Special tutorials both online and off-line were experimented in order to provide extra support for the senior pre-service mathematics teachers at an Australian regional university to improve their learning experience and achieve the best possible learning outcomes in an advanced mathematics course focusing on solving ordinary differential…

In this chapter we are interested in how undergraduate physics students in three countries experience the equations they meet in their education. We asked over 350 students in the USA, Australia and Sweden the same simple question: How do you know when you understand a physics equation? Students wrote free-text answers to this question, and these…

This paper provides evidence that a short, fully online, well-constructed diagnostic test based on research literature can give teachers information about their students' thinking and strategies that is sufficiently accurate to use for formative assessment purposes. The example is a test for students beginning to learn to solve equations. The main…

A textbook model of a contagious disease, the dynamics of which are represented by the SIS epidemic model with saturating treatment, is considered. I show that this model, as originally formulated, is not dimensionally consistent. The model can be fixed by including a dimensional constant [alpha] of value one (with units individuals[superscript…

Chemical equilibrium is one of the most important concepts in chemistry. The changes in properties of the chemical system at equilibrium induced by variations in pressure, volume, temperature, and concentration are always included in classroom teaching and discussions. This work introduces a novel, geometrical approach to understanding the…

The sudden perception of a connection between ideas is exhilarating. We might call these moments 'Aha!' moments. The purpose of this paper is to demonstrate how several different ideas can come together in Year 12 mathematics. The subject Further Mathematics in the Victorian Certificate of Education (VCE) is the Victorian adaptation of General…

Jagadguru Shankaracharya Swami Bharati Krishna Tirtha (commonly abbreviated to Bharati Krishna) was a scholar who studied ancient Indian Veda texts and between 1911 and 1918 (vedicmaths.org, n.d.) and wrote a collection of 16 major rules and a number of minor rules which have collectively become known as the "sutras of Vedic…

Over the past 25 years, performance measurement has gained salience in higher education, and with the explosion of structured data and the impact of business analytics and intelligence systems, there are new angles by which big volumes of data can be analyzed. Using traditional analytical approaches, pairs of reciprocal cohorts are considered as…

In this article the authors analyze the written solutions of some first year undergraduate mathematics students from Victorian universities as they answered tutorial exercise questions relating to complex numbers and differentiation. These students had studied at least Mathematics Methods or its equivalent at secondary school. Complex numbers was…

In an experiment, secondary students from Australia and Malaysia (n = 130) were randomly assigned to one of three approaches (equation, unitary, unitary-pictorial) to learn how to solve challenging percentage-change problems. In line with the differential types of cognitive load associated with the three approaches, the unitary-approach group…

The leap into the wonderful world of differential calculus can be daunting for many students, and hence it is important to ensure that the landing is as gentle as possible. When the product rule, for example, is met in the "Australian Curriculum: Mathematics", sound pedagogy would suggest developing and presenting the result in a form…

Mathematical reasoning is one of the four proficiencies in the Australian Curriculum: Mathematics (AC:M) where it is described as: "[the] capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising" (Australian Curriculum, Assessment and Reporting Authority [ACARA],…

In this study, we investigated students' thinking about the use of letters in algebra. Responses from over 1,400 Australian secondary school students to a set of three algebra items were analysed to determine the prevalence of the "letter as object" misconception. We estimate that 50% to 80% of Year 7 students bring this misconception to…

This paper describes how a simple application of de Moivre's theorem may be used to not only find the roots of a quadratic equation with real or generally complex coefficients but also to pinpoint their location in the Argand plane. This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not…

Traditionally, "z" is assumed to be a complex number and the roots are usually determined by using de Moivre's theorem adapted for fractional indices. The roots are represented in the Argand plane by points that lie equally pitched around a circle of unit radius. The "n"-th roots of unity always include the real number 1, and…