Diferential instructional efectiveness: overcoming the challenge of learning to solve trigonometry problems that involved algebraic transformation skills

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 eficient 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 eficiency for learning to solve trigonometry problems

Expanding the scope of “trans-humanism”: situating within the framework of life and death education – the importance of a “trans-mystical mindset”

Life and death education, as noted from the literatures, has been studied and researched extensively in China, Malaysia, and Taiwan. Our own research undertakings over the past several years, situated in different sociocultural settings have delved into aspects of life and death that could help advance theoretical understanding of the subject matters (e.g., does the meaning of “effective life functioning” connote differing interpretations for different cultural groups?). Situating within the framework of life and death education, we expand the study of trans-humanism by introducing an extended prefix or nomenclature known as “trans-mystical”. Specifically, our philosophized concept of trans-mysticism considers a related concept, which we term as a “trans-mystical mindset”. A trans-mystical mindset, differing from an ordinary mindset, from our philosophical rationalization, is defined as “a person’s higher-order state of consciousness, espousing her perception, judgment, belief, and attempted interpretation of life and death phenomena that are mystifying and fall outside the ordinary boundaries of human psyche.” Our focus of inquiry, as reported in the present article, seeks to advance our proposition: that a trans-mystical mindset, unlike an ordinary mindset, may help a person to rationalize, appreciate, and understand metaphysical contexts, mystical experiences, and the like. This focus, interestingly, serves to highlight an important discourse - namely, that there is a dichotomy in theoretical lenses (i.e., objective reality vs. individual subjectivity) that a person may use to rationalize the significance or non-significance of universal contexts, events, phenomena, etc. (e.g., a person’s experience of “premonition”). As such, then, there is an important question that we seek to consider: whether philosophization, or the use of philosophical psychology, would yield perceived “scientific evidence” to support or to reject the study of metaphysicism, mysticism, and the like? For example, does our philosophization of an “equivalency” between a person’s trans-mystical mindset and her experience of self-transcendence help to normalize and/or to scientize the subject matters of metaphysicism, mysticism, etc.

Solution representations of percentage change problems: the pre-service primary teachers' mathematical thinking and reasoning

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

Evaluating a CALL software on the learning of English prepositions

This study assessed the effectiveness of using a CALL lesson (Computer Assisted Language Learning) over a conventional lesson to facilitate learning of English prepositions at Bario, Malaysia. CALL was developed by the Ministry of Education, Malaysia as support material to enhance learning of English prepositions. Both the conventional and the CALL lessons were matched with the same content except for the medium in which the lesson was being delivered. Students were provided with computers to go through the CALL lesson in a self-regulated manner; while a teacher taught the conventional lesson in a classroom. Test results indicate that students who received the conventional lesson outperformed those who went through the CALL lesson. The Relative Condition Efficiency measurement also showed that the conventional group learned more efficiently than the CALL group. The findings are interpreted from the perspective of cognitive load required in processing the presentation mode of the learning materials

A Case for Cognitive Entrenchment: To Achieve Optimal Best, Taking Into Account the Importance of Perceived Optimal Efficiency and Cognitive Load Imposition

One interesting observation that we may all concur with is that many experts, or those who are extremely knowledgeable and well-versed in their respective domains of functioning, become “mediocre” and lose their “touch of invincibility” over time. For example, in the world of professional football, it has been argued that an elite football coach would lose his/her air of invincibility and demise after 10–15 years at the top. Why is this the case? There are different reasons and contrasting viewpoints that have been offered to account for this observed demise. One notable concept, recently introduced to explain this decline, is known as cognitive entrenchment, which is concerned with a high level of stability in one's domain schemas (Dane, 2010). This entrenchment or “situated fixation,” from our proposition, may act to deter the flexibility and/or willingness of a person to adapt to a new context or situation. Some writers, on this basis, have argued that cognitive entrenchment would help explain the demise of some experts and/or why some students have difficulties adapting to new situations. An initial inspection would seem to indicate that cognitive entrenchment is detrimental, potentially imparting evidence of inflexibility, difficulty, and/or the unwillingness of a person to adapt to new contexts (Dane, 2010). This premise importantly connotes that expertise may constrain a person from being flexible, innovative, and/or creative to ongoing changes. In this analysis, an expert may experience a cognitive state of entrenchment, facilitated in this case by his/her own experience, knowledge, and/or theoretical understanding of a subject matter. Having said this, however, it is also a plausibility that cognitive entrenchment in itself espouses some form of positivity, giving rise to improvement and/or achievement of different types of adaptive outcomes. Drawing from our existing research development, we propose in this conceptual analysis article that personal “entrenchment” to a particular context (e.g., the situated fixation of a football coach to a particular training methodology) may closely relate to three major elements: self-cognizance of cognitive load imposition, a need for efficiency, and the quest for stability and comfort. As we explore later, there is credence to accept the “positivity” of cognitive entrenchment—that by nature, for example, a person would purposively choose the status quo in order to minimize cognitive load imposition, optimize efficiency, and/or to achieve minimum disruption and a high level of comfort, which could then “optimize” his/her learning experiences. We strongly believe that our propositions, which consider eight in this article, are of significance and may, importantly, provide grounding for further research development into the validity of cognitive entrenchment

