**3. The importance of algebra**

The Framework of Achievement Bests is unique for its attempts to explain the "achievement" of optimal functioning. That is, from **Figure 1**, an individual's point of reference is his/her realistic level of best practice (note: as time progresses, an individual's realistic level of best practice becomes his/her historical level of best practice, which is defined as *previous record of accomplishment* in a subject matter). This level of best practice, consequently, serves as a source and/or a reference point by which an individual would use to formulate his/her optimal level of best practice. The *zone of optimization*, which refers to the "difference" or "range" between the realistic level and the optimal level of best practice (i.e., OL-RL), in essence, delves into the process of optimization. In other words, as Phan and colleagues explain, optimization is a psychological process that serves to optimize an individual's internal state of functioning to

The psychological process of optimization, recently updated in terms of theorization [10], varies in terms of *intensity* and *scope* (or volume). The intensity of optimization emphasizes *the extent and amount of resources* (e.g., appropriate instructional design) needed to optimize an individual's state of functioning. The scope (or volume) of optimization, by contrast, focuses on the *amount of effort* and *time* (e.g., the extent to which a student is motivated to invest effort and time) needed to optimize an individual's state of functioning. Optimizing a small zone of optimization in mathematics learning (e.g., knowing how to solve one-step equations such as *x* + 3 = 5, to knowing how to solve one-step equations such as *x* + 4 = −7), for instance, may require only a small amount of effort and time, and/or the amount of resource (e.g., limited scaffolding from a teacher). Both equations (i.e., *x* + 3 = 5 and *x* + 4 = −7) share identical problem structure except that the latter equation has a negative number (i.e., −7), and thus may pose a difficulty for students [12]. By contrast, it will require more resources (e.g., effective instructional design), effort, and time to optimize a large zone of optimization in mathematics learning (e.g., knowing how to solve one-step equations such as *y* + 3 = 7 (**Figure 2(a)**, to know-

*<sup>a</sup>* = 2 (**Figure 2(b)**). This is because the one-step

*y* + 5 = 13

 4 = 2 = 2

4 ÷ 2 = *a*

*y* = 13 – 5 *y* = 8

(+ 5 becomes – 5)

(÷ *a* becomes × *a*)

× *a* (× 2 becomes ÷ 2)

*<sup>a</sup>* = 2 has more solution steps than the one-step equation such as *y* + 3 = 7,

Line 1

Line 2

Line 3

Line 1

Line 2

Line 3

Inverse method

Inverse method

Line 4 2 = *a*

irrespective of the methods (i.e., balance or inverse). The differential efficiency between the balance and inverse methods will be discussed later. Similarly, in physical education,

**Figure 2.** (a) One-step equation involving one operational and two relational lines. (b) One-step equation involving two

"progress" from one level to that of another level.

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

ing how to solve one-step equations such as \_\_4

*y* = 8

× *a* × *a*

÷ 2 ÷ 2

*y* + 5 = 13 (– 5) on both sides

(× *a*) on both sides

4 = 2*a* (÷ 2) on both sides

operational lines and three relational lines.

Line 52 = *a*

equation such as \_\_4

(a) Balance method

Balance method

Line 2 – 5 – 5

Line 1

Line 3

(b)

Line 1

Line 2

Line 3

Line 4

Mathematics educators acknowledge the prominent role of algebra in mathematics learning and curriculum development [30, 31]. They regard algebraic skills as a "gatekeeper" that facilitates the engagement of higher-order mathematical thinking skills [32]. Algebraic skills are useful not only for solving real-life problems (e.g., "If your father wants to increase your weekly allowance of \$20 by 5%, what is your new allowance?" [33] but also enable effective learning in comparable subject areas, such as Physics and Chemistry (e.g., "A solution contains 1.1 g of sodium nitrate, NaNO3 in 250 ml of solution, what is the molarity of this solution?") [34–36].

