Theories of Mathematics Education,Edited by Bharath Sriraman and Lyn English: Common Ground for Scholars and Scholars in the Making

Theories of mathematics education: Seeking new frontiers is the first book in a new Springer series, Advances in Mathematics Education. To some degree the book is based on a collection of previously published articles from special issues of ZDM—The International Journal on Mathematics Education (previously known as Zentralblatt für Didaktik der Mathematik). These articles, dealing with the role and use of theories in and about mathematics education, originally stem from various conferences and meetings such as PME and CERME. For this reason some of the articles in the book are already well known, a few may even be considered to be “modern classics” within theories of mathematics education such as Frank Lester's “On the theoretical, conceptual, and philosophical foundations for research in mathematics education.” What is new—and non-traditional—in the book, however, is its form of presentation and format, the main articles being accompanied by preludes and commentaries by established researchers and newcomers.     

Research,Reform, and Equity in U.S. Mathematics Education

This article discusses mathematics education research in relation to equity and current U.S. reforms. Although mathematics education researchers and reformers give attention to equity, work in this area tends to ignore relevant social and cultural issues. I begin by surveying articles on equity published in recent, mainstream education journals, highlighting the lack of attention given to social class and ethnicity. I discuss the implications of this limited research base. Specifically, I argue that current mathematics education reforms have been shaped by good intentions and existing research, neither of which offers adequate guidance to address the complexities of equity in mathematics classrooms today. Drawing from a study of social class differences in students' experiences in one reform-oriented classroom, I discuss the challenges and dilemmas inherent in sociocultural approaches to research in mathematics education and their potential contributions. I call for research from a sociocultural perspective, focusing on ways in which students from underrepresented groups can struggle when encountering particular instructional approaches, and ways in which teachers and students are able to address such struggles.     

Economy: the absent centre of mathematics education

Social and political turns in mathematics education research have brought into the field postmodern theorisations that researchers have been using to dismantle traditional philosophies of mathematics, to posit mathematics in the sociocultural terrain, and to spell out the role mathematics has in school exclusion. Sociopolitical perspectives constitute a privileged field of research to address the influence of economy on mathematical achievement. However, instead of investigating the role of economy in students’ achievement, sociopolitical studies have been contributing to a disavowal of the economic dimension of school mathematics. This paper synthesises a set of investigations carried out by the author in the last 5 years endeavouring to posit mathematics education in the political and economic spectrum of our time. It takes advantage of the contemporary combination of Hegel’s dialectics, Lacanian psychoanalysis and Marx’s critique of political economy, carried out by Slavoj ?i?ek, to develop a critique of the way research within the so-called ‘sociopolitical turn’ deals with the issue of equity; and marks out the contours of mathematics education’s ideological belonging.     

The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.     

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.     

Searching and exploring properties of geometric configurations using dynamic software

The decline in enrolments and interest in advanced mathematics studies is of growing concern internationally. Previous research suggests that a range of factors can influence students' academic decisions. The focus of the paper is on one of these potential sources of influence— students' perceptions of the tertiary mathematics learning environment. Data from two large-scale surveys (N = 1883) and from a smaller number of interviews (N = 71) with students enrolled in tertiary mathematics courses at five Australian universities are presented and discussed. Collectively, the survey results and the interview data reveal considerable variations in the quality of the teaching and student support available in different mathematics departments. Students' comments were constructive and offered valuable ideas for improving the existing situation, retaining current students and attracting others to mathematics.     

On the theoretical, conceptual, and philosophical foundations for research in mathematics education

The current infatuation in the U.S. with “what works” studies seems to leave education researchers with less latitude to conduct studies to advance theoretical and model-building goals and they are expected to adopt philosophical perspectives that often run counter to their own. Three basic questions are addressed in this article:What is the role of theory in education research? How does one's philosophical stance influence the sort of research one does? And,What should be the goals of mathematics education research? Special attention is paid to the importance of having a conceptual framework to guide one's research and to the value of acknowledging one's philosophical stance in considering what counts as evidence.     

Mark Vygodsky: several episodes from the life of a scholar

The subject of this paper is Mark Vygodsky, a prominent Soviet mathematician, one of the founders of the Soviet school of the history of mathematics. The paper draws on Vygodsky's surviving archive to develop a better understanding of how the life of a scholar unfolded within the context of Soviet reality. Discussion in the paper is confined to several episodes connected with Vygodsky's work as the author of books for secondary school students and as a mathematics teacher educator. It is argued that an examination of these episodes adds to the existing picture of the time when they took place, by shedding light not only on the significance of the political in a field seemingly distant from politics, but also on the position of the mathematics community and of people interested in mathematics education in general.     

The development of instructional guidelines for elementary mathematical writing

Mathematical writing recently has been defined as writing to reason and communicate mathematically. But mathematics instructional resources lack guidance for teachers as to how to implement such writing. The purpose of this paper is to describe how methods of design-based research were used to develop an instructional resource when one does not currently exist. Thirty-four participants—including teachers, mathematics coaches, mathematics curriculum developers, literacy coaches, a mathematician, and academics in elementary mathematics education, mathematics education, writing education, and science education—participated in a multi-step process to recommend, revise, and confirm instructional guidelines for elementary mathematical writing. The development process began with thirty-two recommendations from science writing and language arts writing. Through multiple cycles of feedback, five instructional guidelines and related considerations and techniques for implementation of elementary mathematical writing emerged.     

