse* methods (**Figure 2(a)**). The balance method is popular among Western countries [46], whereas some Asian countries (e.g., Singapore, Korea, and Japan) have introduced and preferred the inverse method in primary mathematics curriculum [47].

In this section of the book chapter, we discuss the characteristics of the balance and inverse methods for effective learning in mathematics. *Differentiation* between the two methods involves clarity and explanation of the solution procedure of one-step equations, which may involve understanding of the difference between *relational* and *operational lines* [2]. A relational line indicates the relationship between the elements on the left side of the equation, which is equaled to the right side of the equation (e.g., Lines 1 and 3 in **Figure 2(a)**). By contrast, an operational line refers to the application of a mathematical operation that changes the state of the equation, and yet at the same time preserves its equality (e.g., Line 2 in **Figure 2(a)**).

#### **6.1. Balance method**

In accordance with **Figure 2(a)**, Line 1 is a relational line and it involves six elements, consisting of *y*, 5, 13, and three concepts. These three concepts are as follows: (i) *y* represents an unknown number, (ii) the "=" sign describes a quantitative relation between elements, with the left side of the equation equals to the right side, and (iii) to find *y*, the learner needs to perform the same operation on both sides in order to balance the equation. A learner is required to coordinate the interaction between the six elements simultaneously. By contrast, Line 2 is an operational line that involves three elements and consists of a number (i.e., −5) and two concepts. The two concepts require the learner to cancel +5 with −5 on the left side of the equation, and to perform 13 − 5 on the right side of the equation in order to maintain the equality of the equation. Interaction between elements occurs on both sides of the equation when the learner performs +5 with −5 on the left side of the equation as well as 13 − 5 on the right side of the equation. Lastly, Line 3 is regarded as a relational line; it consists of three elements such as *y*, 8, and one concept. The concept requires the learner to be able to process Lines 1 and 2 successfully so that he or she would then know that *y* equals to 8 is the solution.

#### **6.2. Inverse method**

the multiple interactive elements. In essence, understanding applies only to high-element interactivity material, but not to low-element interactivity material. Mathematics learning normally involves students learning multiple mathematical concepts simultaneously, which consequently imposes high-element interactivity and high cognitive load [45]. Thus, for optimal learning experience, it is important that we design appropriate instructions that could minimize the burden of the working memory, which in turn would help students learn math-

Based on the Framework of Achievement Bests [4], it is plausible to postulate that appropriate instructional designs and pedagogical practices could serve to optimize students' learning experiences in mathematics. This postulation reflects, in part, our previous research undertakings that involved secondary school students in Australia and Malaysia. We contend that pedagogical practices (i.e., instructional designs) used by teachers are comparative, resulting in perceived differences in terms of effectiveness. Cognitive load imposition [5, 43], as explained, may assist and/or determine the effectiveness of a particular pedagogical approach. In mathematics learning, the two popular methods that facilitate the acquisition of equation-solving skills are the *balance* and *inverse* methods (**Figure 2(a)**). The balance method is popular among Western countries [46], whereas some Asian countries (e.g., Singapore, Korea, and Japan) have introduced and preferred the inverse method in primary mathematics curriculum [47].

In this section of the book chapter, we discuss the characteristics of the balance and inverse methods for effective learning in mathematics. *Differentiation* between the two methods involves clarity and explanation of the solution procedure of one-step equations, which may involve understanding of the difference between *relational* and *operational lines* [2]. A relational line indicates the relationship between the elements on the left side of the equation, which is equaled to the right side of the equation (e.g., Lines 1 and 3 in **Figure 2(a)**). By contrast, an operational line refers to the application of a mathematical operation that changes the state of the equation, and yet at the same time preserves its equality (e.g., Line 2 in **Figure 2(a)**).

In accordance with **Figure 2(a)**, Line 1 is a relational line and it involves six elements, consisting of *y*, 5, 13, and three concepts. These three concepts are as follows: (i) *y* represents an unknown number, (ii) the "=" sign describes a quantitative relation between elements, with the left side of the equation equals to the right side, and (iii) to find *y*, the learner needs to perform the same operation on both sides in order to balance the equation. A learner is required to coordinate the interaction between the six elements simultaneously. By contrast, Line 2 is an operational line that involves three elements and consists of a number (i.e., −5) and two concepts. The two concepts require the learner to cancel +5 with −5 on the left side of the equation, and to perform 13 − 5 on the right side of the equation in order to maintain the equality of the equation. Interaction between elements occurs on both sides of the equation when the

**6. Balance method, inverse method, and element interactivity**

294 New Pedagogical Challenges in the 21st Century - Contributions of Research in Education

ematical concepts.

**6.1. Balance method**

The inverse method differs from the balance method for Line 2, but not for Line 1 and/or Line 3. Line 2 is an operational line and it involves four elements, consisting of *y*, 13, −5, and one concept. This concept requires the use of an inverse operation: move +5 from the left side of Line 1 to become −5 on the right side of Line 2 in order to balance the equation. Element interactivity occurs on one right side of the equation, where −5 interacts with 13. Overall, then, the inverse method incurs only half of the interactive elements as the balance method for the operational line (i.e., Line 2). Consequently, the inverse method imposes lower element interactivity and therefore lower cognitive load than the balance method.

