According to "Standards of Excellence in Teaching Mathematics in Australian Schools," excellent teachers of mathematics need to create "an environment that maximises students' learning opportunities"; so that they can encourage self-directed learning, "model mathematical thinking and reasoning," and provide "purposeful and timely feedback." In the…

The United States of America is a large and diverse country; nevertheless, high schools have a lot in common across the nation. In particular, mathematics teaching in American high schools, while not stereotyped, has some accepted norms which would be seen as unusual here in Australia. In order for the Australians to learn from the Americans, as…

Geometry is an area in which Australian students performed particularly poorly on the 2007 Trends in International Mathematics and Science Study (TIMSS). One innovative area of recreational geometry that has rich potential to engage and challenge a wide variety of students is "impossible geometry." An impossible geometric object is a…

As teachers of first-year college mathematics and science students, the authors are constantly on the lookout for simple classroom exercises that improve their students' analytical and computational skills. In this article, the authors outline a project entitled "Predicting Precipitation in Darwin." In this project, students: (1) analyze and…

A problem given in the Australian Mathematics Competition for the Westpac Awards was stated as follows: With how many zeros does 2008! end? In this article, the author solves this problem, and provides further discussion on the related problems. These problems form a good model that helps students develop a logical thinking process toward problem…

Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…

Recently there have been several news items about possible cancer clusters in the Australian media. The term "cancer cluster" is used when an unusually large number of people in one geographic area, often a workplace, are diagnosed with cancer in a short space of time. In this paper the authors explore this important health issue using probability…

This article presents a mathematical solution to a motorway problem. The motorway problem is an excellent application in optimisation. As it integrates the concepts of trigonometric functions and differentiation, the motorway problem can be used quite effectively as the basis for an assessment tool in senior secondary mathematics subjects.…

This paper presents an investigation by pre-service secondary school teachers in a geometry class of the relationship between the perpendicular distance from the eyeball to the wall (x) and the viewable vertical distance on the wall (y) using a view tube of constant length and diameter. In undertaking the investigation, students used tabular and…

Being a mathematician, the author started to wonder if there are any theorems in mathematics that seem very ordinary on the outside, but when applied, have surprisingly far reaching consequences. The author thought about this and came up with the following unlikely candidate which follows immediately from the definition of the area of a rectangle…

One of the difficulties in any teaching of mathematics is to bridge the divide between the abstract and the intuitive. Throughout school one encounters increasingly abstract notions, which are more and more difficult to relate to everyday experiences. This article examines a familiar approach to thinking about negative numbers, that is an…

This article presents a problem set which includes a selection of probability problems. Probability theory started essentially as an empirical science and developed on the mathematical side later. The problems featured in this article demonstrate diversity of ideas and different concepts of probability, in particular, they refer to Laplace and…

This paper discusses some common iterations of complex functions. The presentation is such that similar processes can easily be implemented and understood by undergraduate students. The aim is to illustrate some of the beauty of complex dynamics in an informal setting, while providing a couple of results that are not otherwise readily available in…

The study of the number of intersection points of y = a[superscript x] and y = log[subscript a]x can be an interesting topic to present in a single-variable calculus class. In this article, the authors present a classroom presentation outline involving the basic algebra and the elementary calculus of the exponential and logarithmic functions. The…

In this article, the author describes the format of the Con Test, an Australian television game show which followed the same general rules and game play as the UK show PokerFace. At the end of each round a contestant needs to decide whether or not he or she should fold. A contestant needs to know how likely it is that he or she is in last place.…