Calculus at the secondary school level has traditionally represented the peak of school mathematics in Australia, and has been available only to the most capable students. Until recently, many calculus curricula have focused on developing standard techniques, such as those concerned with differentiation and integration, with an emphasis on…

A well-developed understanding of rate is foundational to conceptual understanding of introductory calculus. Many students achieve procedural competence with the application of rules for differentiation without developing an awareness of the connection between derivative and rate. In addition, rate-related reasoning is needed to make informed…

A derivative is the limit of a quotient. It is an abstraction of division. Since division is harder to understand than multiplication, teachers teach it later, hopefully only after a sound understanding of multiplication has been attained. For the same reason, it may make sense to teach integration first, and move on to differential calculus only…

In the teaching of calculus, the algebraic derivation of the derivative (gradient function) enables the student to obtain an analytic "global" gradient function. However, to the best of this author's knowledge, all current technology-based approaches require the student to obtain the derivative (gradient) at a single point by implementing…

Computer algebra systems (CAS) have been around for a number of years, as has dynamic geometry. Symbolic geometry software is new. It bears a superficial similarity to dynamic geometry software, but differs in that problems may be set up involving symbolic variables and constants, and measurements are given as symbolic expressions. Mathematical…

Many students experience considerable difficulties when they learn and then attempt to construct and communicate proofs of conjectures using the principle of mathematical induction. Although research on the pedagogy of mathematical induction has gained only occasional attention since the 1970s, there has been an increasing interest in this field…

In this note, we introduce and discuss convolutions of two series. The idea is simple and can be introduced to higher secondary school classes, and has the potential of providing a good background for the well known convolution of function.

The basic Pythagorean theorem for right-angled triangles is well-known in mathematical terms as a[squared]+b[squared]+c[squared] were "a," "b," and "c" are the lengths of the sides of the triangle with "c" as the hypotenuse. When "a," "b," and "c" are all integers and obey this equation, they are referred to as a Pythagorean triple. One property…

This paper presents a generic method of calculating accurate horizontal and vertical object distances from digital images taken with any digital camera and lens combination, where the object plane is parallel to the image plane or tilted in the vertical plane. This method was developed for a project investigating the size, density and spatial…

This article describes how a crystal ball known as "The Flash Mind Reader" is played. "The Flash Mind Reader" is a mathematics game in which the player is invited to select any-two digit number and then subtract the sum of these two digits from the original number. A chart is provided in which the (adjusted) number they obtained will have a symbol…

Today's generation does not always have its clever ways of writing algebraic equations and expressions. This paper attempts to trace how this system has developed since the dawn of civilization. The author looks at a few snapshots taken at distinct times to illustrate this progress. (Contains 4 tables.)

In this article, the authors consider the exponential curves y = ba[superscript x] where a, b [is a member of] R, a [greater than] 0, a [not equal to] 1, and b [not equal to] 1. The authors show that by fixing one of a or b the points (x[subscript 0],y[subscript 0) of tangency of y = ba[superscript x] and the line passing through the origin lie on…

In this article, the author presents a six by six array in which individuals can obtain 182 in total even if they use a different set of numbers. The author then explain why this is possible. The author uses the k-translation of a sequence for this equation. (Contains 8 figures, 2 tables and 6 footnotes.)

Concerns about the importance of variation in statistics education and a lack of research in this topic led to a preliminary study which explored pre-service teachers' ideas in this area. The teachers completed a written questionnaire about variation in sampling and distribution contexts. Responses were categorised in relation to a framework that…

Problem solving and student-centred learning have received a great deal of attention in mathematics curricula for schools and in some universities. Much of this emphasis developed from the pioneering work of George Polya in heuristics, problem solving, and mathematics education. In this study, the author reviews Polya's work, and some of its later…