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The design principles of cognitive load theory and learning by analogy has independently contributed to our understanding why an instruction will or will not work. In an experimental study involving 97 Year 9 Australian students conducted in regular classrooms, we evaluated the effect of the unguided problem-solving approach, worked examples approach and analogy approach on learning to solve two types of trigonometry problem. These trigonometry problems (sin40° = [X over 6] vs. cos50° = [14 over x]) exhibited two levels of complexity owing to the location of the pronumeral (numerator vs. denominator). The solution procedure of worked examples provided guidance, whereas the unguided problem-solving was without any guidance. Analogical learning placed emphasis on comparing a pair of isomorphic examples to facilitate transfer. Across the three approaches, solving practice problems contributed towards performance on the post-test. However, the worked examples approach and analogy approach were more efficient than the unguided problem-solving approach for acquiring skills to solve practice problems regardless of their complexity. Therefore, the worked examples approach and analogy approach that emphasizes algebraic transformation skills have the potential to reform instructional efficiency for learning to solve trigonometry problems.

Capitalizing on cognitive load theory and learning by analogy, we propose two instructional methods to learn a complex linear equation (e.g. two-step equation) by building on prior knowledge of a simpler linear equation (e.g. one-step equation). We will examine the proposal theoretically in this paper. In line with the design principles of cognitive load theory, we propose to strengthen students' prior knowledge of simpler linear equations before they learn complex linear equations with the aid of worked examples. Because a subset of the complex linear equation shares the same schema as the simpler linear equation, students can draw on their schema for the simpler linear equation to understand the complex linear equation, thus alleviating the limitation on working memory load. Based on the principles of learning by analogy, we place a simpler linear equation and a complex linear equation side-by-side and label the solution procedure of both linear equations to encourage active analogical comparison between these two equations. Making both the simpler linear equation and the complex linear equation visible to learners may help to reduce cognitive load demands in retrieving the simpler linear equation in order to facilitate the learning of the complex linear equation.

"Optimal best practice" is a central feat of human agency. It emphasizes a state of flourishing and reflects, in this case, the paradigm of positive psychology. One research inquiry that is of interest relates to an explanatory account of how a person reaches a state of optimal best. Recent research development has considered an important psychological process, known as "optimization," which may explain a person's achievement of optimal best practice. Having said this, very little is known about the process of optimization. In this article, the authors report on a non-experimental study (N = 352 secondary school students), which focused on the testing of a theoretical model of optimization. Innovatively, derived from existing theorizations and empirical evidence, the authors provide a methodological rationalization of flourishing, which is defined as a "quantitative difference" between a person's current level of best practice (denoted as L[subscript 1]) and his/her optimal level of best practice (denoted as L[subscript 2]). Structural equation modeling (SEM) indicated a few major findings, for example, (i) a positive association between a person's optimal best practice and his/her academic performance in a subject matter, (ii) a person's current level of best practice acts as a determinant of optimal best practice, and (iii) personal resolve, as a psychological optimizing agent, directly influences optimal best practice, and potentially mediating the effects of academic striving and a person's current level of best practice on optimal best practice.

In 2016, in the journal of Education, we introduced a theoretical concept known as 'optimal best,' which subsequently led to our extensive research undertakings, theoretically, methodologically, and empirically. Optimal best, in brief, relates to the maximization, or the fullest potential, of a person's state of functioning (e.g., optimal cognitive functioning in mathematics learning for the topic of Geometry). An important premise arising from this research development relates to the promotion and cultivation of optimal best in different subject matters. Academically, for example, our research inquiries have shown that instilling a high level of 'personal resolve' would, in turn, facilitate the successful achievement and experience of optimal best. Advancing our focus of inquiry, we propose a related concept, which we name as 'goals of best practice' (GsBP) for consideration. Goals of best practice, consisting of two major types (i.e., 'goal of actual best' and 'goal of optimal best') may assist to motivate a person to strive for optimal best. Furthermore, coinciding with this proposed concept, we develop a Likert-scale measure, coined as the 'Goals of Best Practice Questionnaire' (GsBPQ), for consideration. Educators and researchers may use the GsBPQ, in this analysis, to facilitate the zone of 'optimal experience' and, by the same token, to negate the zone of 'sub-optimal experience.' Keywords: Goals of Best Practice, Goal of Actual Best, Goal of Optimal Best, Optimization, Optimal Best, Flow, Flourishing, Introduction Achieving a state of optimal best, or optimal functioning, is an interesting area of research in the field of Educational Psychology. In the early 2000s, Fraillon (2004) provided an [...]

