Beliefs and beyond: affect and the teaching and learning of mathematics

The importance of beliefs for the teaching and learning of mathematics is widely recognized among mathematics educators. In this special issue, we explicitly address what we call “beliefs and beyond” to indicate the larger field surrounding beliefs in mathematics education. This is done to broaden the discussion to related concepts (which may not originate in mathematics education) and to consider the interconnectedness of concepts. In particular, we present some new developments at the conceptual level, address different approaches to investigate beliefs, highlight the role of student beliefs in problem-solving activities, and discuss teacher beliefs and their significance for professional development. One specific intention is to consider expertise from colleagues in the fields of educational research and psychology, side by side with perspectives provided by researchers from mathematics education.     

Teaching values and valued teaching in the mathematics classroom: toward a research agenda

In this paper, we first discuss the teaching of values by focusing on the kinds of values that have been discussed and studied in the other papers in this special journal issue and elsewhere. Then we raise a number of issues about the product-based values in mathematics education, which we identify as teaching values and which can be realized through classroom instruction. In the second section, we discuss the process-based valued teaching methods used to maximize the realization of the teaching values in the classroom. As valued teaching may be perceived differently by different people, in the discussion we analyze how it is seen from both students?? and teachers?? perspectives. We end this paper by discussing a number of methodological issues in studying teaching values and valued teaching as well as offering suggestions for future research.     

Seeking research-grounded solutions to problems of practice: classroom-based interventions in mathematics education

Research on classroom-based interventions in mathematics education has two core aims: (a) to improve classroom practice by engineering ways to act upon problems of practice; and (b) to deepen theoretical understanding of classroom phenomena that relate to these problems. Although there are notable examples of classroom-based intervention studies in mathematics education research since at least the 1930s, the number of such studies is small and acutely disproportionate to the number of studies that have documented problems of classroom practice for which solutions are sorely needed. In this paper we first make a case for the importance of research on classroom-based interventions and identify three important features of this research, which we then use to review the papers in this special issue. We also consider the issue of ‘scaling up’ promising classroom-based interventions in mathematics education, and we discuss a major obstacle that most such interventions find on the way to scaling up. This obstacle relates to their long duration, which means that possible adoption of these interventions would require practitioners to do major reorganizations of the mathematics curricula they follow in order to accommodate the time demands of the interventions. We argue that it is important, and conjecture that it is possible, to design interventions of short duration in mathematics education to alleviate major problems of classroom practice. Such interventions would be more amenable to scaling up, for they would allow more control over confounding variables and would make more practicable their incorporation into existing curriculum structures.     

Mathematical creativity and giftedness: a commentary on and review of theory, new operational views, and ways forward

In this commentary we synthesize and critique three papers in this special issue of ZDM (Leikin and Lev; Kattou, Kontoyianni, Pitta-Pantazi, and Christou; Pitta-Pantazi, Sophocleous, and Christou). In particular we address the theory that bridges the constructs of “mathematical creativity” and “mathematical giftedness” by reviewing the related literature. Finally, we discuss the need for a reliable metric to assess problem difficulty and problem sequencing in instruments that purport to measure mathematical creativity, as well as the need to situate mathematics education research within an existing canon of work in mainstream psychology.     

Dynamics of change of mathematics education in Brazil and a scenario of current research

Mathematics education in Brazil, if we consider what one may call the scientific phase, is about 30 years old. The papers for this special issue focus mainly on this period. During these years, many trends have emerged in mathematics education to address the complex problems facing Brazilian society. However, most Brazilian mathematics educators feel that the separation of research into trends is a theoretical idealization that does not respond to the dynamics of the problems we face. We raise the conjecture that the complexity of Brazilian society, where pockets of wealth coexist with the most shocking poverty, has contributed to the adoption and generation of different strands in mathematics education, crossing the boundaries between trends. At a more micro level, we also raise the conjecture that Brazilian trends in research are interwoven because of the way that Brazilian mathematics educators have experienced the process of globalization over these 30 years. This tapestry of trends is a predominant characteristic of mathematics education in Brazil.     

Paying the Price for Sugar and Spice: Shifting the Analytical Lens in Equity Research

The analytical stance taken by equity researchers in education, the methodologies employed, and the interpretations that are drawn from data all have an enormous impact on the knowledge that is produced about sources of inequality. In the 1970s and 1980s, a great deal of interest was given to the issue of women's and girls' underachievement in mathematics. This prompted numerous different research projects that investigated the extent and nature of the differences between girls' and boys' achievement and offered reasons why such disparities occurred. This work contributed to a discourse on gender and mathematics that flowed through the media channels and into schools, homes, and the workplace. In this article, I consider some of the scholarship on gender and mathematics, critically examining the findings that were produced and the influence they had. In the process, I propose a fundamental tension in research on equity, as scholars walk a fine and precarious line between lack of concern on the one hand and essentialism on the other. I argue in this article that negotiating that tension may be the most critical role for equity researchers as we move into the future.     

