**7. Special features**

Aside from operational and relational lines, the presence of *special features* that involve complex elements also increases the complexity of one-step equations (see Appendix A). Operating with negative numbers is an integral component of middle-school mathematics curriculum. Having said this, operating with negative numbers continually poses challenges for schoolage students [50, 51]. For example, in relation to multiplication, many students struggle with problems that have two negative numbers in algebraic expression problems (e.g., − 4(5*x* − 2) [50]. Furthermore, aside from negative numbers, students also commit errors when operating with fractions [52]. Finally, to compound this difficulty, many students also fail to engage in mathematical reasoning that emphasizes the connection between fraction, percentage, and decimal [53].

On this basis, when the number of operational lines and relational lines is kept constant in one-step equations, operations with special features (see Appendix A) pose an additional challenge for students. For example, the equation 2*x* = 6 shares a similar structural feature with that of the equation 10%*x* = 20 and, consequently, both have the same level of element interactivity. However, 10%*x* = 20 poses a greater challenge than 2*x* = 6, owing to the fact that the latter equation has a percentage (i.e., 10%). The percentage (i.e., 10%) is regarded as a complex element because it comprises not only a number (i.e., 10) but also a percentage sign (i.e., %).

In summary, from the discussion so far, what can we say about the two pedagogical approaches: inverse versus balance? We argue that the inverse method, preferred by many Asian countries, is more effective than the balance method for two major attributes: (i) the number of operational lines and relational lines that exist and (ii) the presence of special features in the equations. Indeed, our previous research undertakings have provided evidence that the inverse method is better than the balance method for complex one-step equations that involve two operational lines and three relational lines. The inverse method, though, is comparable to the balance method for simple one-step equations that involve one operational line and two relational lines [48]. Furthermore, as our research showed, the presence of special features favored the inverse method when the number of operational and relational lines is kept constant [54].

### **8. Mathematical equivalence**

Mathematics education researchers have regarded conceptual and procedural knowledge as essential components of mathematics proficiency [55, 56]. According to Rittle-Johnson, Siegler [57], *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].

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 method for the operational line.