Quality mathematics learning is more than just the acquisition of mastery in different topical themes; rather, it involves successful acquisition of mathematical proficiency, which espouses a number of cognitive attributes—for example, a student's critical insight of a mathematical concept (e.g., productive disposition). Despite the pivotal role of mathematical proficiency in mathematics curriculum, syllabus requirements fall short of highlighting the design of appropriate instructional approaches that could specifically facilitate the acquisition of different mathematical proficiency strands. The present conceptual analysis article discusses the design of comparative instructional approaches that are based on two well-documented learning theories: (1) learning by comparison theory, such as the active comparison of isomorphic example pairs, and (2) cognitive load theory, such as the use of worked examples to reduce the negative impact of cognitive load imposition on learning. We premise that appropriate instructional approaches, informed by the use of both learning by comparison theory and cognitive load theory, may help to facilitate successful acquisition of multifaceted proficiency strands in mathematics learning. As revealed in the latter sections of the article, our proposed theoretical contention is significant, potentially establishing grounding for future research development and to help complement constructivist learning in the acquisition of mathematical proficiency strands. [ABSTRACT FROM AUTHOR]

AbstractAccording to cognitive load theory, an expert learner can access prior knowledge to interpret a diagram with minimum effort. A novice learner who lacks prior knowledge, in contrast, would experience a high level of cognitive load when processing a diagram, which would interfere with learning. On learning to solve percentage problems, we investigated the effect of approach (non-algebra vs. algebra), with and without diagram support for two levels of learner expertise. Participants were 218 Asian students whose mean age was 15.00 (SD = 0.18). Contrary to our hypothesis, the presence or absence of a diagram may bore little, if any, benefit for students who were more knowledgeable. Considering two levels of learner expertise together, as hypothesized, performance outcomes favoring more knowledgeable students were greater for the unitary-pictorial approach than the equation-pictorial approach. Interestingly, the impact of a diagram was more pronounced for more knowledgeable students; however, with relevant prior algebra knowledge, the presence of a diagram (e.g., equation-pictorial approach) can also be helpful for less knowledgeable students. Overall, then, our research undertaking has yielded important implications for teaching and research purposes. [ABSTRACT FROM AUTHOR]

Engagement is a prominent theoretical orientation that has received great attention from educators and researchers. This article provides a literature overview of the engagement construct pertaining to its various definitions, dimensions, and major conceptualisations. In addition, the review sheds light on two major approaches to engagement in the current literature: one entails students' cognitive, behavioural, and emotional engagement (Fredricks, Blumenfeld, & Paris, 2004), and the other approach is characterised by vigor, dedication, and absorption (Schaufeli, Salanova, Gonzalez-Rom, & Bakker, 2002). Further, the association between these two models and students' performance is discussed.

The present study considered "global self-esteem" and "academic self-efficacy" as two contrasting precursors and distinctive pathways that could account for effective schooling experiences: (1) the "social pathway" (e.g., the predictive effect of teacher-student social relationship) and (2) the "personal pathway" (e.g., the predictive effect of a child's recognition of realistic best practice). This theoretical-conceptual model for examination, which we tested with a cohort of secondary school students from Taiwan (N = 750 Year 11) affirmed the direct and potential mediating effects of "social relationships," "personal well-being," "best practice," "positive emotions," and "academic striving." Causal modeling techniques used showed a number of significant findings--for example: the direct effect of teacher-student social relationship on a student's experience of positive emotions, the direct effects of a student's belief of optimal best practice and academic striving on his/her academic achievement, and the potential mediating role of personal well-being between global self-esteem and a student's recognition of realistic best. This evidence, overall, makes substantive theoretical and methodological contributions to the study of positive schooling experiences, using social psychology and educational psychology theories as a basis for further development. One distinction, in this case, entails our intricate conceptualization of two contrasting courses of action that a student may experience. Moreover, from an applied practice point of view, our research inquiry has informed educators, researchers, and stakeholders of different viewpoints that could foster proactive social relationships, subjective well-being, positive emotional functioning, and academic performance at school.

