This article gives an elementary proof of the famous identity [image omitted]. Using nothing more than freshman calculus, the present proof is far simpler than many existing ones. This result also leads directly to Euler's and Neville's identities, as well as the identity [image omitted].

Conventional application of these two calculus staples is stretched here, somewhat recreationally, but also to raise solid questions about the role of limit interchange in analysis--without, however, delving any deeper than first-year Calculus.

Limits are foundational to the central concepts of calculus. However, the authors' experiences with students and educational research abound with examples of students' misconceptions about limits and infinity. The authors wanted calculus students to understand, appreciate, and enjoy their first introduction to advanced mathematical thought. Thus,…

This paper reports on different underlying structures of the derivative schema of three undergraduate students that were considered to be at the trans level of development of the derivative schema (action-process-object-schema). The derivative schema is characterized in terms of the students' ability to explicitly transfer the relationship between…

For almost all students, what happens when they push buttons on their calculators is essentially magic, and the techniques used are seemingly pure wizardry. In this article, the author draws back the curtain to expose some of the mathematics behind computational wizardry and introduces some fundamental ideas that are accessible to precalculus…

In this paper, we explore the development of two grounded theories. One theory is mathematical and grounded in the work of university calculus students' collaborative development of mathematical methods for finding the volume of a solid of revolution, in response to mathematical necessity in problem solving, without prior instruction on solution…

A new proof for the monotonicity of the sequence [image omitted] is given as a special case of a large family of monotomic and bounded, hence convergent sequences. The new proof is based on basic calculus results rather than induction, which makes it accessible to a larger audience including business and life sciences students and faculty. The…

This paper presents details about a summer instructional program created by physics graduate students who sought a chance to help students from the local community and to gain greater experience teaching science and math. The students initiated, designed, and conducted the program, which gave the instructors their first chance to independently…

This submission contains the Proceedings of the 2017 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at McGill University in Montreal, Quebec June 2-6. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning. The aims of the…

The quest to find the equation of a catenary makes an ideal investigation for upper secondary students. In the modelling exercise that follows, no knowledge of calculus is required to gain a fairly good understanding of the nature of the curve. This investigation is best described as a scientific investigation--a "hands on" experience…

Historians of mathematics considered the nineteenth century to be the Golden Age of mathematics. During this time period many areas of mathematics, such as algebra and geometry, were being placed on rigorous foundations. Another area of mathematics which experienced fundamental change was analysis. The drive for rigor in calculus began in 1797…

We describe a modelling activity for students in a course in which modelling with differential equations is appropriate. We have used this model in our coursework for years and have found that it enlightens students as to the model building process and parameter estimation for a linear, first-order, ordinary differential equation. The activity…

After students have constructed a conceptual understanding of a mathematical idea, applying mathematical symbolism to represent and define the idea concisely becomes an easier task. In fact, Moore (1994) found that students "often needed to develop their concept images through examples, diagrams, graphs, and other means before they could…

An online testing system developed for entry-skills testing of first-year university students in algebra and calculus is described. The system combines the open-source computer algebra system "Maxima" with computer scripts to parse student answers, which are entered using standard mathematical notation and conventions. The answers can…

In this article, we suggest an instructional intervention to help students understand statements involving multiple quantifiers in logical contexts. We analyze students' misinterpretations of multiple quantifiers related to the epsilon-N definition of convergence and point out that they result from a lack of understanding of the significance of…