An Examination of Pre-Service Teachers' Content Knowledge on Linear Equations: A Cross-Cultural Study

We examined Australian and Malaysian pre-service teachers' cognitive process in solving linear equations from a cognitive load perspective. Seventy six Australian and Malaysian pre-service teachers were randomly assigned to solve one-step, two-step and multi-step linear equations, and to undertake a concept test. For Australian pre-service teachers, both the one-step and two-step groups outperformed the multi-step group. In contrast, no difference between the one-step, two-step and multi-step groups was observed for Malaysian pre-service teachers. Moreover, all the three groups of Malaysian pre-service teachers outperformed the corresponding groups of Australian pre-service teachers. Regarding the concept test, both Australian and Malaysian pre-service teachers performed better on the inverse operation (e.g., -4 becomes +4) than the balance operation (e.g., +4 on both sides). The inverse operation incus half as many interactive elements as the balance operation, thus imposing lower cognitive load

Learning linear equations: capitalizing on cognitive load theory and learning by analogy

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

Developing Problem-Solving Expertise for Word Problems

Studying worked examples impose relatively low cognitive load because learners’ attention is directed to learn the schema, which is embedded in the worked examples. That schema encompasses both conceptual knowledge and procedural knowledge. It is well documented that worked examples are effective in facilitating the acquisition of problem solving skills. However, the use of worked examples to develop problem-solving expertise is less known. Typically, experts demonstrate an efficient way to solve problems that is quicker, faster, and having fewer solution steps. We reviewed five studies to validate the benefit of worked examples to develop problem-solving expertise for word problems. Overall, a diagram portrays the problem structure, coupled with either study worked examples or complete multiple example–problem pairs, facilitates the formation of an equation to solve words problems efficiently. Hence, an in-depth understanding of conceptual knowledge (i.e., problem structure) might contribute to superior performance of procedural knowledge manifested in the reduced solution steps

Learning to Solve Trigonometry Problems That Involve Algebraic Transformation Skills via Learning by Analogy and Learning by Comparison

The subject of mathematics is a national priority for most countries in the world. By all account, mathematics is considered as being "pure theoretical" (Becher, 1987), compared to other subjects that are "soft theoretical" or "hard applied." As such, the learning of mathematics may pose extreme difficulties for some students. Indeed, as a pure theoretical subject, mathematics is not that enjoyable and for some students, its learning can be somewhat arduous and challenging. One such example is the topical theme of Trigonometry, which is relatively complex for comprehension and understanding. This Trigonometry problem that involves algebraic transformation skills is confounded, in particular, by the location of the pronumeral (e.g., x)—whether it is a numerator sin30° = x/5 or a denominator sin30° = 5/x. More specifically, we contend that some students may have difficulties when solving sin30° = x/5, say, despite having learned how to solve a similar problem, such as x/4 = 3. For more challenging Trigonometry problems, such as sin50° = 12/x where the pronumeral is a denominator, students have been taught to “swap” the x with sin30° and then from this, solve for x. Previous research has attempted to address this issue but was unsuccessful. Learning by analogy relies on drawing a parallel between a learned problem and a new problem, whereby both share a similar solution procedure. We juxtapose a linear equation (e.g., x/4 = 3) and a Trigonometry problem (e.g., sin30° = x/5) to facilitate analogical learning. Learning by comparison, in contrast, identifies similarities and differences between two problems, thereby contributing to students’ understanding of the solution procedures for both problems. We juxtapose the two types of Trigonometry problems that differ in the location of the pronumeral (e.g., sin30° = x/5 vs. cos50° = 20/x) to encourage active comparison. Therefore, drawing on the complementary strength of learning by analogy and learning by comparison theories, we expect to counter the inherent difficulty of learning Trigonometry problems that involve algebraic transformation skills. This conceptual analysis article, overall, makes attempts to elucidate and seek clarity into the two comparative pedagogical approaches for effective learning of Trigonometry

Algebra word problem solving approaches in a chemistry context : equation worked examples versus text editing

Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three equation steps each of which comprises essential units of relevant information. In an experiment, students were randomly assigned to either text editing or equation worked examples condition in a regular classroom setting to learn how to solve algebra word problems in a chemistry context. The equation worked examples group outperformed the text editing group for molarity problems, which were more difficult than dilution problems. Empirical evidence supports the theoretical rationale in using equation worked examples to facilitate students' construction of a coherent problem structure so as to develop problem skills and expertise to solve molarity problems