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## Commentary on the Chapter by Gabriele Kaiser, Maren Hoffstall and Anna B. Orschulik, “Gender Role Stereotypes in the Perception of Mathematics—Results of an Empirical Study with Secondary Students in Germany”

### Towards Equity in Mathematics Education (2012): 141-144 , January 01, 2012

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Evidence reported by Kaiser, Hoffstall and Orschulik suggests that German girls and boys hold stereotypical attitudes regarding gender and mathematics, and that these biases strengthen during adolescence. These results raise important questions about the sources of students’ beliefs, including their view of girls as succeeding in mathematics through diligence and boys as succeeding through ingenuity. Addressing these disparities must involve grappling with differences in boys’ and girls’ classroom behaviors, their self-perceptions in relation to mathematics versus other subjects, and societal beliefs about the nature of mathematics, itself.

## Preface to “Israeli Jewish and Arab Students’ Gendering of Mathematics”

### Towards Equity in Mathematics Education (2012): 199-201 , January 01, 2012

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The pursuit of equity in educational outcomes, and gender equity in particular, was a driving force of the research reported in the article that follows. The underlying assumption was that social science can elucidate contributing social and cultural variables that, if acted upon, can change the outcome of educational endeavours. In adopting this perspective the alternative view, that educational advantage is largely inherited and therefore closed to innovative change, was rejected.

## Mathematics Education for Adults: Can It Reduce Inequality in Society?

### Towards Equity in Mathematics Education (2012): 283-301 , January 01, 2012

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Adult education in mathematics is considered from a number of perspectives. It is a key element in the general concept of lifelong learning. Lifelong learning is essential in a rapidly changing economic and technological world. For example, quickly changing production conditions in companies lead to quickly changing demands on skills possessed by company employees. Skills that are taught in schools and institutions of vocational education also need to be adapted to fulfil the requirements of a changing job environment. Lifelong learning is a process that corrects omissions in basic education. In this sense, lifelong learning is viewed as an opportunity to reduce societal inequalities. However, not only is economic life changing, but conditions within society are changing, too. In this essay it will be argued that lifelong learning is undoubtedly a prerequisite for making democratic participation possible for all members of a society. All adult learners have a learning history that is intimately connected with their experience of learning at school. The school experience has a great influence on learning as adults, particularly if the adult is forced to, acquire new skills because of unemployment. In this chapter we begin with a discussion of the general conditions of lifelong learning. We continue with an account of the role played by mathematics in societies. We then conclude with problems faced by adult learners of mathematics.

## From the Known to the Unknown: Pattern, Mathematics and Learning in Papua New Guinea

### Towards Equity in Mathematics Education (2012): 415-428 , January 01, 2012

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During the late 1990s, the Papua New Guinean Department of Education introduced a new elementary school mathematics curriculum that utilised the country’s rich and diverse cultural traditions. The resulting changes saw patterns, one of a family of practices related to the decorative arts, take on a prominent role as a tool for understanding number, space, time, measurement, all of which form the basis of mathematics. Drawing primarily on the author’s own anthropological fieldwork, this chapter examines the culture of pattern in community life in order to understand its selection as a cultural resource for mathematics learning. It will demonstrate that while pattern is not spoken about, people are nevertheless especially adept at engaging with it. Since Papua New Guinea is full of patterns, and pattern plays such a robust role in the mathematics curriculum, the chapter demonstrates how pattern can be understood as an expression of the mathematical mind at work.

## Commentary on the Chapter by Gabriele Kaiser, Maren Hoffstall, and Anna B. Orschulik, “Gender Role Stereotypes in the Perception of Mathematics: Results of an Empirical Study with Secondary Students in Germany”

### Towards Equity in Mathematics Education (2012): 145-149 , January 01, 2012

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Since the mid-1970s gender differences favouring males in mathematics achievement at all levels of schooling and participation at post-compulsory levels of schooling have declined around the world. Yet gender differences in mathematics self-efficacy that favour males have remained intransigent, and the gender stereotyping of mathematics proficiency and interest persist. The study by Kaiser, Hoffstall and Orschulik is the latest in a series of studies commencing in the mid-1970s (Fennema and Sherman 1977) to provide evidence of higher self-concept in mathematics for male students compared to female students, and male stereotyped views of mathematics learning and achievement, especially among senior secondary aged students.

## Israeli Jewish and Arab Students’ Gendering of Mathematics

### Towards Equity in Mathematics Education (2012): 203-225 , January 01, 2012

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In English-speaking, Western countries, mathematics has traditionally been viewed as a “male domain”, a discipline more suited to males than to females. Recent data from Australian and American students who had been administered two instruments (Leder and Forgasz 2002) tapping their beliefs about the gendering of mathematics appeared to challenge this traditional, gender-stereotyped view of the discipline. The two instruments were translated into Hebrew and Arabic and administered to large samples of grade 9 students attending Jewish and Arab schools in northern Israel. The aims of this study were to determine if the views of these two culturally different groups of students differed and whether within group gender differences were apparent. The quantitative data alone could not provide explanations for any differences found. However, in conjunction with other sociological data on the differences between the two groups in Israeli society more generally, possible explanations for any differences found were explored. The findings for the Jewish Israeli students were generally consistent with prevailing Western gendered views on mathematics; the Arab Israeli students held different views that appeared to parallel cultural beliefs and the realities of life for this cultural group.