**iii.** *Germane cognitive load*, which entails an investment of cognitive resources to assist in the learning of relevant aspects of the instructional material. We can change the design of an instruction in order to increase germane cognitive load. For example, one way to improve problem-solving skills is to provide learners with variability practice, involving the identification of a category of problems that share a similar problem structure but have

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

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

293

Recent research development on cognitive load theory has highlighted an important concept, known as *element interactivity*, that exists across the three types of cognitive load [43]. Element interactivity, in this case, emphasizes the *interaction that exists between elements within a learning material*. An element refers to anything that requires learning (e.g., a number, a symbol, a concept, a procedure, etc.) [44]. Under this conceptualization of cognitive load theory, the level of element interactivity determines the extent to which a particular type of cognitive load would exert its influence on the design of an instruction. Why is this the case? There are three possible reasons as to why this is the case: (i) the level of element interactivity determines the intrinsic nature of the material and, thus, the intrinsic cognitive load, (ii) the level of element interactivity determines, in part, the appropriateness and/or inappropriateness of an instruction and its extraneous cognitive load, and (iii) the level of element interactivity determines the beneficial design feature of an instruction and, thus, its corresponding germane cognitive load. Because cognitive resources, in the case of germane cognitive load, facilitate in the learning of relevant aspects of instructional material, germane cognitive load is not an independent

source of cognitive load; rather, it is incorporated in the intrinsic cognitive load.

Learning material reflects low-element interactivity knowledge if we can learn each element independently of another element [1]. In mathematical numeracy, a student can learn to recognize a number (e.g., 5) independently of another number (e.g., 9). Learning individual numbers therefore constitutes low-element interactivity knowledge, as each number is independent and may be learned in isolation. Moreover, because a student can learn to recognize individual numbers sequentially (e.g., "5" and then "6"), minimal working memory resources are involved when a student learns to recognize each number. Manipulation of multiple interactive elements simultaneously, by contrast, reflects high-element interactivity knowledge. In the case of learning how to solve a simple one-step equation, such as *x* − 7 = 13, a student would need to understand the role of the variable *x*, and the quantitative relationship between the elements, by which the left-hand side of the equation is equaled to the right-hand side. Manipulating multiple interactive elements simultaneously in order to solve an equation

Because there is limited association between individual numbers, a student can learn to recognize a vast number of numerals individually, via memorization (e.g., rote learning). However, since multiple elements within a linear equation interact, a student must learn these elements simultaneously rather than individually. On this basis, learning to solve an equation may pose a challenge for a student because it requires him/her to understand the relation between

**5. Element interactivity, learning, and understanding**

would impose a heavy cognitive load.

different contexts [42].

Regarding the use of algebra to solve real-life problems, successful problem solvers differ from unsuccessful problem solvers in their use of *schematic knowledge* to set up an equation that could then generate a solution [36, 37]. However, unless the problem solvers possess *adequate equation-solving skills*, they are unlikely to obtain the solution even if they have succeeded in setting up an equation pertaining to the schematic knowledge. Since equation solving is an integral component of the algebra problem-solving process [38], it is timely that we propose *appropriate instructions* that could facilitate optimal learning experiences of equation solving. On this basis, reflecting the Framework of Achievement Bests, we contend that appropriate instructional designs could serve to optimize students' understanding of equation solving.

Referring to our previous mention, a conceptualization that involves a focus on instructional designs could in effect provide evidence that attests to the explanatory power of the Framework of Achievement Bests. In this analysis, we contend that appropriate instructional designs could serve as educational mechanisms to optimize students' learning experiences in mathematics. Appropriateness of an instructional design is determined, in part, from its inverse association with the *negative impact of cognitive load imposition* [5, 6]. In our recent studies, for example, we proposed a theoretical position, which posits that optimal instructions that impose low cognitive load may generate positive emotions, resulting in an increase in motivation to learn equation solving. By contrast, however, suboptimal instructions are more likely to result in high cognitive load imposition, which may then generate negative emotions and a decline in motivation to learn equation solving.
