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

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

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

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

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

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

and three relational lines [1, 48, 49].

**7. Special features**

decimal [53].

is kept constant [54].

**8. Mathematical equivalence**

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

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 solving, prior studies have revealed the superiority of the inverse method over the balance method for solving complex one-step equations, but not for simple one-step equations [48]. This testament has credence, given that the complex one-step equations have more operational lines (2 vs. 1) and relational lines (2 vs. 3) than the simple one-step equations. Moreover, the superiority of the inverse method over the balance method also extends to one-step equations involving special features (e.g., 12%*x* = 28).

*inference problem contexts* [61]. Investing germane cognitive load to assist learning is in accordance with *deliberate practice*, whereby engagement in practice activities serves to assist learners to develop *expertise* in the domain [62]. Having said this, difficulties may arise, as van Gog, Ericsson [62] argued, whereby learners' lack of motivation may deter their willingness to invest germane cognitive load, and/or to engage in deliberate practice activities with a view

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

http://dx.doi.org/10.5772/intechopen.70568

299

A review of the empirical literature indicates that, to date, research development into the relationship between germane cognitive load and students' motivational beliefs is inconclusive. The *Goal-based Scenarios* (GBS) technique used in multimedia instructional designs, for example, is advantageous by motivating learners to study the instructional material, which then leads to improved understanding of the material, in total [63]. Nevertheless, despite this pedagogical initiative, there is little, if any, association between students' motivational beliefs and their perceived increase in germane cognitive load. In another study, however, Rey and Buchwald [64] found that the *probability of success*, a subdimension of motivation, was partially associated with the investment of cognitive load in learning. On this basis, evidence pertaining to the relationship between motivational beliefs and investment of germane cognitive load is inconclusive and requires further research development. For example, in a recent development, we proposed a theoretical model that conceptualized the relationships between optimal and suboptimal instructional designs (e.g., varying levels of element interactivity), and levels of best practice (varying levels of motivation) in the domain of percentage problems. Our theorization, as shown in **Figure 3**, is holistic and seeks to illuminate the combined effects of cognitive (e.g., cognitive load imposition), affective (e.g., a heightened state of anxiety), and motivational (e.g., personal self-efficacy beliefs) dimensions of effective learning to

Note:

+

+

Internal Personal Processes

Cognitive Load Imposition

Extraneous Intrinsic Germaine

Persistence Effort Effective

**Figure 3.** Proposed relationships between instructional designs, cognitive load, motivational processes, and achievement

Function

+

+ve = positive association, or predictive effect ve = negative association, or predictive effect Single arrow = predictive effect Double arrow = association

Achievement Bests

Realistic Optimal

Competence

Test Scale

+

to improve learning.

Unitary Unitary Pictorial Equation Equation Pictorial

bests. Source: Adapted from Ref. [4].

facilitate optimal best in the percentage problems.

Instructional Design

In relation to the acquisition of conceptual knowledge in equation solving, we also found that the inverse method is better than the balance method when the "=" sign concept is applied to the operational line, but not to the relational line [1]. Thus, the inverse method is better than the balance method in facilitating the acquisition of both procedural and conceptual knowledge of one-step equations. This evidence provides empirical support for our proposition, regarding the *alignment between optimal instructions* (i.e., the inverse method) and *the demonstration of competence not only for simple one-step equations* (i.e., a realistic level of best practice) *but also for complex one-step equations* (i.e., an optimal level of best practice). At the same time, we propose an analogous *alignment between suboptimal instructions* (i.e., the balance method) and *the demonstration of competence for simple one-step equations* (i.e., realistic level of best practice).

An important question, certainly, entails the constructive application of the Framework of Achievement Bests in the context of academic learning. The Framework of Achievement Bests may provide grounding to assist educators in their teaching practices. This application may take into consideration the impact of cognitive load theory [5, 43], and its subsequent influence on the development of appropriate instructional designs. For example, the use of the inverse method is likely to assist middle-school students to achieve an optimal level of best practice to solve complex one-step equations. Consequently, competence in solving complex one-step equations may enable middle-school students to apply such skills to solve real-life problems. Consider a problem that reflects a real-life situation, for example: "Sally wants to invite her friends to her birthday party. She has 15 lollies and she wants to give three lollies for each friend. How many friends should Sally invite for her birthday?" A student, in this case, could use algebra to "set up" the equation – for instance: 15/*x* = 3, solve for *x*. Because this equation involves two operational lines and three relational lines, it is obvious then that the balance method would reflect high cognitive load imposition, and subsequently hinder students' learning. The inverse method, by contrast, would associate with low cognitive load imposition, enabling students to solve such equation.

