When studying correlations, how do the three bivariate correlation coefficients between three variables relate? After transforming Pearson's correlation coefficient r into a Euclidean distance, undergraduate students can tackle this problem using their secondary school knowledge of geometry (Pythagoras' theorem and similarity of triangles).…

Suppose that we are given a rectangular box in 3-space. Given any two points on the surface of this box, we can define the surface distance between them to be the length of the shortest path between them on the surface of the box. This paper determines the pairs of points of maximum surface distance for all boxes. It is often the case that these…

Fibonacci's forgotten number is the sexagesimal number 1;22,7,42,33,4,40, which he described in 1225 as an approximation to the real root of x[superscript 3] + 2x[superscript 2] + 10x - 20. In decimal notation, this is 1.36880810785...and it is correct to nine decimal digits. Fibonacci did not reveal his method. How did he do it? There is also a…

Positive sums count. Alternating sums match. Alternating sums of binomial coefficients, Fibonacci numbers, and other combinatorial quantities are analyzed using sign-reversing involutions. In particular, we describe the quantity being considered, match positive and negative terms through an Involution, and count the Exceptions to the matching rule…

Motivated by centaurs jumping around a circular stadium, we derive Kronecker's Approximation Theorem, which in turn provides elementary solutions to difficult problems in the theory of Diophantine approximations.

Using only fairly simple and elementary considerations--essentially from first year undergraduate mathematics--we show how the classical Stokes' theorem for any given surface and vector field in R[superscript 3] follows from an application of Gauss' divergence theorem to a suitable modification of the vector field in a tubular shell around the…

This volume, a comprehensive survey and critical analysis of today's issues in mathematics education, distills research to build knowledge and capacity in the field. The compendium is a valuable new resource that provides the most comprehensive evidence about what is known about research in mathematics education. The 38 chapters present five…

This study adds momentum to the ongoing discussion clarifying the merits of visualization and analysis in mathematical thinking. Our goal was to gain understanding of three calculus students' mental processes and images used to create meaning for derivative graphs. We contrast the thinking processes of these three students as they attempted to…

Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the…

In June 2018 the Canadian Mathematics Education Study Group/Groupe Canadien d'étude en didactique des mathématiques (CMESG/GCEDM) held its 42nd meeting in the idyllic setting of Squamish, British Columbia. This meeting marked the first time CMESG/GCEDM had been in British Columbia since 2010 and the first time it had been held at Quest University.…

The seventeen-year-old Christiaan Huygens was the first to prove that a hanging chain did not take the form of the parabola, as was commonly thought in the early seventeenth century. We will examine Huygen's geometrical proof, and we will investigate the later history of the catenary.

A stopper is called "universal" if it can be used to plug pipes whose cross-sections are a circle, a square, and an isosceles triangle, with the diameter of the circle, the side of the square, and the base and altitude of the triangle all equal. Echoing the well-known result for equal cubes that is attributed to Prince Rupert, we show that it is…

If you break a stick at two random places, the probability that the three pieces form a triangle is 1/4. How does this generalize? To answer this question, we give a method for finding the probability that n randomly chosen points in a given interval fall within a specified distance of one another. We use this method to provide solutions to…

Parametric differentiation and integration under the integral sign constitutes a powerful technique for calculating integrals. However, this topic is generally not included in the undergraduate mathematics curriculum. In this note, we give a comprehensive review of this approach, and show how it can be systematically used to evaluate most of the…

How fast does a tank drain? Of course this depends on the shape of the tank and is governed by a physical principle known as Torricelli's law. This note investigates some connections between tank shape and a mathematical function related to the time required for the tank to drain completely. The techniques employed provide some interesting…