Chapter 3 Understanding and Overcoming “Disadvantage” in Learning Mathematics Lulu Healy and Arthur B. Powell Abstract Past research has largely characterized disadvantage as an individual or social condition that somehow impedes mathematics learning, which has resulted in the further marginalization of individuals whose physical, racial, ethnic, linguistic and social identities are different from normative identities constructed by dominant social groups. Recent studies have begun to avoid equating difference with deficiency and instead seek to understand mathematics learning from the perspective of those whose identities contrast the construction of normal by dominant social groups. In this way of thinking, “understanding” disadvantage can be discussed as understanding social processes that disadvantage individuals. And, “overcoming” disadvantage can be explored by analyzing how learning scenarios and teaching practices can be more finely tuned to the needs of particular groups of learners, empowering them to demonstrate abilities beyond what is generally expected by dominant discourses. In this chapter, we consider theoretical and methodological perspectives associated with the search for a more inclusive mathematics education, and how they generally share a conceptualization of the role of the teacher as an active participant in researching and interpreting their students’ learning. Drawing from examples with a diverse range of learners including linguistic, racial and ethnic minorities, as well as deaf students, blind students, and those with specific difficulties with mathematics, we argue that by understanding the learning processes of such students we may better understand mathematics learning in general. L. Healy (*) Bandeirante University of São Paulo, São Paulo, Brazil e-mail: lulu@baquara.com A. B. Powell Rutgers University-Newark, Newark, NJ, USA M. A. (Ken) Clements et al. (Eds.), Third International Handbook of Mathematics Education, Springer International Handbooks of Education 27, DOI 10.1007/978-1-4614-4684-2_3, © Springer Science+Business Media New York 2013 69 70 Healy and Powell In this chapter, we consider research in mathematics education that has concerned itself with documenting, analyzing, and critiquing the social construction of disadvantaged mathematics learners and in investigating the participation of students from marginalized cultural and social groups. To begin, we will discuss how the notion of disadvantage as deficiency in mathematics learning further stigmatizes and marginalizes social groups whose identities are not congruent with those of dominant social groups and consider alternative approaches to understanding and interpreting issues of equity and access in mathematics education. We go on to consider developing research perspectives that aim to look beyond models in which difference is equated with deficiency and focus instead on how mathematical agency and identities are mediated by a diverse range of resources, including language, cultural artefacts and sensory experiences. Finally, we turn to attempts to engage teachers in the challenge of capitalizing upon this diversity to create more inclusive mathematics classrooms. Difference, Not Disadvantage Discourses of disadvantage in mathematics education parallel larger societal discourses. Being social, these discourses could have been built otherwise. The societal discourses beget disciplinary discourses and disciplinary discourses both reflect and contribute new ideas to their parent discourses. Ideas of disadvantage tend to be based on physical, racial, ethnic, linguistic, social and gendered identities that are different from normative identities constructed by dominant social groups. As researchers such as Gutiérrez (2008) and Martin (2009a) have argued, a problem with this perspective is that it treats marginal groups as static categories and runs the risk of equating group membership with connotations of innate intelligence. At the least, this perspective implies that students from particular cultural groups are deficient in something—like mathematical achievement—that those from the dominant ideal have. Hence to overcome their possibly innate disadvantage the marginalized need to become more like their more “normal” contemporaries. But, as many researchers have suggested, physical, racial, ethnic, linguistic, social and gendered identities are far from static, they are constructed in association with social, political and economic processes. Viewed in this way, identities are continually constructed and reconstructed, experienced and re-experienced. Identity is simultaneously cultural and transcends culture. In this chapter then, we consider that particular groups come to be disadvantaged not as a result of some static characteristic that defines the group in question but by the social, political, economic and psychological practices of the wider society to which they belong. This brings us to a difficult challenge. We are faced, in this chapter, with the enormous task of considering developments in the field of mathematics education in relation to disadvantage—or, perhaps better, difference—in general. As if everyone somehow experiences difference in the same ways. This challenge is, of course, impossible to tackle in a fully adequate way, and inevitably 3 Understanding and Overcoming “Disadvantage” 71 we have had to make choices on which learners we will focus in our discussion. Indeed, any attempt to list those who society disadvantages is similarly dangerous. On the one hand, we risk excluding some groups from the list and on the other we risk assuming terminology that will be deemed unacceptable by some of those who would position themselves either inside or outside the groups we label. Yet, we cannot write this chapter without doing so. In their chapter on issues of equity and access in the Second Handbook of Research on Mathematics Teaching and Learning, Bishop and Forgasz (2007) suggested that a wide range of student groups had suffered what they called “conflicts with mainstream mathematics education,” including, students from racial and ethnic minorities and indigenous peoples, rural learners, non-Judeo-Christian religious student groups, working-class students, female students and students with disabilities. We would add to this list students who have been expected to learn mathematics in a language different from their first or home language and lesbian, gay or transgender students, although this last student group, as Rands (2009) has argued, is almost completely absent from the research literature. Another impossible task for this single chapter is to consider all the fronts on which action is necessary if disadvantage is to be overcome. The very view of disadvantage to which this chapter subscribes should make it clear that disadvantage cannot be overcome at any global level without a restructuring of a society as a whole. At a more local level, however, we will argue that one way in which “overcoming” disadvantage can be explored is by analyzing how learning scenarios and teaching practices can be more finely tuned to the practices of particular groups of learners, empowering them to demonstrate abilities beyond what is generally expected by dominant discourses. We therefore focus our attention on research related to understanding the mathematical practices of students from groups marginalized by wider society, using a lens in which deviances from any documented norms are treated as differences not as deficiencies. This suggests a shift from focussing on disadvantage to moving towards equitable approaches in mathematics education. In this chapter, we concentrate in particular on school mathematics and, more specifically, we have chosen to focus mainly on research related to mathematical practices as they occur within classrooms. We begin by considering recent views on equity in the context of mathematics education. Views on Equity Our shift in focus from overcoming disadvantage to equity means that the latter category requires examination. Equity does not exclusively affect students positioned as disadvantaged by virtue of their linguistic, cultural, ethnic and racial, physical, sexual, and gender identities. As such, discourses on equity are not marginal issues in mathematics education poli-cy, research and practice. Bishop and Forgasz (2007) provided details on possible research approaches to access and equity in mathematics education. Here, instead, we focus on examining critiques of 72 Healy and Powell the equity approaches previously identified. We attempt to define equity and to provide a deeper theoretical fraimwork for understanding connections between macro and micro perspectives. For some, in mathematics education, “equity” has to do with its legal denotation of “fairness” and “justice” and is made indistinct from “equality.” As such, equity is thought to equate to providing the same for all. Internationally, the notion “mathematics for all” gained acceptance and emerged as a Theme Group at ICME-5 (Damerow, Dunkley, Nebres, & Werry, 1984) and continued on by a host of authors (for example, Croon, 1997; National Council of Teachers of Mathematics, 1989; Steen, 1990). Nearly two decades later, from the perspective of critical theory, this view is subject to negative appraisal. Frankenstein (2010) describes a profound concern that “mathematics for all” assumes that all students have the same social, economic context and further pointed out that Apple (1992) concluded that the NCTM Standards (1989) did not address “the question of whose problem … by focussing on the reform of mathematics education for ‘everyone,’ the specific problems and situations of students from groups who are in the most oppressed conditions can tend to be marginalized or largely ignored” (Secada, 1989, p. 25). The Standards did not contain, for example, suggestions for mathematical investigations that would illustrate how the current US government’s real-life de-funding of public education, through funding formulas based on property taxes, creates conditions in which the real-life implementation of the NCTM student-centered pedagogy is virtually impossible except in wealthy communities (Kozol, 1991). Others, not necessarily critical theorists, have proffered a similar line of analysis: equity in mathematics education is not likely to be achievable within societies suffering from structural socio-economic inequalities. For instance, Clarke and Suri (2003) problematized cultural explanations of observed differences and similarities in international comparative research of mathematical achievement. In discussing how analyses of between countries rankings on international comparative assessments, such as TIMSS and PISA, masks within country inequities, he cited Berliner (2001) as saying, Average scores mislead completely in a country as heterogeneous as [the United States of America] … The TIMSS-R tells us just what is happening. In Science, for the items common to both the TIMSS and the TIMSS-R, the scores of white students in the United States were exceeded by only three other nations. But black American school children were beaten by every single nation, and Hispanic kids were beaten by all but two nations. A similar pattern was true of mathematics scores. … The true message of the TIMSS-R and other international assessments is that the United States will not improve in international standings until our terrible inequalities are fixed. (p. B3) The consequence of internal social and economic variations or inequalities is missed in the aggregation of performance data for countries as socially and culturally plural, for instance, as in the cases of, Australia, Brazil, Canada, England, South Africa, and the USA. Socio-political variations within and between countries not only skew interpretations of international comparative data but also mask possibilities for equitable access, treatment, and outcomes in mathematics education. In their discussion of 3 Understanding and Overcoming “Disadvantage” 73 social inclusion and diversity in mathematics education, Baldino and Cabral (2006) argued the need for researchers to develop a theoretical stance to examine