Interpretations of a constructivist philosophy in mathematics teaching

This thesis is a research biography which reports a study of mathematics teaching. It involves research into the classroom teaching of mathematics of six teachers, and into their associated beliefs and motivations. The teachers were selected because they gave evidence of employing an investigative approach to mathematics teaching, according to the researcher's perspective. A research aim was to characterise such an approach through the practice of these teachers. An investigative approach was seen to be embedded in a radical constructivist philosophy of knowledge and learning. Observations and analysis were undertaken from a constructivist perspective and interpretations made were related to this perspective. Research methodology was ethnographic in form, using techniques of participant-observation and informal interviewing for data collection, and triangulation and respondent validation for verification of analysis. Analysis was qualitative, leading to emergent theory requiring reconciliation with a constructivist theoretical base. Rigour was sought by research being undertaken from a researcher-as-instrwnent position, with the production of a reflexive account in which interpretations were accounted for in terms of their context and the perceptions of the various participants including those of the researcher. Research showed that those teachers who could be seen to operate from a constructivist philosophy regularly made high level cognitive demands which resulted in the incidence of high level mathematical processes and thinldng skills in their pupils. Levels of interpretation within the study led to the identification of investigative teaching both as a style of mathematics teaching and as a form of reflective practice in the teaching of mathematics. These forms were synthesised as a constructivist pedagogy and as an epistemology for practice which may be seen to forge links between the theory of mathematics teaching and its practice. The research is seen to have implications for the teaching of mathematics, and for the development of mathematics teaching itself through professional development of mathematics teachers

Building and sustaining inquiry communities in mathematics teaching development: teachers and didacticians in collaboration

Teachers and didacticians both bring areas of expertise, forms of knowing and relevant experience to collaboration in mathematics teaching development. The notion of inquiry community, provides a theoretical and practical foundation for development. Within an inquiry community all participants are researchers (taking a broad definition). With reference to a research and development project in Norway (Learning Communities in Mathematics – LCM) this chapter explains the theoretical notions, discusses how one community was conceived and emerged in practice and addresses the issues contingent on emergence and sustaining of inquiry practices. In doing so it provides examples of collaborative activity and the reciprocal forms of expertise, knowing and experience that have contributed to community building. It illuminates issues and tensions that have been central to the developmental process and shows how an activity theory analysis can help to navigate the complexity in characterizing development

Learning communities in mathematics: creating an inquiry community between teachers and didacticians

This paper reports on a project designed to develop inquiry communities between teachers and didacticians, aimed at improving the learning of mathematics in classrooms, and at studying the processes, practices, issues and outcomes in and of the project. Theoretical notions of inquiry and community underpin the project. The focus here is largely methodological, tracing the origination and development of the project and decisions taken through its first phases. The project is seen to be situated within a 'developmental' research paradigm in which research both studies the developmental process and contributes centrally to it. Issues in the interpretation of inquiry practices in mathematics learning and teaching and the building of communities at various levels are seen as important outcomes. The roles and relationships of teachers and didacticians emerge as key concepts in the developmental process

Mathematics teaching development as a human practice: identifying and drawing the threads

The didactic triangle links mathematics, teachers and students in a consideration of teaching– learning interactions in mathematics classrooms. This paper focuses on teachers and teaching in the development of fruitful learning experiences for students with mathematics. It recognises primarily that teachers are humans with personal characteristics, subject to a range of influences through the communities of which they are a part, and considers aspects of teachers’ personhood, identity and agency in designing teaching for the benefit of their students. Teaching is seen as a developmental process in which inquiry plays a central role, both in doing mathematics in the classroom and in exploring teaching practice. The teacher-as-inquirer in collaboration with outsider researchers leads to growth of knowledge in teaching through development of identity and agency for both groups. The inclusion of the outsider researcher brings an additional node into the didactic triangle

Developing teaching of mathematics to engineering students: teacher research, student epistemology and mathematical competence

