Barbara Jaworski

Investigating mathematics teaching : a constructivist enquiry

An Investigative Approach: Why and How? Constructivism: A Philosophy of Knowledge and Learning Working with Two Teachers: Defining the Study The Research Process Interlude 1: From Phase 1 to Phase 2 Clare: Origins of the Teaching Triad Mike: Significant Episodes and the Teaching Triad Interlude 2: From Phase 2 to Phase 3 Ben: Affirming the Teaching Triad Investigative Mathematics Teaching: Characteristics and Tensions Reflection and Development Epilogue.

Developing Mathematics Teaching: Teachers, Teacher Educators, and Researchers as Co-Learners

This chapter addresses mathematics teaching and teacher development from the perspective of teachers and educators working together in co-learning partnerships to improve the learning of mathematics in school classrooms. From a recognition of the complexities involved, in theory and in practice, the need for and nature of a co-learning partnership is discussed. Reference is made to a particular example in which mathematics teachers and educators have worked together, sometimes as researchers, for the development of teachers and educators, jointly and in parallel. Significance here lies in the complementary roles, relationships, and learning outcomes of the various participants. From these illustrative examples, some general principles for co-learning partnerships are suggested and issues raised, followed by questions and areas for further research.

Tackling Complexity In Mathematics Teaching Development: Using the Teaching Triad as a Tool for Reflection and Analysis

This paper reports research that attempts tomake sense of the complexity of mathematicsteaching and its development at secondaryschool level. The research was conducted inpartnership between two teachers and twoeducator/researchers over one school term intwo U.K. schools. A theoretical construct, theteaching triad, was used as an analyticaldevice (by the researchers) and as a reflectiveagent for teaching development (by theteachers). The focus of analysis was theinteractions between teacher and students atwhole class and small group level. Both micro-and macro-analyses were undertaken. We presentdetails of the processes involved in examplesfrom the teaching of one teacher as shetranslated theoretical aims into classroompractice. The use of the triad allowed accessto complexity, involving both psychological andsociological elements, and to the position of asincere teacher with respect to competingforces in the educational system. Thepotential of the triad for teacher and teachingdevelopment is discussed.

Mentoring in mathematics teaching

A mentors eye view, Anne Watson a focus on learning to teach, Peter Gates mathematics and mentoring, Susan E. Sanders working together - roles and relationships in the mentoring process, Rita Noider et al reflective practice, Stephen Lerman planning for learning, Pat Perks and Stephanie Prestage interpreting the mathematics curriculum, Doug French the wider curriculum, Barrie Crosson and Simon Haines evaluation and judgement, Maggie Crosson and Christine Shill mentoring, co-mentoring and the inner mentor, Barbara Jaworski and Anne Watson.

Video as a Tool for Teachers’ Professional Development

This paper will discuss the production and use of classroom video‐tape as folows: 1) The academic (as opposed to technical) production of video‐tape of mathematics classrooms. 2) The use of video‐tape as an inservice education tool for teachers of mathematics. It results from extensive experience and research by members of the Centre for Mathematics Education at the Open University, UK, in filming real mathematics lessons of teachers of pupils in the 5 to 18 years age range, and the use of the material produced for teachers’ professional development. This use is two‐fold, including both face to face workshops with teachers and as a component of distance learning materials. The paper will not discuss technical details of filming or tape production 1 The two stages described are independent of each other. It is possible to produce video‐tape which others will use. It is possible to use videotape which others have produced. However, in what is described below, each stage has been enhanced by experience gaine...

INSIDERS AND OUTSIDERS IN MATHEMATICS TEACHING DEVELOPMENT: THE DESIGN AND STUDY OF CLASSROOM ACTIVITY

This paper addresses the development of mathematics teaching to improve mathematics learning for students of mathematics at all levels. It focuses on the design of activity for classroom intervention in which teachers are centrally involved along with didacticians. Research and development are closely inter-related with both teachers (insiders) and didacticians (outsiders) taking research roles. Three models or paradigms are described and compared: design research; learning study; and co-learning inquiry. A theoretically-based, innovative, iterative design process is indicated for reliable developmental outcomes. It is suggested that teachers need to be partners in design and study. The nature of partnership is raised as a question for further exploration in connection with the dissemination and sustainability of approaches to development.