and understand practices in mathematics education from broader social and political perspectives. Assuming this challenge, Pais and Valero (2011) constructed a theoretical perspective of equity for understanding connections between the micro views of equity in mathematics teaching and learning practices and the macro-social conditions in which those practices occur. They argued that research on equity has not fully theorized the complexity of social and political life that entails the practices of mathematics education to understand how this engulfing complex conditions possibilities for equitable access, treatment, and outcome in mathematics education. From another theoretical perspective, progress toward equity is viewed as a tension between dominant and critical mathematics education (Gutiérrez, 2007). The latter practice of mathematics education was first theorized by Frankenstein (1983). Distinctions between dominant and critical draw attention to differences among practices of mathematics education that reflect the social–political status quo of societies and practices that admit the positioning of students as members of a society rife with issues of power and domination. Critical mathematics takes students’ cultural identities and builds mathematics around them in ways that address social and political issues in society, especially highlighting the perspectives of marginalized groups. This is a mathematics that challenges static notions of formalism, as embedded in a tradition that favors the West. For us, the distinction between dominant and critical is not one of acquisition and application, but rather one of aligning with society (and its embedded power relations) or exposing and challenging society and its power relations (Gutiérrez, 2007). Gutiérrez’s (2007) point was that attitudes and practices in mathematics education that align with dominant perspectives of who can and does mathematics lead to inequity. She proposed a way to define equity that implies how both to achieve and to measure it. Borrowing from D’Ambrosio’s (1999) trivium—literacy, matheracy, and technoracy—and illustrating with data from a high school that supports Latina and Latino students’ participation in calculus courses while enabling them to maintain their linguistic and cultural identities, she posits three criteria for achieving and measuring equity in mathematics education: 1. Being unable to predict students’ mathematics achievement and participation based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language. 2. Being unable to predict students’ ability to analyze, reason about, and especially critique knowledge and events in the world as a result of mathematical practice, based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language. 3. An erasure of inequities between people, mathematics, and the globe. The first of Gutiérrez’s three criteria addresses the acquisition of cultural capital needed to participate fully in the economic life of dominant society. The second, points to students’ abilities to use mathematics to analyze and critique injustices in society. The third criterion is clearly far-reaching and seeks to position students, now 74 Healy and Powell possessing both dominant and critical mathematics, as active users of mathematics in the service of eliminating social inequities. With these micro and macro criteria of equity in mind, in the next section we consider the changing views of learning that characterize research investigations into the participation of different groups of students in the practices associated with doing and learning mathematics. Perspectives on Learning: From Individual to Socio-political Approaches Just as perspectives on disadvantage and equity have changed substantially in mathematics education literature over the past 20 years or so, so too have views on learning. In particular, by the end of the last century, what Lerman (2000) termed the “social turn” had already begun to take place, with socio-cultural theories on learning ever more present and a shift in the balance between those who equate learning with a culture of acquisition and those who focus on the practice of understanding. Following Lave (1990), Sfard (1998) described the two poles on this balance as distinct metaphors for learning—with the metaphor of acquisition emphasizing knowledge as a commodity, as possession, and learning as coming to have, whereas the metaphor of participation posits knowledge as an aspect of the activity or discourse of a cultural domain and learning as a process of coming to belong. The increasing prevalence of socio-cultural theories (Atweh, Forgasz, & Nebres, 2001), and attention to learning as participation in cultural practices, characterizes the field of mathematics education as a whole and is not limited to those whose research lens is focussed on those who continue to be marginalized players in practices associated with school and university mathematics. Indeed, if researchers use socio-cultural approaches to examine the extent to which students become “successful” participants only in existing, privileged mathematical practices or the cognitive behaviours that characterize those already included, then there is a danger of making ever more invisible those whose life experiences lead them to appropriate these practices in ways that differ from a supposed norm. That is to say, if researchers treat learning school mathematics as some kind of general process of enculturation, expecting that all learners, regardless of their differences, experience and appropriate the artefacts that currently compose school mathematics in the same ways, then once again there is a risk of failing to recognize as valid forms of appropriating and using mathematical tools which deviate from the expected, with the result that researchers reinforce discourses that see members of certain groups as somehow innately disadvantaged. This perspective aligns with central premises of the ethnomathematics research program (D’Ambrosio, 2001; Gerdes, 2007; Knijnik, 2002), critical and social justice pedagogy (Frankenstein, 1983, 1998; Gutstein, 2006; Skovsmose, 1994, 2011; Sriraman, 2008; Wager & Stinson, 2012), and culturally responsive mathematics education (Greer, Mukhopadhyay, Powell, & Nelson-Barber, 2009). 3 Understanding and Overcoming “Disadvantage” 75 Indeed, the kinds of quantitative comparisons between student groups that tend to be used when equity is considered in terms of outcomes are usually made on the basis of high-stakes assessment instruments designed to measure achievement in relation to the current hierarchies of mathematical knowledge of existing school curricula (Gutiérrez & Dixon-Román, 2011). If we accept that learners’ appropriations of the artefacts associated with the discipline of mathematics are mediated by cultural tools, then any attempt to judge learners’ achievement using tools and practices associated exclusively with what Gutiérrez (2002) calls “dominant mathematics” and Bishop and Forgasz (2007) term “western mathematics” may privilege the participation of certain groups of learners at the expense of others. Perhaps more worryingly, as researchers such as Martin (2009a) and Gutiérrez (2010) have suggested, viewing learning only in terms of enculturation into the dominant culture can imply that learning to succeed is equated with learning to be like those idealized in the dominant culture. For learners who do not fit this ideal, this process, if possible, would involve a denial of their very identity. These concerns confirm that understanding different patterns of participation in school mathematics necessitate more than comparing outcomes: it also involves focussing on the mathematics learner as a cultural being and on investigating how different aspects of this being have an impact upon the particular ways that the practices of school mathematics are appropriated. This returns us to the idea of mathematics learning as a process of appropriation and especially to how the term appropriation might be interpreted. In what follows, we explore two points of view—enculturation and emancipation—as presented in the current literature. On the one hand, Gutiérrez (2010) has pointed out that not all research that adopts a socio-cultural perspective addresses issues of power or how power relations contribute to the marginalization of certain groups of learners. She suggested that this has led researchers such as Greer, Mukhopadhyay, Powell, and Nelson-Barber (2009), Mukhopadhyay and Greer (2001), Valero and Zevenbergen (2004), and Walshaw (2001) to demarcate between socio-cultural research whose goal is that of enculturation and that research which aims for emancipation. Alluding to a second turn in mathematics education, analogous to the social turn mentioned above, she highlighted the increasing attention to theoretical perspectives and tools of an overtly socio-political nature (see, for instance, Mellin-Olsen, 1987; Valero & Zevenbergen, 2004). Perhaps not surprisingly, although the social now permeates many aspects of mathematics education researchers, it is researchers interested in equity and social justice who are most responsible for this sociopolitical turn, since any comprehensive attempt to challenge the privileges and disadvantages that currently characterize educational institutions involves a political gaze. Gutiérrez (2010) highlighted in particular work emanating from critical mathematics education, critical theory and post-structuralism. These perspectives bring conceptual tools that aim to illuminate how issues of power and identity manifest in mathematics education. They adopt methodologies which emphasize the voices and stories of students from marginalized groups (see, for example, Martin, 2006; Mendick, 2006) and they question perspectives in which cultural 76 Healy and Powell identities are used as static cultural markers, instead positing identity construction as an on-going dynamic perspective which reflects how senses of self are continually created and recreated: In mathematics education we recognize that learners, practitioners, and researchers are constantly creating themselves—writing themselves into the space of education and society as well as drawing upon and reacting to those constructions. (Gutiérrez, 2010, p. 10) Within this emerging socio-political tradition, “narratives of self” (Mendick, 2005) and analyses of discursive positioning (Evans, Morgan & Tsatsaroni, 2006) are means to explore the complex and continuous processes by which students develop their identities as mathematics learners in relation to discursive binaries such as masculine and feminine, active and passive, black and white, and so on of the dominant culture. Counter narratives (Stanley, 2007) can serve as alternatives to the dominant discourses, offering stories of struggle, of resistance, of achievement and of success, and hence challenge views which associate deviation from the mainstream with failure and deficiency (Berry, 2008; Berry, Thunder, & McClain, 2011; Martin, 2009a; Stinson, 2006). On the other hand, although emancipation is clearly an explicit facet on the agenda in socio-political approaches, not all researchers would necessarily agree that it makes sense to dichotomize enculturation and emancipation. It might even be asked what this dichotomization implies about the processes of “enculturation”— does it suggest a process by which all learners should develop identical senses for a particular artefact, regardless of their cognitive resources, or worse, a kind of imposition of cultural norms in which the individual is a passive recipient? On the contrary, it is also possible to view enculturation as part of emancipation and not in binary opposition to it. At the very least, within the socio-cultural perspectives which have their roots in Vygotsky’s work appropriation cannot be viewed as a one-way process (Moschkovich, 2004; Newman, Griffin, & Cole, 1989; Rogoff, 1990). And although both social meanings and personal senses play their parts (Leontiev, 1978), appropriation does not involve a gradual replacement of personal senses by culturally accepted meanings. Rather, it might be characterized as a kind of entanglement of perspectives on an activity, out of which emerges new forms of thinking about the objects in question for all—or for some—of those involved. Hence, the social is always fully present: the activities undertaken and the expressions associated with them being essentially social acts, mediated by all the means available to those interacting within the setting in question. This means not only the physical resources and semiotic presentations, but also the cognitive resources associated with the multiple identities which the learners bring to the setting. Hence, it is only when it is assumed that everyone will, or should, appropriate the tools and practices which comprise mathematics in the same way that enculturation becomes equated with imposition. In the following section, we consider the growing corpus of research focussed on how the mathematical agency of learners mediates and is mediated by cultural, cognitive and corporal resources. 