Building bridges within mathematics education: Teaching, research, and instructional design

Within mathematics education, classroom teachers, educational researchers, and instructional designers share the common goals of understanding and improving the teaching and learning of mathematics. Teachers work to help students learn; researchers study how people learn and teach mathematics; and designers develop instructional materials to support teachers and students. Each community (of teachers, of researchers, and of designers) develops its own perspectives, methods, and expertise. Too seldom, however, do practitioners have the opportunity to share their knowledge across communities. This first-person, retrospective case study speaks to the challenges and rewards of building bridges among these three communities by charting the evolution of an instructional activity (using graphing software to explore slope) through four cycles of teaching, research, and design. Initially separate, the three perspectives of teacher, researcher, and designer begin to interact as the worksite moves from the university laboratory to the author's classroom and then to other teachers’ classrooms. Many of these interactions are fruitful, resulting in new insights and strategies that strengthen the final product and inform the practitioner. At the same time, some tensions arise, particularly between teaching and research, highlighting fundamental differences between these fields. Lessons from this case study suggest implications for collaborations among teachers, researchers, and designers.     

Progress and stagnation of gender equity: contradictory trends within mathematics research and education in Sweden

During the last decade women in Sweden have reduced men’s lead in participation in mathematics education and in professional careers as mathematicians. However, the development is uneven and slow overall. In some areas and at the highest levels women have increased their participation only marginally. Why, one may ask, is progress so slow after almost 20 years of active work from the Women and Mathematics movement in Sweden and within a society in which gender equity is highly valued at the societal and political levels? The development is described in quantitative measures going back 20 years. Several concrete and successful initiatives from the last decade intended to “de-gender” mathematics and to involve women and men alike in mathematics are described. In contrast a gender-blind position or a view of women as problems in mathematics seems to reign within some influential bodies.     

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.     

The onto-semiotic approach to research in mathematics education

In this paper we synthesize the theoretical model about mathematical cognition and instruction that we have been developing in the past years, which provides conceptual and methodological tools to pose and deal with research problems in mathematics education. Following Steiner’s Theory of Mathematics Education Programme, this theoretical framework is based on elements taken from diverse disciplines such as anthropology, semiotics and ecology. We also assume complementary elements from different theoretical models used in mathematics education to develop a unified approach to didactic phenomena that takes into account their epistemological, cognitive, socio cultural and instructional dimensions.     

Balancing mathematics education research and the NCTM standards

The release of the Principles and Standards for School Mathematics in the United States by the National Council of Teachers of Mathematics (NCTM) brought to the forefront the debate of whether research should determine the validity of the espoused Standards? Or conversely whether the Standards should influence the research agenda of the mathematics education community? How should university teacher educators address this issue? Should pre-service and practicing teachers blindly, accept the Standards as well as the research, or do we cultivate the critical thinking skills that will allow preparing teachers to resolve this dilemma? In this article a university mathematics educator and an idealistic pre-service elementary teacher try to resolve the dilemma of balancing the Standards with research and personal beliefs about the teaching and learning of mathematics.     

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.     

The emergence of women on the international stage of mathematics education

In this article, I consider the history of the International Commission on Mathematical Instruction (ICMI) from its inception until the International Congress on Mathematical Education (ICME) held in 1969. In this period, mathematics education developed as a scientific discipline. My aim is to study the presence and the contribution of women (if any) in this development. ICMI was founded in 1908, but my history starts before then, at the end of the nineteenth century, when the process of internationalization of mathematics began, thanks to the first International Congress of Mathematicians. Already in those years, the need for internationalizing the debate on mathematics teaching was spreading throughout the mathematical community. I use as my main sources of information the didactics sections in the proceedings of the International Congresses of Mathematicians and the proceedings of the first ICME. The data collected are complemented with information from the editorial board of two journals that for different reasons are linked to ICMI: L’Enseignement Mathématique and Educational Studies in Mathematics. In particular, as a result of my analyses, I have identified four women who may be considered as pioneer women in mathematics education. Some biographical notes on their professional life are included in the paper.     

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.     