Capitalising on current research progress, we explored via means of structural equation modelling (SEM), the operational nature of three newly developed optimising concepts: "personal resolve," "effective functioning," and "academic striving." We conceptualised self-efficacy as a source of origin, and effort expenditure and motivation towards learning as adaptive outcomes of the three mentioned concepts. Correlational responses from Taiwanese university students (N = 1010) affirmed our hypothesis regarding the operational nature of personal resolve, effective functioning, and academic striving -- for example -- (i) self-efficacy as a source of origin of personal resolve, effective functioning, and academic striving, and (ii) the effects of personal resolve and academic striving on effort, and the effects of effective functioning and academic striving on motivation towards learning. This significant evidence, overall, contributes to the study of optimal achievement best.

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.

Between two popular teaching methods (i.e., balance method vs. inverse method) for equation solving, the main difference occurs at the operational line (e.g., +2 on both sides vs. -2 becomes +2), whereby it alters the state of the equation and yet maintains its equality. Element interactivity occurs on both sides of the equation in the balance method, but only on one side in the case of the inverse method. Thus, the balance method imposes twice as many interacting elements as the inverse method for each operational line. In two experiments, secondary students were randomly assigned to either the balance method or the inverse method to learn how to solve one-step, two-step, and three-or-more-step linear equations. Test results indicated that the interaction between method and type of equation favored the inverse method for equations involving higher element interactivity. Hence, by managing element interactivity, the efficiency of instruction for equation solving can be improved.

The present study focused on an examination of both "global" and "domain-specific self-esteems" in secondary mathematics learning. The extent to which self-esteem, in general, would account and explain educational success through "social relationships with teachers" and "peers", and "personal interest in learning tasks" is the main inquiry of this correlational investigation. Two hundred and eighty-three year 10 students (128 girls, 155 boys) were asked to respond to a number of Likert-scale inventories. Causal modeling procedures, aided by the statistical software M"Plus" 7.3, were used to analyze the data and to test the hypothesized model. A series of a posteriori analyses yielded a modified model for discussion, producing a number of key findings, namely: (a) the differential influences of both global (e.g., [right arrow]relationship with teachers) and domain-specific (e.g., [right arrow]relationship with peers) self-esteems, (b) the positive influences of social relation with teachers (e.g., [right arrow]exam result) and interest in learning tasks (e.g., [right arrow]end-of-school term grade), and (c) the mediating functioning of relationship with peers and interest in learning tasks. This evidence, overall, provides additional theoretical insights into the operational nature and trajectories of effective learning.

In an experiment, secondary students from Australia and Malaysia (n = 130) were randomly assigned to one of three approaches (equation, unitary, unitary-pictorial) to learn how to solve challenging percentage-change problems. In line with the differential types of cognitive load associated with the three approaches, the unitary-approach group outperformed both the unitary group and the equation group across Australia and Malaysia. In support of cross-cultural findings, the Malaysian students outperformed the Australian students for the equation approach but not the unitary approach nor the unitary-pictorial approach. The Australian students, in contrast, learned better with the unitary-pictorial approach. This study, overall, reveals the "gap" between the Asian and Western countries in the use of problem-solving approaches across different cultural settings.

The present longitudinal research investigation explored the differential effects of "contextualised self-efficacy beliefs" (i.e. task, course, global) on the concepts of "personal resolve" and "effective functioning," and two adaptive outcomes, namely: "school experience" and "academic achievement." 291 (141 girls, 150 boys) Year 7 secondary school students participated in the study, which spanned the course of four time points. Subsequent SEM analyses produced the following results, for example: (i) Time 1 task self-efficacy positively influenced Time 2 personal resolve and Time 2 effective functioning, (ii) Time 2 personal resolve positively influenced Time 3 contextualised self-efficacy beliefs, (iii) Time 2 effective functioning positively influenced Time 4 school experience, and Time 4 academic achievement, and (iv) Time 3 task-specific self-efficacy positively influenced Time 4 academic achievement and Time 4 school experience. This evidence, collectively, provides grounding for further research development (e.g. the importance of effective functioning) and educational practices for implementation.