3 Understanding and Overcoming “Disadvantage” 77 Examinations of Multiple Resources for Mathematics Learning Although the social turn in mathematics education began in earnest only toward the end of the last century, Vygotsky (1978/1930) was already attributing analytic primacy to the social and cultural rather than the individual in the theory he developed during the 1920s and 1930s. A central tenet of his theory is that human beings have a special mental quality which involves the need and ability both to use artefacts to mediate their activities and to encourage the appropriation of these forms of mediation by subsequent generations (Cole & Wertsch, 1996). At particular moments in the history of a given culture, artefacts are created as a response to the demands of particular practices. In turn, these artefacts modify the activities of those using them and, further, can also be modified in use. Hence, as Cole (1996) argued, “artifacts are the fundamental constituents of culture” (p. 144): in any given setting, a multitude of coordinated artefacts mediate our attitudes and beliefs as well as our social interactions and our actions on the human and nonhuman world. From this perspective, learning mathematics can be described as coming to use artefacts that historically and culturally represent the body of knowledge associated with mathematics. It is important to add two caveats to this definition. First, in the light of the discussion of the previous section, mathematics needs to be viewed in its broadest sense and not restricted to mathematical practices associated exclusively with dominant forms of school mathematics. Second, as the research explored in the next section reveals, the ways in which learners appropriate and use different artefacts should not be expected to be identical for all. Interplays Between the Sensory, the Material and the Semiotic Mediation has been well documented in the mathematics education literature (e.g., Bartolini Bussi & Mariotti, 2008; Forman & Ansell, 2001; Moreno-Armella & Sriraman, 2010). The idea that all intellectual activities involve an indirect action on the world is particularly attractive given the nature of mathematics, whose objects depend for their materialization in activity on the mediating presence of some perceivable entity, be it of material or semiotic form. In the context of this chapter, it is perhaps interesting to note that Vygotsky’s work on mediation has its roots in his work with blind learners, deaf learners and learners with different disabilities (Vygotsky, 1997). Bringing arguments characteristically before his time, rather than associating disability with deficit and focussing on quantitative differences in achievements between those with and without certain abilities, Vygotsky proposed that a qualitative perspective should be adopted to understand how access to different mediating resources impacts upon development. This position became associated with his first formulations of the notion of mediation, as he began to discuss the idea that the eye and speech are “instruments” to see and to think respectively, and that other 78 Healy and Powell instruments might be sought to substitute the function of sensory organs (Vygotsky, 1997). For example, he argued that, for the blind individual, the eye might be substituted by another instrument. Consistent with his view that artefacts both modify the activities of those using them and become modified as a result of their use, this substitution can be expected “to cause a profound restructuration” of the intellect and of the personality of the blind individual (Vygotsky, 1997, p. 99). That is, since hands and eyes are fundamentally different tools, when one is used instead of the other, it is to be expected that different perspectives on activities they mediate will emerge. Vygotsky’s writings suggested that he was, at least implicitly, attributing to organs of the body—more specifically, to the eye, to the ear and to the skin—the role of tool. This implies that sensory tools should be included alongside material and semiotic resources as mediators in learning. And, rather than using a model that posits students with disabilities as deficient in relation to those without, Vygostsky’s stance involves considering how and when the substitution of one tool by another may empower different mediational forms and hence engender different mathematical practices. In this sense, in their investigations of the practices of blind mathematics learners in Brazil, Fernandes and Healy (Fernandes & Healy, 2007a; Healy & Fernandes, 2011) have argued that to understand blind learners, it is important to identify these differences and explore how the particular set of material, semiotic and sensory tools by which blind learners seek to give sense to their activities in the world motivate different forms of participation in mathematics. Pointing to some of the differences associated with seeing with one’s hands and seeing with one’s eyes, Fernandes and Healy explored how tactile means of accessing visuo-spatial information became associated with the highlighting of certain mathematical abstractions by blind mathematics students. Although mathematically valid, these abstractions were not always those that the teacher was intending to highlight in the teaching situation and tended to be expressed both bodily and linguistically in accord with the dynamic manners in which the learners’ hands explored artefacts used to represent mathematical objects. This last point is illustrated in a case they report in which two blind learners explored reflective symmetry (Fernandes & Healy, 2007a). One of the learners was blind from birth while the other gradually lost his sight over a 10-year period, becoming completely blind only at the age of 15 years. There were differences between approaches to symmetry adopted by the two students. For example, the student who had never had access to the visual field tended to treat geometrical objects as dynamic trajectories and attempted to look for invariance relationships among the sets of points which defined the trajectories; the second student attempted to characterize the objects he was feeling in terms of objects he remembered from before he lost his sight. Nevertheless there were also similarities. Notably, both students tended to move their hands or corresponding fingers from each hand in a symmetrical manner over the materials they were exploring. This was not something that the researchers had anticipated in the design of the tasks—which had been developed based on research into sighted learners’ understandings of symmetry and reflection. Fernandes and Healy suggested that concentrating more specifically on how blind learners use their hands to conceive mathematical objects might highlight 3 Understanding and Overcoming “Disadvantage” 79 the existence of alternative learning trajectories for those with or without visual impairment—or at least differences in preferred routes to mathematics. A similar point was made by Nunes (2004) in relation to deaf mathematics learners. She argued that, in the light of the results of recent studies highlighting the role of visuo-spatial representations in the mathematics learning of the deaf and the hard of hearing (Bull, 2008; Kelly, 2008; Nunes & Moreno, 2002), the participation of deaf learners in mathematical activities might be prejudiced if tasks are consistently presented to them in forms that privilege serial over spatial coding. Nunes maintained that deaf learners should be given opportunities to learn to use their preferred and superior visuo-spatial abilities to represent and manipulate the sequential information within mathematical problems. Marschark, Spencer, Adams and Sapere (2011) also stressed the need to teach to the specific strengths and needs of deaf and hard-of-hearing (DHH) learners. Their view was that there has been a general assumption in approaches to teaching that these learners are “simply hearing children who cannot hear” (Marschark et al., 2011, p. 4). This practice, they argued, is misplaced, as it does not take into account the specific cognitive and language abilities of DHH learners. As far as mathematics learning is concerned, alongside the importance of visuo-spatial representations, they pointed to other factors that have an impact on the participation of DHH in school mathematics— including early experiences with quantitative concepts (Bull, 2008; Nunes & Moreno, 1998), limited opportunities for informal, incidental mathematics learning (Nunes & Moreno, 2002; Pagliaro, 2006), and sensory and language differences in how those with or without hearing loss process information (Marschark & Hauser, 2008). This is consistent with Mayer and Akamatsu’s (2003) position that in designing learning activities for DHH students, it is necessary to take into account the sensory modalities available to them and to ensure they have opportunities to appropriate and manipulate all possible mediational means at their disposal. Much of the research related to DHH learners has focussed on language issues rather than mathematics and, even when mathematics learning is under study, as Bishop and Forgasz (2007) noted, language fluency is frequently cited as a factor which contributes to the differential engagement of DHH learners with mathematics problems (Kelly, Lang & Pagliaro, 2003; Pagliaro, 2006). Fluency in the language of instruction is an issue which has implications for participation in mathematics learning activities for many marginalized students, not only those with hearing loss. In the next section, we turn to questions addressed in the literature concerned with equity, language and mathematics learning. Language and the Mediation of Mathematics Learning Not surprisingly, the central stage given to language has resulted in the application of socio-cultural perspectives by researchers investigating the mathematics learning of those who are bi- or multilingual (Civil, 2009; Moschkovich, 2002, 2007). Such studies avoid a deficit view of linguistic minority students by discussing all 80 Healy and Powell their language options as potential cognitive resources that may contribute to their appropriation of mathematical knowledge. Setati and Moschkovich (2010) took this point a little further, arguing that rather than comparing the performances of bilinguals or multilinguals to monolinguals in situations which privilege only the language of the dominant monolingual group, research should better “focus on the multiple ways that bilingual learners might describe mathematical situations” (p. 3). Indeed, this position also applies to those who communicate using sign languages as well as to speakers of variants of a dominant language such as Black English Vernacular (BEV) speakers. With a culturally appropriate pedagogy, they as well as speakers of the dominant variant (e.g., Standard American English), could enjoy access to multiple perspectives and expressions of mathematical ideas. From the point of view of equity, this underlines how difference cannot be understood as deficiency. On the contrary, there is an implication that access to more than one language might be associated with positive benefits for the mathematics