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.

Problem solving for the 21st century

Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, problem solving, here in a broad sense to cover a range of activities that challenge and extend one’s thinking. In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving. In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum development.

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.

Mathematical modelling in the early school years

In this article we explore young children's development of mathematical knowledge and reasoning processes as they worked two modelling problems (the Butter Beans Problem and the Airplane Problem). The problems involve authentic situations that need to be interpreted and described in mathematical ways. Both problems include tables of data, together with background information containing specific criteria to be considered in the solution process. Four classes of third-graders (8 years of age) and their teachers participated in the 6-month program, which included preparatory modelling activities along with professional development for the teachers. In discussing our findings we address: (a) Ways in which the children applied their informal, personal knowledge to the problems; (b) How the children interpreted the tables of data, including difficulties they experienced; (c) How the children operated on the data, including aggregating and comparing data, and looking for trends and patterns; (c) How the children developed important mathematical ideas; and (d) Ways in which the children represented their mathematical understandings.

Young children's early modelling with data

An educational priority of many nations is to enhance mathematical learning in early childhood. One area in need of special attention is that of statistics. This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling activities. Such modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (i.e., identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Results are reported from the first year of a three-year longitudinal study in which three classes of first-grade children and their teachers engaged in activities that required the creation of data models. The theme of “Looking after our Environment,” a component of the children’s science curriculum at the time, provided the context for the activities. Findings focus on how the children dealt with given complex attributes and how they generated their own attributes in classifying broad data sets, and the nature of the models the children created in organising, structuring, and representing their data.

Interdisciplinary modelling

This article examines one approach to promoting creative and flexible use of mathematical ideas within an interdisciplinary context in the primary curriculum, namely, through modelling. Three classes of fifth-grade children worked on a modelling problem, The First Fleet (Australia’s settlement), situated within the curriculum domains of science and studies of society and environment. Reported here are the cycles of development displayed by one group of children as they worked the problem, together with the range of models created across the classes. Children developed mathematisation processes that extended beyond their regular curriculum, including identifying and prioritising key problem elements, exploring relationships among elements, quantifying qualitative data, ranking and aggregating data, and creating and working with weighted scores. Aspects of Goldin’s (2000, 2007) affective structures also appeared to play an important role in the children's mathematical developments.

Curriculum design innovation in flexible science teaching

In this paper you will be introduced to a number of principles which can be used to inform good teaching practice and rigorous curriculum design. Principles relate to: * Application of a common sequence of events for how learners learn; * Accommodating different learning styles; * Adopting a purposeful approach to teaching and learning; * Using assessment as a central driving force in the curriculum and as an organising structure leading to coherence of teaching and learning approach; and * The increasing emphasis that is being placed on the development of generic graduate competencies over and above discipline content knowledge. The principles are particularly significant in relation to adult learning. The paper will use three specific applications as illustrations to help you to learn how these principles can be applied. The illustrations are taken from a second year subject in supercomputing that uses scientific case studies. The subject has been developed (with support from Silicon Graphics Inc. and Intel) to be taught entirely via the Internet.

Towards an intelligent learning system for the natural born cyborg

We propose to design a Custom Learning System that responds to the unique needs and potentials of individual students, regardless of their location, abilities, attitudes, and circumstances. This project is intentionally provocative and future-looking but it is not unrealistic or unfeasible. We propose that by combining complex learning databases with a learner’s personal data, we could provide all students with a personal, customizable, and flexible education. This paper presents the initial research undertaken for this project of which the main challenges were to broadly map the complex web of data available, to identify what logic models are required to make the data meaningful for learning, and to translate this knowledge into simple and easy-to-use interfaces. The ultimate outcome of this research will be a series of candidate user interfaces and a broad system logic model for a new smart system for personalized learning. This project is student-centered, not techno-centric, aiming to deliver innovative solutions for learners and schools. It is deliberately future-looking, allowing us to ask questions that take us beyond the limitations of today to motivate new demands on technology.

The 10% rule. Maximising learning through collaborative game design

This paper presents a retrospective view of a game design practice that recently switched from the development of complex learning games to the development of simple authoring tools for students to design their own learning games for each other. We introduce how our ‘10% Rule’, a premise that only 10% of what is learnt during a game design process is ultimately appreciated by the player, became a major contributor to the evolving practice. We use this rule primarily as an analytical and illustrative tool to discuss the learning involved in designing and playing learning games rather than as a scientifically and empirically proven rule. The 10% rule was promoted by our experience as designers and allows us to explore the often overlooked and valuable learning processes involved in designing learning games and mobile games in particular. This discussion highlights that in designing mobile learning games, students are not only reflecting on their own learning processes through setting up structures for others to enquire and investigate, they are also engaging in high-levels of independent inquiry and critical analysis in authentic learning settings. We conclude the paper with a discussion of the importance of these types of learning processes and skills of enquiry in 21st Century learning.

Milk : students building mobile learning games in everyday places

This paper explores how mobile games can transform everyday places into dynamic learning spaces filled with information and inspiration. It discusses the motivation inherent in playing games and creating games for others, and how this stimulates an iterative process of creation and reflection and evokes a natural desire to engage in learning. The use of MiLK at the Adelaide Botanic Gardens is offered as a case in point. MiLK is an authoring tool that allows students and teachers to create and share SMS games for mobile phones. A group of South Australian high school students used MiLK to play a game, create their own games and play each other’s games during a day at the gardens. This paper details the learning processes involved in these activities and how the students reflected on their learning, conducted peer assessment, and engaged in a two-way discussion with their teacher about new technologies and their implications for learning. The paper concludes with a discussion of the needs and requirements of 21st Century learners and how MiLK can support constructivist and connectivist teaching methods that engage learners and may produce an appropriately skilled future workforce.