The development, in an introductory differential equations course, of boundary value problems in parallel with initial value problems and the Fredholm Alternative. Examples are provided of pairs of homogeneous and nonhomogeneous boundary value problems for which existence and uniqueness issues are considered jointly. How this heightens students'…

The paper examines the roles and purposes of proof mentioned by university research faculty when reflecting on their own teaching and teaching at their institutions. Interview responses from 14 research mathematicians and statisticians who also teach are reported. The results suggest there is a great deal of variation in the role and purpose of…

Augustus De Morgan (1806-1871) was a significant Victorian Mathematician who made contributions to mathematics history, mathematical recreations, mathematical logic, calculus, and probability and statistics. He was an inspiring mathematics professor who influenced many of his students to join the profession. One of De Morgan's significant books…

Successful outcomes for a "Transition Course in Mathematics" have resulted from two unique design features. The first is to run the course as a "survey course" in mathematics, introducing sophomore-level students to a broad set of mathematical fields. In this single mathematics course, undergraduates benefit from an introduction of proof…

In this paper, we present data from an exploratory study that aimed to investigate the ways in which, and the extent to which, undergraduates enrolled in a transition-to-proof course considered examples in their attempted proof constructions. We illustrate how some undergraduates can and do use examples for specific purposes while successfully…

Mathematical statements involving both universal and existential quantifiers occur frequently in advanced mathematics. Despite their prevalence, mathematics students often have difficulties interpreting and proving quantified statements. Through task-based interviews, this study took a qualitative look at undergraduate mathematics students'…

Biangular coordinates specify a point on the plane by two angles giving the intersection of two rays emanating from two fixed poles. This is a dual of Cartesian coordinates wherein a point on the plane is described by two distances. Biangular coordinates, first written about in 1803 in France, were subsequently studied in Britain at the end of the…

This submission contains the Proceedings of the 2016 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at Queen's University in Kingston, Ontario. 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 Study…

We calculate the number of subgroups in a direct product of finite cyclic groups by applying the fundamental theorem of finite abelian groups and a well-known structure theorem due to Goursat. We also suggest ways in which the results can be generalized to a direct product of arbitrary finite groups.

We give an alternative to the standard method of reduction or order, in which one uses one solution of a homogeneous, linear, second order differential equation to find a second, linearly independent solution. Our method, based on Abel's Theorem, is shorter, less complex and extends to higher order equations.

Four different families of linear recurrences are derived for Catalan numbers. The derivations rest on John Riordan's 1973 generalization of Catalan numbers to a set of polynomials. Elementary differential and integral calculus techniques are used and the results should be of interest to teachers and students of introductory courses in calculus…

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].

This study presents the results of the analysis of a group of teacher-made test questions for statistics courses at the university level. Teachers were asked to submit tests they had used in their previous two semesters. Ninety-seven tests containing 978 questions were gathered and classified according to the SOLO taxonomy (Biggs & Collis,…

We consider the expected range of a random sample of points chosen from the interval [0, 1] according to some probability distribution. We then use the notion of convexity to derive an upper bound for this expected range which is valid for all possible choices of this distribution. Finally we show that there is only one distribution for which this…

When playing pool or billiards, a player often has the opportunity to make a "straight-in" shot, that is, one in which the cue ball, the object ball, and the target (e.g., a pocket) are collinear. With the distance from the cue ball to the target assumed fixed, the relative difficulty is here explored of shots taken at varying positions of the…