"Cognitive" technology tools have been described as "technologies that help transcend the limitation of the mind... in thinking, learning and problem solving activities" (Pea 1985, p. 168). These tools also respond to a user's commands and make mathematical actions more overtly apparent (Zbiek et al. 2007). By definition, these cognitive…

"Saxon Math" is a textbook series covering grades K-12 based on incremental development and continual review of mathematical concepts to give students time to learn and practice concepts throughout the year. The series is aligned with standards of the National Council of Teachers of Mathematics (NCTM) and various states, and can be…

This paper is the second part of a study of plane transformations using a complex setting. The first part was devoted to homotheties and translations, now attention is turned towards plane isometries. The group theoretic properties of plane isometries are easy to derive and images of classical geometrical objects by these transformations are…

This paper sets out to examine how narrative modes of thinking play a part in the claiming of mathematical territories as our own, in navigating mathematical landscapes and in conversing with the mathematical beings that inhabit them. We begin by exploring what constitutes the narrative mode, drawing principally on four characteristics identified…

One of the most interesting episodes in the saga of angle trisection was Archimedes' discovery that he could trisect an angle if he could make two marks on his straightedge. This note gives a method of trisection without marks; of course, one does have to "cheat" a little.

The problem of angle trisection continues to fascinate people even though it has long been known that it can't be done with straightedge and compass alone. However, for practical purposes, a good iterative procedure can get you as close as you want. In this note, we present such a procedure. Using only straightedge and compass, our procedure…

This article explores phenomena related to fitting polynomials with data sets with equally spaced x-values.

The well-known "hats" problem, in which a number of people enter a restaurant and check their hats, and then receive them back at random, is often used to illustrate the concept of derangements, that is, permutations with no fixed points. In this paper, the problem is extended to multiple items of clothing, and a general solution to the problem of…

The "matching" hats problem is a classic exercise in probability: if "n" people throw their hats in a box, and then each person randomly draws one out again, what is the expected number of people who draw their own hat? This paper presents several extensions to this problem, with solutions that involve interesting tricks with iterated…

The sum of the divisors of a positive integer is one of the most interesting concepts in multiplicative number theory, while the number of ways of expressing a number as a sum is a primary topic in additive number theory. In this article, we describe some of the surprising connections between and similarities of these two concepts.

When a dog (in this case, Tim Pennings' dog Elvis) is in the water and a ball is thrown downshore, it must choose to swim directly to the ball or first swim to shore. The mathematical analysis of this problem leads to the computation of bifurcation points at which the optimal strategy changes.

In "Just Perfect: Part 1," the author defined a perfect number N to be one for which the sum of the divisors d (1 less than or equal to d less than N) is N. He gave the first few perfect numbers, starting with those known by the early Greeks. In this article, the author provides an extended list of perfect numbers, with some comments about their…

Several tasks used in research studies are presented with assessment rubrics and examples of the development of student understanding. The tasks focus on students' appreciation of variation in several contexts and illustrate the need to discuss variation in the classroom and to ask students specifically about it during assessment.

This paper provides surface area and volume formulas for surfaces of revolution in R[superscript n]. In addition the authors illustrate how to obtain the formulas for volume and surface areas of revolution about the x- or y-axis in two different ways: a "heuristic" argument and a rigorous calculation using "cylindrical" coordinates. In the last…

This article presents an example of skimming and skipping stone motion in mathematical terms available to students studying A-level mathematics. The theory developed in the article postulates a possible mathematical model that is verified by experimental results.