## Commentary on the Chapter by Wolfgang Schlöglmann, “Mathematics Education for Adults: Can It Reduce Inequality in Society?”

### Towards Equity in Mathematics Education (2012): 303-307 , January 01, 2012

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Adults, education, Bildung, inequality, lifelong learning, and mathematics are the key terms in the chapter written by Schlöglmann. In the 1990s, he was one of the pioneers that cultivated the borderland between mathematics education, adult education, and vocational education as a subfield of mathematics education research (see Wedege 2000). Together with Jungwirth and Maasz at the University of Linz, he conducted a large empirical study exploring “the state of mathematics education within the adult education system in Austria” (Jungwirth et al. 1995, p. 13). In this study, the authors made an important distinction between courses where mathematics is explicitly taught and courses where mathematical concepts and methods are used implicitly. In order to label the latter they constructed the term “Mathematikhaltige Weiterbildung” (translation: “Mathematics-containing continuing education”) presumably to remind people that mathematics in vocational training, as in the workplace itself, is integrated with other subjects and vocational competences. Elsewhere I claimed that within the scientific domain of mathematics education they paved the way for research on vocationally oriented adult education, where mathematics is an integral part (Wedege 2000).

## Commentary on the Chapter by Graeme Were, “From the Known to the Unknown: Pattern, Mathematics and Learning in Papua New Guinea”

### Towards Equity in Mathematics Education (2012): 429-432 , January 01, 2012

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Papua New Guinea (PNG) is an anthropologist’s paradise—several hundred still existing languages, people still living in remote villages with memories of first contact with European civilisations, a country still trying to define itself in the context of the modern world, and many cultures still waiting to be documented and understood. Part of its push to modernisation relates to formal school education and Were’s chapter is situated in the context of a PNG Department of Education curriculum development concerning elementary mathematics education. The thrust which interests Were is the plan “to deliver a community-based mathematics education programme in the local vernacular that was sensitive to a child’s existing knowledge of local cultural traditions.” This was indeed a bold initiative, one which few developing countries have successfully realised due to an array of challenges.

## Adolescent Girls’ Construction of Moral Discourses and Appropriation of Primary Identity in a Mathematics Classroom

### Towards Equity in Mathematics Education (2012): 63-86 , January 01, 2012

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This qualitative study examines the way three American young adolescent girls who come from different class and racial backgrounds construct their social and academic identities in the context of their traditional mathematics classroom. The overall analysis shows an interesting dynamic among each participant’s class and racial background, their social/academic identity and its collective foundation, the types of ideologies they repudiate and subscribe to, the implicit and explicit strategies they adopt in order to support the legitimacy of their own position, and the ways they manifest their position and identity in their use of language referring to their mathematics classroom. Detailed analysis of their use of particular terms, such as “I,” “we,” “they,” and “should/shouldn’t” elucidates that each participant has a unique view of her mathematics classroom, developing a different type of collective identity associated with a particular group of students. Most importantly, this study reveals that the girls actively construct a social and ideological web that helps them articulate their ethical and moral standpoint to support their positions. Throughout the complicated appropriation process of their own identity and ideological standpoint, the three girls made different choices of actions in mathematics learning, which in turn led them to a different math track the following year largely constraining their possibility of access to higher level mathematical knowledge in the subsequent schooling process.

## Students’ Attitudes, Engagement and Confidence in Mathematics and Statistics Learning: ICT, Gender, and Equity Dimensions

### Towards Equity in Mathematics Education (2012): 151-179 , January 01, 2012

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In this chapter the findings of five studies are reported. Two research instruments were used: the *Mathematics and Technology Attitudes Scale* (MTAS), and the *Survey of Attitudes Toward Statistics Scale* (SATS). The aims, methods, data analyses, selected findings and conclusions are presented, as well as implications for the teaching and learning of mathematics and statistics. The studies involved samples from Australia and Greece. Findings from the three MTAS studies revealed that there is a complex nexus of relationships between secondary mathematics students’ mathematics confidence, confidence with technology, attitude to learning mathematics with technology, affective engagement and behavioural engagement, achievement, and gender. Findings from the SATS studies indicated that male Greek tertiary students had more positive attitudes toward statistics than female students; there was no gender gap for the Australian tertiary students. Secondary students’ attitudes towards ICT use for mathematics learning require further scrutiny in order to bring about gender equity and to facilitate improved outcomes for all students. Gender and cultural sensitivity are paramount in the instructional planning, decision making, and implementation of secondary mathematics and tertiary statistics.

## Preface to “Ethnomathematics and Philosophy”

### Towards Equity in Mathematics Education (2012): 227-229 , January 01, 2012

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We who undertake ethnomathematical studies still have a philosophical problem. I still believe that a version of mathematical relativity is one of our basic assumptions. That is, the study of ethnomathematics rests on the idea that there can be more than one form of mathematics—it is neither absolute nor Platonist. Postmodern writing is exposing more about the contingent nature of mathematics, and the historicity of mathematics is becoming more accepted. Subjectivity and objectivity are becoming blurred (Brown 2011; Radford et al. 2008). But we are still a long way from wide agreement on a philosophical position that “allows” ethnomathematical investigations.

