In this paper we examine the roles that metonymy may play in student reasoning. To organize this discussion we use the lens of a structured derivative framework. The derivative framework consists of three layers of process-object pairs, one each for ratio, limit, and function. Each of the layers can then be illustrated in any appropriate context,…

There is a little-known but very simple generalization of the standard result that for uncorrelated random variables with common mean [mu] and variance [sigma][superscript 2], the expected value of the sample variance is [sigma][superscript 2]. The generalization justifies the use of the usual standard error of the sample mean in possibly…

It has been widely acknowledged that there is some discrepancy in the teaching of vector calculus in mathematics courses and other applied fields. The curl of a vector field is one topic many students can calculate without understanding its significance. In this paper, we explain the origin of the curl after presenting the standard mathematical…

Using the modular ring Zeta[subscript 4], simple algebra is used to study diophantine equations of the form (x[cubed]-a=y[squared]). Fermat challenged his contemporaries to solve this equation when a = 2. They were unable to do so, although Fermat had devised a rather complicated proof himself. (Contains 2 tables.)

The discrete dynamical system of absolute differences defined by the map [psi]( x[subscript 1] , x[subscript 2] , x[subscript 3] , x[subscript 4] ) = ([vertical line] x[subscript 2] - x[subscript 1] [vertical line], [vertical line] x[subscript 3] - x[subscript 2] [vertical line], [vertical line] x[subscript 4] - x[subscript 3] [vertical line],…

The importance of the mathematical notion of concavity in relation to thermodynamics is stressed and it is shown how it can be useful in increasing the enthusiasm of physics' students for their mathematics' courses.

For students to develop meaningful conceptions of fractions and fraction operations, they need to think of fractions in terms other than as just whole-number combinations. In this article, we suggest two powerful images for thinking about fractions that move beyond whole-number reasoning. (Contains 5 figures.)

A triple (x,y,z) of natural numbers is called a Primitive Pythagorean Triple (PPT) if it satisfies two conditions: (1) x[squared] + y[squared] = z[squared]; and (2) x, y, and z have no common factor other than one. All the PPT's are given by the parametric equations: (1) x = m[squared] - n[squared]; (2) y = 2mn; and (3) z = m[squared] +…

The chain rule is one of the hardest ideas to convey to students in Calculus I. It is difficult to motivate, so that most students do not really see where it comes from; it is difficult to express in symbols even after it is developed; and it is awkward to put it into words, so that many students can not remember it and so can not apply it…

Pascal's Triangle is, without question, the most well-known triangular array of numbers in all of mathematics. A well-known algorithm for constructing Pascal's Triangle is based on the following two observations. The outer edges of the triangle consist of all 1's. Each number not lying on the outer edges is the sum of the two numbers above it in…

Math can sometimes seem like a strange language from foreign land--one communicated in symbols, numbers, and geometric figures. When teachers talk about mathematical concepts, even familiar, garden variety words such as "parallel," "power," "even," "odd," "multiply," "difference," "product," "positive," and "negative" take on brand-new meanings.…

The author discusses the historical development of the thermometer with the view of helping children understand the role that mathematics plays in society. A model thermometer that is divided into three sections, each displaying one of the three temperature scales used today (Fahrenheit, Celsius and Kelvin) is highlighted as a project to allow…

Three activities are presented that are designed to develop an understanding of equivalence. Equivalent fractions have the same value, but may be expressed with a different denominator or different notation. "Decimal Fraction Dominoes" focuses on the equivalence of commonly occurring fractions, decimal fractions, percentages and their…

This article examines the mathematics underlying the construction of perspective images of three-dimensional objects. Through hands-on applications and the use of standard secondary content, the article presents perspective art in a away that is accessible to secondary teachers and their students.

This study reports on how students can be led to make meaningful connections between such structures on a set as a partition, the set of equivalence classes determined by an equivalence relation and the fiber structure of a function on that set (i.e., the set of preimages of all sets {b} for b in the range of the function). In this paper, I first…