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.     

Inhibitory control and mathematics learning: definitional and operational considerations

The topic of inhibition in mathematics education is both well timed and important. In this commentary, we reflect on the role of inhibition in mathematics learning through four themes that relate to how inhibition is defined, measured, developed, and applied. First, we consider different characterizations of inhibition and how they may shape the ways that inhibition is conceptualized and studied in mathematics contexts. Second, we discuss methods that researchers use to study inhibition and how differences across these methods may constrain researchers’ conclusions or what these differences may imply for students’ use of inhibition when solving authentic mathematics problems. Third, we consider the relationship between intuition and mathematics content knowledge, including how this relationship may vary for students with different levels of knowledge. We end with a discussion of inhibition’s practical educational relevance, in which we offer a set of questions that may inform future conversations or research in the field.     

Paying the Price for "Sugar and Spice": Shifting the Analytical Lens in Equity Research

The analytical stance taken by equity researchers in education, the methodologies employed, and the interpretations that are drawn from data all have an enormous impact on the knowledge that is produced about sources of inequality. In the 1970s and 1980s, a great deal of interest was given to the issue of women's and girls' underachievement in mathematics. This prompted numerous different research projects that investigated the extent and nature of the differences between girls' and boys' achievement and offered reasons why such disparities occurred. This work contributed to a discourse on gender and mathematics that flowed through the media channels and into schools, homes, and the workplace. In this article, I consider some of the scholarship on gender and mathematics, critically examining the findings that were produced and the influence they had. In the process, I propose a fundamental tension in research on equity, as scholars walk a fine and precarious line between lack of concern on the one hand and essentialism on the other. I argue in this article that negotiating that tension may be the most critical role for equity researchers as we move into the future.     

Online mathematics teacher education: overview of an emergent field of research

Online mathematics teacher education is characterized as an emergent area of research in mathematics education. We identify some key topics that require further research: communities and networks of teachers in online environments; sustainability of these communities and kinds of organizational structures; knowledge-building practices in technology-mediated work group interactions; and online interactions among teachers. The emergence of new research issues also gives rise to new theoretical approaches or the adaptation of existing theoretical perspectives that are presented in this special issue. We summarize some of these theoretical perspectives and attempt to show how online environments have changed them, as well as some theoretical problems that remain to be solved.     

Trends in researching the socioeconomic influences on mathematical achievement

We introduce the topic of socioeconomic influences on mathematical achievement through an overview of existing research reports and articles. International trends in the way the topic has emerged and become increasingly important in the international field of mathematics education research are outlined. From this review, there is a discussion about what appears to be neglected in previous work in this area and how the papers in this issue of ZDM provide information about some of these neglected areas. The main argument in this article is that socioeconomic influences on mathematical achievement should not be considered as a taken-for-granted fact that is accepted uncritically. Instead, it is suggested that the relationship between multiple socioeconomic influences and various understandings of mathematical achievement are historically contingent ways of understanding exclusions and inclusions in mathematics education practices. Research is not simply “evidencing” the facts of these relationships; research is also implicated in constructing the ways in which we think about these. Thus, mathematics education researchers could devise more nuanced approaches for understanding the social, political and historical constitution of these relationships.     

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.     

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM—The International Journal on Mathematics Education—by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers’ and didacticians’ engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice.     

The state-of-art in mathematical creativity

Creativity is a topic which is often neglected within mathematics teaching. Usually teachers think that it is logic that is needed in mathematics in the first place, and that creativity is not important and learning mathematics. On the other hand, if we consider a mathematician who develops new results in mathematics. we cannot overlook his/her use of the creative potential. Thus, the main questions are as follows: What methods could be used to foster mathematical creativity within school situations? What scientific knowledge, i.e. research results, do we have on the meaning of mathematical creativity?     

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.     

Interviewing in mathematics education research: Choosing the questions

With the development of qualitative methodologies, interviewing has become one of the main tools in mathematics education research. As the first step in analyzing interviewing in mathematics education we focus here on the stage of planning, specifically, on designing the interview questions. We attempt to outline several features of interview questions and understand what guides researchers in choosing the interview questions. Our observations and conclusions are based on examining research in mathematics education that uses interviews as a data-collection tool and on interviews with practicing researchers reflecting on their practice.     

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.     

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).