In social sciences, the use of stringent methodological approaches is gaining increasing emphasis. Researchers have recognized the limitations of cross-sectional, non-manipulative data in the study of causality. True experimental designs, in contrast, are preferred as they represent rigorous standards for achieving causal flows between variables. The Solomon four-group design, for example, is ideal for its positioning to account for, and factor out, confounded influences of predictors on outcomes. However, in daily life settings, it is often difficult to emulate true experimental conditions. Identified limitations include financial resources, logistic difficulties, time constraint, and small sample sizes in social science research settings. There are, of course, other experimental designs that are noteworthy for consideration. Time series and single-case designs, quasi in nature, are effective alternatives for educators and researchers to consider in their research foci. This article examines the different experimental designs that may be implemented in naturalistic classroom settings. In particular, one important inquiry of our theoretical discussion pertains closely to the conceptualization of two innovative designs that we have made, consequently as a result of our research development and examination of the literature: "a sequential, multiple time series multi-group design" and a "multi time series, multi-group single-case design." These experimental designs are innovative and enable comparisons for "within" and "between" differences under different experimental conditions.

We recently developed the "Framework of Achievement Bests" to explain the importance of effective functioning, personal growth, and enrichment of well-being experiences. This framework postulates a concept known as "optimal achievement best," which stipulates the idea that individuals may, in general, strive to achieve personal outcomes, reflecting their maximum capabilities. "Realistic achievement best," in contrast, indicates personal functioning that may show moderate capability without any aspiration, motivation, and/or effort expenditure. Furthermore, our conceptualization indicates the "process of optimization," which involves the optimization of achievement of optimal best from realistic best. In this article, we explore the "Framework of Achievement Bests" by situating it within the context of student motivation. In our discussion of this theoretical orientation, we explore in detail the impact of instructional designs for effective mathematics learning as an "optimizer" of optimal achievement best. Our focus of examination of instructional designs is based, to a large extent, on cognitive load paradigm, theorized by Sweller and his colleagues. We contend that, in this case, cognitive load imposition plays a central role in the structure of instructional designs for effective learning, which could in turn influence individuals' achievements of optimal best. This article, conceptual in nature, explores varying efficiencies of different instructional approaches, taking into consideration the potency of cognitive load imposition. Focusing on mathematical problem solving, we discuss the potentials for instructional approaches to influence individuals' striving of optimal best from realistic best.

Assimilating multiple interactive elements simultaneously in working memory to allow understanding to occur, while solving an equation, would impose a high cognitive load. "Element interactivity" arises from the interaction between elements within and across operational and relational lines. Moreover, operating with special features (e.g. negative pronumeral) poses additional challenge to master equation solving skills. In an experiment, 41 8th grade students (girls = 16, boys = 25) sat for a pre-test, attended a session about equation solving, completed an acquisition phase which constituted the main intervention and were tested again in a post-test. The results showed that at post-test, students performed better on one-step equations tapping low rather than high element interactivity knowledge. In addition, students performed better on those one-step equations that contained no special features. Thus, both the degree of element interactivity and the operation with special features affect the challenge posed to 8th grade students on learning how to solve one-step equations.

The formation of self-efficacy, according to Bandura's (1997) social-cognitive theory, is an important area of inquiry. This theoretical tenet posits the importance of enactive learning experience, followed by lesser influences of vicarious experience, verbal persuasion, and emotional and physiological states. Quantitative research, predominantly, has produced clear and consistent evidence that supports this position. We argue that the elementary school years may indicate differently, whereby children's limited cognitive maturity and learning experiences could compel them to rely on other psychosocial informational sources. To date and to our knowledge, very few studies, if any, have explored the sustained influence of enactive learning experience across time. In this study, consequently, we tested a sequential predictive model that involved the differential influences of the 4 major informational sources on self-efficacy and then self-efficacy on academic achievement. Three time points of data (N = 328, Year 6) were collected across the calendar year, and Mplus 7.3 (Muthén & Muthén, 1998-2012) was used to assist us in our structural modeling testing. At Time 1, only enactive learning experience and vicarious experience positively influenced self-efficacy. At Time 2, after controlling for prior variance of Time 1 corresponding factors, only enactive learning experience remained significant. At Time 3, after controlling for autoregressive paths, enactive learning experience remained significant, and both verbal persuasion and emotional and physiological states positively influenced self-efficacy. The impact of self-efficacy on academic achievement was significant across the 3 occasions (ßs = 0.20-0.46).

