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.     

Gender perspectives in mathematics education: intentions of research in Denmark and Norway

A framework is presented for analyzing gender perspectives in mathematics education (structural, symbolic, personal and interactional gender), and the Danish and Norwegian researchers’/teachers’ work within the field of gender and mathematics is presented with reference to these four perspectives. Furthermore, the gender issue in TIMSS and PISA is briefly discussed. The main thread through the article is the researchers’ willingness and intentions of investigating the gender perspectives in mathematics education. However, so far, these research intentions have not been realized in Denmark and Norway.     

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.     

Sociocultural perspectives in research on and with mathematics teachers: a zone theory approach

Sociocultural theories view teacher learning as changing participation in social practices that develop their professional identities rather than as acquisition of new knowledge or beliefs that are internal to the individual. Although sociocultural research on mathematics teacher education has tended to focus on understanding teachers’ learning, this article argues that sociocultural perspectives can also guide more interventionist research involving changing classroom practice. The approach illustrated here uses an adaptation of Valsiner’s zone theory to analyse teacher learning and development in two separate research studies. In one study the aim was to understand how teachers incorporated digital technologies into their practice, while the other study helped teachers implement an investigative approach to working mathematically consistent with a new syllabus. In both studies, productive tensions between teachers’ beliefs, contexts, and goals were a trigger for learning and development.     

A global survey of international perspectives on modelling in mathematics education

In this article we survey the current debate on modelling and, describe different perspectives on this debate. We relate these perspectives with earlier perspectives and show similarities and differences between these different approaches.     

A comprehensive theory of representation for mathematics education

Representation is a difficult concept. Behaviorists wanted to get rid of it; many researchers prefer other terms like “conception” or “reasoning” or even “encoding;” and many cognitive science resarchers have tried to avoid the problem by reducing thinking to production rules.There are at least two simple and naive reasons for considering representation as an important subject for scientific study. The first one is that we all experience representation as a stream of internal images, gestures and words. The second one is that the words and symbols we use to communicate do not refer directly to reality but to represented entities: objects, properties, relationships, processes, actions, and constructs, about which there is no automatic agreement between two persons. It is the purpose of this paper to analyse this problem, and to try to connect it with an original analysis of the role of action in representation. The issue is important for mathematics education and even for the epistemology of mathematics, as mathematical concepts have their first roots in the action on, and in the representation of, the physical and social world; even though there may be a great distance today between that pragmatical and empirical source, and the sophisticated concepts of contemporary mathematics.     

Game show mathematics: Specializing,conjecturing, generalizing,and convincing

This article describes the authors’ use of three game shows – Survivor, The Biggest Loser, and Deal or No Deal? – to determine to what degree students engaged in mathematical thinking: specializing, conjecturing, generalizing, and convincing ( Burton, 1984). Student responses to the task of creating winning strategies to these shows were collected and analyzed. The data showed that students generally did not engage in the process of mathematical thinking unless directed to do so and the effects this had on the students’ responses is discussed.     

Theories in practice: mathematics teaching and mathematics teacher education

Whilst research on the teaching of mathematics and the preparation of teachers of mathematics has been of major concern in our field for some decades, one can see a proliferation of such studies and of theories in relation to that work in recent years. This article is a reaction to the other papers in this special issue but I attempt, at the same time, to offer a different perspective. I examine first the theories of learning that are either explicitly or implicitly presented, noting the need for such theories in relation to teacher learning, separating them into: socio-cultural theories; Piagetian theory; and learning from practice. I go on to discuss the role of social and individual perspectives in authors’ approach. In the final section I consider the nature of the knowledge labelled as mathematical knowledge for teaching (MKT). I suggest that there is an implied telos about ‘good teaching’ in much of our research and that perhaps the challenge is to study what happens in practice and offer multiple stories of that practice in the spirit of “wild profusion” (Lather in Getting lost: Feminist efforts towards a double(d) science. SUNY Press, New York, 2007).     

Immigrant parents’ perspectives on their children’s mathematics education

This paper draws on two research studies with similar theoretical backgrounds, in two different settings, Barcelona (Spain) and Tucson (USA). From a sociocultural perspective, the analysis of mathematics education in multilingual and multiethnic classrooms requires us to consider contexts, such as the family context, that have an influence on these classrooms and its participants. We focus on immigrant parents' perspectives on their children's mathematics education and we primarily discuss two topics (1) their experiences with the teaching of mathematics, and (2) the role of language (native language and second language). The two topics are explored with reference to the immigrant student's or their parents' former educational systems (the “before”) and their current educational systems (the “now”). Parents and schools understand educational systems, classroom cultures and students' attainment differently, as influenced by their sociocultural histories and contexts.     

Cristoforo Borri and the epistemological status of mathematics in seventeenth-century Portugal

This paper argues that the epistemological promotion of mathematics by the Jesuit Cristoforo Borri, while he was teaching at the Coimbra Jesuit College in the late 1620s, played a decisive role in the updating of cosmological ideas in 17th-century Portugal. The paper focuses on Borri's position on the celebrated quaestio de certitudine mathematicarum and on his understanding of the classification of sciences. It argues that by conferring on mathematics the status of Aristotelian causal science, Borri made it possible to integrate mathematical data into the philosophical debate, particularly with regard to the new cosmology.     

Computing devices,mathematics education and mathematics: Sexton’s omnimetre in its time

Material objects can tell us much about mathematical practice. In 1899, Albert Sexton, a Philadelphia mechanical engineer, received the John Scott Medal of the Franklin Institute for his invention of the omnimetre. This inexpensive circular slide rule was one of a host of computing devices that became common in the United States around 1900. It is inscribed “NUMERI MUNDUM REGUNT”. In part because of instruments such as the omnimetre, numbers increasingly ruled the practical world of the late 19th and early 20th century. This changed not only engineering, but mathematics education and mathematical work.     

Classroom-based interventions in mathematics education: relevance, significance, and applicability

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth 2013); (2) Development of Ways of Thinking (Jahnke and Wambach 2013; Lehrer, Kobiela and Weinberg 2013; Mariotti 2013; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides 2013; Tabach, Hershkowitz and Dreyfus 2013); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart 2013); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann 2013).     

The mutual influence of theory and practice in mathematics education: implications for research and teaching

Questions related on how to connect theory and practice in school mathematics have been under debate for several years. Also, different forms of co-operation between academic researchers and school teachers are widely discussed. In the search for boundary conditions to mediate knowledge between the two poles, there is evidence that any conception which assigns to “theory” the place of instructin, “practice” is doomed to fail, and the necessity of developing the notion of cooperation comes as a consequence. Following this assumption, existing literature provides interesting contributions supporting the idea of blending mathematical content with pedagogical knowledge. This contribution focuses on the role that theoretical models, as emerged from the observation of students at work, can play on instructing practice. In particular, we will approach algebraic thinking and refer to a theoretical model based on the distinction between sense and denotation of algebraic expressions. We will then discuss how this theoretical model can shed light on students’ difficulties when solving equations and inequalities. Finally, we will point out how findings coming from research can suitably orient teachers and promote further development.