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

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 con-

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

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

*Human cognitive architecture*, compromising of both working and long-term memory, is central to the importance of *cognitive load theory* [5]. The working memory is severely limited in its capacity to process unfamiliar information [39]; however, this limitation disappears when familiar information is retrieved from the long-term memory for processing. By contrast, long-term memory has an unlimited capacity, which enables it to store a large amount of information for an infinite period. Cognitive load theory, proposed by Sweller [5, 40], is an instructional theory that attempts to explain why a specific instruction will or will not work.

**i.** *Extraneous cognitive load*, which is imposed by an inappropriate instruction. We can change the design of inappropriate instruction to reduce extraneous cognitive load.

**ii.** *Intrinsic cognitive load*, which is imposed by the inherent complexity of a learning unit (or a material). We can change either the design of the instruction [33, 41] or the knowledge

in 250 ml of solution, what is the molarity of this solu-

tains 1.1 g of sodium nitrate, NaNO3

and a decline in motivation to learn equation solving.

**4. Cognitive load theory: a theoretical overview**

Three types of cognitive load affect the design of a specific instruction:

base of a learner to reduce intrinsic cognitive load.

tion?") [34–36].

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 would impose a heavy cognitive load.

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 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 mathematical concepts.

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

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

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

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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 inter-

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,

successfully so that he or she would then know that *y* equals to 8 is the solution.

activity and therefore lower cognitive load than the balance method.

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

**6.2. Inverse method**
