Fay Turner

Developing primary mathematics teaching : reflecting on practice with the Knowledge Quartet

This book helps readers to become better, more confident teachers of mathematics by enabling them to focus critically on what they know and what they do in the classroom. Building on their close observation of primary mathematics classrooms, the authors provide those starting out in the teaching profession with a four-stage framework which acts as a tool of support for developing their teaching: -Making sense of foundation knowledge – focusing on what teachers know about mathematics -Transforming knowledge – representing mathematics to learners through examples, analogies, illustrations, and demonstrations -Connection – helping learners to make sense of mathematics through understanding how ideas and concepts are linked to each other -Contingency – what to do when the unexpected happens Each chapter includes practical activities, lesson descriptions, and extracts of classroom transcripts to help teachers reflect on effective practice. Video versions of these lessons are also available on a companion website.

The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers’ Mathematics Knowledge

In this chapter we present some findings from a study which evaluated the effectiveness of one classroom-based approach to the development of elementary mathematics teaching. This approach drew on earlier research into teachers’ mathematical content knowledge at the University of Cambridge, when a framework for the analysis of mathematics teaching - the Knowledge Quartet - was developed. The chapter begins with a rationale for our focus on teachers’ content knowledge in action in the classroom and a brief description of the study which led to the development of the Knowledge Quartet. It then proceeds to a report of the longitudinal study in which this framework was used to identify and develop a group of beginning teachers’ mathematics content knowledge for teaching.

Growth in teacher knowledge: individual reflection and community participation

Barbara Jaworski (2001) posed the question “In terms of teacher education, do we see a teacher’s growth of knowledge as a personal synthesis from experience or as deriving from interactions within social settings in which teachers work?” (p. 298) Evidence from my four year study suggests that participants’ growth of knowledge for mathematics teaching has been influenced by individual reflection as well as by participation in communities of practice, with the interaction between the two being dependent on individual contexts. In this paper I present some findings from the case studies of Amy and Kate.

Using the Knowledge Quartet to develop mathematics content knowledge: the role of reflection on professional development

In this paper I report findings from a four year study of beginning elementary school teachers which investigated development in their mathematical knowledge for teaching (MKT). The study took a developmental research approach, in that the teachers and the researcher collaborated to develop the mathematics teaching of the teachers, while also trying to understand how such development occurred and might be facilitated. The Knowledge Quartet (KQ) framework was used as a tool to support focused reflection on the mathematical content of teaching, with the aim of promoting development in mathematical content knowledge. Although I focused primarily on whether and how focused reflection using the KQ would promote development, it was impossible to separate this from other influences, and in this paper I discuss the ways in which reflection was found to interrelate with other areas of influence. I suggest that by helping the teachers to focus on the content of their mathematics teaching, within the context of their experience in classrooms and of working with others, the KQ framework supported development in the MKT of teachers in the study.

Beginning elementary teachers’ use of representations in mathematics teaching

A teacher’s ability to use representations appropriately is a key aspect of what Shulman (1986) termed ‘pedagogical content knowledge’ (PCK). This was also an important aspect of teacher content knowledge found by Rowland, Huckstep and Thwaites (2005). They observed how subject matter knowledge (SMK) and (PCK) can be seen to underpin the pedagogical decisions made by prospective teachers. They identified 18 codes with which to label such instances, and from these codes they developed a framework of four dimensions as lenses through which to view teaching; foundation, transformation, connection and contingency. Choice of representation was a key code within the transformation dimension, encompassing the ways in which the teacher’s own knowledge is transformed to make it accessible to the learner. The Knowledge Quartet (KQ) framework formed the basis for reflection on, and discussion of, mathematics lessons in my four-year longitudinal study of beginning teachers. Findings presented here come from the first two years of this study. Mathematics lessons of eleven PGCE student teachers were observed during their final placement, then two lessons each from nine of these during their first year of teaching. Post-lesson discussions were informed by the KQ framework. More systematic analysis using the 18 codes from the four dimensions of the KQ was later carried out and participants sent summaries of these. Throughout the project participants wrote reflections based on the KQ framework, and in the final two years used it independently to reflect on their own mathematics teaching. The use of representations was frequently found to be an issue in the teaching of participants. Difficulties frequently arose in the use of number lines and hundred squares. Evidence from the study suggests that beginning teachers chose representations based on superficial attractiveness and aesthetic appeal rather than on how the mathematics they wanted the children to learn might be mediated through that representation. Number lines were used inappropriately, confusing the ordinal and cardinal aspects of number, and moving quickly from practical activities to the use of iconic representation. Hundred squares were used to demonstrate procedures for calculations without assessing children’s understanding of the structure of such grids. Beginning teachers focused on direction and distance of movement around the grid, rather than on the possibilities afforded for calculations by the base-ten number notation. There is evidence that the Knowledge Quartet has focused the attention of the beginning teachers on their choice of representations. Qualitative data from group

The transfer of content knowledge in a cascade model of professional development

A cascade model of professional development presents a particular risk that ‘knowledge’ promoted in a programme will be diluted or distorted as it passes from originators of the programme to local trainers and then to the target teachers. Careful monitoring of trainers’ and teachers’ knowledge as it is transferred through the system is therefore imperative. This paper focuses on the transfer of content knowledge through an in-service teacher professional development programme and offers an innovative methodology for investigating knowledge transfer, i.e. through insights gained during a mentoring process. The findings suggest that this methodology facilitated assessment of knowledge transfer because it involved the identification of knowledge in practice. The focus on knowledge in practice appeared to avoid a deficit model of trainers’/teachers’ knowledge and revealed that content knowledge was generally being successfully transferred throughout the system. A detailed analysis of different aspects of content knowledge transfer suggested various foci for additional training.

CERME7 Working group 17: From a study of teaching practices to issues in teacher education

We report on a pilot project that has investigated the hypothesis that, in addition to subject and pedagogical knowledge, much of what experienced teachers know is what we call attention-dependent knowledge, and that it is this knowledge that enables them to respond effectively to what happens during lessons. A study of mathematics lessons taught by six teachers has led to some further conjectures about the role of attention-dependent knowledge in teaching, and about the interplay between different knowledge sources in planning and teaching.From a study of teaching practices to issues in teacher education : Introduction to the the papers and posters of WG17