**2. Achieving optimal functioning**

Achieving optimal best in different subject matters is a central feat of human agency. This personal attribute emphasizes an internal state of determination and resilience to achieve optimal functioning. *Optimal functioning*, in contrast to the experience of stagnation, places emphasis on an individual's quest to fulfill his/her personal and psychological needs. Importantly, perhaps, the accomplishment of optimal best indicates the maximization in capability that an individual may demonstrate [4]. In the context of schooling, for example, optimal best for a Year 8 student may involve his/her understanding of mathematics equations involving special features (e.g., 10%*x* = 20, solve for *x*) (Appendix A). This level of exceptionality of mathematics learning, as mentioned, reflects the student's fullest potential for the stipulated time point.

Optimal best, in essence, coincides with the theoretical tenets of positive psychology [7, 8], which emphasize the *importance of human proactivity*, *personal fulfillment*, and the *aspiration to lead fruitful and meaningful lives*. Optimal best, consequently, indicates the development and manifestation of virtues, inner strengths, and resilience, and the achievement of exceptionality. These attributes and/or characteristics are positive, in nature. In recent years, researchers have advanced the study of optimal best, theoretically, methodologically, and empirically. Phan and colleagues [3, 4], for example, have developed the *Framework of Achievement Bests*, detailing an underlying internal mechanism that could explain how an individual reaches a state of optimal functioning. This framework is significant as it contributes to existing work [7–9], and advances the inquiry into the tenets of optimal best.

#### **2.1. The Framework of Achievement Bests**

performance at the end of the school term. Ineffective learning, by contrast, may result in loss of interest, engagement in maladaptive outcomes, and superficial learning. This recognition places emphasis on a need for educators and researchers, alike, to focus on motivational initiatives, pedagogical strategies, and educational programs that could foster *engagement of* and

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

Our postulated positioning, based on previous research undertakings in the area of mathematics [1, 2], is that *appropriate instructional designs* may serve to facilitate and promote effective learning. Instructional designs, an important element of pedagogical practices in the teaching and learning processes, are central to the achievement of effective learning. We contend that, in this case, a particular instructional design is efficient when it imposes minimal *cognitive load on an individual's processing of information*. Instructional designs that impose high levels of cognitive load, by contrast, are ineffective and inefficient for implementation and practice. This line of research development, in general, has notable implications for us to consider, especially in relation to Teacher Education Pre-service preparation and training. This chapter then, in accordance with the scope of the edited book, explores the importance of comparative instructional designs in the context of mathematics learning. Drawing from our previous work, we focus on the development of a conceptualization that emphasizes on the choosing of an appropriate instructional design for implementation. This conceptualization, in particular, focuses on the *achievement of optimal best in mathematics* [3, 4], taking into consideration *the negative impact of cognitive load imposition* [5, 6]. Furthermore, arising from this discussion, we consider methodological and theoretical issues for continuing research

Achieving optimal best in different subject matters is a central feat of human agency. This personal attribute emphasizes an internal state of determination and resilience to achieve optimal functioning. *Optimal functioning*, in contrast to the experience of stagnation, places emphasis on an individual's quest to fulfill his/her personal and psychological needs. Importantly, perhaps, the accomplishment of optimal best indicates the maximization in capability that an individual may demonstrate [4]. In the context of schooling, for example, optimal best for a Year 8 student may involve his/her understanding of mathematics equations involving special features (e.g., 10%*x* = 20, solve for *x*) (Appendix A). This level of exceptionality of mathematics learning, as mentioned, reflects the student's fullest potential for the stipulated time point.

Optimal best, in essence, coincides with the theoretical tenets of positive psychology [7, 8], which emphasize the *importance of human proactivity*, *personal fulfillment*, and the *aspiration to lead fruitful and meaningful lives*. Optimal best, consequently, indicates the development and manifestation of virtues, inner strengths, and resilience, and the achievement of exceptionality. These attributes and/or characteristics are positive, in nature. In recent years, researchers have advanced the study of optimal best, theoretically, methodologically, and empirically. Phan and colleagues [3, 4], for example, have developed the *Framework of Achievement Bests*,

*preference for* effective learning.

development into the area of instructional designs.

**2. Achieving optimal functioning**

The Framework of Achievement Bests, developed by Phan and colleagues [3, 4, 10], explores the personal experience of optimal functioning. Optimal functioning, according to the authors, is defined as an internal state of experience and accomplishment that reflects maximization in capability (e.g., a Year 8 student's indication to learn and understand linear equations that involve multiple solution steps (e.g., 4/*x* = 11, solve for *x*). "What is the best that I can accomplish?", in this instance, is a question that indicates an individual's self-awareness of his/her potential best practice.