#### **10. Cognitive load and motivation**

*Worked example* is one of the popular instructional designs that has extensively been researched [59]. The merit of worked example depends largely on its design. For example, we could use illustrations (e.g., a diagram) to represent the problem situation of the specific problem, which in turn would increase germane cognitive load and hence improve students' problemsolving skills [33, 41]. Aside from worked example, other pedagogical strategies to increase germane cognitive load include the *incorporation of self-explanation* [60], and *high contextual*  *inference problem contexts* [61]. Investing germane cognitive load to assist learning is in accordance with *deliberate practice*, whereby engagement in practice activities serves to assist learners to develop *expertise* in the domain [62]. Having said this, difficulties may arise, as van Gog, Ericsson [62] argued, whereby learners' lack of motivation may deter their willingness to invest germane cognitive load, and/or to engage in deliberate practice activities with a view to improve learning.

solving, prior studies have revealed the superiority of the inverse method over the balance method for solving complex one-step equations, but not for simple one-step equations [48]. This testament has credence, given that the complex one-step equations have more operational lines (2 vs. 1) and relational lines (2 vs. 3) than the simple one-step equations. Moreover, the superiority of the inverse method over the balance method also extends to one-step equa-

In relation to the acquisition of conceptual knowledge in equation solving, we also found that the inverse method is better than the balance method when the "=" sign concept is applied to the operational line, but not to the relational line [1]. Thus, the inverse method is better than the balance method in facilitating the acquisition of both procedural and conceptual knowledge of one-step equations. This evidence provides empirical support for our proposition, regarding the *alignment between optimal instructions* (i.e., the inverse method) and *the demonstration of competence not only for simple one-step equations* (i.e., a realistic level of best practice) *but also for complex one-step equations* (i.e., an optimal level of best practice). At the same time, we propose an analogous *alignment between suboptimal instructions* (i.e., the balance method) and *the demonstration of competence for simple one-step equations* (i.e., realistic level of

An important question, certainly, entails the constructive application of the Framework of Achievement Bests in the context of academic learning. The Framework of Achievement Bests may provide grounding to assist educators in their teaching practices. This application may take into consideration the impact of cognitive load theory [5, 43], and its subsequent influence on the development of appropriate instructional designs. For example, the use of the inverse method is likely to assist middle-school students to achieve an optimal level of best practice to solve complex one-step equations. Consequently, competence in solving complex one-step equations may enable middle-school students to apply such skills to solve real-life problems. Consider a problem that reflects a real-life situation, for example: "Sally wants to invite her friends to her birthday party. She has 15 lollies and she wants to give three lollies for each friend. How many friends should Sally invite for her birthday?" A student, in this case, could use algebra to "set up" the equation – for instance: 15/*x* = 3, solve for *x*. Because this equation involves two operational lines and three relational lines, it is obvious then that the balance method would reflect high cognitive load imposition, and subsequently hinder students' learning. The inverse method, by contrast, would associate with low cognitive load

*Worked example* is one of the popular instructional designs that has extensively been researched [59]. The merit of worked example depends largely on its design. For example, we could use illustrations (e.g., a diagram) to represent the problem situation of the specific problem, which in turn would increase germane cognitive load and hence improve students' problemsolving skills [33, 41]. Aside from worked example, other pedagogical strategies to increase germane cognitive load include the *incorporation of self-explanation* [60], and *high contextual* 

tions involving special features (e.g., 12%*x* = 28).

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

imposition, enabling students to solve such equation.

**10. Cognitive load and motivation**

best practice).

A review of the empirical literature indicates that, to date, research development into the relationship between germane cognitive load and students' motivational beliefs is inconclusive. The *Goal-based Scenarios* (GBS) technique used in multimedia instructional designs, for example, is advantageous by motivating learners to study the instructional material, which then leads to improved understanding of the material, in total [63]. Nevertheless, despite this pedagogical initiative, there is little, if any, association between students' motivational beliefs and their perceived increase in germane cognitive load. In another study, however, Rey and Buchwald [64] found that the *probability of success*, a subdimension of motivation, was partially associated with the investment of cognitive load in learning. On this basis, evidence pertaining to the relationship between motivational beliefs and investment of germane cognitive load is inconclusive and requires further research development. For example, in a recent development, we proposed a theoretical model that conceptualized the relationships between optimal and suboptimal instructional designs (e.g., varying levels of element interactivity), and levels of best practice (varying levels of motivation) in the domain of percentage problems. Our theorization, as shown in **Figure 3**, is holistic and seeks to illuminate the combined effects of cognitive (e.g., cognitive load imposition), affective (e.g., a heightened state of anxiety), and motivational (e.g., personal self-efficacy beliefs) dimensions of effective learning to facilitate optimal best in the percentage problems.

**Figure 3.** Proposed relationships between instructional designs, cognitive load, motivational processes, and achievement bests. Source: Adapted from Ref. [4].
