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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 6 vs. cos50° = 14 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. [ABSTRACT FROM AUTHOR]

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 L1) and his/her optimal level of best practice (denoted as L2). 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 MPlus 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., →relationship with teachers) and domain-specific (e.g., →relationship with peers) self-esteems, (b) the positive influences of social relation with teachers (e.g., →exam result) and interest in learning tasks (e.g., →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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [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. [ABSTRACT FROM AUTHOR]

Eighth grade students in Australia ( N = 60) participated in an experiment on learning how to solve percentage change problems in a regular classroom in three conditions: unitary, pictorial, and equation approaches. The procedure involved a pre-test, an acquisition phase, and a post-test. The main goal was to test the relative merits of the three approaches from a cognitive load perspective. Experimental results indicated superior performance of the equation approach over the unitary or pictorial approach especially for the complex tasks. The unitary approach required students not only to process the interaction between numerous elements within and across solution steps, but also to search for critical information, thus imposing high cognitive load. The pictorial approach did not provide a consistent approach to tackling various percentage change problems. Coupled with the need to coordinate multiple elements within and across solution steps, and the need to search for relevant information in the diagram, this approach imposed high cognitive load. By treating the prior knowledge of percentage quantity as a single unit, the equation approach required students to process two elements only. Empirical evidence and theoretical support favor the equation approach as an instructional method for learning how to solve percentage change problems for eighth graders. [ABSTRACT FROM AUTHOR]

Within the cognitive load theory framework, we designed and compared three alternative instructional solution formats that can be derived from a common static hierarchical network representation depicting problem structure. The interactive-solution format permitted students to search in self-controlled manner for solution steps, static-solution format displayed all solutions steps, and no-solution format did not have solution steps. When we matched instructional time across the formats, in relation to the complex molarity problems rather than the dilution problems, differential transfer performance existed between the static-solution or no-solution formats and the interactive-solution format, but not between the static-solution format and no-solution format. The manner in which learners interact with the static-solution and no-solution formats depends on their level of expertise in the chemistry domain. With considerable learner expertise, provision of solution steps may be redundant incurring extraneous cognitive load. Absence of the solution steps may not have left sufficient cognitive capacity for germane cognitive load as some beginning learners lacked the prior knowledge to deduce the solution steps. Searching for solution steps presumably incurred extraneous cognitive load which interfered with learning and hence, in the interactive-solution format, it outweighed the benefit of engaging in self-regulated interaction with the content. Hence, cognitive load theory is a promising tool to predict the mental load associated with learning from the three alternative computer-based instructional formats. [ABSTRACT FROM AUTHOR]

A correction is presented to the article "Managing Element Interactivity in Equation Solving," that appeared February 24, 2017 issue of the journal.