The conceptualisation of mathematics competencies in the international teacher education study TEDS-M

The main aim of the international teacher education study Teacher Education and Development Study in Mathematics (TEDS-M), carried out under the auspices of the International Association for the Evaluation of Educational Achievement (IEA), was to understand how national policies and institutional practices influence the outcomes of mathematics teacher education. This paper reports on the definition of effective mathematics teacher education in TEDS-M, distinguishing between mathematics content knowledge and mathematics pedagogical content knowledge as essential cognitive components of mathematics teachers’ professional competencies. These competence facets were implemented as proficiency tests based on extensive coordination and validation processes by experts from all participating countries. International acceptance of the tests was accomplished whereas, by necessity, national specifications had to be left out, as is common in comparative large-scale assessments. In this paper, the nature of the TEDS-M tests for the primary study is analysed and commented on detail. The aims are to increase our understanding of mathematics content knowledge and mathematics pedagogical content knowledge, which are still fuzzy domains, to provide a substantive background for interpretations of the test results and to examine whether some educational traditions may be more accurately reflected in the test items than others. For this purpose, several items that have been released by the IEA are presented and elaborately analysed in order to substantiate the test design of TEDS-M. Our main conclusion is that the overall validity of the TEDS-M tests can be regarded as a given, but that readers have to be aware of limitations, amongst others from a continental European point of view.     

Textbook research in mathematics education: development status and directions

This paper presents a survey study aiming to systematically examine, analyse and review relevant research focusing on mathematics textbooks and hence identify future directions in this field of research. The literature surveyed is selected from different data sources, including mainly journal articles, research theses and conference proceedings. The survey revealed that important progress has been made over the last few decades in mathematics textbook research, though the major achievement has been concentrated in the areas of textbook analysis (including textbook comparison), and the use of textbooks in teaching and learning. It is overall no longer true that the textbook research in mathematics is “scattered, inconclusive, and often trivial” as described six decades ago; however, the development of research on mathematics textbooks has been unbalanced in different areas. Following the review and discussion, the paper proposes five needed directions for advancing the research in this field.     

Problem solving in Chinese mathematics education: research and practice

This paper is an attempt to paint a picture of problem solving in Chinese mathematics education, where problem solving has been viewed both as an instructional goal and as an instructional approach. In discussing problem-solving research from four perspectives, it is found that the research in China has been much more content and experience-based than cognitive and empirical-based. We also describe several problem-solving activities in the Chinese classroom, including “one problem multiple solutions,” “multiple problems one solution,” and “one problem multiple changes.” Unfortunately, there are no empirical investigations that document the actual effectiveness and reasons for the effectiveness of those problem-solving activities. Nevertheless, these problem-solving activities should be useful references for helping students make sense of mathematics.     

Exploring MLD in mathematics education: Ten years of research

Mathematical learning disabilities or difficulties (MLD) are an increasing source of educational inequalities. This article explores research about MLD in Mathematics Education over the past ten years. The methodology focuses on specific, validated keywords. These keywords are used to identify articles in leading journals in mathematics education. Our work makes several new contributions to the field of mathematics education, notably: a reusable methodology and keywords for a literature review; an exhaustive list of articles about MLD in leading mathematics education journals; a discussion of the definitions and features of MLD used in these articles; and a tool to classify research dealing with MLD (categories that characterize students with MLD). We also highlight some unexplored dimensions regarding MLD in Mathematics Education research.     

A survey of advanced mathematics topics: a new high school mathematics class

The number of students pursuing undergraduate degrees in mathematics is decreasing. Research reveals students who pursue mathematics majors complained about inadequate high school preparation in terms of disciplinary content or depth, conceptual grasp, or study skills. Unfortunately, the decrease in the number of students studying advanced mathematics occurs at a time when the world's technological drive demands students have improved critical thinking and problem-solving skills. This paper suggests one solution for this alarming problem: a high school class offered to seniors as a means of preparing them for the rigours of college level mathematics while simultaneously increasing their motivation to pursue advanced mathematics. This paper provides the course scope, goals, structure, and analysis of how the curriculum aligns to professional standards. Although this programme has not currently been field tested, the authors are convinced of its impact. Once implemented and properly taught, the proposed Survey of Advanced Mathematics Topics class could increase the quantity and quality of students pursuing studies in mathematics at the university level.