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.     

In the books there are golden houses: mathematics assessment in East Asia

In this paper, some fundamental issues on mathematics assessment and how they are related to the underlying cultural values in East Asia are discussed. Features of the East Asian culture that impact on mathematics assessment include the pragmatic nature of the culture, the social orientation of East Asian people, and the lop-sided stress on the utilitarian function of education. East Asians stress the algorithmic side of mathematics, and mathematics is viewed more as a set of techniques for calculation and problem solving. The notion of fairness in assessment is of paramount importance, and there is a great trust in examination as a fair method of differentiating between the able and the less able. The selection function of education and assessment has great impact on how mathematics is taught, and assessment constitutes an extrinsic motivation which directs student learning. Finally, the strengths and weaknesses of these East Asian values are discussed.     

中美大学数学教育研究之比较

论证了大学数学教育是高等教育的核心,数学教育本质上是一种素质教育,数学的应用遍及自然科学和社会科学.数学是大学理工科各专业知识的基础;探讨了大学数学教育研究的重要性,对中美大学数学教育的研究进行了对比,指出我们在研究大学数学教育方面的不足之处,提出了我们在大学数学教育研究方面的任务和目标.     

立足教育创新瞩目大学数学教育

在回顾我国非数学专业数学教育、教学改革的基础上,对当今的大学数学教育进行了分析和展望.对大学数学教育的地位及作用进行了探讨.认为大学数学教育,要体现教育创新思想,要在改革中求生存,以成绩促发展.     

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.     

Mathematics education and democracy

In this paper, we investigate the relationship between mathematics education and the notions of education for all/democracy. In order to proceed with our analysis, we present Marx’s concept of commodity and Jean Baudrillard’s concept of sign value as a theoretical reference in the discussion of how knowledge has become a universal need in today’s society and ideology. After, we engage in showing mathematics education’s historical and epistemological grip to this ideology. We claim that mathematics education appears in the time period that English becomes an international language and the notion of international seems to be a key constructor in the constitution of that ideology. Here, we draw from Derrida’s famous saying that “there is nothing beyond the text”. We conclude that a critique to modern society and education has been developed from an idealistic concept of democracy.     

The relationship between mathematicians’ pedagogical goals,orientations, and common teaching practices in advanced mathematics

Despite mathematics educators’ research into more effective modes of teaching, lecture is still the dominant mode of instruction in undergraduate mathematics courses. Surveys suggest this is because most mathematicians believe this is the best way to teach. This paper answers a call by mathematics education researchers to explore mathematicians’ needs and goals concerning teaching. We interviewed eight mathematicians about findings in the mathematics education research literature concerning common pedagogical practices of instructors of advanced mathematics classes: “chalk talk,” the presentation of formal and informal content, and teacher questioning. We then analyzed the responses for resources, orientations, and goals that might influence the participants to engage in these practices. We describe how participants believed common lecturing practices allowed them to achieve their goals and aligned with their orientations. We discuss these findings in depth and consider what implications they may have for researchers that aim to change mathematicians’ teaching practices.     

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.     

Introducing the Emerging Discipline of Statistics Education

Increasing attention has been given over the last decade by the statistics, mathematics and science education communities to the development of statistical literacy and numeracy skills of all citizens and the enhancement of statistics education at all levels. This paper introduces the emerging discipline of statistics education and considers its role in the development of these important skills. The paper begins with information on the growing importance of statistics in today's society, schools and colleges, summarizes unique challenges students face as they learn statistics, and makes a case for the importance of collaboration between mathematicians and statisticians in preparing teachers to teach students how to understand and reason about data. We discuss the differences and interrelations between statistics and mathematics, recognizing that mathematics is the discipline that has traditionally included instruction in statistics. We conclude with an argument that statistics should be viewed as a bridge between mathematics and science and should be taught in both disciplines.     

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.     

Online distance mathematics education in Brazil: research, practice and policy

In this article, we address online distance mathematics education research and practice in Brazil, which are relative newcomers to the educational scene. We present the national context of education in Brazil, highlighting the organization of the educational system, and also a summary of national legislation on distance education and an overview of digital inclusion in the country. We outline the potential and relevance of distance education for the Brazilian educational system and show how it could intervene in the system. With respect to research and practice in online mathematics education, we present support for research, examples of studies and highlight different aspects being addressed, including its essential components. In addition, we discuss the synergy between distance education and teacher education, and mathematics distance education and modeling, as well as other initiatives in the national scenario.     

Preservice Secondary Mathematics Teachers' Conceptualizations of Equity: Access and Power as Seen Through Vignette Responses

Attention to equity in the mathematics education field has been growing in recent years. We have evidence that many novice secondary mathematics teachers do not feel prepared to teach in regards to diverse populations. We need to know more about how secondary preservice mathematics teachers (PSMTs) conceptualize equitable environments. This study investigates 30 secondary PSMTs' proposed responses to two hypothetical vignettes from mathematics department conversations regarding calculator usage and mathematical discourse, respectively, utilizing two of Gutiérrez's four dimensions of equity: Access and Power. Results suggest these PSMTs considered equity, equality, and creating a classroom that invites participation among other factors when thinking of an equitable approach with respect to calculator usage. When considering mathematical discourse, PSMTs cited the need to “model” proper use of mathematical language as well as to allow students to themselves verbalize it. Implications mathematics education and teacher education more broadly are to integrate equity and equality discussions in methods courses and to include strategies to facilitate productive discourse.     

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 capital in the educational field: Bourdieu and beyond

Mathematics education needs a better appreciation of the dominant power structures in the educational field: Bourdieu's theory of capital provides a good starting point. We argue from Bourdieu's perspective that school mathematics provides capital that is finely tuned to generationally reproduce the social structures that serve to keep the powerful in power, while ensuring that less powerful groups are led to accept their own failure in mathematics. Bourdieu's perspective thereby highlights theoretical inadequacies in much mathematics education research, insofar as it presumes a consensus about a ‘what works agenda’ for improving achievement for all. Drawing on one case where we manufactured awkward facts, we illustrate a Bourdieusian interpretation of mathematics capital as reproductive, and the crucial role of its cultural arbitrary. We then criticise the Bourdieusian concept of ‘mathematical capital’ as the value of mathematical competence in practice and propose to extend his tools to include the contradictory ‘use’ and ‘exchange’ values of mathematics instead: we will show how this conceptualisation goes ‘beyond Bourdieu’ and helps explain how teaching-learning might (ideally) produce ‘cultural use value’ in mathematical competence, while still recognising the contradictions teachers and learners face. Finally, we suggest how critical education research generally can benefit from this theoretical framework: (1) in exposing the interest of the dominant classes; but also (2) in researching critical pedagogic alternatives that challenge orthodoxy in educational policy and practice both in mathematics education and more generally.     

Trends of the history of mathematics education in Brazil

The text aims at characterizing and analyzing the production of the history of mathematics education in Brazil. The study takes the presentations from the last National Seminar of History of Mathematics as a starting point. Such event gathered researchers, in a more significant and comprising way, interested in historical studies of mathematics and its teaching. The characterization points at the existence of four different tendencies: a production that considers the studies about history of mathematics education as part of the research about history of mathematics; a trend of opinions that takes into account the pedagogical use of history in mathematics education and establishes conditions for the research in the history of mathematics education; studies that use oral history for mathematics teachers’ training courses; and, finally, another trend that treats the history of mathematics education as history, that means, a specificity of historical production, having the mathematics education as an object. It will be shown that the characterization of those trends reveals different ways of representing the past of mathematics education, as well as the relationships Brazilian researches keep with international studies about this subject.     

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.     

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.     

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.