Search

Good mathematics stands the test of time. As culture changes, we often ask different questions, bringing new perspectives, but modern mathematics stands on ancient discoveries. Isaac Newton’s discovery of calculus (along with Leibniz) may seem old but is predated by Archimedes’ findings. Current mathematics students should be familiar with parabolas and simple curves; in our introductory calculus courses, we teach them to compute the areas under such curves. Our modern approach derives its roots from Newton’s work; however, we have filled in many of the gaps in the pursuit of mathematical rigor. What many students may not know is that Archimedes solved the area problem for parabolas long before the use of algebraic expressions became mainstream. Archimedes used the geometry of the ancient Greeks, which gave him a vastly different perspective. In this paper, we provide both Archimedes’ and Newton’s proofs involving the quadrature of the parabola, trying to remain true to their original texts as much as feasible. [ABSTRACT FROM AUTHOR]

Little is known about instructional means by which the aesthetic experience of mathematics can be enhanced for undergraduate learners. This paper presents and discusses an iterative lesson design process towards creating an opportunity for students to appreciate the beauty of an unexpected solution to a challenging calculus problem. The lesson design draws on insights from both mathematics education research on aesthetics and research on aesthetic appreciation in music. The data were collected over the course of five lessons with different groups of calculus students in which the intended problem was presented in two different ways. In addition, stimulated-recall interviews were conducted with nine students who took part in the later lessons and exhibited strong emotions regarding the problem. The data suggest that the students' aesthetic response to the problem was essentially conditioned by the extent of their surprise as a result of revealing a clever solution to the problem after being exposed to repeated failed attempts. Implications for practice are drawn. [ABSTRACT FROM AUTHOR]

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 Livescribe pen 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. [ABSTRACT FROM AUTHOR]

In this work, a faculty member takes a journey along with students as they enhance their understanding of how people solve mathematical problems through a mainly qualitative statistical project. Student authors of this paper registered for a problem solving seminar led by the faculty author, and then created and analyzed self-built assessment tools to explore problem solving techniques. Here we share our findings and recommendations, which we hope will inspire others to explore novel pedagogical techniques in the teaching of mathematical problem solving. We incorporate into our presentation our voices, reflecting on how we and others solve problems. [ABSTRACT FROM AUTHOR]

When a new mathematical idea is presented to students in terms of abstract mathematical symbols, they may have difficulty to grasp it. This difficulty arises because abstract mathematical symbols do not directly refer to concretely perceivable objects. But, when the same content is presented in the form of a graph or a gesture that depicts that graph, it is often much easier to grasp. The process of solving a complex mathematical problem can also be facilitated with the use of a graphical representation. Transforming a mathematical problem or concept into a graphical representation is a common problem solving strategy, and we may view it as a kind of mathematical metaphor, in the sense that a certain representation of a mathematical problem is described in terms of a visual representation of that problem. Furthermore, since a graphical representation is visual, it can be depicted by gestures. Therefore, visual and motor systems can be actively employed to process a given problem and find a solution for it. In this way, mathematical metaphor offers us a way to employ a wider range of cognitive resources to understand mathematics. [ABSTRACT FROM AUTHOR]

Mathematical expertise demands effective thinking and learning methods, and these techniques transfer well to other domains. In this article, I discuss how my own training as a pure mathematician influenced my performance in three disparate domains: electrical engineering, art appreciation, and learning Italian. In electrical engineering, the focus is on how mathematical reasoning and thinking processes impact knowledge acquisition and problem solving. Appreciating and analyzing art raises the question, “How do we know for certain?” Acquiring fluency in another language is akin to gaining mathematics proficiency, and here, I explore the human side of persistence. The article combines narrative, reflection, analysis, and teaching ideas to suggest how, when teaching our subject, the mathematics community might pass on our core strength: our thinking and learning methodology. [ABSTRACT FROM AUTHOR]

In this article I describe Adelphi University's Mathematics Orientation Seminar, a new course that was introduced into the mathematics major to help students find their passion in mathematics and to strengthen the educational community within our department. I discuss quantitative and qualitative results of surveys among students in the Mathematics Orientation Seminar in Fall 2016 and Fall 2017, which suggest that this might be a useful course for other institutions to utilize within any major. Finally, I explore faculty perspectives and describe what I believe to be the final version of this course. [ABSTRACT FROM AUTHOR]

Gyöorgy Pólya's problem solving method has influenced generations of mathematicians and non-mathematicians alike. Though almost all math teachers have come across Pólya's problem solving method, his ideas are not regularly implemented in the classroom. Few studies have examined the effectiveness of his approach in teaching remedial math. In this article we revisit this once well-known teaching method and show how it can be used in basic skills math classes to ease student fears of math, and potentially change their common misconceptions of the subject. [ABSTRACT FROM AUTHOR]