Supporting Mathematical Discussions: the Roles of Comparison and Cognitive Load

Fraction Ball impact on student and teacher math talk and behavior

We assessed the impacts of Fraction Ball-a novel suite of games combining the benefits of embodied guided play for math learning-on the math language production and behavior of students and teachers. In the Pilot Experiment, 69 fifth and sixth graders were randomly assigned to play four different Fraction Ball games or attend normal physical education class. The Efficacy Experiment was implemented to test improvements made through co-design with teachers with 160 fourth through sixth graders. Researchers observed and coded for use of math language and behavior. Playing Fraction Ball resulted in consistent increases of students' and teachers' use of fraction (SDs = 0.98-2.42) and decimal (SDs = 0.65-1.64) language and number line arithmetic, but not in whole number, spatial language, counting, instructional gesturing, questioning, and planning. We present evidence of the math language production in physical education and value added by Fraction Ball to support rational number language and arithmetic through group collaboration.

Differential instructional effectiveness: 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° = $$\frac{x}{6}$$ x 6 vs. cos50° = $$\frac{14}{x}$$ 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.

Exploring students' mathematical discussions in a multi-level hybrid learning environment

The research described in this paper focused on the issue of describing and understanding how mathematical discussion develops in a hybrid learning environment, and how students participate in it. The experimental plan involved several classes working in parallel, with pupils and teachers interacting both in their real classrooms and in a digital environment with other pupils and teachers. The research was based on a rich set of data collected from the M@t.abel 2020 project, which was developed in Italy during the Covid health crisis. Based on Complementary Accounts Methodology, the data analysis presented in this paper involved specialists from the fields of mathematics education and inclusive education. In the study we considered the complexity of learning and the different elements that have an impact on students' activity and participation, when they are engaged in mathematical discussions within the multilevel-digital environment that emerged due to the pandemic. These parallel analyses showed that 'mathematical discussion in the classroom' is a complex (and sometimes chaotic) phenomenon wherein different factors interweave. A complementary approach assists in developing a global vision for this dynamic phenomenon and in highlighting local episodes that are crucial in this interplay of factors. It is precisely in these episodes that the role of the teacher is fundamental: these episodes appear as catalysts for the different variables, with the teacher acting as mediator.

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.

How Can Cognitive-Science Research Help Improve Education? The Case of Comparing Multiple Strategies to Improve Mathematics Learning and Teaching

The current article focuses on efforts to understand how a basic learning process—comparison—can be harnessed to improve learning, especially mathematics learning in schools. To harness the power of comparison in instruction, we must investigate three core decisions: what, when, and how to compare. Comparing different strategies for solving the same problem or easily confusable problem types is particularly effective for supporting mathematics learning. Comparing examples early in the learning process can be challenging, but delaying comparison can reduce procedural flexibility. Indeed, comparison is resource demanding, so it is more impactful when carefully supported (e.g., side-by-side visual presentation, explanation prompts). To bridge from research to practice, we communicated research findings to teachers and policymakers and developed curricular materials, instructional routines, and professional-development materials to help math teachers leverage these learning processes. We conclude this review with key open questions.

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.

The Use of a Diagnostic Competence Model About Children’s Operation Sense for Criterion-Referenced Individual Feedback in a Large-Scale Formative Assessment

We provide evidence of validity for a newly developed diagnostic competence model of operation sense, by both (a) describing the theoretically substantiated development of the competence model in close association with its use within a large-scale formative assessment and (b) providing empirical evidence for the theoretically described cognitive levels of competences. The competence model describes students’ operation sense on four distinct levels. On each level, the model elaborates on the characteristics of tasks that students on this level are able to answer correctly. Moreover, the model explains this by referring to two kinds of cognitive processes that are supposed to be necessary to respond to these kinds of tasks successfully. In a validation study, about 85% of the variance in the item difficulties was explained by the four, a priori allocated, levels of operation sense. We discuss the relevance of the validation of the diagnostic competence model for the provision of criterion-referenced feedback in a large-scale formative assessment, including suggestions for teachers’ subsequent support activities, and the contributions of the model to the state of research about operation sense.

Using Relational Reasoning Strategies to Help Improve Clinical Reasoning Practice

Clinical reasoning-the steps up to and including establishing a diagnosis and/or therapy-is a fundamentally important mental process for physicians. Unfortunately, mounting evidence suggests that errors in clinical reasoning lead to substantial problems for medical professionals and patients alike, including suboptimal care, malpractice claims, and rising health care costs. For this reason, cognitive strategies by which clinical reasoning may be improved-and that many expert clinicians are already using-are highly relevant for all medical professionals, educators, and learners.In this Perspective, the authors introduce one group of cognitive strategies-termed relational reasoning strategies-that have been empirically shown, through limited educational and psychological research, to improve the accuracy of learners' reasoning both within and outside of the medical disciplines. The authors contend that relational reasoning strategies may help clinicians to be metacognitive about their own clinical reasoning; such strategies may also be particularly well suited for explicitly organizing clinical reasoning instruction for learners. Because the particular curricular efforts that may improve the relational reasoning of medical students are not known at this point, the authors describe the nature of previous research on relational reasoning strategies to encourage the future design, implementation, and evaluation of instructional interventions for relational reasoning within the medical education literature. The authors also call for continued research on using relational reasoning strategies and their role in clinical practice and medical education, with the long-term goal of improving diagnostic accuracy.