Theories of Mathematics Education : Seeking New Frontiers

This inaugural book in the new series Advances in Mathematics Education is the most up to date, comprehensive and avant garde treatment of Theories of Mathematics Education which use two highly acclaimed ZDM special issues on theories of mathematics education (issue 6/2005 and issue 1/2006), as a point of departure. Historically grounded in the Theories of Mathematics Education (TME group) revived by the book editors at the 29th Annual PME meeting in Melbourne and using the unique style of preface-chapter-commentary, this volume consist of contributions from leading thinkers in mathematics education who have worked on theory building. This book is as much summative and synthetic as well as forward-looking by highlighting theories from psychology, philosophy and social sciences that continue to influence theory building. In addition a significant portion of the book includes newer developments in areas within mathematics education such as complexity theory, neurosciences, modeling, critical theory, feminist theory, social justice theory and networking theories. The 19 parts, 17 prefaces and 23 commentaries synergize the efforts of over 50 contributing authors scattered across the globe that are active in the ongoing work on theory development in mathematics education.

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.

Robust outdoor visual localization using a three-dimensional-edge map

Visual localization systems that are practical for autonomous vehicles in outdoor industrial applications must perform reliably in a wide range of conditions. Changing outdoor conditions cause difficulty by drastically altering the information available in the camera images. To confront the problem, we have developed a visual localization system that uses a surveyed three-dimensional (3D)-edge map of permanent structures in the environment. The map has the invariant properties necessary to achieve long-term robust operation. Previous 3D-edge map localization systems usually maintain a single pose hypothesis, making it difficult to initialize without an accurate prior pose estimate and also making them susceptible to misalignment with unmapped edges detected in the camera image. A multihypothesis particle filter is employed here to perform the initialization procedure with significant uncertainty in the vehicle's initial pose. A novel observation function for the particle filter is developed and evaluated against two existing functions. The new function is shown to further improve the abilities of the particle filter to converge given a very coarse estimate of the vehicle's initial pose. An intelligent exposure control algorithm is also developed that improves the quality of the pertinent information in the image. Results gathered over an entire sunny day and also during rainy weather illustrate that the localization system can operate in a wide range of outdoor conditions. The conclusion is that an invariant map, a robust multihypothesis localization algorithm, and an intelligent exposure control algorithm all combine to enable reliable visual localization through challenging outdoor conditions.

Experiments in outdoor operation of RatSLAM

This paper shows initial results in deploying the biologically inspired Simultaneous Localisation and Mapping system, RatSLAM, in an outdoor environment. RatSLAM has been widely tested in indoor environments on the task of producing topologically coherent maps based on a fusion of odometric and visual information. This paper details the changes required to deploy RatSLAM on a small tractor equipped with odometry and an omnidirectional camera. The principal changes relate to the vision system, with others required for RatSLAM to use omnidirectional visual data. The initial results from mapping around a 500 m loop are promising, with many improvements still to be made.

Visual localisation in outdoor industrial building environments

This paper presents a vision-based method of vehicle localisation that has been developed and tested on a large forklift type robotic vehicle which operates in a mainly outdoor industrial setting. The localiser uses a sparse 3D edgemap of the environment and a particle filter to estimate the pose of the vehicle. The vehicle operates in dynamic and non-uniform outdoor lighting conditions, an issue that is addressed by using knowledge of the scene to intelligently adjust the camera exposure and hence improve the quality of the information in the image. Results from the industrial vehicle are shown and compared to another laser-based localiser which acts as a ground truth. An improved likelihood metric, using peredge calculation, is presented and has shown to be 40% more accurate in estimating rotation. Visual localization results from the vehicle driving an arbitrary 1.5km path during a bright sunny period show an average position error of 0.44m and rotation error of 0.62deg.

Meeting and greeting : activities in public outdoor spaces outside high-density urban residential communities

This paper examines the opportunities for social activities in public outdoor spaces associated with high-density residential living. This study surveyed activities in outdoor spaces outside three high-density residential communities in Brisbane. Results indicated that activity patterns in public outdoor space outside residential communities are different to general urban public outdoor space. This broadly but not fully supports current theories concerning activities in public space. That is some environmental factors have impacts on the level of social interaction. The relationship between outdoor space and a residential building may have a significant impact on the level of social activities. As a consequence, a new classification of activities in public space is suggested. In improving the level of social contact in public outdoor space outside a residential community, the challenge is how to encourage people to leave their comfortable homes and spend a short time in these public spaces. For residential buildings and public space to be treated as an integrated whole, the outdoor open spaces close to and surrounding these buildings must have a more welcoming design.

Wide-angle visual feature matching for outdoor localization

Wide-angle images exhibit significant distortion for which existing scale-space detectors such as the scale-invariant feature transform (SIFT) are inappropriate. The required scale-space images for feature detection are correctly obtained through the convolution of the image, mapped to the sphere, with the spherical Gaussian. A new visual key-point detector, based on this principle, is developed and several computational approaches to the convolution are investigated in both the spatial and frequency domain. In particular, a close approximation is developed that has comparable computation time to conventional SIFT but with improved matching performance. Results are presented for monocular wide-angle outdoor image sequences obtained using fisheye and equiangular catadioptric cameras. We evaluate the overall matching performance (recall versus 1-precision) of these methods compared to conventional SIFT. We also demonstrate the use of the technique for variable frame-rate visual odometry and its application to place recognition.