Professional Development and Knowledge of Mathematics Teachers

Stefan Zehetmeier, Despina Potari, Miguel Ribeiro, Stefan Zehetmeier, Despina Potari, Miguel Ribeiro

• Publisher: Routledge
• Year: 2020
• Print ISBN: 9780367442408
• eBook ISBN: 9781000298116
• Topic: Education
• Subtopic: Education General

Mathematics teaching and professional development of mathematics teachers are areas where research has increased substantially in recent years. In this dynamic field, mathematics teaching practices, pedagogical knowledge of mathematics teachers and professional development via collaboration between mathematics teachers have emerged as vital domains of inquiry.

Professional Development and Knowledge of Mathematics Teachers addresses the underlying characteristics of mathematics teacher education, and those professional development contexts that have a positive impact on teachers' professional learning. Recognizing the impact of broader institutional settings on mathematics teaching and teacher professional development, the editors suggest bridging the gaps between theoretical practices and methodological approaches in the field by focusing on and conceptualizing the following relational factors:

• The study of mathematics teaching and classroom situations

• Researching teacher and teacher educator knowledge, since these issues inform the quality of mathematics teaching directly

• Mathematics teacher education and professional development, focusing on design principles and the impact they have on teacher professional learning

Combining central issues of mathematics teaching, knowledge and professional development, the chapters in this volume address each of the above factors and provide profound considerations on both theoretical and practical levels. This book will be an essential resource for researchers, teachers and students working in the fields of mathematics teaching and mathematics teacher professional development.

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MATHEMATICS TEACHING AND MATHEMATICS TEACHER PROFESSIONAL DEVELOPMENT

Stefan Zehetmeier, Despina Potari & Miguel Ribeiro

Mathematics teaching and mathematics teacher professional development are areas where research has increased substantially in recent years. In this ongoing field of research, many issues need further investigation. We need to better understand the underlying characteristics of mathematics teacher education and the professional development contexts that have a positive impact on teachers’ professional learning. This includes in particular research about mathematics teaching practices and mathematics teacher knowledge, as well as about teacher professional development and collaboration. Moreover, research about mathematics teaching and mathematics teacher professional development relates to the classroom level as well as to the school and the broader institutional settings within an educational system. Research-informed professional development structures for teachers and teacher educators, especially at large scale, seem to be crucial for the improvement of mathematics teaching at school. In sum, further discussion is needed on how to connect particular research findings and how to link research with mathematics teacher education and classroom practices.

Researching this bridge between theory and practice encompasses a complex range of possible topics. To reduce this complexity, we may follow a path with several concrete stations, all of which are central to mathematics teaching and mathematics teacher professional development. Thus, this path can serve as a guiding line along which research may be conceptualized and located.

This path may start from the study of mathematics teaching and classroom situations as a first station (1). The path’s stations may continue with researching teacher and teacher educator knowledge as a second station (2), since these issues inform the quality of mathematics teaching directly. Following this path may lead to a third station (3) regarding research on mathematics teacher education and professional development, focusing on design principles and the impact they have on teacher professional learning.

This book provides theoretical and empirical studies which follow this path of research. The chapters provide profound considerations on both theoretical and practical levels regarding each of these concrete stations: (1) mathematics teaching and classroom situations; (2) mathematics teacher and mathematics teacher educator knowledge and (3) mathematics teacher education and professional development. For each station, this book provides three chapters: the first chapter provides theoretical frameworks which are suited to analyze the content of this station, while the other two present research projects going deep into the respective realm. In sum, this book provides diverse perspectives and approaches which contribute to our knowledge and understanding about mathematics teaching and mathematics teacher professional development.

1. Research on mathematics teaching and classroom situations

The first chapter written by Hamsa Venkat and Jill Adler, “Mediating mathematics in instruction: Trajectories towards generality in ‘traditional’ teaching”, provides two theoretical frameworks (the Mediating Primary Mathematics – MPM – framework at the primary level, and the Mathematical Discourse in Instruction – MDI – framework at the secondary level) which have been developed in the course of researching and developing mathematics teaching in primary and secondary schools in South Africa. In their chapter, they highlight the wide applicability of mathematical tools and results across various examples and situations. In particular, they illustrate through these frameworks the meaning of teaching for generality in the context of “traditional” instruction, where learners often imitate taught procedures. Finally, they discuss the rationales underlying the similarities and differences in the two formulations.

The second chapter, “The role of teachers’ knowledge in the use of learning opportunities triggered by mathematical connections”, focuses on the relationship between extra- and intra-mathematical connections and the role of a teacher’s knowledge in the use of learning opportunities. Genaro de Gamboa, Edelmira Badillo, Miguel Ribeiro, Miguel Montes and Gloria Sánchez-Matamoros provide insights into empirical research on non-standard measurement in second grade of primary school and the connections related to the mathematical foundations of length measurement. Learning opportunities stemming from such connections are described and analyzed. In this chapter, the authors highlight that extra-mathematical connections are strongly based on intra-mathematical connections. In particular, different types of knowledge can help teachers to make the most of the learning opportunities arising from such connections.

