Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework

The article contributes to the ongoing discussion on ways to deal with the diversity of theories in mathematics education research. It introduces and systematizes a collection of case studies using different strategies and methods for networking theoretical approaches which all frame (qualitative) empirical research. The term ‘networking strategies’ is used to conceptualize those connecting strategies, which aim at reducing the number of unconnected theoretical approaches while respecting their specificity. The article starts with some clarifications on the character and role of theories in general, before proposing first steps towards a conceptual framework for networking strategies. Their application by different methods as well as their contribution to the development of theories in mathematics education are discussed with respect to the case studies in the ZDM-issue.     

Theories of Mathematics Education: A global survey of theoretical frameworks/trends in mathematics education research

In this article we survey the history of research on theories in mathematics education. We also briefly examine the origins of this line of inquiry, the contribution of Hans-Georg Steiner, the activities of various international topics groups and current discussions of theories in mathematics education research. We conclude by outlining current positions and questions addressed by mathematics education researchers in the research forum on theories at the 2005 PME meeting in Melbourne, Australia.     

Towards theoretical foundations of mathematics education

Although research in mathematics education developed in the last decades as a vivid scientific field, the nature and the place of theories in the field are still under discussion. H.-G. Steiner contributed to this debate in several ways. He not only intervened in the scientific debate but also played an active role in organizing the discipline and the scientific discussion at the international level. This paper attempts to give a synthetic view of the evolution of mathematics education with respect to theory by focusing on the French situation of research in mathematics education and paying a particular attention to the role and contribution of H.-G. Steiner.     

Research in mathematics education— Who benefits?

For a long time mathematics education tries hard to win recognition as an academic discipline. In related classroom research and curriculum development one can find not rarely theories and methods in use that are adapted from other (and well established) disciplines. However, in many cases these adaptations do not serve the researcher's goals, yet more, their effects can contradict the stated purposes. The article discusses a few fundamental problems related to empirical research in mathematics education (e.g. the role of the teachers in experimental/control groups), using as concretization a printed research report that is chosen deliberately (and made nameless, therefore).     

A social perspective on technology-enhanced mathematical learning: from collaboration to performance

This paper documents both developments in the technologies used to promote learning mathematics and the influence on research of social theories of learning, through reference to the activities of the International Commission on Mathematical Instruction (ICMI), and argues that these changes provide opportunity for the reconceptualization of our understanding of mathematical learning. Firstly, changes in technology are traced from discipline-specific computer-based software through to Web 2.0-based learning tools. Secondly, the increasing influence of social theories of learning on mathematics education research is reviewed by examining the prevalence of papers and presentations, which acknowledge the role of social interaction in learning, at ICMI conferences over the past 20 years. Finally, it is argued that the confluence of these developments means that it is necessary to re-examine what it means to learn and do mathematics and proposes that it is now possible to view learning mathematics as an activity that is performed rather than passively acquired.     

Theories of Mathematics Education,Edited by Bharath Sriraman and Lyn English: Common Ground for Scholars and Scholars in the Making

Theories of mathematics education: Seeking new frontiers is the first book in a new Springer series, Advances in Mathematics Education. To some degree the book is based on a collection of previously published articles from special issues of ZDM—The International Journal on Mathematics Education (previously known as Zentralblatt für Didaktik der Mathematik). These articles, dealing with the role and use of theories in and about mathematics education, originally stem from various conferences and meetings such as PME and CERME. For this reason some of the articles in the book are already well known, a few may even be considered to be “modern classics” within theories of mathematics education such as Frank Lester's “On the theoretical, conceptual, and philosophical foundations for research in mathematics education.” What is new—and non-traditional—in the book, however, is its form of presentation and format, the main articles being accompanied by preludes and commentaries by established researchers and newcomers.     

Connecting social perspectives on mathematics teacher education in online environments

This article explores theoretical issues underpinning the design and use of online learning environments in mathematics teacher education. It considers the contribution of social theories of learning to conceptualising technology-mediated interaction, focusing specifically on community of practice models and the notion of digital mathematics performance. The article begins by introducing social perspectives on collaboration. Because of the diversity of theories within this broad research paradigm, the next section outlines networking strategies that have been proposed for connecting theoretical approaches. There follows a discussion of studies that illustrate the community of practice and performance-based approaches to research into online mathematics teacher education. The main purpose of the article is to show how these approaches could be connected by examining the same teaching and learning scenarios through different theoretical lenses. The final section identifies implications of this exploration for the design of online learning environments in mathematics teacher education to capitalise on the affordances of Web-based technologies.     

Combining theories in research in mathematics teacher education

In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each of the two theories and our suggestions for combining them, formulating the Shulman–Fischbein framework. Then, we present two research segments that illustrate the potential of the implementation of the Shulman–Fischbein framework to the study of mathematics teachers’ ways of thinking. We conclude with general comments on possible contributions of combining theories that were developed in mathematics education and in other domains to mathematics teacher education.     

Mathematics Education as a Design Science

We propose re-conceptualizing the field of mathematics education research as that of a design science akin to engineering and other emerging interdisciplinary fields which involve the interaction of “subjects”, conceptual systems and technology influenced by social constraints and affordances. Numerous examples from the history and philosophy of science and mathematics and ongoing findings of M&M research are drawn to illustrate our notion of mathematics education research as a design sicence. Our ideas are intended as a framework and do not constitute a, “grand” theory (see Lester. 2005, this issue). That is, we provide a framework (a system of thinking together with accompanying concepts, language, methodologies, tools, and so on) that provides structure to help mathematics education researchers develop both models and theories, which encourage diversity and emphasize Darwinian processes such as: (a) selection (rigorous testing), (b) communication (so that productive ways of thinking spread throughout relevant communities), and (c) accumulation (so that productive ways of thinking are not lost and get integrated into future developments).     

Hans-Georg Steiner: a life dedicated to the development of didactics of mathematics as a scientific discipline

In our introductory paper to this special issue we follow two goals. First of all, we take on the challenge to give an account of more than 40 years of academic work by one of the leading members of our discipline by looking at Hans Georg Steiner’s contributions to the development of didactics of mathematics as a scientific discipline in Germany as well as internationally. Therefore, we try to highlight major research interests, publications and conferences during his early years in Münster, Karlsruhe and Bayreuth as well as during the 20 years at the IDM in Bielefeld. Closely linked to these periods of this life and work are specific research interests, professional contacts and friendships. Hence, the second goal of our paper is to emphasise Hans-Georg Steiner’s relationships with national and international colleagues (many of whom became friends) and their shared interests and collaborations in the development of mathematics education through a selection of invited papers that address different stages and professional foci in the life of Hans-Georg Steiner. These papers are organised in four sections: (1) Revisiting the New Math reform, (2) Developing specific research domains in didactics of mathematics, (3) Discussing theories of mathematics education (TME), and (4) Reflecting on goals and results of mathematics education.     

Diversity of theories in mathematics education—How can we deal with it?

This article discusses the central question of how to deal with the diversity and the richness of existing theories in mathematics education research. To do this, we propose ways to structure building and discussing theories and we contrast the demand for integrating theories with the idea of networking theories.     

Mathematics education research embracing arts and sciences

As a young field in its own right (unlike the ancient discipline of mathematics), mathematics education research has been eclectic in drawing upon the established knowledge bases and methodologies of other fields. Psychology served as an early model for a paradigm that valorized psychometric research, largely based in the theoretical frameworks of cognitive science. More recently, with the recognition of the need for sociocultural theories, because mathematics is generally learned in social groups, sociology and anthropology have contributed to methodologies that gradually moved away from psychometrics towards qualitative methods that sought a deeper understanding of issues involved. The emergent perspective struck a balance between research on individual learning (including learners’ beliefs and affect) and the dynamics of classroom mathematical practices. Now, as the field matures, the value of both quantitative and qualitative methods is acknowledged, and these are frequently combined in research that uses mixed methods, sometimes taking the form of design experiments or multi-tiered teaching experiments. Creativity and rigor are required in all mathematics education research, thus it is argued in this paper, using examples, that characteristics of both the arts and the sciences are implicated in this work.     

On the teaching and learing of Dienes' principles

Zoltan Dienes' principles of mathematical learning have been an integral part of mathematics education literature and applied both to the teaching and learning of mathematics as well as research on processes such as abstraction and generalization of mathematical structures. Most extant textbooks of cognitive learning theories in mathematics education include a treatment of Dienes' seminal contributions. Yet, there are no available studies at the tertiary level on how students internalise the meaning of Dienes' principles. This paper explores post-graduate mathematics education student's understanding of Dienes' principles and their ability to reflexively apply the principles to their own thinking on structurally similar problems. Some implications are offered for university educators engaged in the training of future researchers in the field.     

Conception of expert mathematics teacher: a comparative study between Hong Kong and Chongqing

This paper reports the similarities and differences in how “expert mathematics teacher” is conceptualized by mathematics educators in Hong Kong and Chongqing, two cities in China which share similar but different cultural and social backgrounds. Thirty-seven mathematics education researchers, school principals with mathematics education background, and mathematics teachers were interviewed on their perceptions of expert mathematics teacher. It is found that in both cities an expert mathematics teacher should have a profound knowledge base in mathematics, teaching, and students; strong ability in teaching; and a noble personality and a spirit of life-long learning. As for differences, an expert mathematics teacher should have the ability to conduct research, mentor other teachers, and have profound knowledge of examination and educational theories in Chongqing. These attributes were not found in Hong Kong. These similarities and differences are discussed, and relevant social and cultural factors in the two contexts are examined.     

