Surveying theories and philosophies of mathematics education

Any theory of thinking or teaching or learning rests on an underlying philosophy of knowledge. Mathematics education is situated at the nexus of two fields of inquiry, namely mathematics and education. However, numerous other disciplines interact with these two fields which compound the complexity of developing theories that define mathematics education. We first address the issue of clarifying a philosophy of mathematics education before attempting to answer whether theories of mathematics education are constructible? In doing so we draw on the foundational writings of Lincoln and Guba (1994), in which they clearly posit that any discipline within education, in our case mathematics education, needs to clarify for itself the following questions: (1) What is reality? Or what is the nature of the world around us? (2) How do we go about knowing the world around us? [the methodological question, which presents possibilities to various disciplines to develop methodological paradigms] and, (3) How can we be certain in the “truth” of what we know? [the epistemological question]

Politicizing mathematics education : has politics gone too far? Or not far enough?

In this chapter we tackle increasingly sensitive questions in mathematics education, those that have polarized the community into distinct schools of thought as well as impacted reform efforts.

Thai students' engagement with cooperative learning in mathematics classrooms

Over the past decade, Thai schools have been encouraged by the Thai Ministry of Education to introduce more student-centred pedagogies such as cooperative learning into their classrooms (Carter, 2006). However, prior research has indicated that the implementation of cooperative learning into Thai schools has been confounded by cultural traditions endemic within Thai schools (Deveney, 2005). The purpose of the study was to investigate how 32 Grade 3 and 32 Grade 4 students enrolled in a Thai school engaged with cooperative learning in mathematics classrooms after they had been taught cooperative learning strategies and skills. These strategies and skills were derived from a conceptual framework that was the outcome of an analysis and synthesis of social learning, behaviourist and socio-cognitive theories found in the research literature. The intervention began with a two week program during which the students were introduced to and engaged in practicing a set of cooperative learning strategies and skills (3 times a week). Then during the next four weeks (3 times a week), these cooperative learning strategies and skills were applied in the contexts of two units of mathematics lessons. A survey of student attitudes with respect to their engagement in cooperative learning was conducted at the conclusion of the six-week intervention. The results from the analysis of the survey data were triangulated with the results derived from the analysis of data from classroom observations and teacher interviews. The analysis of data identified four complementary processes that need to be considered by Thai teachers attempting to implement cooperative learning into their mathematics classrooms. The paper concludes with a set of criteria derived from the results of the study to guide Thai teachers intending to implement cooperative learning strategies and skills in their classrooms.

Students as decoders of graphics in mathematics

This paper reports on students’ ability to decode mathematical graphics. The findings were: (a) some items showed an insignificant improvement over time; (b) success involves identifying critical perceptual elements in the graphic and incorporating these elements into a solution strategy; and (c) the optimal strategy capitalises on how information is encoded in the graphic. Implications include a need for teachers to be proactive in supporting students’ to develop their graphical knowledge and an awareness that knowledge varies substantially across students.

Individual differences in learning from worked examples by senior secondary mathematics students

The primary purpose of this research was to examine individual differences in learning from worked examples. By integrating cognitive style theory and cognitive load theory, it was hypothesised that an interaction existed between individual cognitive style and the structure and presentation of worked examples in their effect upon subsequent student problem solving. In particular, it was hypothesised that Analytic-Verbalisers, Analytic-Imagers, and Wholist-lmagers would perform better on a posttest after learning from structured-pictorial worked examples than after learning from unstructured worked examples. For Analytic-Verbalisers it was reasoned that the cognitive effort required to impose structure on unstructured worked examples would hinder learning. Alternatively, it was expected that Wholist-Verbalisers would display superior performances after learning from unstructured worked examples than after learning from structured-pictorial worked examples. The images of the structured-pictorial format, incongruent with the Wholist-Verbaliser style, would be expected to split attention between the text and the diagrams. The information contained in the images would also be a source of redundancy and not easily ignored in the integrated structured-pictorial format. Despite a number of authors having emphasised the need to include individual differences as a fundamental component of problem solving within domainspecific subjects such as mathematics, few studies have attempted to investigate a relationship between mathematical or science instructional method, cognitive style, and problem solving. Cognitive style theory proposes that the structure and presentation of learning material is likely to affect each of the four cognitive styles differently. No study could be found which has used Riding's (1997) model of cognitive style as a framework for examining the interaction between the structural presentation of worked examples and an individual's cognitive style. 269 Year 12 Mathematics B students from five urban and rural secondary schools in Queensland, Australia participated in the main study. A factorial (three treatments by four cognitive styles) between-subjects multivariate analysis of variance indicated a statistically significant interaction. As the difficulty of the posttest components increased, the empirical evidence supporting the research hypotheses became more pronounced. The rigour of the study's theoretical framework was further tested by the construction of a measure of instructional efficiency, based on an index of cognitive load, and the construction of a measure of problem-solving efficiency, based on problem-solving time. The consistent empirical evidence within this study that learning from worked examples is affected by an interaction of cognitive style and the structure and presentation of the worked examples emphasises the need to consider individual differences among senior secondary mathematics students to enhance educational opportunities. Implications for teaching and learning are discussed and recommendations for further research are outlined.

Modelling the foil rolling process

A review of the main rolling models is conducted to assess their suitability for modelling the foil rolling process. Two such models are Fleck and Johnson's Hertzian model and Fleck, Johnson, Mear and Zhang's Influence Function model. Both of these models are approximated through the use of perturbation methods. Decrease in the computation time resulted when compared with the numerical solution. The Hertzian model was approximated using the ratio of the yield stress of the strip to the plane-strain Young's Modulus of the rolls as the small perturbation parameter. The Influence Function model approximation takes advantage of the solution of the well-known Aerofoil Integral Equation to gain an insight into how the choice of interior boundary points affects the stability of numerical solution of the model's equations. These approximations require less computation than their full models and, in the case of the Hertzian approximation, only introduces a small error in the predictions of roll force roll torque. Hence the Hertzian approximate method is suitable for on-line control. The predictions from the Influence Function approximation underestimates the predictions from the numerical results. Better approximation of the pressure in the plastic reduction regions is the main source of this error.

Implementing a Pattern and Structure Mathematics Awareness Program (PASMAP) in kindergarten

This paper provides an interim report of a large empirical evaluation study in progress. An intervention was implemented to evaluate the effectiveness of the Pattern and Structure Mathematical Awareness Program (PASMAP) on Kindergarten students’ mathematical development. Four large schools (two from Sydney and two from Brisbane), 16 teachers and their 316 students participated in the first phase of a 2-year longitudinal study. Eight of 16 classes implemented the PASMAP program over three school terms. This paper provides an overview of key aspects of the intervention, and preliminary analysis of the impact of PASMAP on students’ representation, abstraction and generalisation of mathematical ideas.