We used a 2 (prior knowledge: low vs. high) × 4 (instructional approach: unitary vs. unitary‐pictorial vs. equation vs. equation‐pictorial) ANOVA to examine the relationship between instructional approach, student prior knowledge, and personal belief of best practice for learning of the find‐whole percentage problems, which pose a challenge for many middle school students. The unit percentage concept is central to both the unitary approach and the unitary‐pictorial approach, where the latter has a diagram to scaffold the unit percentage concept. Both the equation approach and the equation‐pictorial approach, in contrast, are algebra approaches that integrate relevant information to form an equation, allowing learners to solve for an unknown (e.g., x). Furthermore, the equation‐pictorial approach relies on the proportional concept, scaffolded by a diagram to form an equation. A student's best practice, reflected by what is known as the 'actual – optimal bests dichotomy', details her belief in capability to perform task complexity (i.e., simple task vs. complex task). The concept of element interactivity within cognitive load theory framework predicts differential instructional efficiency: equation‐pictorial > equation approach > unitary‐pictorial > unitary. Our findings (N = 218 secondary students) showed that performance outcomes favored high prior knowledge students for complex problems and, to a lesser extent, practice problems and simple problems. Low prior knowledge students benefitted most from the equation‐pictorial approach, and they invested higher mental effort than high prior knowledge students across three approaches (unitary, unitary‐pictorial, equation) but not the equation‐pictorial approach. Importantly, cognitive load imposition, by proxy of students' mental effort, was unrelated to students' belief in optimal best. [ABSTRACT FROM AUTHOR]

The degree of element interactivity determines the complexity and therefore the intrinsic cognitive load of linear equations. The unpacking of linear equations at the level of operational and relational lines allows the classification of linear equations in a hierarchical level of complexity. Mapping similar operational and relational lines across linear equations revealed that multi-step equations are derivatives of two-step equations, which in turn, are derivatives of one-step equations. Hence, teaching and learning of linear equations can occur in a sequential manner so that learning new knowledge (e.g. two-step equations) is built on learners' prior knowledge (e.g. one-step equations), thus reducing working memory load. The number and nature of the operational line as well as the number of relational lines also affects the efficiency of instructional method for linear equations. Apart from the degree of element interactivity, the presence of complex element (e.g. fraction, negative pronumeral) also increases the complexity of linear equations and thus poses a challenge to the learners.

We examined the use of balance and inverse methods in equation solving. The main difference between the balance and inverse methods lies in the operational line (e.g. +2 on both sides vs -2 becomes +2). Differential element interactivity favours the inverse method because the 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 an experimental study, 63 students (mean age = 13) were randomly allocated to either balance or inverse group to undertake a pre-test, study an instruction sheet, complete acquisition equations, sit for a post-test and a concept test. Procedural knowledge was assessed on performance on practice equations and post-test, whereas conceptual knowledge was assessed on performance on the concept test. The inverse group outperformed the balance group on practice equations but not the post-test. Both the balance and inverse groups scored higher on the inverse concept test than the balance concept test. Positive association between performance on procedural knowledge and performance on conceptual knowledge was found for the inverse group but not the balance group. Overall, the evidence obtained indicates a number of educational implications for implementation.

Central to equation solving is the maintenance of equivalence on both sides of the equation. However, when the process involves an interaction of multiple elements, solving an equation can impose a high cognitive load. The balance method requires operations on both sides of the equation, whereas the inverse method involves operations on one side only. In an experiment, middle school students (N?=?71) were randomly assigned to the balance and inverse methods to complete a pre-test, an acquisition phase and a post-test. Pre-test and post-test comparisons found that the inverse group outperformed the balance group for those equations that involved high element interactivity. Instructional efficiency measures further confirmed that the balance method imposed higher cognitive load. The inverse method was capable of reducing cognitive load due to the interacting elements.