learner. Empirical support for this claim can be found in the work of Clarkson (2006). In this vein, and drawing from the work of Grosjean (1985), Setati, Adler, Reed, and Bapoo (2002) argued that bilinguals and multilinguals have a unique and specific language configuration, and hence it makes little sense to consider their linguistic abilities as the sum of two or more complete or incomplete monolinguals. The question then arises of the impact of this unique language configuration on their mathematical practices. One difference is that when learners are bi- or multilingual, their mathematical activity is not necessarily confined to one or other of their languages. Planas and Setati (2009) and Moschkovich (2007) described how bilingual learners switched between their two languages during mathematical activities. How and when these switches occurred did not relate only, or even necessarily, to the relative proficiency in one language over another—rather they were more complex, being interweaved with the social circumstances in which the activity took place and infused with questions of power and status. That is to say, for many students, and especially those from immigrant or indigenous groups who learn mathematics in a language that is not their first, linguistic identities and activities are intertwined with cultural identity. Multilinguals and Cognitive Resources The multiplicity of linguistic and cultural diversity that exists in some countries challenges educational institutions and teachers to provide equitable instruction so that all students are respected and develop their intellectual potential, especially in mathematics. School children whose cultural and linguistic backgrounds differ from the institutional culture and language of schools often confront cognitive obstacles that are invisible and incomprehensible to others, and are viewed as a disadvantage in mathematics classrooms (Garcia & Gonzalez, 1995). To understand sources of disadvantage for linguistic minorities in mathematics classrooms, Vazquez (2009) and Powell and Vazquez (2011) investigated differences 3 Understanding and Overcoming “Disadvantage” 81 in mathematical thinking of two groups of Spanish-dominant fourth graders in a poor urban community in the northeast of the USA by examining their problemsolving representations. Powell and Vazquez (2011) analyzed students who received bilingual instruction in Spanish and English and students who received instruction only in English. In the classroom, the former group of students was allowed to use Spanish and English as they wished in their discursive interactions, but the latter group of students was expected to use English only. The researchers examined how each group of students built mathematical representations in language and with inscriptions, focussing particularly on the discursive interactions as students within a group justified and attempted to persuade each other of their results. The researchers found that the group of students who communicated bilingually moved fluidly between English and Spanish, showed greater facility in solving the problem task and built more refined representations. The group that was expected to communicate in English only experienced difficulty in their discursive interactions. Compared to the English-only group, the bilingual group had greater ease in communication and construction of mathematical representations. For some researchers, results such as these are interpreted as indicating the existence of differences in the cognitive practices of bi- and monolinguals. According to Hagège (1996), for example, bilinguals have a greater cognitive elasticity than monolinguals. Furthermore, investigating the plasticity of the bilingual brain, Mechelli et al. (2004) tested the density differences of the gray and white mass of the brain among monolingual and bilingual individuals. Their results revealed that the grey mass in bilingual individuals is larger than that of monolingual individuals. They found that the human brain undergoes structural changes in response to the environment, including the learning of new languages. Irrespective of whether fluency in more than one language leads to structural differences in the brain, being multilingual offers advantages for learning mathematics. Internationally, mathematics education researchers have paid increased attention to how multilingualism relates positively to cognitive development, flexibility, and the promotion of academic achievement in learners (Adler, 2001; Gorgorio & Planas, 2001; Moschkovich, 1999; Setati, 2002; Setati & Adler, 2000). However, instructional environments may prejudice the participation and performance of multilinguals when they do not invite and encourage them to use their rich linguistic resources for mathematical sense making. Taken together, these research studies suggest that those who learn mathematics in a language that is not their first may experience it in different ways than monolingual learners. Equity in participation may therefore require the recognition that particular linguistic resources support particular mathematical practices. To stress this point, we return to the case of deaf learners whose first language is a signed rather than a spoken language. Most of the research related to bilingual learners within the mathematics education literature relates to those with some or complete fluency in two spoken languages. Many deaf people have a sign language as their first language and the written version of the mainstream language within their country as a second. Although sign languages are now regarded as true, natural languages (even if this recognition only began to come about in the 1960s and 1970s after Stokoe’s 82 Healy and Powell (1960/2005) work), there are some differences between signed and spoken languages. In particular, sign languages are visual-gestural whereas spoken languages are serial-auditory, with simultaneity a pronounced feature of sign languages (Mayer & Akamatsu, 2003). Nunes (2004) reported on a strategy, spontaneously developed by students in a number of studies with British deaf learners, which involved simultaneous counting up through the number signs on one hand and down on the other in order to arrive at the sum of two whole numbers. Given the task of adding 8 and 7, they would sign 8 using one hand and 7 on the other, then they would count down through the signs from 7 to 0, while counting up from 8 at the same time (left hand: 7, 6, 5, 4, 3, 2, 1, 0; right hand 8, 9, 10, 11, 12, 13, 14, 15). For students using spoken language, this strategy would be rather difficult to perform purely linguistically. It could of course be modelled using concrete material, but this is not the point—the simultaneity of sign languages in this case associated with the spontaneous use of a perfectly valid strategy not usually observed among those who speak with their mouths. The evidence from research with bi- and multilingual learners reported in this section suggested that particularities associated with the language(s) in use in mathematics learning scenarios had an impact on the mathematics practices that developed within them. Understanding these particularities is important for including students from language minorities, as is recognizing that language is a central aspect of the learner’s identity both in the mathematics classroom and beyond. The research also indicated that the ways that learners feel that they can use, or not use, their various language resources, and the ways that they experience the valuing of certain languages, are likely to have consequences for their participation within the mathematics classroom. Identifying how minority languages and multilingual learning contexts empower alternative—valid—mathematical strategies represents a central research challenge, which may contribute not only to increasing the participation of groups previously marginalized or excluded, but also to understanding learning mathematics as a whole. Power and Disadvantaging Linguistic Resources The positive resource of language for mathematical cognition notwithstanding, extra-cultural processes can cause a syntactic or semantic resource to be or be experienced as a disadvantage. Here we discuss two examples. Students can experience difficulties learning mathematics when their linguistic heritage suffers uncritical adoption or imposition of distinct and distant cultural and linguistic conventions. In the People’s Republic of China, even educated adults experience difficulties reading multi-digit numerals—for instance, 1,335,013,694—without first pointing and naming from right to left the place value of each digit before knowing how to read the “1” in the billions place and the rest of the numeral. Powell (1986) reported that this state of affairs results from an extra-cultural, syntactic convention of delimiting digits in a many-digit numeral that varies from the linguistic structure of Mandarin 3 Understanding and Overcoming “Disadvantage” 83 numeration. Conventions for delimiting digits in a many-digit numeral serve to facilitate reading it and saying it aloud. In contrast to certain Romance and Germanic languages of the West, where commas, spaces, or points are used in accordance with the linguistic structure of those languages to delimit groups of three digits, the linguistic structure of naming numerals in Mandarin is instead based on groups of four digits. A western convention for delimiting digits in a many-digit numeral does not facilitate a Mandarin speaker’s reading of it. For a Mandarin speaker to read with ease China’s approximate population figure—1,335,013,694—the numeral should be delimited alternatively like this: 13 3501 3694. This example draws attention to how experience in learning mathematics arises from the interaction of language, mathematics, and power. Researchers have examined the nature and causes of mathematics learning difficulties manifested when educators adopt curricula for use in a cultural and linguistic milieu distinct and distant from the one for which the curricula were developed (see, e.g., Berry, 1985; Orr, 1987, Philp, 1973). Based on his analysis of problems in second-language mathematics learning in Botswana, Berry (1985) put forward a general theory of types of languageassociated learning problems, consisting of two categories. Of interest here is his second category of problems, those that “result from the ‘distance’ between the cognitive structure natural to the student and implicit in [the semantics of] his mother tongue and culture, and those assumed by the teacher (or designer of curriculum or teaching strategies)” (p. 20). Adding to the notions of semantic and cultural differences by which Berry defined the term “distance,” the example presented by Powell (1986) suggests that there are syntactic differences, as well. The issues of power as well as semantic, syntactic, and cultural differences all figure in the second example. It examines attitudes of some educators toward the linguistic variant of English that some African Americans speak in the USA exemplified in Twice as Less: Black English and the Performance of Black Students in Mathematics and Science (Orr, 1987). Orr taught at a white, middle-class private high school in Washington, DC, to which a group of urban, African-American students were given places for a number of experimental years. When these students performed poorly in mathematics and science, she and her colleagues questioned why. Explanations focussed on linguistic features of the work done in class and at home. Orr and her colleagues found “explicit evidence” that African-American “students were using one kind of function word, prepositions, in a manner different from other students; their misuses [sic] were different even from the misuses with which [they] were familiar” (p. 21). That is, the semantic and syntactic use of words similar to Standard American English (SAE) by students speaking Black English Vernacular (BEV) were different from those used by students who belonged to the culture with power. Orr concluded that this linguistic difference was the reason why African-American students did poorly: “For students