#### **6.3. Differential element interactivity between the balance and inverse methods**

For both the balance and inverse methods of mathematics learning, understanding can only occur when learners simultaneously assimilate multiple interactive elements that arise within each line, and across the three lines of the solution procedure. For each relational line, the level of element interactivity arises from the interaction of elements within and between the left side and right side of the equation. Because the level of element interactivity is caused by the intrinsic nature of the equation, there is no differential element interactivity between the balance and inverse methods. By contrast, differential element interactivity between the balance and inverse methods favors the inverse method for the operational line. Interaction between elements occurs on both sides of the equation for the balance method, but only on one side of the equation for the inverse method. In other words, the balance method incurs twice as many interactive elements as the inverse method for each operational line. Nevertheless, for a simple one-step equation (e.g., *y* + 3 = 7) that consists of one operational line and two relational lines (**Figure 2(a)**), the total cognitive load required to process the level of element interactivity would expect to be low for both the balance and inverse methods. Indeed, research has shown that the inverse method is not better than the balance method for one-step equations that consist of one operational and two relational lines in the solution procedure [1, 48, 49].

The inverse method, as shown, is comparable with the balance method for simple one-step equations that involve one operational line and two relational lines (e.g., *y* + 3 = 7). The inverse method, however, is more advantageous when complex one-step equations consisting of two operational lines and three relational lines are involved (e.g., **Figure 2(b)** [48]. Compared with simple one-step equations, the level of element interactivity of the complex one-step equations for both the balance and inverse methods has increased because of an increase in both operational lines (2 vs. 1) and relational lines (3 vs. 2). Nonetheless, the ratio of the interactive elements between the balance and inverse methods remains the same (i.e., 2:1), irrespective of the number of operational line. Having said this, the total number of interactive elements for two operational lines is twice the total number of interactive elements for one operational line, irrespective of whether it is the balance method or inverse method. Consequently, as revealed by prior studies, differential element interactivity between the balance and inverse methods favors the inverse method for complex one-step equations that involve two operational lines and three relational lines [1, 48, 49].

*conceptual knowledge* refers to the principle that governs a domain, and *procedural knowledge* refers to a sequence of actions to obtain a solution. The extent to which students have acquired procedural knowledge of one-step equations is reflected in their ability to solve one-step equations [1]. However, the acquisition of conceptual knowledge for one-step equations is concerned with students' understanding of the mathematical equivalence (i.e., "=" sign concept) with respect to both relational and operational lines [1]. Apparently, the relational understanding of the equal sign ("=") is critical to a student's success in solving equations [58].

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

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We recently explored the issue of equal sign with reference to the two comparative pedagogical approaches, balance versus inverse. Using a two-group pretest-posttest experimental design, we found that the inverse group had no advantage over the balance group with regard to students' understanding of the equal sign for the relational line [1]. For example, presented with an equation such as *x* + 6 = 11, students could justify that the "=" sign indicated "balance, equal, etc." There are two ways of presenting the "=" sign concept with respect to the operational line: (i) balance method: *x* + 3 = 5, *x* + 3 − 3 = 5 − 3 and (ii) inverse method: *x* + 3 = 5, *x* = 5 − 3. When students were asked to judge whether a pair of equations was equivalent (e.g., balance method: *x* + 3 = 5, *x* + 3 − 3 = 5 − 3), both the balance and inverse groups performed better when the pair of equations was presented using the inverse method [1]. This evidence suggests that, in general, the differential element interactivity favors the inverse

Experience of optimal best in mathematics learning, according to Phan et al. [4], may involve demonstration of competence for not only the simple percentage problems but also percentage problems that are more complex. A realistic level of best practice, by contrast, reflects the demonstration of competence for simple percentage problems only. This conceptualization of achievement bests is significant and highlights variations in personal functioning in different subject domains of academia. A realistic level of best practice serves as a point of selfreference for determination and/or aspiration of an optimal level of best practice. A student's determination of his/her level of optimal best, in part, depends on what he/she is capable of, at present. In the context of mathematics learning, we postulate pedagogical practices (e.g., an appropriate instructional design), involving the impact of element interactivity and cognitive load imposition, that could associate with differing levels of best practice. For example, in relation to our discussion so far, we conceptualize that an optimal instructional design devised to assist in the achievement of optimal best in complex percentage problems would impose a lower level of element interactivity. Suboptimal instructional designs devised for a realistic level of best practice in simple percentage problems, by contrast, would impose a higher level of element interactivity. Furthermore, as noted, the level of element interactivity

As discussed, the balance method imposes twice as many interactive elements as the inverse method for each operational. In regard to the acquisition of procedural knowledge in equation

method for the operational line.

**9. Achieving optimal best for one-step equations**

is directly proportionate to the degree of cognitive imposition [5, 6].
