The open-ended assessment item supports students in achieving mathematical understanding that the National Council of Teachers of Mathematics (NCTM) advocates. The items encourage students to demonstrate their understanding in creative and informative ways.

Explores the understanding that students who are training to be secondary mathematics teachers have of elementary concepts in mathematical analysis. Research involved probing students' conceptual understanding of convergence and continuity and associated reasoning. Concludes that even mathematically well-qualified students have difficulties with…

The origin of the function concept is usually traced to Galileo's work on motion. We argue that specific proto-function concepts appeared in the work of Tartaglia a century before the publication of Galileo's Two New Sciences. The study of Tartaglia's ideas can be used in the classroom as a historical introduction to various function concepts,…

A problem on a state's high school exit exam asked for the longest straw that would fit in a box. The examiners apparently wanted the length of a diagonal of the box, but the figure accompanying the question suggested otherwise--that the radius of the straw be considered. This article explores that more general problem.

How does artificial gravity affect the path of a thrown ball? This paper contrasts ball trajectories on the Little Prince's asteroid planet B-612 and Arthur C. Clarke's rotating-drum spacecraft of 2001, and demonstrates curve balls with multiple loops in the latter environment.

We show that there is a link between a standard calculus problem of finding the best view of a painting and special tangent lines to the graphs of exponential functions. Surprisingly, the exponential function with the "best view" is not the one with the base "e." A similar link is established for families of functions obtained by composing…

In a well-known calculus problem, an open top box is to be made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. The task is to find the dimensions of the box of maximum volume. Typically, the length of the sides of the corners that produces the largest volume turns out to be an irrational…

As a precursor to lessons on prime decomposition and reducing fractions, rules are generally presented for divisibility by 2, 3, 5, 9, and 10 and sometimes for those popular composites such as 4 and 25. In our experience students often ask: "What about the one for 7?" and we are loathe to simply state that there isn't one. We have yet to see a…

In terms of modern pedagogy, having visual interpretation of trigonometric functions is useful and quite helpful. This paper presents, pictorially, an easy approach to prove all single angle trigonometric identities on the axes. It also discusses the application of axial representation in calculus--finding the derivative of trigonometric functions.

In this work, we present an algorithm for computing logarithms of positive real numbers, that bears structural resemblance to the elementary school algorithm of long division. Using this algorithm, we can compute successive digits of a logarithm using a 4-operation pocket calculator. The algorithm makes no use of Taylor series or calculus, but…

Although variability is of fundamental concern and interest to statisticians, often this does not get communicated to students who are taught instead to view variability as a nuisance parameter. A brief survey of a few case studies, as well as a recounting of some history, shows that variability is worthy of study in its own right, and examination…

In this article, the authors begin by asking a simple question: To what extent can art education be related to mathematics education? One reason for asking this is that there is, on the one hand, a significant body of claims that assert that mathematics is an art, and, on the other hand, work in art that has a mathematical basis. Observations of…

The notions of "abstract" and "concrete" are central to the conceptualization of mathematical knowing and learning. Much of the literature takes a dualist approach, leading to the privileging of the former term at the expense of the latter. In this article, we provide a concrete analysis of a scientist interpreting an unfamiliar graph to show how…

This note uses some elementary concepts in linear algebra to prove some well-known results in two- and three-dimensional geometry.

The topic of random numbers is investigated in such a way as to illustrate links between mathematics, physics and computer science. First, the generation of random numbers by a classical computer using the linear congruential generator and logistic map is considered. It is noted that these procedures yield only pseudo-random numbers since…