Alf Coles and Barry Orr reflect on what it looked like when Alf taught a year 7 lesson. In this article, Alf was asked by Barry (student) if he would take a lesson with the Y7 class that he had taken over from another teacher in the department in order to see what someone else might do with his lesson plans. Alf describes what some moments with…

In his famous quadrature of the parabola, Archimedes found the area of the region bounded by a parabola and a chord. His method was to fill the region with infinitely many triangles each of whose area he could calculate. In his solution, he stated, without proof, three preliminary propositions about parabolas that were known in his time, but are…

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…

Using the power series solution of a differential equation and the computation of a parametric integral, two elementary proofs are given for the power series expansion of (arcsin x)[squared], as well as some applications of this expansion.

A simple problem relating to birds chasing each other gives rise to a homogeneous differential equation. The solution draws on student skills in differential equations and basic co-ordinate geometry.

Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y , and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given…

A formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved. The formula is applied to the study of the geometry of n-simplices.

This note could find use as enrichment material in a course on the classical geometries; its preliminary results could also be used in an advanced calculus course. It is proved that if a , b and c are positive real numbers such that a[squared] + b[squared] = c[squared] , then cosh ( a ) cosh ( b ) greater than cosh ( c ). The proof of this result…

Given two circles C 1 and C 2 in a plane such that neither one of the two circles is contained in the other, there are either four common tangents when the circles do not intersect at all or the circles have three common tangents when they touch each other externally or only two common tangents when the circles intersect exactly at two points. The…

An integral, either definite or improper, cannot always be computed by elementary methods, such as reversed usage of differentiation formulae. Graphical properties, in particular symmetries, can be useful to compute the integral, via an auxiliary computation. We present graded examples, then prove a general result. (Contains 4 figures.)

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…

This article describes explorations using The Geometer's Sketchpad for algebra and calculus content.

The geometry textbooks of the new system of secondary education in Iran differed dramatically from the old ones considering the aims, the visions, the content, the approach, and the educational purposes. Four hundred eighty mathematics teachers participated in a nationwide professional development program conducted by the author to facilitate the…

This article begins with a comparison of two groups of teachers, working on the same tasks in Analytic Geometry. One group has only basic experience in CAS-assisted problem solving, and the other group has extensive experience. The comparison is discussed in terms of the interplay between reflection, operative knowledge and execution. The findings…

The interpretive framework for the study of learning introduced in this article and called "commognitive" is grounded in the assumption that thinking is a form of communication and that learning mathematics is tantamount to modifying and extending one's discourse. These basic tenets lead to the conclusion that substantial discursive change, rather…