The Framework of Achievement Bests draws comparison with Fraillon's [27] theorization of *optimization*, which is a psychological process that focuses on an individual's optimal best from some point of self-reference. Phan and colleagues' [3, 4, 10] conceptualization of achievement bests depicts two major levels of best practice: (i) *realistic level of best practice* (i.e., denoted as RL), which entails what an individual is realistically capable of accomplishing, at present (e.g., what can I actually do, at present, in Algebra?; how much do I know…..?), and (ii) *optimal level of best practice* (i.e., denoted as OL), which is defined as an individual's accurate indication of projected accomplishment that is exceptional, in nature (i.e., as of today, what is the best that I can do for this topical theme, realistically?). Reaching an optimal level of best practice from a realistic level of best practice reflects, in this case, a state of flourishing or optimal experience. **Figure 1** illustrates the Framework of Achievement Bests, in its totality [4, 11].

**Figure 1.** The Framework of Achievement Bests. Source: Adapted from Ref. [1].

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

optimizing an individual's functioning in a physical activity (e.g., running from 500 m to 5 km) may require much more effort, time, and resources. On this basis, the "quantitative" difference between the realistic level of best practice and the optimal level of best practice, which in this case reflects its complex nature, determines the intensity and scope of optimization.

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

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

291

In essence, the psychological process of optimization encompasses the utilization of resources, and the expenditure of time and effort in order to optimize an individual's internal state of functioning. Optimization, based on existing theorizations, involves more than just personal scaffolding from a capable authority figure [13]. According to Phan and colleagues' representation, there are three types of "mechanisms" [4] that operate to optimize a person's state of

**i.** The initiation and execution of *optimizing agents* (i.e., *psychological mechanisms*, *educational practices*, and/or *psychosocial mechanisms*) that operate to influence the internal personal processes for learning and performance. There are three types of optimizing agents, namely (i) *psychological mechanisms*, such as a person's self-efficacy beliefs for learning [14, 15], hope [16, 17], and motivation, in general [18], (ii) *educational practices*, such as instructional efficiency and appropriate pedagogical approach that enable better comprehension and understanding of the instructional materials [1, 4], and (iii) *psychosocial factors*, such as the impact of the home social environment that may shape a student's

**ii.** Upon the positive influences of optimizing agents, internal personal processes of *persistence* [20–22], *effort expenditure* [20, 23, 24], and *effective functioning* [25–27] are activated. This activation, in turn, plays a central role in motivating an organism to reach an optimal

Research development emphasizing the operational nature of optimization is in its early stage of evolution. In terms of empirical research, for example, a few researchers have used quantitative methodological designs to study the explanatory functioning of the three mentioned mechanisms [11, 28]. Phan and colleagues have used the Optimal Outcome Questionnaire (OOQ) [29] to explore the importance of "profiling of best practice" [4, 11], and the predictive and explanatory effects of the different types of optimizing mechanisms [28]. Aside from empirical research, it is also possible to explore optimal best practice and the operational nature of optimization from the perspective of conceptualization, using authentic contexts. In this section of the book chapter, we provide an in-depth discussion of a conceptualization of

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

functioning. The sequencing of this psychological process is as follows:

state of functioning [19].

level of functioning.

optimal best in the area of mathematics learning.

**3. The importance of algebra**

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 knowing how to solve one-step equations such as \_\_4 *<sup>a</sup>* = 2 (**Figure 2(b)**). This is because the one-step equation such as \_\_4 *<sup>a</sup>* = 2 has more solution steps than the one-step equation such as *y* + 3 = 7, 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 operational lines and three relational lines.

optimizing an individual's functioning in a physical activity (e.g., running from 500 m to 5 km) may require much more effort, time, and resources. On this basis, the "quantitative" difference between the realistic level of best practice and the optimal level of best practice, which in this case reflects its complex nature, determines the intensity and scope of optimization.

In essence, the psychological process of optimization encompasses the utilization of resources, and the expenditure of time and effort in order to optimize an individual's internal state of functioning. Optimization, based on existing theorizations, involves more than just personal scaffolding from a capable authority figure [13]. According to Phan and colleagues' representation, there are three types of "mechanisms" [4] that operate to optimize a person's state of functioning. The sequencing of this psychological process is as follows:


Research development emphasizing the operational nature of optimization is in its early stage of evolution. In terms of empirical research, for example, a few researchers have used quantitative methodological designs to study the explanatory functioning of the three mentioned mechanisms [11, 28]. Phan and colleagues have used the Optimal Outcome Questionnaire (OOQ) [29] to explore the importance of "profiling of best practice" [4, 11], and the predictive and explanatory effects of the different types of optimizing mechanisms [28]. Aside from empirical research, it is also possible to explore optimal best practice and the operational nature of optimization from the perspective of conceptualization, using authentic contexts. In this section of the book chapter, we provide an in-depth discussion of a conceptualization of optimal best in the area of mathematics learning.