Jeppe Skott, Dorte Moeskær Larsen and Camilla Hellsten Østergaard in their chapter, “Learning to teach to reason: Reasoning and proving in mathematics teacher education”, provide an empirical intervention study which addresses the problems of reasoning and proving in mathematics teacher education in Denmark. School mathematics and teacher education aim at emphasizing proving “why” rather than proving “that” when teaching reasoning and proving in schools. From this background, authors outline the background, framework and results of their empirical study. In particular, they highlight that teachers face problems with reasoning and proving and have difficulties selecting adequate classroom situations. The authors suggest a dual emphasis on both proving that and proving why in mathematics teacher education.

2. Research on teacher and teacher educator knowledge

In her theoretical chapter, “The role of frameworks in researching knowledge and practices of mathematics teachers and teacher educators”, Ronnie Karsenty discusses the notion of “framework” and its utilization for conceptualizing knowledge and practices of mathematics teachers and mathematics teacher educators. She focuses on double-level use of frameworks, referring to frameworks that can serve for the purpose of researching both teachers and teacher educators. This chapter explores in particular if and how frameworks from classroom level can be transferred to the level of professional development. Moreover, a specific case of framework adaptation from teacher level to facilitator level is presented.

The second chapter of station (2), “Parallel stories: teachers and researchers searching for mathematics teachers’ specialized knowledge”, written by José Carrillo, focuses on the relationship between mathematics teachers and mathematics education research. It describes the joint learning trajectory, which develops when working in the teacher professional development realm. A particular focus is on teacher knowledge as a central fostering factor promoting professional development. The author reviews the Mathematics Teacher’s Specialized Knowledge (MTSK) model and gives an empirical example of using this model when analysing one particular teacher’s classroom activities in Spain.

The following empirical chapter of Diana Vasco Mora and Nuria Climent Rodríguez, “The specialized knowledge and beliefs of two university lecturers in linear algebra”, uses the aforementioned MTSK model to analyse university mathematics lecturers’ knowledge and beliefs in teaching linear algebra. In particular, the chapter focuses on the use of knowledge and beliefs as personal resources. Using qualitative research and an instrumental case study design, the authors found evidence that elements of content knowledge, pedagogical content knowledge and beliefs appear to be consistent with each other. The study’s results point to the central role of examples used by the lectures when supporting students to overcome mathematical difficulties.

3. Research on mathematics teacher education and professional development

Laurinda Brown and Brent Davis in the first chapter of this station, “Using the discourses of learning in education mapping to analyse research into mathematics teacher education and professional development”, deal with the question of theoretical perspectives on learning. To do this, they use a theoretical framework for analysing mathematics teacher education and professional development: the Discourses on Learning in Education. In particular, they use two case studies (concerning Variation Theory and Professional Learning Community) to illustrate their analysis. The authors argue that any theoretical or analytical framework for analysing mathematics teacher education and professional development should be based on images and metaphors used by researchers and theorists to characterize learning and learners. Moreover, they suggest that such a framework should be more about description than prescription, to be able to expand understanding and communication.

Content-specific Design Research is provided by the chapter “Promoting and investigating teachers’ professionalization processes towards noticing and fostering students’ potentials: A case of content-specific Design Research for teachers” written by Susanne Prediger, Susanne Schnell and Kim-Alexandra Rösike. Against the backdrop of a specific teacher professional development content (called “noticing and fostering students’ mathematical potentials”), they present the conduction and analysis of design experiments. Particular focus is on the specification of what teachers have to learn and the respective design of professional development. Moreover, exemplary outcomes regarding teachers’ noticing and fostering students’ potentials are discussed.

Bärbel Barzel and Rolf Biehler, in the chapter “Theory-based design of professional development for upper secondary teachers – focusing on the content specific use of digital tools”, provide research results regarding the effects of professional development programmes on teachers’ beliefs and knowledge in Germany. One exemplary programme focuses on digital tools fostering process-related competences (such as modelling, flexible use of representations and problem-solving), another programme is related to probability and statistics as content. In particular, these programmes aim at supporting teachers in implementing process and content-related competences and digital tools in their classrooms.

