**11. Cognitive load imposition, internal personal processes, and achievement bests**

An important focus of inquiry for development entails the potential associations between cognitive load impositions, internal personal processes of learning, and levels of best practice. This development, reflected in our recent conceptualization [4], indicates a concerted effort to integrate three major strands of research, namely cognitive processes, motivational beliefs and affective dimensions, and achievement bests. We urge researchers and educators to consider this theoretical model for research development. This empirical validation is worth noting and may indicate significance regarding the impact of an integration of different strands of inquiries. For example, in relation to affective responses, Ashcraft and Kirk [65] found that heightened anxiety levels negatively influenced the working memory capacity to process different types of mathematic-learning tasks. It is plausible to assume that a proportion of the work memory resources is used to "counter" the heightened state of anxiety, and on this basis, very little is left for processing of information. Similar evidence has been reported in a simulation training study in the area of Medical Education [66]. In this study by Fraser et al. [66], the authors found that negative affective responses (e.g., anxiety) increased extraneous cognitive load imposition, which then led to a decrease in the working memory capacity for learning. However, from the results, the relationship between cognitive load imposition, positive emotions, and learning outcomes was less predictable.

affective responses (e.g., happiness), consequently as a result of a learner's ability to understand and to demonstrate the mastery of complex one-step equations, and (iii) reflects, correspondingly, a high score on the Optimal Best Subscale of the Optimal Outcomes Questionnaire (e.g., I feel positive when I am asked to solve complex one-step equations). We recently developed, as mentioned, the Optimal Outcomes Questionnaire [29], which has two subscales, the Realistic Best Subscale (i.e., consists of eight items) and the Optimal Best Subscale (i.e., consists of eight items). Aside from focusing on the importance of "profiling" of best practice [11, 28], we contend that this questionnaire could measure and assess students' motivational levels and affective responses as a result of their exposures to different instructional designs. This recognition, which we recommend for further research advancement, indicates the importance of *diagnostic assessment* of motivational levels and achievement bests

**Solution procedure for one-step equations that have special features in the test items**

+2 + 2 *a* = −1

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

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

301

0.5 = 5 × 0.5 × 0.5 *x* = 2.5

÷3 ÷ 3 *m* = 2/3

÷2 ÷ 2 0.5 = *p*

−6 − 6 −*q* = 4 ÷(−1) ÷ (−1)

*q* = −4

*<sup>a</sup>* = 2 × *a* × *a* 4 = 2*a* ÷2 ÷ 2 2 = *a*

*Note:* The solution procedure of those equations marked by \* has two operational lines (e.g., −6 on both sides, and ÷(−1) on both sides) and thus impose higher element interactivity than other equations that have one operational line (e.g., +2

÷10 % <sup>÷</sup> <sup>10</sup>% *x* = 200

**Equation type Balance method**

*Negative numbers a*–2 = 3

*A percentage* 10 % *x* = 20

*A fraction or a decimal as a solution* 3*m* = 2

*Pronumeral on the right side* 1 = 2*p*

*\*Negative pronumeral* 6 − *q* = 10

*\*Pronumeral as a denominator* \_\_4

*A decimal number* \_\_\_*x*

that arise from varying levels of cognitive load imposition.

**Appendix**

on both sides).

Our theorization, as shown in **Figure 3**, has a number of proposed associations for consideration. Central to our conceptual model is the recognition and inclusion of the two major theories: cognitive load imposition [5, 43] and achievement bests [4, 11]. Importantly, of course, a focus of inquiry may involve the use of both theories to inform the development of appropriate pedagogical practices (e.g., an instructional design) to promote effective learning experiences. For example, suboptimal instructional designs (e.g., the balance method), which directly associate with negative cognitive load imposition, could have adverse effects on motivational beliefs and achievement of optimal best (e.g., the achievement of realistic best practice only, which, in this case, may involve simple onestep equations).

In relation to what we have discussed so far, it is evident that in the context of mathematics learning, comparative instructional designs may have differing effects on students' understanding. Future research undertakings may pursue this inquiry, delving into the relationships between comparative instructional designs (e.g., balance vs. inverse) and levels of best practice. This postulation, emphasizing two contrasting associations (i.e., balance method ↔ simple one-step equations vs. inverse method ↔ complex one-step equation, where ↔ = closely aligned association), is of value for testament, especially when we consider its potentials to influence motivational beliefs and affective responses. Our argument, overall, based on previous research development, is that *the inverse method is superior to the balance method for effective learning*. This conviction, we contend, draws to the fact that the inverse method (i) imposes low cognitive load imposition, enabling a learner to understand both simple and complex one-step equations, (ii) elicits positive affective responses (e.g., happiness), consequently as a result of a learner's ability to understand and to demonstrate the mastery of complex one-step equations, and (iii) reflects, correspondingly, a high score on the Optimal Best Subscale of the Optimal Outcomes Questionnaire (e.g., I feel positive when I am asked to solve complex one-step equations).

We recently developed, as mentioned, the Optimal Outcomes Questionnaire [29], which has two subscales, the Realistic Best Subscale (i.e., consists of eight items) and the Optimal Best Subscale (i.e., consists of eight items). Aside from focusing on the importance of "profiling" of best practice [11, 28], we contend that this questionnaire could measure and assess students' motivational levels and affective responses as a result of their exposures to different instructional designs. This recognition, which we recommend for further research advancement, indicates the importance of *diagnostic assessment* of motivational levels and achievement bests that arise from varying levels of cognitive load imposition.