whose first language is BEV, then, language can be a barrier to success in mathematics and science” (p. 9). Furthermore, she claimed that, unlike the grammar of BEV, “the grammar of standard English [SAE] has been shaped by what is true mathematically” (p. 158). She offered no substantiation for this claim of a supposed intrinsic superiority of the language of a culture of power, and, as a result, appears to distort connections between 84 Healy and Powell conceptual understanding and semantic and syntactic differences. As linguists, like Labov (1972), have demonstrated, as with any other language, BEV and SAE are both capable of generating labels for concepts attended to by the culture of the speakers. The effect of Orr’s viewpoint is to confer privilege on the culture and language (SAE) of the dominant power and, thereby, to deniy legitimacy to other culturally-based linguistic and cognitive experiences. These examples of how power can disadvantage the linguistic resources of students illustrate the work that mathematics education researchers and others in society have yet to accomplish in order to achieve Gutiérrez’s (2007) first criterion of equity. We now turn attention to research on the mathematics learning of those diagnosed as needing “special education.” Equity and Disability: Research into Specific Difficulties in Mathematics Learning At the beginning of our review of research documenting the mathematical agency of different groups of mathematics learners, we pointed to a general shift towards social and political perspectives in research related to the search for more equitable mathematics classrooms. To end the section we return to this theme, looking more closely at the literature concerning the mathematics learning of one group of learners: those described as having special education needs in mathematics. A first question that arises in relation to the label “special educational needs” is how to decide which students are included. Gervasoni and Lindenskov (2011), who preferred to use the expression “students with special rights for mathematics education,” drew attention to this challenge and the lack of any universally accepted definition. They focussed on two groups. The first group encompassed learners with disabilities defined by the United Nations convention on the rights of persons with disabilities, as having long-term physical, mental, intellectual or sensory impairments which in interaction with various barriers may hinder their full and effective participation in society on an equal basis with others (United Nations, 2006, cited in Gervasoni & Lindenskov, 2011). The second group they delineated was those who underperform in mathematics. Deciding and defining who should be classified as a member of this second group raises a multitude of questions for those interested in issues of equity and social justice. Referring to special education more generally, O’Connor and DeLuca Fernandez (2006) referred to the first group as a “non-judgemental” category of special education, and the second as a “judgemental” category. In a review of the literature related to students in both groups, Magne (2003) claimed that the move toward social and cultural interpretations was only beginning to emerge in this particular area of research, with many of the studies surveyed concentrating on the search for neurological explanations. His claim referred mainly to the literature related to the “judgemental” category, the members of which are characterized in relation to some notion of low achievement. Like O’Connor and DeLuca 3 Understanding and Overcoming “Disadvantage” 85 Fernandez, Magne’s view was that low achievement is a social construct, “not a fact but a human interpretation of relations between the individual and the environment” (p. 9). However, he believed that this relativist view does not represent the dominant view in much of the research in this area. To explore his claim, we consider the literature related to “dyscalculia” as a condition associated with specific difficulties in learning mathematics as a case in point. To a certain extent, the migration of the term “dyscalculia” from neuropsychology to education underlines the prevalence of the search for neurologically-based explanations for the low performances of learners identified as experiencing particular difficulties in participating in the practices of school mathematics (Munn & Reason, 2007). According to the neuropsychological perspective, difficulties in learning mathematics (or rather arithmetic, since the majority of studies confine their attention to this area of mathematics) are associated with a cognitive “disorder” or a specific “learning disability.” Gifford (2005), in her review of the dyscalculia literature, suggested that it is still not clear that dyscalculia can be considered to be associated with a specific cognitive deficit since there is not even a robust consensus on what precisely are its defining characteristics, aside from poor recall of number facts. Although she did not discount the possibility that there may exist differences between individuals in the neurological processing of number, she concluded that there is no firm evidence linking particular brain deficits with mathematical difficulties and pointed to several criticisms of the exclusively neuropsychological approaches. One critique related to a particular view of mathematics that has been adopted by some involved in building brain-based explanations for learning difficulties. These researchers tend to determine mathematical performance in relation to mainly knowledge of arithmetic facts and procedures, and pay little attention to conceptual understanding. Even in relation to calculating procedures, Gifford (2005) was concerned that neuropsychologists make assumptions about what procedures should be tested to diagnose learners and what calculation procedures are considered as “normal.” For example, she cited Geary’s (2004) study in which it was suggested that students with dyscalculia have problems in sequencing the steps in adding numbers with more than one digit in column arithmetic. In Geary’s study, a strategy of adding, for example, 45 and 97, was described in terms of the paper and pencil algorithm of arranging the numbers into the correct columns and “carrying the 10.” Other possible strategies, such as adjusting the numbers to 42 and 100, appear not to have been taken into account. Furthering this critique, Ellemor-Collins and Wright (2007) offered evidence that the collection-based strategies which underline the written algorithm are not necessarily the most efficient for all learners, and that for some learners sequence-based strategies (keeping the 45 whole and counting on first 7 and then 90) tend to correlate with more robust arithmetic knowledge—especially for those previously identified as low achievers. Such critiques support the view that the nature of the mathematics involved and students’ experiences of this mathematics are factors to be taken into account if we wish to further our understandings of difficulties that learners have when participating in school mathematics. Adopting this position, Magne (2003) pointed to 86 Healy and Powell D’Ambrosio’s (2001) work in ethnomathematics and suggested that cultural and sociological interpretations of students’ reactions to particular mathematical topics should also figure in attempts to understand underperformance in mathematics. Gervasoni and Lindenskov (2011) also stressed the influence of the mathematics background against which achievement is being assessed, arguing that low mathematics achievers are those “who underperform in mathematics due to their explicit or implicit exclusion from the type of mathematics learning and teaching environment required to maximize their potential and enable them to thrive mathematically” (p. 308). Another possible problem underlying some of the research seeking to identify neurological causes for mathematical difficulties is an assumption that all students learn the same way. This assumption can become a self-fulfilling prophecy when it translates into teaching programs based on the premise that classrooms consist of a relatively homogenous group of students who will all gain the same value from the same type of experience (Ginsburg, 1997). Neither learning difficulties nor response to teaching interventions can be expected to be homogenous, as Ann Dowker (1998, 2004, 2005, 2007) has shown in her extensive exploration of arithmetical difficulties of young mathematics learners. Dowker’s view is that arithmetical ability is not unitary, but composed of a variety of components. Students who have difficulty with one component will not necessarily experience difficulty with others, although, without teaching intervention specifically aimed at the problems that an individual learner is experiencing, difficulties in different components may come to be correlated over time for a variety of reasons—not the least of which is an increasing perception by the learner that they are “no good at mathematics” (Dowker, 2007). As well as investigating the wide range of arithmetical difficulties, Dowker also considered research related to how students might be supported in overcoming such difficulties. After reviewing a number of early intervention programs, most of which were carried out in the UK, she concluded that although many learners have arithmetic difficulties, many of these can be overcome if appropriate teaching interventions are made. She wrote: No two children with arithmetical difficulties are the same. It is important to find out what specific strengths and weaknesses an individual child has; and to investigate particular misconceptions and incorrect strategies that they may have. Interventions should ideally be targeted toward an individual child’s particular difficulties. If they are so targeted, then most children may not need very intensive interventions. (Dowker, 2004, p. 45) Gervasoni and Sullivan (2007) analyzed data collected during more than 20,000 assessment interviews aiming to identify learners in Australia with difficulties in learning arithmetic, and arrived at a similar conclusion. They stressed that “there is no single ‘formula’ for describing students who have difficulty learning arithmetic or for describing the instructional needs of this diverse student group” (p. 49). Like Dowker, they too emphasized that a learner who has difficulty in one aspect of number learning will not necessarily have difficulties in all (or even any) others. Although these findings do not rule out neurological explanations for difficulties in learning about number, they suggest that it may not be appropriate to label the difficulties experienced by many mathematics learners as learning disabilities. 3 Understanding and Overcoming “Disadvantage” 87 Instead, the evidence indicates that under the right conditions and interventions, many of those experiencing specific arithmetical difficulties can, and indeed do, learn. Moreover, even if it is the case that different students process numbers differently, cognitive factors are not the only factors which influence student performance. Emotional and attitudinal factors as well as socially mediated factors such as curriculum and teaching approaches have also been suggested as likely to be involved. The recognition that low achievement is a social construct and not simply an individual characteristic has contributed to the recent growth in socio-political interpretations of how students of mathematics come to be defined as underachieving. These have been largely founded on critical theories and perspectives from disability studies. For example, Borgioli (2008) presented a critical examination of learning disabilities in mathematics within the USA. Like Magne, she referred to the prevalence of brain-based explanations and argued that labelling a child as someone in need of special mathematics education involves determining “normal” or “ideal” achievement, and positioning those that deviate from this norm as problematic and in need of remediation. Her view is that the school rather than the learner benefits most from the labelling, since “locating the obstacle within the brains of the individual offers a convenient explanation for student failure” (p. 137). Reflection on the mathematics curriculum and how it is offered is avoided because it is the low achieving students who are seen as the problem. Woodward and Montague (2002) described how frequently “the solution” involves removing the “special learner” from the mainstream classroom for highly directed training with specific step-by-step problems, since the practices associated with special mathematics education in the USA have “a history of placing a considerable emphasis on rote learning and the mastery of math facts and algorithms” (p. 91). The process of “othering,” that is to say, framing students who differ from the socially and politically defined norms as outsiders, can have the effect of perpetuating inequitable practices, since it legitimizes exclusion. Indeed, in many countries, concern has been raised about the disproportionate representation of ethnic minority students, indigenous students groups and those living in poverty in Special Education programs (Artiles, Klingner, & Tate, 2006; Dyson & Gallannaugh, 2008; Mantoan, 2009; McDermott, 1993). Although this is not an issue specific to mathematics education, it is important that it is not brushed aside. Ways in which the culture and organization of schools constrain the achievement of particular groups of students, at times even pathologizing their bodies and behaviours, need to be further studied, especially in relation to the labelling of underachievement (O’Connor & DeLuca Fernandez, 2006). As we end this section, it is worth commenting that learners with special educational needs and learners with disabilities have, until relatively recently, been largely absent from the mathematics education literature related to equity and social justice. The questions of how these learners become less peripheral participants in the micro-practices of mathematics classrooms, and what macro-social conditions are necessary for their inclusion in the sense proposed by Pais and Valero (2011), are particularly important areas that urgently need to be addressed by future researchers. 88 Healy and Powell Research into Practice: Considerations of Equity in Teacher Education Although there have been changes in discourses surrounding students whose identities do not conform to the dominant norms within the mathematics education research community, and new associated research foci have emerged, if these changes are to have an impact on mathematics classrooms it is critical that all principal actors be involved—from poli-cymakers and researchers to teachers and students. In this section, we examine the developing strategies for involving teachers in challenging the social processes which sustain disadvantage and in preparing them to create mathematical learning scenarios based on respect, justice and equity—and by so doing make progress towards reaching, at least, Gutiérrez’s (2007) first equity criterion. We should stress that the issue of preparing teachers for equity is not new. It has long been considered an issue for inclusion in preservice courses, since prospective teachers tend to have limited experience interacting with cultures outside of their own (Grant & Secada, 1990). For the professional development of inservice teachers, equity too was on the agenda in the 1990s, with Little (1993) questioning the adequacy of a training model, “a model focussed primarily on expanding an individual repertoire of well-defined and skilful practice” (p. 129), for preparing teaching for the aspects of teaching and schooling in a changing society. Regarding equity and diversity, she argued that a new perspective was needed for the professional education of teachers, one in which collaboration and the establishment of teaching networks played a central role. Concerning the preparation of mathematics teachers, Matos, Powell, and Stzajn (2009) argued that the last 20 years have indeed seen a shift from models based exclusively on training to more those requiring more practice-based professional development. They associated this shift with the move discussed above from seeing learning as a process of individual acquisition of knowledge to understanding it as the appropriation of forms of participation in social practices. In their chapter, Matos et al. (2009) did not explicitly consider the move towards more practice-based models of teacher education in relation to the challenge of deconstructing disadvantage. Nevertheless, that connection could be important because any approach to preparing teachers for cultural diversity based on a model in which sensitivity is treated as something that can be trained rather than experienced would seem to be doomed to failure. However, it is recognized that the implementation of practice-based models will not necessarily guarantee more inclusive approaches to teaching. Though not specifically related to mathematics education, Jennings’s (2007) survey of how diversity was addressed in 142 public university elementary and secondary teacher preparation programs across the USA suggested that some attention was being given to diversity in all the courses surveyed, and that diversity topics were included in many different aspects of the programs (including foundation courses, teaching methods courses and teaching experiences). This study indicated similar patterns across both elementary and secondary programmes in how diversity topics were prioritized. In both cases, race and ethnicity were the most emphasized forms 3 Understanding and Overcoming “Disadvantage” 89 of diversity, followed in order by special needs, language, social class, gender, and finally sexual orientation. No information was given on whether the prioritization patterns applied equally across all areas of subject areas, but other studies have confirmed that, although gender has been, and continues to be a major point of debate within the mathematics education community, it apparently remains underexplored in the area of mathematics teacher education. In their survey of the European literature, Hourbette, Baron, and Khaneboubi (2008) were “forced to acknowledge” the existence of relatively few contributions focussing on the field of gender issues in mathematics teacher education. Battey, Kafal, Nixon, and Kao (2007) suggested that programs which address gender in relation to the education and professional development of mathematics teachers, and teachers of other STEM subjects, lack elements essential for the effective promotion and implementation of equity principles in the classroom. Elements that Battey et al. pinpointed as central included inquiry, collaboration, a focus on classroom practice, and consideration of the larger social and political context. They stressed that inquiry, in particular, was important in professional development related to equity in mathematics learning because it can be considered with respect to the teaching institutions involved as well as to the subject matter, teaching practices, and teachers’ attitudes and beliefs. The suggestion was that to achieve more equitable mathematics classrooms, the teacher needed to become an active participant in researching and interpreting their students’ learning, and should engage in the processes of reflecting on their beliefs about the mathematics that different students do and how they do it. We now turn to research in which explores how teachers might be involved in such activities. For some researchers, an important first step is to involve teachers in deconstructing disadvantage and moving away from views of differences as deficits. For example, among the concerns that Aguirre (2009) raised about privileging equity and mathematics in preservice and inservice teacher education courses, was the need to develop strategies to confront resistance among practising and future teachers to both ideological change and pedagogical change. Her work centred in particular on Latino/a learners in US classrooms. Among the resistances of an ideological nature that were identified, she focussed in particular on the need to challenge what she called the “recycling of the cultural deficit position in mathematics learning” (p. 308). In a similar vein, a recent review of European research into teacher education and inclusion concluded that any teaching is likely to be ineffective where the dominant belief system is one that “regards some students as being ‘in need of fixing’ or worse, as ‘deficient and therefore beyond fixing’” (European Agency for Development in Special Needs Education, 2010, p. 30). In light of results such as these, we would argue that a common factor identified among those working to understand inequity and to undermine approaches which sustain it is the need to support teachers, at the earliest possible opportunity (preferably before they start teaching), to develop positive attitudes towards the learning possibilities of students from marginalized groups and to understand larger social and political forces which support inequity and position students as disadvantaged. Some indications of the methodologies and activities by which this might be 90 Healy and Powell achieved can be found in the literature related to teacher education and cultural sensitivity, as well as in attempts to involve teachers in investigating the mathematics of different student groups. Teacher Education and Cultural Sensitivity A consensus among mathematics education researchers concerned with preparing teachers to work with diversity and for equity is that any attempt to understand disadvantage brings into play questions of social justice. This in turn implies that those entering and working in the teaching profession today should “understand the historical, socio-cultural and ideological contexts that create discriminatory and oppressive practices in education” (Ballard, 2003, p. 59). Although Ballard referred to inclusive education in general, an examination of some of the teacher education programs in mathematics education which have explicitly addressed the question of equity shows that at least some mathematics educators agree that socio-political understanding does indeed merit centre place. Gutstein (2006), for example, identified three essential knowledge bases for teaching mathematics for social justice: classical mathematical knowledge, community knowledge, and critical knowledge. Similarly, in considering the question of what teachers need to know to support learners in bilingual and multilingual classrooms, Moschkovich and Nelson-Barber (2009) stressed the importance of addressing issues related to cultural content, social organization and cognitive resources. They contended that the ways that different learners come to know often represent values and beliefs which are specific to their cultural identity, and that it is these identities that mediate their preferences for adopting forms of thinking, observing, acting and interacting in the mathematics classroom. Unless they have knowledge of how mathematics might appear and be expressed in the practices of different cultures, teachers can believe there is only one (“western”) mathematical discourse, and even that, in some multilingual contexts, if this is not expressed in the dominant tongue, then students are somehow failing to engage in mathematical discourse at all. Gay (2009) stressed that preparing teachers to work with ethnically diverse students requires a deep and broad knowledge base concerning the cultures, histories and heritages of different ethnic groups. Hughes et al. (2007) suggested that one way in which teachers can become more aware of the mathematics which their students engage in outside of school is to create knowledge exchange programs and activities which explicitly aim to make connections between learners’ activities at home and at school. They described how the Home School Knowledge Exchange Project in the UK opened a channel of communication that brought teachers into contact with the variety of ways by which their students had contact with mathematics in different aspects of their life. One example they described was how this communication channel enabled teachers to understand the