On Similarities and Differences Between Proving and Problem Solving

A link between proving and problem solving has been established in the literature [5, 21]. In this paper, I discuss similarities and differences between proving and problem solving using the Multidimensional Problem-Solving Framework created by Carlson and Bloom [2] with Livescribepen data from a previous study [13]. I focus on two participants’ proving processes: Dr. G, a topologist, and L, a mathematics graduate student. Many similarities between the framework and the proving processes of Dr. G and L were revealed, but there were also some differences. In addition, there were some distinct differences between the proving actions of the mathematician and that of the graduate student. This study suggests the feasibility of an expanded framework for the proving process that can encompass both the similarities and the differences found

Mathematical Problem-Solving via Wallas’ Four Stages of Creativity: Implications for the Undergraduate Classroom

The central theme in this article is that certain problem-solving frameworks (e.g., Polya, 1957; Carlson & Bloom, 2005) can be viewed within Wallas’ four stages of mathematical creativity. The author attempts to justify the previous claim by breaking down each of Wallas’ four components (preparation, incubation, illumination, verification) using both mathematical creativity and problem-solving/proving literature. Since creativity seems to be important in mathematics at the undergraduate level (Schumacher & Siegel, 2015), the author then outlines three observations about the lack of fostering mathematical creativity in the classroom. Finally, conclusions and future research are discussed, with emphasis on using technological advances such as Livescribe™ pens and neuroscience equipment to further reveal the mathematical creative process

Does Content Matter in an Introduction-to-Proof Course?

Introduction-to-proof courses are becoming more prevalent in mathematics departments as more recognize the need to support students while they transition from courses focused on computation (such as calculus) to proof-intensive courses (such as real analysis). In such introduction courses, there are some common proving techniques to teach (induction, contradiction, and contraposition to name a few), but the content varies from institution to institution. This note adds to the discussion on content in such courses by analyzing two prior studies, one using a coding scheme designed to illuminate step-by-step justifications in a proof, and the other focused on interviews with course instructors. Our analysis of the literature shows that there may be reason to believe that content-based introduction-to-proof courses inadvertently overemphasize specific mathematical-area reasoning, which may not translate effectively to subsequent proof-based courses in different content areas. Simply put, while some mathematicians may be convinced of this, a real analysis, number theory, or abstract algebra course may not be the most effective introduction-to-proof course for students to transition to other proof-based courses

Productive Failures: From Class Requirement to Fostering a Support Group

Mistakes occur frequently in mathematics. Reframing mistakes into positive moments can be psychologically important in a student’s educational journey. We investigated two tertiary math classes that explicitly valued mistakes through a pedagogical requirement called “productive failure”. For a percentage of their grade, students demonstrated how they made mistakes in their problem solving and, most importantly, how they overcame those mistakes. Through interviews, video-stimulated recalls, and evaluations of the course all from students, we initially looked for affectual responses to the pedagogical allowance and student-led demonstration. Many of the responses, both benefits and drawbacks of the productive failure, were interpreted by the research group to resemble the psychology literature on peer-led support groups. Descriptions of both productive failure and support groups, as well as quotes from the students, aim to shed light on psychological benefits of valuing mistakes. Finally, we believe that productive failures benefitted many students because it made the human aspect of mathematics more explicit

Special Issue Call for Papers: Creativity in Mathematics

The Journal of Humanistic Mathematics is pleased to announce a call for papers for a special issue on Creativity in Mathematics. Please send your abstract submissions via email to the guest editors by March 1, 2019. Initial submission of complete manuscripts is due August 1, 2019. The issue is currently scheduled to appear in July 2020

Inquiry as an Entry Point to Equity in the Classroom

Although many policy documents include equity as part of mathematics education standards and principles, researchers continue to explore means by which equity might be supported in classrooms and at the institutional level. Teaching practices that include opportunities for students to engage in active learning have been proposed to address equity. In this paper, through aligning some characteristics of inquiry put forth by Cook, Murphy and Fukawa-Connelly with Gutiérrez\u27s dimensions of equity, we theoretically explore the ways in which active learning teaching practices that focus on inquiry could support equity in the classroom

Silicon Differently Affects Apoplastic Binding of Excess Boron in Wheat and Sunflower Leaves

Monocots and dicots differ in their boron (B) requirement, but also in their capacity to accumulate silicon (Si). Although an ameliorative effect of Si on B toxicity has been reported in various crops, differences among monocots and dicots are not clear, in particular in light of their ability to retain B in the leaf apoplast. In hydroponic experiments under controlled conditions, we studied the role of Si in the compartmentation of B within the leaves of wheat (Triticum vulgare L.) as a model of a high-Si monocot and sunflower (Helianthus annuus L.) as a model of a low-Si dicot, with the focus on the leaf apoplast. The stable isotopes 10B and 11B were used to investigate the dynamics of cell wall B binding capacity. In both crops, the application of Si did not affect B concentration in the root, but significantly decreased the B concentration in the leaves. However, the application of Si differently influenced the binding capacity of the leaf apoplast for excess B in wheat and sunflower. In wheat, whose capacity to retain B in the leaf cell walls is lower than in sunflower, the continuous supply of Si is crucial for an enhancement of high B tolerance in the shoot. On the other hand, the supply of Si did not contribute significantly in the extension of the B binding sites in sunflower leaves. © 2023 by the authors

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements