Theories of mathematics education: European perspectives, commentaries and viable research directions

We briefly comment on different perspectives on (1) the role of mathematics education theories, (2) the issue of plurality and healthy heterogeneity versus consolidation; (3) underlying inquiry systems or the implicit role of philosophy in theories of mathematics education. This paper also outlines developments within the European research scene on theory usage in mathematics education research, which complement the discussion at the 29th PME research forum in Melbourne.     

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.     

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.     

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.     

Traveling down the road: from cognitive neuroscience to mathematics education … and back

In this commentary paper to the special issue on “Cognitive Neuroscience and Mathematics Education”, we reflect on the connection between cognitive neuroscience and mathematics education from an educational research point of view. The current issue highlights that cognitive neuroscience offers a series of tools, methodologies and theories to investigate cognitive processes that take place during mathematical thinking and learning. This might complement and extend our knowledge that has been obtained on the basis of behavioral data only, the common approach in educational research. At the same time, we note that the existing neuroscientific studies have investigated mathematical performance in relative isolation from the educational context. The characteristics of this context have, however, a large influence on mathematical performance and its correlated brain activity, an issue that should be addressed in future research. We contend that traveling back and forth from cognitive neuroscience to mathematics education might yield a better understanding of how mathematical learning takes place and how it can be influenced.     

Research,Reform, and Equity in U.S. Mathematics Education

This article discusses mathematics education research in relation to equity and current U.S. reforms. Although mathematics education researchers and reformers give attention to equity, work in this area tends to ignore relevant social and cultural issues. I begin by surveying articles on equity published in recent, mainstream education journals, highlighting the lack of attention given to social class and ethnicity. I discuss the implications of this limited research base. Specifically, I argue that current mathematics education reforms have been shaped by good intentions and existing research, neither of which offers adequate guidance to address the complexities of equity in mathematics classrooms today. Drawing from a study of social class differences in students' experiences in one reform-oriented classroom, I discuss the challenges and dilemmas inherent in sociocultural approaches to research in mathematics education and their potential contributions. I call for research from a sociocultural perspective, focusing on ways in which students from underrepresented groups can struggle when encountering particular instructional approaches, and ways in which teachers and students are able to address such struggles.     

Research, Reform, and Equity in U.S. Mathematics Education

This article discusses mathematics education research in relation to equity and current U.S. reforms. Although mathematics education researchers and reformers give attention to equity, work in this area tends to ignore relevant social and cultural issues. I begin by surveying articles on equity published in recent, mainstream education journals, highlighting the lack of attention given to social class and ethnicity. I discuss the implications of this limited research base. Specifically, I argue that current mathematics education reforms have been shaped by good intentions and existing research, neither of which offers adequate guidance to address the complexities of equity in mathematics classrooms today. Drawing from a study of social class differences in students' experiences in one reform-oriented classroom, I discuss the challenges and dilemmas inherent in sociocultural approaches to research in mathematics education and their potential contributions. I call for research from a sociocultural perspective, focusing on ways in which students from underrepresented groups can struggle when encountering particular instructional approaches, and ways in which teachers and students are able to address such struggles.     

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.     

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.     

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.     

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).     

A contemporary analysis of the six “Theories of Mathematics Education” theses of Hans-Georg Steiner

In this contribution we discuss the six theses presented by Hans-Georg Steiner (1987), which were instrumental in the community becoming interested in theories and philosophies of mathematics education. We discuss overlooked aspects of this seminal paper particularly in light of recent developments in the field of mathematics education. Nearly 20 years later, we reflect on the development of Steiner’s program for theory development and examine if any progress has been made at all on the open questions that Steiner (1987) posed to the community.     

刍议大学数学教学活动的问题与对策

当前各高校的大学数学教学或多或少存在一些不协调的地方,如理科类学生课程深度过大,文科类学生数学能力弱,考核试卷的难度控制不好等等.针对大学数学的教学法及大众化教育的分析研究,结合教学实践,提出大学数学教学改革的一些对策.     

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.     

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.     

Representing context in video records of practice for urban mathematics teacher education

Mathematics education research has not sufficiently theorized about mathematics teacher knowledge and practice, teacher learning, and teacher education in ways that are reflective of the specificities of the sociopolitical contexts of schooling. In the USA, this is particularly important for urban mathematics education. This paper examines the affordances and challenges of representing context in video records of practice, particularly in the urban context, for use in the preparation of mathematics teachers for urban settings. The discussion, grounded in current research and theory relevant to representations of teaching, urban education, and mathematics teacher education, takes up three key issues: how is a focus on the urban context relevant to the design of video records of practice for mathematics teacher education? How can video records support prospective teachers’ understandings of the sociopolitical contexts of mathematics teaching? How does a focus on the urban context impact the meaning teachers make of video records?     

Dialogue on mathematics education: two points of view on the state of the art

On many fronts, the field of mathematics education does not speak with a single voice. There appears to be no firm consensus regarding the scientific character of mathematics education, the research methodologies it deems legitimate, the kinds of questions it addresses, the appropriate preparation for its practitioners, and its relationship with other disciplines, including, ironically, mathematics itself. Our field seems to be going through a new phase of self-definition, a crisis from which we shall have to decide who we are and what direction we are going. The authors of the present paper themselves tend towards different positions on these questions. The paper, then, takes the form of a letter in which one of us raises issues about the current state of mathematics education and the other responds. We see this as an attempt to initiate a dialogue on our field, which we consider urgently needed.