A research project “Engineering Students Understanding Mathematics” (ESUM) has shown tensions between teachers’ theoretical perspectives and design of teaching and students strategic approach to learning. In my research team we seek to improve our teaching of engineering students and to develop their conceptual understandings of mathematics. Given our findings on student perspectives, and the institutional affordances and constraints, we are exploring a range of resources and teaching approaches using theoretical perspectives of inquiry community, documentational genesis and activity theory

Mathematics teacher educator learning and development

This chapter serves as an introduction to Volume 4 as a whole. The stated aim of the volume is to open up the practice of mathematics teacher educators to scrutiny and critique: locating our practices internationally, identifying issues both local and global, and seeing ourselves as practitioners alongside teacher practitioners with whom we work. The chapter begins with a brief discussion of mathematics teacher educator knowledge, learning and development, locating this both historically and theoretically. It goes on to address the three sections: Challenges to and theory in mathematics teacher education; Reflection on developing as a mathematics teacher educator; Working with prospective and practising teachers: what we learn; what we come to know; and presents a short account of each chapter. The chapter ends with a vision of teacher educator practice for the future

Mathematical competence framework : an aid to identifying understanding?

Research into the teaching of mathematics to engineering students to promote their conceptual understanding (Jaworski and Matthews 2011) has shown the problematic nature of planning for and identifying understanding. I review the project briefly and introduce the idea of competencies from the Danish project, KOM (translated as Competencies and Mathematical Learning). Through the medium of designing inquirybased tasks for students and use of the competency framework for analysis of tasks, I consider the relevance of such a competency-based analysis and its usefulness (or otherwise) for recognising student understanding. This leads to important questions for further research of a developmental nature

Developing mathematics teaching through collaborative inquiry

This chapter addresses theory and practice in a developmental research project in which mathematics teachers and didacticians worked together to develop mathematics teaching. The mathematics teachers were from 8 schools in Norway ranging from lower primary to upper secondary. Didacticians were academics in mathematics education in a university. Both were practitioners in their own areas of practice and in the project both were researchers. The project sought to know more about how mathematics teaching can develop to enhance learning experiences for students in mathematics classrooms. It was called Learning Communities in Mathematics, LCM , and involved teachers and didacticians in inquiry communities exploring together and evaluating possibilities for classrooms and students (Jaworski, Fuglestad, Bjuland, Breiteig, Goodchild & Grevholm, 2007). Over a period of four years, including three phases of school-related activity each of one school year, 14 didacticians (including 5 doctoral students) and 35 teachers were involved in the project

Teaching mathematics to address fundamental human rights

Mathematics is one of the most important subjects in the curriculum, central to so many areas of life and academic disciplines. Yet students – and people widely – struggle with mathematics possibly more than with any other subject. It is the right of every human being to know and understand mathematics relative to the context and purpose for which it is needed. These statements have profound implications for the teaching of mathematics. In this short monograph (a written version of my inaugural lecture for the Donders Chair at the University of Utrecht) I address the following questions about the role of teaching: .What does it mean to teach mathematics? .What are the characteristics of “good” teaching of mathematics? .How does/can “good teaching” develop? Because it is hard to address “what is good teaching?” in a simple way at the outset, I will start with another question: How can we teach mathematics for the effective learning of our students

Research practice into/influencing mathematics teaching and learning development: towards a theoretical framework based on co-learning partnerships

This paper addresses issues linking research into the classroom teaching and learning of mathematics with the growth of knowledge in mathematics teaching, developments in the practice of teaching and the enhanced learning of mathematics by students in classrooms. A basic premise is that research promotes development. The paper considers both insider and outsider research and co-learning between teachers and educators in promoting classroom inquiry. Through a consideration of elements of theory such as knowledge and inquiry in teaching and of learning as knowledge growth through research/ inquiry leading to enhancement of students’ learning of mathematics, a framework is suggested. Its purposes include analysis of a research project’s contribution to teaching development and conceptualization of research which has teaching development as one of its aims. Use of the framework is exemplified through its application to reports of three mathematics education research projects in the public domain. A brief afterword links the framework to concepts in activity theory