Researchers have speculated that children find it more difficult to acquire conceptual understanding of the inverse relation between multiplication and division than that between addition and subtraction. We reviewed research on children and adults' use of shortcut procedures that make use of the inverse relation on two kinds of problems:…

Given a set of oriented hyperplanes P = {p1, . . . , pk} in R[superscript n], define v : R[superscript n] [right arrow] R by v(X) = the sum of the signed distances from X to p[subscript 1], . . . , p[subscript k], for any point X [is a member of] R[superscript n]. We give a simple geometric characterization of P for which v is constant, leading to…

The usual definition of the Riemann integral as a limit of Riemann sums can be strengthened to demand more of the function to be integrated. This super-Riemann integrability has interesting properties and provides an easy proof of a simple change of variables formula and a novel characterization of derivatives. This theory offers teachers and…

We construct semigroups with any given positive rational commuting probability, extending a Classroom Capsule from November 2008 in this Journal.

We give an example of cross coursing in which a subject or approach in one course in undergraduate mathematics is used in a completely different course. This situation crosses falling body modelling in an upper level differential equations course into a modest discrete dynamical systems unit of a first-year mathematics course. (Contains 1 figure.)

This is a report of a study of students' understanding of infinite series. It has a three-fold purpose: to show that students may construct two essentially different notions of infinite series, to show that one of the constructions is particularly difficult for students, and to examine the way in which these two different constructions may be…

The author tells the story of an exploration he undertook, what he learned, and the questions he was able to answer as a result. Thinking, on the part of the learner, is complex, far from explicit, and some might say intangible. But, by recognising certain "clues" it might be possible to begin to understand how different types of thinking…

An idempotent satisfies the equation x[superscript 2] = x. In ordinary arithmetic, this is so easy to solve it's boring. We delight the mathematical palette here, topping idempotents off with modular arithmetic and a series of exercises determining for which n there are more than two idempotents (mod n) and exactly how many there are.

Finding a series expansion, such as Taylor series, of functions is an important mathematical concept with many applications. Homotopy perturbation method (HPM) is a new, easy to use and effective tool for solving a variety of mathematical problems. In this study, we present how to apply HPM to obtain a series expansion of functions. Consequently,…

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of…

The golden ratio, one of the most beautiful numbers in all of mathematics, arises in some surprising places. At first glance, we might expect that a General checking his troops' progress would be nothing more than a basic distance-rate-time problem. However, further exploration reveals a multi-faceted problem, one in which the ratio of rates…

The Venn diagram is suggested as a graphical solution to conjunction fallacies and a modification of it is suggested to more fully communicate set relations.

In recent years there have been calls for ambitious mathematics teaching which places student thinking and reasoning at the centre of instruction. Drawing on a larger study concerning implementation of practice-based pedagogies within our initial teacher education mathematics programme, this paper examines the range of opportunities for…

Learning to teach is a complex matter, and many different models of pre-service teacher education have been used to support novice teachers' preparation for the classroom. More recently there have been calls for a focus on core high-leverage teaching practices and for novice teachers to engage in representations, decompositions, and approximations…

To be able to support meaningful mathematical experiences, preservice elementary school teachers (PSTs) must learn mathematics in deep and meaningful ways (Ma 1999). They need to experience investigating and making sense of the mathematics they will be called on to teach. To expand their own--often limited--views of what it means to teach and…