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MEDIATING MATHEMATICS IN INSTRUCTION

Trajectories towards generality in “traditional” teaching

Hamsa Venkat & Jill Adler

1. Introduction

Ole Skovsmose’s (2011, p. 18) conjecture that “90% of research in mathematics education concentrates on the 10% of the most affluent classroom environments in the world” has been important to our research and development work in South Africa. Skovsmose outlines the implications of this skew in the empirical sites of mathematics education research for the theoretical frameworks available for take-up in the field. Specifically, he notes the lack of need to deal with issues of widespread poverty, unemployment and hunger, with consequences in terms of developmental delays, of classrooms with limited access to electricity and few educational resources, of large classes and overcrowded classrooms and of authoritarian instructional forms. Some combinations of these issues continue to be realities in many government schools in post-apartheid South Africa, in spite of increased expenditure on education and successes relating to getting near universal access to primary education and improvements in provision of basic resources such as national workbooks.

In supporting and studying classrooms in these realities, we share the commitment to expand access to richer and more connected mathematical experiences that underpins much of the research in Skovsmose’s dominant contexts. Our own experiences and broader research in South African schools paint a picture of mathematics classrooms with a different “base” from these dominant contexts that our research and development efforts have had to take into account. In this chapter, our focus is on frameworks we have developed for analyzing and supporting mathematics teaching that have sought this contextual salience while retaining aspirations for high-quality mathematics teaching that would be recognizable in the international research base. While the frameworks differ across primary and secondary mathematics, they share common bases in socio-cultural theory. Key tenets in focus are a view of instruction as mediating learning via a range of mediating means, and as goal-directed activity towards learning mathematics as a network of scientific concepts, in which generality and increasingly complex disciplinary thinking are sought. These tenets are broadly shared in the international mathematics education community. Our emphasis on the teacher’s handling of mathematics within the two frameworks places our work at the content-specific end of Charalambous and Praetorius (2018) generic- to content-specific continuum and more instruction- than interaction-facing than the frameworks that have received wide attention in better-resourced contexts. The latter aspect reflects classroom cultures in which teacher-directed whole-class instruction remains highly prevalent.

Our focus in this chapter is on the different ways in which we have considered mathematical generality across the two frameworks (the Mediating Primary Mathematics (MPM) framework at the primary level, and the Mathematical Discourse in Instruction (MDI) framework at the secondary level). In the ERME Topic Conference paper that was the precursor to this chapter, our focus was on the analysis of instructional talk as a key mediating form across these two frameworks, and how quality towards generality was configured as a goal within this strand (Venkat & Adler, 2017). In this chapter, we home in on the notion of generality more specifically and consider the ways in which generality is considered in relation to teachers’ work across mediating forms in the MPM and MDI models within tasks/examples and their broader instructional discourse. While the frameworks themselves and the aspects in focus within them have been written about elsewhere (Venkat & Askew, 2017, 2018; Adler & Ronda, 2015, 2017), our attention here is on the theoretical strands in the international literature we have drawn from to consider generality, across primary and secondary levels, with emphasis on continuities and adaptations to existing theoretical formulations based on an anchoring in a ground of largely traditional instructional forms. In the South African context, “traditional” forms commonly include a teacher-directed “transmissionist” form of instruction, with chorused responses to mainly routine problems. In this chapter, we explore overlaps and differences in the theoretical derivations that underlie the ways in which we work with generality.

Generality can frequently feel like a distant goal in the South African context, given the evidence of low performance in mathematics at all levels of the schooling system. This raises questions about why we chose to develop frameworks oriented towards generality as a key goal. Our response would be that in a South African terrain where we have highlighted frequent disconnection and incoherence in mathematics instruction (Venkat & Adler, 2012; Adler & Venkat, 2014), this theoretical orientation provided a useful and important counterpoint and aspiration that could guide and focus our research and professional development activity. But a goal of generality in isolation of a trajectory for its achievement was likely to lead us down a rabbit hole of deficit analyses that would defeat our development objectives. Thus, across the two frameworks, our attention was on trajectories towards generality that could inform our teacher development activity as well as our r...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• List of figures
• List of tables
• List of contributors
• 1 Mathematics teaching and mathematics teacher professional development
• 2 Mediating mathematics in instruction: trajectories towards generality in “traditional” teaching
• 3 The role of teachers’ knowledge in the use of learning opportunities triggered by mathematical connections
• 4 Learning to teach to reason: reasoning and proving in mathematics teacher education
• 5 The role of frameworks in researching knowledge and practices of mathematics teachers and teacher educators
• 6 Parallel stories: teachers and researchers searching for mathematics teachers’ specialized knowledge
• 7 The specialized knowledge and beliefs of two university lecturers in linear algebra
• 8 Using the discourses of learning in education mapping to analyse research into mathematics teacher education and professional development
• 9 Promoting and investigating teachers’ professionalization processes towards noticing and fostering students’ potentials: a case of content-specific Design Research for teachers
• 10 Theory-based design of professional development for upper secondary teachers – focusing on the content-specific use of digital tools
• Index