## Commentary on the Chapter by Wolfgang Schlöglmann, “Mathematics Education for Adults: Can It Reduce Inequality in Society?”

### Towards Equity in Mathematics Education (2012): 309-313 , January 01, 2012

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The four “new societal goals” promulgated by the National Council of Teachers of Mathematics in 1989 provide a lens to look at Wolfgang Schlöglmann’s chapter “Mathematical Education for Adults: Can It Reduce Inequality in Society?”.

## Preface to “Equity in Mathematics Education: Unions and Intersections of Feminist and Social Justice Literature”

### Towards Equity in Mathematics Education (2012): 31-37 , January 01, 2012

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In recent years, research examining results from tests of mathematics performance has generally documented small or reduced gaps between male and female students (e.g., Else-Quest et al. 2010; Hyde et al. 2008; McGraw et al. 2006). However, this is not always the case. For example, research exploring the gender gap among high-achieving high school students, using data from the American Mathematics Competitions, has indicated that the gender gap widens dramatically at very high percentiles and that the highest-achieving girls are concentrated in a very small set of elite schools (Ellison and Swanson 2010). A study by Fryer and Levitt (2010) documented the emergence of a substantial mathematics gender gap in the early years of schooling in the United States, documenting that girls lose more than two-tenths of a standard deviation relative to boys over the first six years of school, across every strata of society.

## Commentary on the Chapter by Anastasios Barkatsas, “Students’ Attitudes, Engagement and Confidence in Mathematics and Statistics Learning: ICT, Gender, and Equity Dimensions”

### Towards Equity in Mathematics Education (2012): 181-185 , January 01, 2012

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The chapter by Barkatsas summarises the findings of five small-scale studies which employ similar questionnaire instruments and statistical methods to assess facets of attitude towards mathematics and statistics. The first three of these studies focus on attitudes to mathematics and to technology use in studying mathematics, as found in student populations in secondary schools in Victoria, Australia or Athens, Greece. The last two focus on attitudes to statistics as found in student populations in tertiary education in one or other of these same locations. Although there is a gender dimension to all of the studies, this is not highly developed or strongly theorised. Nevertheless, the empirical findings merit some comment, particularly where they can be related to the main theoretical hypothesis advanced early in the chapter, a posited mechanism through which use of technology in studying mathematics affects attitudes to the subject and achievement in it. For that reason, I will focus my remarks on the set of three studies that address this.

## Heteroglossia in Multilingual Mathematics Classrooms

### Towards Equity in Mathematics Education (2012): 315-332 , January 01, 2012

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What does linguistic or cultural diversity look like in a mathematics classroom? How does such diversity influence the teaching or learning of mathematics? In this chapter, I address these and related questions. Specifically, I draw on Bakhtin’s notion of heteroglossia to analyse the literature on teaching and learning mathematics in linguistically diverse classrooms. Based on this analysis, I describe and discuss four tensions that arise in linguistically diverse mathematics classrooms: tensions between school and home languages; between formal and informal language in mathematics; between language policy and mathematics classroom practice; and between a language for learning mathematics and a language for getting on in the world. These tensions can all be traced to an underlying tension between what Bakhtin calls centripetal and centrifugal forces in language. I conclude by considering some of the implications of my analysis for equity in mathematics teaching.

## Introduction

### Towards Equity in Mathematics Education (2012): 1-3 , January 01, 2012

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The issue of equity remains one of the most difficult and persistent problematics in the theory and practice of mathematics education. In the broadest sense, *equity* encompasses matters involving regimes of inclusions and exclusions that both enable and constrain the lived conditions—by gender, cultural identity, and diversity—of individuals in complex societies. Undoubtedly, this is why the presumption of *access* is foregrounded by Bishop and Forgasz (2007) in their synthesis of research on equity in mathematics education. They claimed that “without *access* to mathematics education there can be no equity” (p. 1146). This view offers a powerful counterdiscourse to *inequity* which, as a concept, is represented by marked (negative) differences in outcomes among groups within a given context or setting. Inequities are manifest in educational situations that involve achievement, enrolment, and attitudes. While differential outcomes implicitly convey inequity of some kind, they are distinguished from *diversity*, a concept that relates to the complex of interacting ways in which groups of individuals are identified and categorized. In this book, diversity encompasses: gender; race, ethnicity and culture; nationality and language background; socioeconomic status; exceptionality; and physical and learning disabilities.

## Commentary on the Chapter by Anastasios Barkatsas, “Students’ Attitudes, Engagement and Confidence in Mathematics and Statistics Learning: ICT, Gender, and Equity Dimensions”

### Towards Equity in Mathematics Education (2012): 187-191 , January 01, 2012

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Barkatsas’ chapter contributes to the discussion on gender issues in the affective aspects of learning statistics (using SATS) and learning mathematics with ICT (using MTAS). My response to the chapter has two parts: contributions of the findings to the original theoretical basis of the instruments, and implications of the studies on equitable policy and practice.

Since the studies provided evidence of the validity and reliability of both the MTAS and the SATS instruments, a next step would be to investigate how the findings of the studies relate back to the theoretical basis of the instruments.

