Developing mathematical proficiency

It has long been recognised that successful mathematical learning comprises much more than just knowledge of skills and procedures. For example, Skemp (1976) identified the advantages of teaching mathematics for what he referred to as “relational” rather than “Instrumental” understanding. More recently, Kilpatrick, Swafford and Findell (2001) proposed five “intertwining strands” of mathematical proficiency, namely Conceptual Understanding, Procedural Fluency, Strategic Competence, Adaptive Reasoning, and Productive Disposition. In Australia, the new Australian Curriculum: Mathematics (F–10), which will be implemented from 2013, has adapted and adopted the first four of these proficiency strands to emphasise the breadth of mathematical capabilities that students need to acquire through their study of the various content strands. This paper addresses the question of what types of classroom practice can provide opportunities for the development of these capabilities in elementary schools. It draws on data from a number of projects, as well as the literature, to provide illustrative examples. Finally, the paper argues that developing the full set of capabilities requires complex changes in teachers’ pedagogy

Generic versus subject specific pedagogy - should mathematics be in the centre?

Research into pedagogy and school change is a high priority in Australia and many other countries. This paper, which includes some preliminary findings from the Improving Middle Years Mathematics and Science: The role of subject cultures in school and teacher change (IMYMS) project, argues that, while there are key features that are common to quality learning environments across all subject areas, generic formulations of pedagogy fail to take account of the extent to which the disciplines being taught shape pedagogy or the contribution of Pedagogical Content Knowledge (PCK) to effective teaching - i.e. that there really is a need to put &quot;mathematics in the centre &quot;.<br /

Problematising the mathematics curriculum : some examples from Australia and Japan

Problem solving is often seen as being the core of mathematics. While there are many examples of teaching for and about problem solving, there are relatively few examples of teaching mathematical content through problem solving. This paper uses data from three, apparently quite different, mathematics lessons from Australia and Japan to explore different ways in which mathematics can be taught successfully through problem solving and to analyse some of the characteristics of such lessons. It also attempts to identify some of the supports and constraints for adopting a problem solving approach to the teaching of mathematics that exist in the quite different contexts of Japan and Australia.<br /

Lesson study - how it could work for you

There is growing worldwide interest in Japanese Lesson Study as a form of professional development, with adaptations of Lesson Study taking place in hundreds of schools clusters in USA, large-scale adoption in the UK, and smaller scale implementation in Australia, and many other countries, including Chile, Indonesia, Malaysia, Mexico, Peru, Philippines, Singapore, Thailand, and Vietnam. This presentation will use classroom video to illustrate the typical Japanese structured-problem-solving research lessons that form the basis for Lesson Study, and discuss how they are planned, the role of the teacher, and the use of Lesson Study as a means of professionaldevelopment

Assessment as a strategic tool for enhancing learning in teacher education: a case study

This small exploratory case study describes an attempt to integrate the academic and practical aspects of a teacher education course in order to promote deep understanding of children\u27s ways of understanding mathematics. The assessment regime of the course was used as a strategic tool for engaging students, and the assessment tasks themselves were used as the means of generating genuine integration, or case knowledge, of the content of the course. The results indicate that the approach was effective in achieving the aims of the course, and student reaction to the approach was extremely positive.<br /

Progressive discourse in mathematics classes - the task of the teacher

This paper uses data from two mathematics lessons to explore the nature of progressive discourse and examine critical features of teacher actions that contribute to mathematics classrooms functioning as communities of inquiry. Features found to promote progressive discourse include a focus on the conceptual elements of the curriculum and the use of complex, challenging tasks that problematised the curriculum; the orchestration of student reporting to allow all students to contribute to progress towards the community\u27s solution to the problem; and a focus on seeking, recognizing, and drawing attention to mathematical reasoning and justification, and using this as a basis for learning. <br /

Teachers\u27 pre-conceptions of young children\u27s conceptions of number

This paper uses data from several sources to argue that while teachers of young children are reasonably accurate in their predictions of when the majority of children will achieve mastery of specific objectives in mathematics, they are much less likely 10 be aware of the conceptions of high achieving children, and that, as a result, their classroom activities constrain these students\u27 learning.<br /

Effective mathematics learning environments : primary teachers\u27 beliefs and practices

A first priority for changing teaching practice is to make problematic for teachers aspects of their current practice. As part of a project on improving mathematics and science teaching in the middle years of schooling, teachers were asked to rate their practice against components of effective teaching and learning and to rate each of these according to their perceived importance. Findings suggest that primary teachers endorsed the components as representing effective practice, scoring most components higher than their actual practice. Gaps were particularly evident for items relating to challenging students conceptually and higher-order thinking, with these becoming the basis for some of the action planning for change.<br /

Using activity theory in researching young children\u27s use of calculators

Constructivist and socio-cultural perspectives in mathematics education highlight the crucial role that activity plays in mathematical development and learning. Activity theory provides a socio-cultural lens to help analyse human behaviour, including that which occurs in classrooms. It provides a framework for co-ordinating constructivist and socio-cultural perspectives in mathematics learning. In this paper, we adopt Cole and Engestr&ouml;m\u27s (1991) model of activity theory to examine the mediation offered by the calculator as a tool for creating and supporting learning processes of young children in the social environment of their classroom. By adopting this framework, data on young children\u27s learning outcomes in number, when given free access to calculators, can be examined not only in terms of the mediating role of the calculator, but also within the broader context of the classroom community, the teachers\u27 beliefs and intentions, and the classroom norms and the division of labour. Use of this model in a post hoc situation suggests that activity theory can play a significant role in the planning of future classroom research.<br /

Japanese lesson study : teacher professional development through communities of inquiry

Japanese Lesson Study has come under increasing attention from educators in the West and throughout South-East Asia since it was revealed outside Japan through the release of the TIMSS Video Study. In this paper we argue that Japanese Lesson Study provides a model for large scale, sustainable professional development. In particular, we draw on our own experience of Japanese Lesson Study and the research literature to describe its characteristic features and examine some of the cultural assumptions that underpin its implementation.<br /