The beliefs of ‘Tomorrow's Teachers’ about mathematics: precipitating change in beliefs as a result of participation in an Initial Teacher Education programme

Mathematics education research has given increasing attention to the role of affective factors in the learning process. While 'affect' is used to refer to a variety of aspects including feelings, emotions, beliefs, attitudes and conceptions, this paper focuses on 'beliefs' of elementary pre-service teachers. In particular, the study evaluates the effect of participation in a reform-based elementary pre-service teacher education (referred to as Initial Teacher Education (ITE)) programme on participants' 'beliefs about the nature of mathematics'. This was completed using two (sub)scales of the Aiken's Revised Mathematics Scale measuring Enjoyment of Mathematics (E) and belief in the Value of Mathematics (V). Both scales were administered before and after participants completed the mathematics education programme, which consisted of 5 compulsory and consecutive modules. This study reveals that entry-level pre-service teachers report generally positive beliefs about the value of and enjoyment in doing mathematics. The findings challenge previous research, which report the tendency of teachers' beliefs to be resistant to change while in teacher education and suggest that it is possible for ITE mathematics education programmes to stimulate improvement in beliefs and attitudes among participants. Particular programme features are identified as instrumental in this positive change in beliefs about mathematics.     

Race,SES, and mathematics achievement conflicts: how policy and political climate affect academic achievement

Socioeconomic status (SES) has been widely used as a determining factor to explain educational processes and outcomes such as mathematics academic achievement. Research has documented the links between SES and mathematics academic achievement. However, further understanding the complex relationship between contextual factors, such as policy, and its implications for these processes within an ideologically patriarchal society is paramount. After decades of United States school policy and reform—with the most recent focus on Common Core Standards—there continue to be inconsistent notions of what the “real” issues are and how to address those issues. This paper sets out to explore one specific case—House Bill 2281 (HB 2281) and, in effect, the banning of the Mexican–American Studies program in one school district in the US—in understanding the implications of policy in the shaping of the public education system. Implications for mathematics education research are explored.     

Digital game‐based learning for K–12 mathematics education: A meta‐analysis

Digital games (e.g., video games or computer games) have been reported as an effective educational method that can improve students' motivation and performance in mathematics education. This meta‐analysis study (a) investigates the current trend of digital game‐based learning (DGBL) by reviewing the research studies on the use of DGBL for mathematics learning, (b) examines the overall effect size of DGBL on K‐12 students' achievement in mathematics learning, and (c) discusses future directions for DGBL research in the context of mathematics learning. In total, 296 studies were collected for the review, but of those studies, only 33 research studies were identified as empirical studies and systematically analyzed to investigate the current research trends. In addition, due to insufficient statistical data, only 17 out of the 33 studies were analyzed to calculate the overall effect size of digital games on mathematics education. This study will contribute to the research community by analyzing recent trends in significant DGBL research, especially for those who are interested in using DGBL for mathematics 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.     

Beliefs and Perspectives of First‐Year,Alternative Preparation,Elementary Teachers in Urban Classrooms

Results from an earlier study ( Hart, 2002 ) suggested that a group of 14 teachers participating in an alternative preparation program for elementary teachers had developed beliefs that were consistent with current thinking in mathematics education. The current study follows 8 of those teachers into their first year of teaching in an urban classroom. Qualitative data were collected from three sources: reflection logs, mathematics case discussions, and field notes made during classroom observations. This provided a triangulation of perspectives: the teachers' views of themselves, the teachers' views of others, and the university faculty member's view of them. Also, teachers completed the Standards Belief Instrument (Zollman & Mason, 1996) at the end of Phase I and at the end of Phase II. Results from the instrument and the qualitative analysis suggest that the teachers maintained a strong reform perspective in their beliefs, but they were unable to consistently implement pedagogy that was consistent with those beliefs.     

On the teaching and learing of Dienes' principles

Zoltan Dienes' principles of mathematical learning have been an integral part of mathematics education literature and applied both to the teaching and learning of mathematics as well as research on processes such as abstraction and generalization of mathematical structures. Most extant textbooks of cognitive learning theories in mathematics education include a treatment of Dienes' seminal contributions. Yet, there are no available studies at the tertiary level on how students internalise the meaning of Dienes' principles. This paper explores post-graduate mathematics education student's understanding of Dienes' principles and their ability to reflexively apply the principles to their own thinking on structurally similar problems. Some implications are offered for university educators engaged in the training of future researchers in the field.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

Time for telling stories: narrative thinking with dynamic geometry

In his work on human cognition, Bruner (The culture of education, Harvard University Press, Cambridge, 1996) distinguishes between narrative and paradigmatic modes of thinking. While the latter is closely associated with mathematics, Bruner’s writings suggest that the former contributes non-trivially to the learning of mathematics. In this paper, we argue that the very nature of dynamic mathematical representations—being intrinsically temporal, occurring over time—offer very different opportunities for narrative thinking than do the static diagrams and pictures traditionally available to learners. Using examples from our research, we analyse these opportunities both in terms of their potential for enhancing understanding and for their relation to the kind of paradigmatic thinking that usually constitutes mathematical knowledge.     

"The Real World As We Have Seen It": Latino/a Parents' Voices On Teaching Mathematics For Social Justice

This article describes the views of Latino/a parents who supported social justice mathematics curriculum for their children in a 7th-grade Chicago public school classroom in which I was the teacher. The parents viewed dealing with and resisting oppression as necessary parts of their lives; they also saw mathematics as integral and important. Because (mathematics) education should prepare one for life—and injustice, resistance, and mathematics were all interconnected parts of life—an education made sense if it prepared children to be aware of and respond to injustices that they faced as members of marginalized communities. Such education may be unusual, but it was congruent with the parents' core values and worth standing up for.