## Commentary on the Chapter by Richard Barwell, “Heteroglossia in Multilingual Mathematics Classrooms”

### Towards Equity in Mathematics Education (2012): 333-338 , January 01, 2012

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In his chapter, Barwell draws on the Bakhtinian notion of co-existing voices, *heteroglossia*, to point to the experience of four tensions in linguistically diverse mathematics classrooms, namely: 1) school versus home languages, 2) formal versus informal language in mathematics, 3) language policy versus mathematics classroom practices, and 4) a language for learning mathematics versus a language for getting on in the world. To further explore Barwell’s points, I bring up the metaphor of *orchestration*. My idea of orchestration refers to the manner in which various voices are individually used to produce coordinated results. As in the case with instruments in an orchestra, the final production requires successful individual practices with instruments that successfully interact with each other. My main argument here is that heteroglossia needs adequate practical orchestration, on the part of all members in a class: they all need to learn what the others can and cannot do, what distributions of work sound more adequate at each time, why is it that some of the ideas and contributions sound different depending on who introduces them, etc. By interpreting orchestration as something that includes all participants, not only the actions and choices by the teacher, it becomes clearer that students and teachers have lots in common.

## Gender Differences in Mathematics and Science Achievement Across the Distribution: What International Variation Can Tell Us About the Role of Biology and Society

### Towards Equity in Mathematics Education (2012): 441-468 , January 01, 2012

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This study examines international data on mathematics and science achievement to illuminate the ways in which macrostructural factors influence gender differences in mathematics, particularly among high achievers. Examining the importance of macrostructural factors for gender differences not only helps us better understand the role that larger contexts play in contributing to these differences, but international variation also provides a unique opportunity to present simple and powerful arguments for the continued importance of social factors vis-à-vis biological considerations. We show that there is considerable international variation in gender differences, and that gender differences among high achievers in both mathematics and science literacy are related to gender inequality in the labor market and differences in the overall status of men and women. We conclude by discussing the implications of our findings for mathematics education and suggesting fruitful avenues for future research.

## Commentary on the Chapter by Paul Dowling and Jeremy Burke, “Shall We Do Politics or Learn Some Maths Today? Representing and Interrogating Social Inequality”

### Towards Equity in Mathematics Education (2012): 105-109 , January 01, 2012

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To start on a personal note, I consider the opportunity to respond to the chapter by Dowling and Burke to be an honour and privilege indeed. In the past I have always found Dowling and his colleagues’ writings to be a challenge to many of our widely accepted views and practices in mathematics education—even for those mathematics educators, may I add, writing from sociopolitical or sociocultural perspectives. Reading this chapter at this particular time was opportune for me. It spoke directly to findings of a recent research project I have participated in as well as some writing I am currently undertaking on the Australian national curriculum. Reading this chapter, I found myself agreeing with the majority of arguments developed, with the exception of one (minor?) point I will address in the last paragraph below. Here I will restrict my comments to two points that the chapter raised for me: the construction of the relationship between mathematics and the world (in particular the social world), and the corresponding tasks for mathematics education (in particular with respect to the agenda of social justice).

## Preface to “Adolescent Girls’ Construction of Moral Discourses and Appropriation of Primary Identity in a Mathematics Classroom”

### Towards Equity in Mathematics Education (2012): 57-62 , January 01, 2012

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My research interest in gender and equity issues in mathematics education has a long history stemming from my dissertation research in which I examined the intersection of race, gender, and class in a middle school mathematics classroom. Though I completed my dissertation as an ethnographic study in 2002, I was deeply puzzled by the contrasting perceptions and experiences manifested in each participating girl’s profile. I wondered if further analysis would provide more powerful means to unravel the complexity and subtle dynamics in the girls’ emerging social and academic identities.

## Commentary on the Chapter by Penner and CadwalladerOlsker, “Gender Differences in Mathematics and Science Achievement Across the Distribution: What International Variation Can Tell Us About the Role of Biology and Society”

### Towards Equity in Mathematics Education (2012): 469-474 , January 01, 2012

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In their article, Penner and CadwalladerOlsker argue that sex differences in performance are largely due to societal factors, not biology. While they acknowledge several biological differences between males and females, they reach their conclusion based largely on differences in the variance in achievement across countries from the 1995 dataset of the Third International Mathematics and Science Study (TIMSS). Given there is no reason to believe that there are relevant biological differences across large cross-sections of the world, the country differences must be societal and cultural in origin.

## Moving Towards a Feminist Epistemology of Mathematics

### Towards Equity in Mathematics Education (2012): 15-29 , January 01, 2012

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There is, now, an extensive critical literature on gender and the nature of science, three aspects of which, philosophy, pedagogy and epistemology, seem to be pertinent to a discussion of gender and mathematics. Although untangling the inter-relationships between these three is no simple matter, they make effective starting points in order to ask similar questions of mathematics to those asked by our colleagues in science. In the process of asking such questions, a major difference between the empirical approach of the sciences, and the analytic nature of mathematics, is exposed and leads towards the definition of a new epistemological position in mathematics.

## Commentary on the Chapter by Penner and CadwalladerOlsker, “Gender Differences in Mathematics and Science Achievement Across the Distribution: What International Variation Can Tell Us About the Role of Biology and Society”

### Towards Equity in Mathematics Education (2012): 475-480 , January 01, 2012

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In their chapter, Penner and CadwalladerOlsker combine TIMSS data with country data from the United Nations, World Bank, and other resources to give two primary results related to mathematics performance by sex. First, they cast doubt on the utility of biological causes as an effective explanation of gender differences in mathematics performance on the TIMSS. Second, since gender differences in math scores depend on social factors, they investigate which country-level factors matter most in explaining those differences. The argument provides far greater insight to the question of gender and mathematics than simple comparison studies, establishing an excellent model for future research. At the same time, the results are macro-level results for a field that has struggled historically with macro-level solutions, thus raising the issue of the implications of this work for addressing gender equity.

## Preface to “Doubtful Rationality”

### Towards Equity in Mathematics Education (2012): 347-350 , January 01, 2012

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In talking about “doubtful rationality” I try to move beyond the *modern conception of mathematics*. This conception includes three elements: the celebration of mathematics as being essential for understanding nature; the celebration of mathematics-based contributions to technological development; and the celebration of mathematics as a pure scientific enterprise.

## Research-Based Mathematics Instruction for Students with Learning Disabilities

### Towards Equity in Mathematics Education (2012): 481-502 , January 01, 2012

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The purpose of this chapter is to describe several advances and trends in teaching mathematics to students with learning disabilities. Mathematics disabilities affect between 5% and 8% of students in the general population. These students generally experience serious difficulties in developing requisite concepts, skills, and strategies for success in mathematics and, consequently, are at risk for school failure and poor post-secondary outcomes. Research in special education has identified a number of evidence-based practices that have proven effective for teaching students with learning, attention, and/or behavioral disorders that interfere with success in mathematics. Following a discussion of student characteristics and evidence-based practices, two examples of research programs focusing on mathematical problem-solving interventions for students with learning disabilities are highlighted.

## Doubtful Rationality

### Towards Equity in Mathematics Education (2012): 351-367 , January 01, 2012

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The paper addresses the following two questions: (1) How can we conceptualise possible socio-political roles of mathematics-based rationality? (2) How can we conceptualise possible the socio-political roles of mathematics education? Together the two questions might help to shed some light on how mathematical rationality might assume different forms and become integrated into social and technological development. It is pointed out how mathematics-based rationality forms part of the fabrication of possibilities, strategies, facts, contingencies, and perspectives. There is, however, no inherent quality to be associated with such mathematics-based fabrications. It is considered to what extent the school mathematics tradition could establish a prescription readiness by submitting students to a school-mathematics absolutism; and to what extent this tradition could facilitate differentiated labelling of the students, as well as endorsement of an ethical filter. These aspects could be important functions of the school mathematics tradition in today’s knowledge society. It is considered to what extent mathematics education could prepare people for critical citizenship, which includes a potential for ‘talking back’ to authority. I do not see such preparation as being related to the school mathematics tradition, nor do I see it as linked to the very nature of mathematics. Rather, I am proposing it as a *possible* function of mathematics education.

## Commentary on the Chapter by Graeme Were, “From the Known to the Unknown: Pattern, Mathematics and Learning in Papua New Guinea”

### Towards Equity in Mathematics Education (2012): 433-438 , January 01, 2012

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In this straightforwardly written ethnomathematical report of geographically and socio-culturally situated mathematical practices—patterning of artefacts and social relations among the Nalik of New Ireland—Were houses a subtle critique of the field of ethnomathematics (EM). However, his critique is rendered less potent by not having demonstrated an attention to the conversational patterns in the recent EM literature. As a reader somewhat familiar with a significant portion of the “basic” literature and the ongoing debates in the field (Khan 2008), I appreciated, however, the author pointing me to texts and manuscripts that I had not encountered. In the hope that this commentary and Were’s chapter be of utility to readers I first offer an outline of some complementary patterns in the more recent (ethno)mathematics education literature before returning to the seed that lies dormant within Were’s paper, which is to be found via a consideration of his key motif—pattern(ing).

## Commentary on the Chapter by Marjorie Montague and Asha Jitendra, “Research-Based Mathematics Instruction for Students with Learning Disabilities”

### Towards Equity in Mathematics Education (2012): 503-506 , January 01, 2012

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This is an extremely interesting chapter that describes children’s difficulties in mathematics. It discusses the problem of learning difficulties in mathematics. As the authors point out, mathematical difficulties are indeed very common. This has been found to be so internationally (Butterworth 2005; Bzufka et al. 2000; Gross-Tsur et al. 1996), though there can be problems in defining the point where they are severe, specific and persistent enough to be diagnosed as a specific disability (Mazzocco and Myers 2003). Studies in the UK (e.g. Bynner and Parsons 1997) have shown that persistent numeracy difficulties in adulthood are much commoner and at least as disabling as literacy difficulties.

## Ethnomathematics and Philosophy

### Towards Equity in Mathematics Education (2012): 231-240 , January 01, 2012

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Any concept of ethnomathematics must eventually meet philosophical debates about the nature of mathematics. In particular neo-realist positions are anathema to the idea that mathematics is culturally based, but even modern quasi-empiricist philosophies are challenged by the fundamental relativity implied in ethnomathematical writing.

