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.     

Relationships Between Student,Teacher, and School Characteristics and Mathematics Achievement

A variety of factors contributes to student achievement in mathematics, including but not limited to student behaviors and student, teacher, and school characteristics. The purpose of this study was to explore which of these factors have an impact on student mathematics achievement. The target population for this study was North Carolina Algebra II students. Analyses of variance models were examined for group differences and a Three‐level Hierarchical Linear Modeling method was employed to examine individual predictors of student achievement in mathematics. Statistically, significant differences were found between students of different ethnicities, socioeconomic statuses (SES), and parental education levels. No gender effects were statistically significant. All teacher‐level variables investigated were found to be statistically significant, impacting student achievement in mathematics. School size and SES were not found to significantly contribute to student achievement. More research on the relationships between these factors shown to make statistically significant differences on mathematics achievement is needed to further explain several phenomena that this research reveals.     

Socio-economic and cultural mediators of mathematics achievement and between-school equity in mathematics education at the global level

This paper aims at identifying and understanding how and to what extent socioeconomic and cultural factors mediate mathematics achievement and between-school equity in mathematics education among countries. First, under the assumption that equity and quality of education are independent constructs, the construct of equity-in-quality in mathematics education is developed. Second, the 18 countries that were identified in the previous work of the author Jurdak in (Toward equity in quality in mathematics education. Springer, New York, 2009) as being diverse in mathematics education will serve as study cases to illustrate, compare, and contrast the mediatory role of socio-economic and cultural factors in mathematics achievement and between-school equity in mathematics education. The results show that the differences in mathematics achievement and between-school equity in mathematics education at the country level are associated with, and can be accounted for in some cases, by socioeconomic and cultural factors.     

Toward the problem-centered classroom: trends in mathematical problem solving in Japan

In this paper, I summarize the influence of mathematical problem solving on mathematics education in Japan. During the 1980–1990s, many studies had been conducted under the title of problem solving, and, therefore, even until now, the curriculum, textbook, evaluation and teaching have been changing. Considering these, it is possible to identify several influences. They include that mathematical problem solving helped to (1) enable the deepening and widening of our knowledge of the students’ processes of thinking and learning mathematics, (2) stimulate our efforts to develop materials and effective ways of organizing lessons with problem solving, and (3) provide a powerful means of assessing students’ thinking and attitude. Before 1980, we had a history of both research and practice, based on the importance of mathematical thinking. This culture of mathematical thinking in Japanese mathematics education is the foundation of these influences.     

When time is an implicit variable: An investigation of students’ ways of understanding graphing tasks

In this study, we investigate students’ ways of understanding graphing tasks involving quantitative relationships in which time functions as an implicit variable. Through task-based interviews of students ages 14–16 in a summer mathematics program, we observe a variety of ways of understanding, including thematic or visual association, pointwise thinking, and reasoning parametrically about changes in the two variables to be graphed. We argue that, rather than comprising a hierarchy, these ways of understanding complement one another in helping students discover an invariant relationship between two dynamically varying quantities, and develop a graph of the relationship that captures this invariance. From these ways of understanding, we conjecture several mathematical meanings for graphing that may account for students’ behavior when graphing quantitative relationships.     

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?     

Three modes of STEM integration for middle school mathematics teachers

Of the four subjects in an integrated science, technology, engineering, and mathematics (STEM) approach, mathematics has not received enough focus. This could be in part because mathematics teachers may be apprehensive or unsure about how to implement integrated STEM education in their classrooms. There are benefits to integrated STEM in a mathematics classroom though, including increased motivation, interest, and achievement for students. This article discusses three methods that middle school mathematics teachers can utilize to integrate STEM subjects. By focusing on open‐ended problems through engineering design challenges, mathematical modeling, and mathematics integrated with technology middle school students are more likely to see mathematics as relevant and valuable. Important considerations are discussed as well as recent research with these approaches.     