Teachers’ perspectives on collaboration with didacticians to create an inquiry community

A research and development project, Learning Communities in Mathematics (LCM)1 was designed to create opportunities for ‘co-learning inquiry’ between mathematics teachers in eight schools and didacticians in a university in Norway (UiA). The focus has been on improving mathematics teaching and learning at school levels from lower primary to upper secondary and on the developmental processes and partnerships involved. A central aim was to create a community of inquiry through which aspects of mathematics teaching and learning could be explored, and through which both teachers and didacticians could learn in practice. Theoretically, ‘Community of Inquiry’ derives from ‘Community of Practice’ as expounded by Lave and Wenger, and particularly Wengers concept of ‘belonging’. The project included three, one-year phases of joint activity. At the end of Phase II, didacticians led focus group interviews with teacher teams to gain insights into schools’ and teachers’ perceptions of the project and its activity. We report on insights into how teachers thought about the activities of the project and what an inquiry community looks like in terms of the learning of those involved. We relate this back to the theoretical perspectives of communities of practice and inquiry.

Teachers learning from teachers

There is much debate within mathematics teacher education over ways in which professional and academic foci could be made to complement each other. On the one hand, teachers’ craft knowledge is emphasized, mainly as this relates to the particular and local level of teaching; on the other hand, the importance of academic subject knowledge cannot be denied. In this chapter the focus will be on how to blend and balance the two through activities in which teachers learn from other teachers, particularly the co-learning of teachers and teacher educators. It will discuss professional relationships, reflective practice, community building, and research in practice. Examples of research-based programs involving lesson study (LS) and the Learner’s Perspective Study (LPS) have moved the relevant research in this area to yet another level, in which theory and practice are combined. Projects such as these and others from diverse parts of the world will be presented and discussed.This chapter seeks to provide an integrating theoretical framework for understanding the somewhat disparate and disconnected literatures on “modelling” and “technology” in mathematics education research. From a cultural–historical activity theory, neo-Vygtoskian perspective, mathematical modelling must be seen as embedded within an indivisible, molar “whole” unit of “activity.” This notion situates “technology”—and mathematics, also—as an essential part or “moment” of the whole activity, alongside other mediational means; thus it can only be fully understood in relation to all the other moments. For instance, we need to understand mathematics and technology in relation to the developmental needs and hence the subjectivity and “personalities” of the learners. But, then, also seeing learning as joint teaching–learning activity implies the necessity of understanding the relation of these also to the teachers, and to the wider institutional and professional and political contexts, invoking curriculum and assessment, pedagogy and teacher development, and so on. Historically, activity has repeatedly fused mathematics and technology, whether in academe or in industry: this provides opportunities, but also problems for mathematics education. We illustrate this perspective through two case studies where the mathematical-technologies are salient (spreadsheets, the number line, and CAS), which implicate some of these wider factors, and which broaden the traditional view of technology in social context.

Development of Teaching in and from Practice

In the discussion of “factors”, we review the beliefs, experiences, and structures that have been identified as significant in studies and programs concerned with the development of mathematics teachers and teaching. This discussion frames the second section of the chapter, in which we delineate some of the markers or “benchmarks” of effective teaching practice, focusing in particular on how these have been used by researchers to study teacher development and effectiveness. Finally, in the “issues” section, we review some of the structures—conceptual, institutional, cultural, and so on—that enable and constrain research into and efforts to support teacher development. Further to the established literature, we draw both on papers that were reviewed and accepted as contributions to the study conference and on notes distilled from our discussions during the conference. When the writing from a paper informs this writing, the relevant quotation from the paper is displayed in a box and attributed to the author. A list of authors, e-addresses and titles of the papers can be found at the end of the chapter.

Unifying Complexity in Mathematics Teaching-Learning Development: A Theory-Practice Dialectic

This chapter addresses theory in relation to mathematics teaching and learning development, drawing on a research study to exemplify theoretical perspectives. In particular it addresses difficulties and issues which arise in a developmental process, from both theoretical and practice-based points of view. The areas of theory are those of inquiry, community and critical alignment, which address developmental processes in mathematics learning and teaching; documentational genesis and instrumentation theory, which address the development of knowledge in teaching; and finally the use of a framework from activity theory, which addresses issues and tensions that emerge from observation and analysis in the research. The illustrative research study addresses perceptions of learning and its outcomes between a teaching team and a cohort of engineering students learning mathematics in a university system. Overall the chapter seeks to address complexity in the developmental process and important synergies between theory, practice and research.