A new way of interpreting mathematical history which may allow for a truly relativist mathematics is described, and some evidence is presented to support this view. The kind of studies which would arise from this perspective on mathematics are outlined.

## Preface to “A Socio-Political Look at Equity in the School Organization of Mathematics Education”

### Towards Equity in Mathematics Education (2012): 369-371 , January 01, 2012

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Whenever we place our attention on the issue of equity in mathematics education, we are addressing a broad social, political, economic, and cultural problem that is related to how societies create and set in place mechanisms that differentiate, include, and exclude certain peoples from valued resources, either material or symbolic. My main point in this paper is to provide evidence that it is not possible to assume that “equity” is a matter that only and exclusively emerges and is reproduced inside mathematics classrooms. Since “problems of equity” in mathematics are a part of general problems of equity, inclusion, and exclusion in society at large (Pais and Valero 2011), I intend to focus on how the notions of ability/disability associated with the exclusion of certain children emerge in the practices and discourses inside the school organization. Of course, the school organization, again, is only one site in which mechanisms of inclusion and exclusion operate in society.

## Commentary on the Chapter by Marjorie Montague and Asha Jitendra, “Research-Based Mathematics Instruction for Students with Learning Disabilities”

### Towards Equity in Mathematics Education (2012): 507-513 , January 01, 2012

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In their chapter, Montague and Jitendra highlight that many students with disabilities are not making adequate progress in their mathematical performance. Further, both cognitive and behavioral characteristics appear to interfere with their performance and development in mathematics. In response to these concerns, Montague and Jitendra highlight two instructional approaches—*Solve It!* and Schema-Based Instruction (SBI), that can be used to help students with disabilities improve their mathematical performance.

## Preface to “Cultural Differences, Oral Mathematics and Calculators in a Teacher Training Course of the Brazilian Landless Movement”

### Towards Equity in Mathematics Education (2012): 241-244 , January 01, 2012

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Six years after the publication of the ZDM paper that examined cultural processes involving oral mathematics and its articulations with the use of a calculator, we would like to assign new meanings to what we wrote there using other theoretical tools that now shape our theoretical framework (Knijnik and Wanderer 2008, 2010). In recent papers we showed how we moved from a postmodern perspective to another one, which embraces Foucauldian and later Wittgensteinian ideas (Knijnik 2007; Knijnik and Wanderer 2008; Wanderer 2007). Based on these theorizations we reconceptualized the *Ethnomathematics Perspective*, which is giving support to the studies developed at the Unisinos *Mathematics Education and Society Inter-Institutional Research Group* (Duarte 2009; Giongo 2008; Knijnik and Bocasanta 2010; Knijnik and Wanderer 2010; Wanderer 2007). This *Ethnomathematics Perspective* consists of a toolbox that allows us to analyze the Eurocentric discourses of academic and school mathematics and their effects of truth and to examine the language games that constitute different mathematics and their family resemblances.

## Shall We Do Politics or Learn Some Maths Today? Representing and Interrogating Social Inequality

### Towards Equity in Mathematics Education (2012): 87-103 , January 01, 2012

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In this chapter we shall first introduce a schema that describes strategies of representation in terms of whether representation is explicit or tacit and whether it is oriented in consonance or dissonance with dominant or expected patterns, in this case of social inequality. We shall then use this schema to describe the construction of gender and social class in school textbooks, giving some attention to the contexts of their use. We shall argue that addressing social inequalities demands explicit, dissonant strategies, referred to here as interrogation. However, by reflecting on a particular critical mathematics lesson apparently interrogating racial inequality, we conclude that interrogation itself is likely to lead to misrepresentation where the mathematical activity is foregrounded and mathematics is likely to lose out where it is not. Ultimately, we may be left with the choice of whether to do politics or to teach mathematics.

## A Socio-Political Look at Equity in the School Organization of Mathematics Education

### Towards Equity in Mathematics Education (2012): 373-387 , January 01, 2012

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This paper presents some theoretical tools to help understand the meaning of mathematics education as socio-political practices and the implications of these for researching mathematics education. Taking two cases of schools and students in Denmark and South Africa, the paper illustrates how the theoretical and methodological ideas come into operation when illuminating issues of equity. It is contended that the disadvantaged positioning of some students for participating in mathematics teaching and learning is the result of the routines, ideas, shared meanings, and ways of talking and conceiving mathematics education among the actors in the school organization, inside as well as outside the classroom.

## Neural Correlates of Gender, Culture, and Race and Implications to Embodied Thinking in Mathematics

### Towards Equity in Mathematics Education (2012): 515-543 , January 01, 2012

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In this chapter, I discuss neuroscience research and selected findings that are relevant to mathematics education. What does it mean, for example, to engage in a neuroscientific analysis of symbol reference? I also discuss various research programs in neuroscience that have useful implications in mathematics education research. Further, I provide samples of studies conducted within and outside mathematics education that provide a neural grounding of gender, culture, and race. The chapter closes with three brief implications of neuroscientific work in mathematics education research, in general, and in individual- and intentional-embodied cognition in mathematical thinking and learning, in particular.