Mathematikdidaktik: Die Erforschung theoretischen Wissens in sozialen Kontexten des Lernens und Lehrens

The problem of “defining” mathematics education as a proper scientific discipline has been discussed controversely for more than 20 years now. The paper tries to clarify some important aspects especially for answering the question of what makes mathematics education a specific scientific discipline and a field of research. With this aim in mind the following two dimensions are investigated: On the one hand, one has to be aware that mathematics is not “per se” the object of research in mathematics education, but that mathematical knowledge always has to be regarded as being “situated” within a social context. This means that mathematical knowledge only gains its specific epistemological meaning within a social context and that the development and understanding of mathematical knowledge is strongly influenced by the social context. On the other hand the specificity of the theory-practice-problem poses an essential demand on the scientific work in mathematics education.     

A case for combinatorics: A research commentary

In this commentary, we make a case for the explicit inclusion of combinatorial topics in mathematics curricula, where it is currently essentially absent. We suggest ways in which researchers might inform the field’s understanding of combinatorics and its potential role in curricula. We reflect on five decades of research that has been conducted since a call by Kapur (1970) for a greater focus on combinatorics in mathematics education. Specifically, we discuss the following five assertions: 1) Combinatorics is accessible, 2) Combinatorics problems provide opportunities for rich mathematical thinking, 3) Combinatorics fosters desirable mathematical practices, 4) Combinatorics can contribute positively to issues of equity in mathematics education, and 5) Combinatorics is a natural domain in which to examine and develop computational thinking and activity. Ultimately, we make a case for the valuable and unique ways in which combinatorics might effectively be leveraged within K-16 curricula.     

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.     

Backward Transfer: An Investigation of the Influence of Quadratic Functions Instruction on Students’ Prior Ways of Reasoning About Linear Functions

The transfer of learning has been the subject of much scientific inquiry in the social sciences. However, mathematics education research has given little attention to a subclass called backward transfer, which is when learning about new concepts influences learners’ ways of reasoning about previously encountered concepts. This study examined when and in what ways a quadratic functions instructional unit productively influenced middle school students’ ways of reasoning about linear functions. Results showed that students’ ways of reasoning about essential properties of linear functions were productively influenced. Furthermore, conceptual connections were identified linking changes in students’ ways of reasoning about linear functions to what they learned during the quadratics unit. These findings suggest that it is possible to productively influence learners’ ways of reasoning about previously learned-about concepts in significant respects while teaching them new material and that backward transfer offers promise as a new focus for mathematics education research.     

Using dynamic software in mathematics: the case of reflection symmetry

This study was carried out to examine the effects of computer-assisted instruction (CAI) using dynamic software on the achievement of students in mathematics in the topic of reflection symmetry. The study also aimed to ascertain the pre-service mathematics teachers’ opinions on the use of CAI in mathematics lessons. In the study, a mixed research method was used. The study group of this research consists of 30 pre-service mathematics teachers. The data collection tools used include a reflection knowledge test, a survey and observations. Based on the analysis of the data obtained from the study, the use of CAI had a positive effect on achievement in the topic of reflection symmetry of the pre-service mathematics teachers. The pre-service mathematics teachers were found to largely consider that a mathematics education which is carried out utilizing CAI will be more beneficial in terms of ‘visualization’, ‘saving of time’ and ‘increasing interest/attention in the lesson’. In addition, it was found that the vast majority of them considered using computers in their teaching on the condition that the learning environment in which they would be operating has the appropriate technological equipment.     

Exploring gender factors related to PISA 2003 results in Iceland: a youth interview study

Students’ mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (by comparison) unusual gender differences in mathematics: Iceland was the only country in which the mathematics gender gap favored girls. When data were broken down and analyzed, the Icelandic gender gap appeared statistically significant only in the rural areas of Iceland, suggesting a question about differences in rural and urban educational communities. In the 2007 qualitative research study reported in this paper, the authors interviewed 19 students from rural and urban Iceland who participated in PISA 2003 in order to investigate these differences and to identify factors that contributed to gender differences in mathematics learning. Students were asked to talk about their mathematical experiences, their thoughts about the PISA results, and their ideas about the reasons behind the PISA 2003 results. The data were transcribed, coded, and analyzed using techniques from analytic induction in order to build themes and to present both male and female student perspectives on the Icelandic anomaly. Strikingly, youth in the interviews focused on social and societal factors concerning education in general rather then on their mathematics education.     