Language and communication in mathematics education: an overview of research in the field

Within the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognised, but there is not agreement about what this role (or these roles) might be or even about what the term ‘language’ itself encompasses. In this issue of ZDM, we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this survey paper, we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We identify four broad areas of concern in mathematics education that are addressed by language-oriented research: analysis of the development of students’ mathematical knowledge; understanding the shaping of mathematical activity; understanding processes of teaching and learning in relation to other social interactions; and multilingual contexts. A further area of concern that has not yet received substantial attention within mathematics education research is the development of the linguistic competencies and knowledge required for participation in mathematical practices. We also discuss methodological issues raised by the dominance of English within the international research community and suggest some implications for researchers, editors and publishers.     

Views of cognition: different lenses for ‘looking-in’ on classrooms

The evolution of different theories of cognition over the years has given mathematics education researchers new tools for highlighting particular characteristics of classroom issues to enable more detailed investigation of how students learn mathematics. In the past there has been a strong dominance of the dualistic view separating body and mind. In recent times, however, the body has been given a more central role in shaping the mind. This has led to the situation where some long-standing conundrums of mathematics education have become more tractable to researchers. Similarly, using older ideas in new ways, light is being shone on what is possible at different stages of development or different levels of schooling. In this issue different views of cognition have been mined by the authors of different articles to frame studies and analyses or invent and apply new tools. They bring new lenses for looking-in on classrooms, a fresh view with old lenses and new methodological tools to the fore. Using a small selection of the articles of the issue the empowerment that differing views of cognition have enabled for mathematics education research is demonstrated.     

On the influence of theory on research in mathematics education: the case of teaching and learning limits of functions

After an introduction on approaches, research frameworks and theories in mathematics education research, three didactical research studies on limits of functions with different research frameworks are analysed and compared with respect to their theoretical perspectives. It is shown how a chosen research framework defines the world in which the research lives, pointing to the difficult but necessary task to compare research results within a common field of study but conducted within different frameworks.     

CAS in general mathematics education

A rational discussion of the use of Computer algebra systems (CAS) in mathematics teaching in general education needs an explicit image of (general) mathematics education, an explication of global perspectives and goals on mathematics teaching focusing on general education (chapter 1). The conception of general education according to the «ability of communication with experts» described in chapter 2 can be such an orientation for analysing, considering, classifying and assessing the didactical possibilities of using CAS. CAS are materialised mathematics allowing for more or less exhaustive outsourcing of operative (also symbolically) knowledge and skills to the machine. This frees up space of time as well as mental space for the development of those competences being in our view relevant for general mathematics education. In chapter 3 the idea of outsourcing and the role of CAS for it is discussed more detailed as well as consequences being possible for the CAS-supported teaching of mathematics. Beyond, CAS can be didactically used and reflected as a model of communication between (mathematical) experts and lay-persons (chapter 4). Chapter 5 outlines some research perspectives.     

Economy: the absent centre of mathematics education

Social and political turns in mathematics education research have brought into the field postmodern theorisations that researchers have been using to dismantle traditional philosophies of mathematics, to posit mathematics in the sociocultural terrain, and to spell out the role mathematics has in school exclusion. Sociopolitical perspectives constitute a privileged field of research to address the influence of economy on mathematical achievement. However, instead of investigating the role of economy in students’ achievement, sociopolitical studies have been contributing to a disavowal of the economic dimension of school mathematics. This paper synthesises a set of investigations carried out by the author in the last 5 years endeavouring to posit mathematics education in the political and economic spectrum of our time. It takes advantage of the contemporary combination of Hegel’s dialectics, Lacanian psychoanalysis and Marx’s critique of political economy, carried out by Slavoj ?i?ek, to develop a critique of the way research within the so-called ‘sociopolitical turn’ deals with the issue of equity; and marks out the contours of mathematics education’s ideological belonging.     

Research into teacher knowledge: a stimulus for development in mathematics teacher education practice

In this paper, we document some developments in teacher education practice at one university, brought about by reflection on research into mathematics teacher knowledge. The authors are three members of the Cambridge-based research team who developed the Knowledge Quartet (KQ), a theory of mathematics teacher knowledge, with a focus on classroom situations in which this knowledge is applied. At the same time as being researchers, the authors were elementary mathematics teacher education instructors. They found that the KQ research brought about new awareness of the importance of some components of mathematics didactics, as well as providing new tools for undertaking some aspects of their teacher educator role. The paper explores some of these awarenesses and tools in detail.