## Equity in Mathematics Education: Unions and Intersections of Feminist and Social Justice Literature

### Towards Equity in Mathematics Education (2012): 39-56 , January 01, 2012

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Traditional models of gender equity incorporating deficit frameworks and creating norms based on male experiences have been challenged by models emphasizing the social construction of gender and positing that women may come to know things in different ways from men. This paper draws on the latter form of feminist theory while treating gender equity in mathematics as intimately interconnected with equity issues by social class and ethnicity. I integrate feminist and social justice literature in mathematics education and argue that to secure a transformative, sustainable impact on equity, we must treat mathematics as an integral component of a larger system producing educated citizens. I argue the need for a mathematics education with tri-fold support for mathematical literacy, critical literacy, and community literacy. Respectively, emphases are on mathematics, social critique, and community relations and actions. Currently, the integration of these three literacies is extremely limited in mathematics.

## Cultural Differences, Oral Mathematics, and Calculators in a Teacher Training Course of the Brazilian Landless Movement

### Towards Equity in Mathematics Education (2012): 245-259 , January 01, 2012

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This paper discusses aspects of a two-year study of a teacher-training course for adult mathematics education organized by a Brazilian landless peoples’ social movement. It takes ethnomathematics as a theoretical framework in which cultural differences are central. The paper analyzes some of the oral mathematics practices that mark the landless peoples’ culture studied. In particular, it discusses a pedagogical process involving the articulation of oral mathematics practices with the use of the calculator, focusing on how pre-service teachers give meaning to their experience and on how cultural differences operated in this setting.

## Commentary on the Chapter by Richard Barwell, “Heteroglossia in Multilingual Mathematics Classrooms”

### Towards Equity in Mathematics Education (2012): 339-342 , January 01, 2012

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Mathematics classrooms are sites of encounter for different voices, perspectives, and ideas. Those differences become even more visible when the object of difference is language. In his chapter, Barwell draws on Bakhtin’s concept of heteroglossia to explore the tensions that underpin multilingual classrooms. He enquires about how those tensions influence the teaching and learning of mathematics and the implications that they may have for equity in mathematics teaching. In my comments, I would like to dwell upon the question of language in the mathematics classroom and on some issues about equity.

## Looking for Gold: Catering for Mathematically Gifted Students Within and Beyond ZDM

### Towards Equity in Mathematics Education (2012): 389-406 , January 01, 2012

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ZDM—The International Journal on Mathematics Education has a forty year long history of sustained publications. There is pride in the Journal’s tradition of “publication of themed issues that aim to bring the state-of-the-art on central sub-domains within mathematics education” (Kaiser and Sriraman 2010, p. 143). In this chapter I trace the scope and themes of ZDM publications in which the education and needs of mathematically gifted students are discussed and compare the findings with those reported in the broader mathematics education research literature.

## Commentary on the Chapter by Ferdinand Rivera, “Neural Correlates of Gender, Culture, and Race and Implications to Embodied Thinking in Mathematics”

### Towards Equity in Mathematics Education (2012): 545-550 , January 01, 2012

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In his chapter *Neural correlates of gender, culture, and race and implications to embodied thinking in mathematics*, Rivera discusses the fascinating and vivid, yet also hotly debated, association between research in neuroscience and mathematics education, by sketching various neuroscientific studies conducted within and outside mathematics education. Against the background of this volume on gender, culture, and diversity, Rivera pays specific attention to findings from the fields of cognitive and social-affective neuroscience, the latter of which is also gradually, but slowly, shedding further light on the neural correlates of gender, culture, and race. Echoing a more general movement to connect research in (mathematics) education and neuroscience (e.g., Ansari and Coch 2006; Howard-Jones 2008; Stern 2005), Rivera suggests that these neuroscientific findings may offer new perspectives on mathematics education research in general, and on individual and intentional embodied cognition in mathematical thinking and learning in particular.

## Front Matter - Towards Equity in Mathematics Education

### Towards Equity in Mathematics Education (2012) , January 01, 2012

## Preface to “Moving Towards a Feminist Epistemology of Mathematics”

### Towards Equity in Mathematics Education (2012): 9-14 , January 01, 2012

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In “*Moving towards a feminist epistemology of mathematics*,” Leone Burton (1995/2008) argued for a feminist epistemology that is parallel to the epistemologies of other STEM fields, but a case unto itself because of the nature of mathematics. Her proposal defined knowing in mathematics in relation to five categories including social relatedness, the aesthetics of mathematical thinking, intuition and insight, recognition and celebration of different approaches and ways of thinking, and global applications. These categories incorporate processes mathematicians employ as well as the experience of the learner in constructing mathematical meaning.

## Commentary on the Chapter by Dowling and Burke, “Shall We Do Politics or Learn Some Maths Today? Representing and Interrogating Social Inequality”

### Towards Equity in Mathematics Education (2012): 111-114 , January 01, 2012

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This chapter posits the following conundrum: Is it possible to structure a mathematics lesson that integrates questioning of social inequality with worthwhile mathematical material? Dowling and Burke answer this question negatively. They conclude that a lesson must either privilege mathematics, providing all students with appropriate and rigorous mathematical content, or a lesson must privilege political motivations, providing for a full and extensive discussion of critical issues; according to the authors, it is not possible to accomplish both goals at the same time.