An “inverse” relationship between mathematics identities and classroom practices among early career elementary teachers: The impact of accountability

This qualitative case study guided by portraiture examines the relationships between three early career elementary teachers’ beliefs about themselves in relation to mathematics (mathematics identities) and their classroom practices. Through autobiographical inquiry, reflective practice, classroom observations, interviews, and artifacts, findings show that all three second grade teachers appeared to have an “inverse” relationship between their mathematics identities and their classroom practices. In this relationship, as negative as they felt about themselves with regards to mathematics, they expended that much more effort to ensure that their students would have positive experiences with it and not be stigmatized by it as they had been. Accountability to schools, students, and parents, to increase student achievement appeared to play an important role in this relationship. Implications for preservice teacher education, inservice professional development, and research on beliefs and practices are discussed.     

Mathematics textbooks and their use in English, French and German classrooms:

After a through review of the relevant literature in terms of textbook analysis and mathematics teachers' user of textbooks in school contexts, this paper reports on selected and early findings from a study of mathematics textbooks and their use in English, French and German mathematics classrooms at lower secondary level. The research reviewed in the literature section raises important questions about textbooks as representations of the curriculum and about their role as a link between curriculum and pedagogy. Teachers, in tunr, appear to exercise control over the curriculum as it is enacted by using texts in the service of their own perceptions of teaching and learning. The second and main part of the paper analyses the ways in which textbooks vary and are used by teachers in classroom contexts and how this influences the culture of the mathematics classroom. The findings of the research demonstrate that classroom cultures are shaped by at least two factors: teachers' pedagogic principles in their immediate school and classroom context; and a system's educational and cultural traditions as they develop over time. It is argued that mathematics classroom cultures need to be understood in terms of a wider cultural and systemic context, in order for shared understandings, principles and meanings to be established, whether for promotion of classroom reform or simply for developing a better understanding of this vital component of the mathematics education process.     

Factors Associated With Mathematics Achievement and Participation in Advanced Mathematics Courses: An Examination of Gender Differences From an International Perspective

This paper reports results of an exploratory study examining factors that might be associated with achievement in mathematics and participation in advanced mathematics courses in Canada, Norway, and the United States of America (USA). These factors, which were not directly related to schooling accounted for large degrees of variability, 24% to 39%, in mathematics achievement scores. Confidence in mathematics was the strongest predictor of achievement for students from Canada and Norway, whereas for the students from the USA, parents' highest education level was the highest predictor of achievement. Student home environment related variables were stronger predictors of achievement for females than for males in all three countries. The participation in advanced mathematics courses could be predicted with 72% to 76% accuracy by the same variables. In all of the three countries, the strongest predictors of participation in advanced mathematics courses were students' attitudes toward mathematics. Parents' education level, a socioeconomic related variable, was one of the strongest predictors of participation for Canadian female students and all students from the USA.     

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.     

Pre-service Teachers’ Developing Conceptions about the Nature and Pedagogy of Mathematical Modeling in the Context of a Mathematical Modeling Course

Adopting a multitiered design-based research perspective, this study examines pre-service secondary mathematics teachers’ developing conceptions about (a) the nature of mathematical modeling in simulations of “real life” problem solving, and (b) pedagogical principles and strategies needed to teach mathematics through modeling. Unlike other studies that have focused on single-topic and lesson-sized research sites, a course-sized research site was used in this study. Having been through several iterations over three teaching semesters, the 15-week long course was implemented with 25 pre-service secondary mathematics teachers. Findings revealed that pre-service teachers developed ideas about the nature of mathematical modeling involving what mathematical modeling is, the relationship between mathematical modeling and meaningful understanding, and the nature of mathematical modeling tasks. They also realized the changing roles of teachers during modeling implementations and diversity in students’ ways of thinking. The researchers’ conceptual development, on the other hand, involved realizing the critical aspect of the “teacher role” played by the instructor during modeling implementations, and the need for more experience of modeling implementations for pre-service teachers.