If a curriculum developer's goal is to create a single linear sequence of tasks that lead to the development of some important mathematical concept, then some researchers have suggested that these sequences should follow progressions similar to stages of development that have been identified in Piaget-like research on the relevant concept(s).…

Suggests that there is continued poverty in student understanding of proof. Reports on how proof attitudes could be inculcated in students by offering them a course design that is somewhat faithful to the historical genesis of modern mathematics. Contends that such a design can enable students to discover a sense of proof for themselves. (Contains…

We present a lottery project for lower level mathematics courses that demonstrates financial decision making in a context familiar to students. Students use elementary principals of probability and financial mathematics to compare the expected values of buying Powerball lottery tickets to some basic savings plans. Then, they compare the expected…

Most serious rock climbers are familiar with a counter-intuitive fact about their sport: The force experienced by a falling climber due to the rope as it arrests his fall does not depend simply on the length of the fall, but rather on a ratio called the fall-factor. This article explains, using elementary physics and simple differential equations,…

As with natural numbers, a greatest common divisor of two Gaussian (complex) integers "a" and "b" is a Gaussian integer "d" that is a common divisor of both "a" and "b". This article explores an algorithm for such gcds that is easy to do by hand.

The authors give a rigorous treatment of the differentiability of the exponential function that uses only differentiable calculus. It can thus make "early transcendental" courses complete.

What may have been the birth of a new calculus problem took place when the author noticed that two coffee cups, one convex and one concave, fit nicely together, and he wondered which held more coffee. The fact that their volumes were about equal led to the topic of this article: complementary surfaces of revolution with equal volumes.

This article has its genesis in an MAA mini-course on origami, where a way to get a parabola by folding paper was presented. This article discusses the methods and mathematics of other curves obtained by paper-folding.

Consider the series [image omitted] where the value of each a[subscript n] is determined by the flip of a coin: heads on the "n"th toss will mean that a[subscript n] =1 and tails that a[subscript n] = -1. Assuming that the coin is "fair," what is the probability that this "harmonic-like" series converges? After a moment's thought, many people…

This paper examines the mathematical work of the French bishop, Nicole Oresme (c. 1323-1382), and his contributions towards the development of the concept of graphing functions and approaches to investigating infinite series. The historical importance and pedagogical value of his work will be considered in the context of an undergraduate course on…

In "Beyond Pascals Triangle" the author demonstrates ways of using "Pascallike" triangles to expand polynomials raised to powers in a fairly quick and easy fashion. The recursive method could easily be implemented within a spreadsheet, or simply by using paper and pencil. An explanation of why the method works follows the several examples that are…

We find infinite series in calculus to be one of the most confusing topics our students encounter. In this note, we look at some issues that our students find difficult or ambiguous involving the Ratio Test, the Root Test, and also the Alternating Series Test. We offer some suggestions and some examples, which could be a supplement to the set of…

Within statistics instruction, students are often requested to sketch the curve representing a normal distribution with a given mean and standard deviation. Unfortunately, these sketches are often notoriously imprecise. Poor sketches are usually the result of missing mathematical knowledge. This paper considers relationships which exist among…

Two simple mathematical models for how individual vehicles follow each other along a stretch of road are discussed. The resulting difference equations can be used as applications of techniques taught at A-level and first year undergraduate level, and as an introduction to the behaviour of the logistic map.

This note considers functions of two variables which are continuous on a possibly unbounded closed region in [vertical bar]R[squared], and the functions of one variable obtained by integrating out the other variable over this region. The question of continuity of these functions is investigated, as are connections with joint density and marginal…