## Preface to “Immigrant Parents’ Perspectives on Their Children’s Mathematics Education”

### Towards Equity in Mathematics Education (2012): 261-266 , January 01, 2012

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In the article originally published in 2005 we addressed immigrant parents’ perspectives of the teaching of mathematics in two geographic contexts, Barcelona (Spain) and Tucson (USA). The article illuminates two specific aspects of marginalization of those who move to a new country with hopes of a better life, including education, for their children. Since then we have expanded upon our work in these two specific aspects—parents’ perceptions about the teaching and learning of mathematics and the role of language in mathematics instruction. We argue that these two issues are central to the marginalization of immigrant parents’ and therefore, their children in the area of mathematics education. Therefore it is critical for educators to understand and address these issues to improve the mathematics education of immigrant students.

## Commentary on the Chapter by Gilah Leder, “Looking for Gold: Catering for Mathematically Gifted Students Within and Beyond ZDM”

### Towards Equity in Mathematics Education (2012): 407-410 , January 01, 2012

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Gilah Leder thoroughly reviews, within the constraints imposed by the structure of this volume, what seems to be a very rich but not fully institutionalized area of research enterprise. An important (and, probably, inevitable) decision about the reviewing strategy was “[not to attempt] to provide a unique definition of mathematically gifted students or its pseudonyms, [but to accept] the diversity of definitions used in different publications… at face value” (Leder, this volume, p. 390). As a result, the review includes publications that, generally speaking, are concerned with *supporting* or *characterizing* those students who seem to be insufficiently challenged by a regular mathematical curriculum or show at least above-average achievements when studying mathematics in different educational settings. In fact, the chapter invites the reader to critically consider whether such a broad population can be referred to as “mathematically gifted students” and, more importantly, whether results of numerous studies utilizing the term “mathematically gifted students” or its pseudonyms can be perceived as a body of accumulated knowledge or as “tantalizing allusions to the needs and development of mathematically gifted students” (Leder, this volume, p. 403).

## Commentary on the Chapter by Ferdinand Rivera, “Neural Correlates of Gender, Culture, and Race and Implications to Embodied Thinking in Mathematics”

### Towards Equity in Mathematics Education (2012): 551-555 , January 01, 2012

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What differences in gender, culture, and race can be attributed to the biological evolution of the species, and what differences in gender, culture, and race can be attributed to social interaction? Such a polarized question is well posed only if these areas of attribution, biology and sociology, are independent of each other. A moment’s reflection may suggest that biological evolution and social interaction in the human species are to a large extent co-dependent. There are many distinctions that are but distinctions in thought and perception, and not independent in fact—and there are those who take the co-dependent arising of all things as a matter of fact. Be this as it may, some differences may predate others. For instance, sexual differences in humans may predate social interaction. However, the biological emergence of sexual differences in species may also be seen as an important affordance, perhaps even a *sine qua non*, for social interaction. Beyond and even within the constraints of kinship, social interaction, for better or for worse, must be predicated, for the most part of our evolutionary history, on biological differences of various kinds, covering wide spectra of attractions and repulsions. Through the means of agriculture and industry, and with continued advances in science, mathematics, engineering, and technology, humans have evolved to a point where social interaction can take place in purely semiotic and ideational terms, independently of biological characteristics. Whereas symbols and signs are visible, ideas are not (Merleau-Ponty 1968). Or are they?

## Gender Role Stereotypes in the Perception of Mathematics: An Empirical Study with Secondary Students in Germany

### Towards Equity in Mathematics Education (2012): 115-140 , January 01, 2012

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Findings from international educational studies such as TIMSS or PISA show that gender differences in mathematical achievements are now only marginal, or have completely vanished in many countries. Nevertheless, beliefs about the higher achievements in mathematics by boys are still widespread. In this chapter the results of a study of students’ views of gender-role stereotyping in mathematics are presented. For this study, 1244 German students from lower and upper secondary level (11–17 year olds) were tested using instruments developed by Forgasz and Leder concerned with gendered views of mathematics and mathematics education. The study clearly shows that as they grow up, boys as well as girls in contemporary Germany still have doubts concerning the equal mathematical performances of girls and boys, and internalise gendered stereotypes of girls being less talented and interested in mathematics, influenced by gender-role stereotypes within society, where mathematics is still described as a male domain.

## Immigrant Parents’ Perspectives on Their Children’s Mathematics Education

### Towards Equity in Mathematics Education (2012): 267-282 , January 01, 2012

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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 students’ 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.

## Commentary on the Chapter by Gilah Leder, “Looking for Gold: Catering for Mathematically Gifted Students Within and Beyond ZDM”

### Towards Equity in Mathematics Education (2012): 411-413 , January 01, 2012

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The title of Leder’s article “Looking for Gold” seems particularly apt given the difficulties she identified in finding recent material that focussed specifically on mathematically gifted and talented students. Over the past 20 years in ZDM, as Leder states, there have been “…tantalizing allusions to the needs and development of mathematically gifted students but few in-depth discussions of these issues.” Leder’s review is comprehensive and canvasses many concerns. The paucity of topical research, however, into the nature of mathematical giftedness, the effects of being mathematically gifted, or ways in which students with mathematical talent should be taught, especially at the high school level, seems surprising given the importance and documented need for quality mathematics students.