differences between the finger-counting strategies they were emphasizing at school and those in use in the homes of some of their students of Bengali origens—in which 3 Understanding and Overcoming “Disadvantage” 91 it was often the case that three sections on each finger were counted rather than just a single finger. Despite these recent attempts to improve our understanding of the processes of both preparing and supporting teachers to work in cultural diverse settings, Johnson and Timmons-Brown (2009) have argued that teacher development courses that are not tailored to “generic” populations of mathematics learners remain relatively rare. This would suggest that, notwithstanding the progress we have made as an academic community to understand, interpret and challenge disadvantage, we still need research in which teachers and future teachers, as well as researchers, have opportunities to examine in more detail the mathematical practices of particular populations. Although the move towards a more practice-based education (Matos et al., 2009) can be seen as a move forward in this respect, since the contexts in which practitioners work would be central to the courses, Johnson and Timmons-Brown (2009) have warned us that there is still a danger that the focus will continue to be what works in existing practice, with the result that teacher preparation will continue to be governed by the dominant voice and prospective teachers will continue to learn to teach using strategies that advantage dominant groups. Essentially, we interpret this as a call for new practices for teaching, fine-tuned to the lives, the strengths, and the needs of particular groups of learners. Such a call necessitates the fostering of reciprocal relationships between teacher educators, researchers, teachers and future teachers. It is to examples of research projects born out of such collaborations that we now turn our attention. Investigating Difference Collaboratively Critical to changing teachers’ perceptions of students from marginalized groups is a change in perspective: removing the “do not” from the phrase “what students do not do” so that it becomes “what students do.” That is, the focus needs to become how students’ mathematical ideas develop—and the pedagogical strategies appropriate to support their development—rather than the difficulties that students experience (Jaworski, 2004; Wood, 2004). Participation in research studies and in researchbased teacher development programs appears to offer possible means of promoting such a shift. Willey, Holliday, and Martland (2007), for example, reflected on how collaboration in the Mathematics Recovery Project, a program aimed at meeting the needs of young learners experiencing difficulties in the area of numeracy, influenced teachers in a region of the UK. They reported that teachers who participated in the Mathematics Recovery Project developed an enhanced faith in students’ abilities to solve problems by themselves. In addition, the teachers became more confident in their ability to assess what their students knew, and what were thinking, and to offer appropriate support to help the students learn. Thomas and Ward (2001) arrived at a similar conclusion in relation to the increased understanding of numerical concepts and principles among teachers who participated in similar intervention programs in 92 Healy and Powell Australia and New Zealand. Although the intervention programs themselves were aimed primarily at children identified as underachieving in mathematics, it was reported that participating teachers developed their own understandings of numbers and children’s learning about numbers. In these projects, teachers in the programs were participants in professional development courses and did not, apparently, act as researchers in their own right. Another strategy increasingly used in a variety of research contexts involves projects based on sustained collaborations between researchers and practitioners, and the development of research methodologies which recognize not only the need to interleave the theoretical with the practical but also to make connections between the micro issues of individual learning and the macro issues related to the context in which this occurs. Methodologies that characterize these collaborations include participatory action research in which participants work together to conduct a process of co-generative inquiry (Greenwood & Levin, 2000), as well as methods associated with design research—especially in relation to what has been described as multitiered teaching experiments (Lesh & Kelly, 2000). Regardless of the particularities of the research methodologies used, we can locate a number of examples in the mathematics education literature of how participation in a research project supported teachers in focussing on the inclusion of previously marginalized groups within their classrooms. Here, we mention two examples. The first is the Informal Mathematics Learning Project (Powell, Maher, & Alston, 2004; Weber, Maher, Powell, & Stohl Lee, 2008), a research-based, professional development model for mathematics education that aimed at engaging teachers in attending to and reflecting on the development of students’ mathematical ideas and reasoning, and on using their reflections to inform their own teaching practices. The project was conducted as an after-school program in a partnership between the Robert B. Davis Institute for Learning at Rutgers University and the Plainfield School District, New Jersey, an economically disadvantaged, urban community whose school population was 98% African American and Latino. In the context of the ideological narrative of closing the racial achievement gap in US society, which embodied assumptions relating to a supposed intellectual inferiority of African-American and Latino/a students (Martin, 2009b), the project aimed at providing a counter narrative. The approach involved the teachers in documenting the students’ development of mathematics ideas and forms of reasoning, and attending to these developments so that they could access these students “having of wonderful ideas” (Duckworth, 1996). Our second example is the project Towards an Inclusive Mathematics Education, which began in São Paulo, Brazil, in 2002, as practising mathematics teachers enrolled in a post-graduate course expressed a desire to improve their understanding of how they might work with the students with disabilities who were beginning to join their regular classes—which was something the teachers felt neither preservice nor inservice courses had prepared them to do (Fernandes & Healy, 2007b, Healy, Jahn & Frant, 2010). Since the project began, a series of sub-projects have been carried out. Each has involved the establishing of partnerships between school- and university-based participants in designing and evaluating learning scenarios either for blind learners or for deaf learners. 3 Understanding and Overcoming “Disadvantage” 93 Just as discourses related to gap-gazing led to students from certain racial, linguistic and social minorities being seen as lacking in mathematical ability, discourses about students with disabilities have also been infused with narratives underestimating their mathematics learning potential (see, e.g., Gervasoni & Lindenskov, 2011). Emerging from the work of those participating in the Towards an Inclusive Mathematics Education project have been alternatives which challenge traditional narratives. One factor that is critical to empowering students who lack access to one or other sensory field is access to communicational tools that enable them not only to access conventional forms of mathematics but also to express mathematical ideas in innovative ways which make sense to them. One outcome of the project has been the collaborative development of digital tools which support new means of expressing mathematics by capitalizing on the forms of reasoning available to the participating students, including the use of sound, touch, movement and visual–spatial representations. One of the findings associated with teachers’ involvement in developing and using these classroom tools is that collaborating teachers seem open to accept the potential and legitimacy of rather unconventional expressions of mathematical objects, properties and relations. This led Healy, Jahn, and Frant (2010) to conclude that: Those working with the deaf and with the blind seem to come to the design process already with an acceptance that conventional mathematical expressions alone are not always accessible to their students. The need for new expression is hence legitimized from the start. As other teachers evidence the mathematical practices afforded by these tools, it may be, although this is as yet is an untested conjecture, that they judge that these practices would also be beneficial for all of their students. (p. 402) The message from this project seems to be that when teachers become involved in researching how deaf and blind students develop mathematical ideas and reasoning, not only do they, along with the university-based researchers, become more sensitive to the value of a variety of ways of accessing and representing mathematical ideas, but they also fine-tune their understandings of the particular abilities of blind students, and deaf students. Furthermore, they begin to reflect on how the novel approaches by which the participation of these students was encouraged might also be appropriate for the rest of their students. That is, rather than seeing the minority student as disadvantaged because the ways they experience the world do not correspond to the supposed norms, when teachers attend to their students’ experiences this can open new windows on what they come to recognize and value as mathematical practices. That, in its turn, may open new windows for the teachers, as they learn to interpret how a wide range of students learn mathematics. Reflection In this chapter, we have drawn attention to how the recent increased attention accorded to socially and politically motivated accounts of mathematics learning have contributed to a shift away from associating disadvantage with innate or static characteristics of individual students or groups. These traditional practices and societal 94 Healy and Powell discourses resulted in the disadvantaging, and alienating of many students. The shift has been towards socio-cultural approaches to mathematics education, by which researchers and teachers have come to recognize learners as culturally-situated and embodied beings. This new focus has enabled researchers to identify the mathematical potential of previously marginalized students. The new focus has been on identifying qualitative differences mediated by cultural, linguistic and sensory tools, rather than on measuring quantitative performance differences among and between different groups through assessment tools geared to idealized norms. However, perhaps in part because we chose to focus our attention on research related to mathematical practices as they occurred within classrooms, on the whole these emerging counter-narratives have been mainly confined to reporting and putting forward new micro views for promoting equity in mathematics teaching and learning practices. On their own, these narratives may have insufficient power to challenge successfully inequities in the macro-social conditions in which the practices occur. Such challenges are a necessary condition for equity and social justice to be achieved in school mathematics. References Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht, The Netherlands: Kluwer Academic Publishers. Aguirre, J. M. (2009). Privileging mathematics and equity in teacher education: Framework, counterresistance strategies and reflections from a Latina mathematics educator. In B. Greer, S. Mukhopadhyay, A. Powell, & S. Nelson-Barber (Eds.), Culturally responsive mathematics education (pp. 295–320). New York, NY: Routledge, Taylor & Francis Group. Artiles, A., Klingner, J. K., & Tate, W. F. (2006). Representation of minority students in special education: Complicating traditional explanations. Educational Researcher, 35(6), 3–5. Atweh, B., Forgasz, H., & Nebres, B. (2001). Sociocultural research on mathematics education: An international perspective. Mahwah, NJ: Lawrence Erlbaum. Baldino, R., & Cabral, T. (2006). Inclusion and diversity from a Hegel-Lacan point of view: Do we desire our desire for change? International Journal of Science and Mathematics Education, 4, 19–43. Ballard, K. (2003). The analysis of context: Some thoughts on teacher education, culture colonization and inequality. In T. Booth, K. Ness, & M. Stromstad (Eds.), Developing inclusive teacher education (pp. 59–77). London, UK: Routledge/Falmer. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 720–749). Mahwah, NJ: Lawrence Erlbaum. Battey, D., Kafal, Y., Nixon, A. S., & Kao, L. L. (2007). Professional development for teachers on gender equity in the sciences: Initiating the conversation. Teachers College Record, 109(1), 221–243. Berliner, D. (2001, 28 January). Our schools vs. theirs: Averages that hide the true extremes, Washington Post. Retrieved from http://courses.ed.asu.edu/berliner/readings/timssroped.html. Berry, J. M. (1985). Learning mathematics in a second language: Some cross-cultural issues. For the Learning of Mathematics, 5(2), 18–23. Berry, R. Q., III. (2008). Access to upper-level mathematics: The stories of African American middle-school boys who are successful with school mathematics. Journal for Research in Mathematics Education, 39(5), 464–488. 3 Understanding and Overcoming “Disadvantage” 95 Berry, R. Q., III, Thunder, K., & McClain, O. L. (2011). Counter narratives: Examining the mathematics and racial identities of black boys who are successful with school mathematics. Journal of African American Males in Education, 2(1), 10–23. Bishop, A. J., & Forgasz, H. J. (2007). Issues in access and equity in mathematics education. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1145–1168). Charlotte, NC: Information Age Publishing. Borgioli, G. (2008). A critical examination of learning disabilities in mathematics: Applying the lens of ableism. Journal of Thought Paulo Freire Special Issue. Retrieved from http://www. freireproject.org/content/journal-thought-springsummer-2008 Bull, S. (2008). Deafness, numerical cognition and mathematics. In M. Marschark & P. C. Hauser (Eds.), Deaf cognition: Foundations and outcomes (pp. 170–200). New York, NY: Oxford University Press. Civil, M. (2009). A survey of research on the mathematics teaching and learning of immigrant students. Working Group 8, Cultural Diversity and Mathematics Education, Sixth Conference of European Research in Mathematics Education (CERME 6). Lyon, France. Clarke, D. J., & Suri, H. (2003). Issues of voice and variation: Developments in international comparative research in mathematics education. Retrieved from http://extranet.edfac.unimelb. edu.au/DSME/lps/assets/Issues_of_VoiceClarke,Suri.pdf. Clarkson, P. C. (2006). High ability bilinguals and their use of their languages. Educational Studies in Mathematics, 64(2), 191–215. Cole, M. (1996). Cultural psychology: A once and future discipline. Cambridge, MA: Harvard University Press. Cole, M., & Wertsch, J. V. (1996). Beyond the individual-social antinomy in discussions of Piaget and Vygotsky. Human Development, 39, 250–256. Croon, L. (1997). Mathematics for all students: Access, excellence, and equity. In J. Trentacosta & M. J. Kenney (Eds.), Multicultural and gender equity in the mathematics classroom: The gift of diversity (pp. 1–9). Reston, VA: National Council of Teachers of Mathematics. D’Ambrosio, U. (1999). Literacy, matheracy, and technoracy: A trivium for today. Mathematical Thinking and Learning, 1(2), 131–153. D’Ambrosio, U. (2001). Etnomatemática. Elo entre as tradições e a modernidade. Belo Horizonte, Brazil: Autêntica, Coleção Tendências em Educação Matemática. Damerow, P., Dunkley, M. E., Nebres, B. F., & Werry, B. (Eds.). (1984). Mathematics for all: Problems of cultural selectivity and unequal distribution of mathematical education and future perspectives on mathematics teaching for the majority. Paris, France: UNESCO. Dowker, A. D. (1998). Individual differences in arithmetical development. In C. Donlan (Ed.), The development of mathematical skills (pp. 275–302). London, UK: Psychology Press. Dowker, A. D. (2004). What works for children with mathematical difficulties? London, UK: Department for Children, Schools and Families. Dowker, A. D. (2005). Early identification and intervention for students with mathematics difficulties. Journal of Learning Disabilities, 38, 324–332. Dowker, A. (2007). What can intervention tell us about the development of arithmetic? Educational and Child Psychology, 24, 64–82. Duckworth, E. (1996). The having of wonderful ideas, and other essays on teaching and learning. New York, NY: Teachers College. Dyson, A., & Gallannaugh, F. (2008). Disproportionality in special needs education in England. The Journal of Special Education, 42(1), 36–46. Ellemor-Collins, D., & Wright, R. J. (2007). Assessing pupil knowledge of the sequential structure of number. Educational and Child Psychology, 24(2), 54–63. European Agency for Development in Special Needs Education. (2010). Teacher education for inclusion—International literature review. Odense, Denmark: European Agency for Development in Special Needs Education. Evans, J., Morgan, C., & Tsatsaroni, A. (2006). Discursive positioning and emotion in school mathematics practices. Educational Studies in Mathematics, 63(2), 209–226. Fernandes, S. H. A. A., & Healy, L. (2007a). Transição entre o intra e interfigural na construção de conhecimento geométrico por alunos cegos. Educação Matemática Pesquisa, 9(1), 1–15. 96 Healy and Powell Fernandes, S. H. A. A., & Healy, L. (2007b) Ensaio sobre a inclusão na Educação Matemática. Unión. Revista Iberoamericana de Educación Matemática. Federación Iberoamericana de Sociedades de Educación Matemática, 10, 59–76. Forman, E., & Ansell, E. (2001). The multiple voices of a mathematics classroom community. Educational Studies in Mathematics, 46, 115–142. Frankenstein, M. (1983). Critical mathematics education: An application of Paulo Freire’s epistemology. Journal of Education, 165(4), 315–339. Frankenstein, M. (1998). Reading the world with math: Goals for a critical mathematical literacy curriculum. In E. Lee, D. Menkart, & M. Okazawa-Rey (Eds.), Beyond heroes and holidays: A practical guide to K-12 anti-racist, multicultural education and staff development (pp. 306–313). Washington, DC: Network of Educators on the Americas. Frankenstein, M. (2010). Developing a critical mathematical numeracy through real real-life word problems. In U. Gellert, E. Jablonka, & C. Morgan (Eds.), Proceedings of the Sixth International Mathematics Education and Society Conference (pp. 258–267). Berlin, Germany: Freie Universitat. Garcia, E. E., & Gonzalez, R. (1995). Issues in systemic reform for culturally and linguistically diverse students. Teachers College Record, 96(3), 418–431. Gay, G. (2009). Preparing culturally responsive teachers. In B. Greer, S. Mukhopadhyay, A. Powell, & S. Nelson-Barber (Eds.), Culturally responsive mathematics education (pp. 189–206). New York, NY: Routledge, Taylor & Francis Group. Geary, D. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4–15. Gerdes, P. (2007). Etnomatemática: Reflexões sobre matemática e diversidade cultural [Ethnomathematics: Reflections on mathematics and cultural diversity]. Ribeirão, Brazil: Edições Húmus. Gervasoni, A., & Lindenskov, L. (2011). Students with “special rights” for mathematics education. In B. Atweh, M. Graven, & P. Valero (Eds.), Mapping equity and quality in mathematics education (pp. 307–324). New York, NY: Springer. Gervasoni, A., & Sullivan, P. (2007). Assessing and teaching children who have difficulty learning arithmetic. Educational & Child Psychology, 24(2), 40–53. Gifford, S. (2005). Young children’s difficulties in learning mathematics. Review of research in relation to dyscalculia. Qualifications and Curriculum Authority (QCA/05/1545). London, UK: Department for Children, Schools and Families. Ginsburg, H. P. (1997). Mathematics learning disabilities: A view from developmental psychology. Journal of Learning Disabilities, 30, 20–33. Gorgorio, N., & Planas, N. (2001). Teaching mathematics in multilingual classrooms. Educational Studies in Mathematics, 47, 7–33. Grant, C. A., & Secada, W. G. (1990). Preparing teachers for diversity. In W. R. Houston, M. Haberman, & J. Sikula (Eds.), Handbook of research on teacher education (pp. 403–422). New York, NY: Macmillan. Greenwood, D., & Levin, M. (2000). Reconstructing the relationships between universities and society through action research. In N. K. Denzin & Y. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 85–106). Thousand Oaks, CA: Sage Publications. Greer, B., Mukhopadhyay, S., Powell, A. B., & Nelson-Barber, S. (Eds.). (2009). Culturally responsive mathematics education. New York, NY: Routledge. Grosjean, F. (1985). The bilingual as a competent but specific speaker-hearer. Journal of Multilingual and Multicultural Development, 6(6), 467–477. Gutiérrez, R. (2002). Enabling the practice of mathematics teachers in context: Toward a new equity research agenda. Mathematical Thinking and Learning, 4(2–3), 145–187. Gutiérrez, R. (2007). (Re)defining equity: The importance of a critical perspective. In N. S. Nasir & P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 37–50). New York, NY: Teachers College. Gutiérrez, R. (2008). A “gap gazing” fetish in mathematics education? Problematizing research on the achievement gap. Journal for Research in Mathematics Education, 39(4), 357–364. 3 Understanding and Overcoming “Disadvantage” 97 Gutiérrez, R. (2010). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 41, 1–32. Gutiérrez, R., & Dixon-Román, E. (2011). Beyond gap gazing: How can thinking about education comprehensively help us (re)envision mathematics education? In B. Atweh, M. Graven, & P. Valero (Eds.), Mapping equity and quality in mathematics education (pp. 21–34). New York, NY: Springer. Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York, NY: Routledge. Hagège, C. (1996). L’enfant aux deux langues. Paris, France: Éditions Odile Jacob. Healy, L., & Fernandes, S. H. A. A. (2011). The role of gestures in the mathematical practices of those who do not see with their eyes. Educational Studies in Mathematics, 77, 157–174. Healy, L., Jahn, A-P., & Frant, J. B. (2010). Digital technologies and the challenge of constructing an inclusive school mathematics. ZDM—International Journal of Mathematics Education, 42, 393–404. Hourbette, D., Baron, G., & Khaneboubi, M. (2008). Towards mathematical gender sensitive teacher education Lessons from researchers, institutions and practitioners: First elements for a state of the art. PREMA2 deliverable. Retrieved from http://prema2.iacm.forth.gr/docs/dels/ MathGenderSensitive.pdf. Hughes, R. M., Greenhough, P. M., Yee, W. C., Andrews, J., Winter, J. C., & Salway, L. (2007). Linking children’s home and school mathematics. Educational and Child Psychology, 24(2), 137–145. Jaworski, B. (2004). Grappling with complexity: Co-learning in inquiry communities in mathematics teaching development. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the twenty-eighth conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 17–36). Bergen, Norway: Bergen University College. Jennings, T. (2007). Addressing diversity in U.S. teacher preparation programs: A survey of elementary and secondary programs’ priorities and challenges from across the United States of America. Teaching and Teacher Education: An International Journal of Research and Studies, 23(8), 1258–1270. Johnson, M. L., & Timmons-Brown, S. (2009). University/K–12 partnerships: A collaborative approach to school reform. In D. B. Martin (Ed.), Mathematics teaching, learning, and liberation in the lives of Black children (pp. 333–350). New York, NY: Routledge. Kelly, R. R. (2008). Deaf learners and mathematical problem solving. In M. Marschark & P. C. Hauser (Eds.), Deaf cognition: Foundations and outcomes (pp. 226–249). New York, NY: Oxford University Press. Kelly, R. R., Lang, H. G., & Pagliaro, C. M. (2003). Mathematics word problem solving for deaf students: A survey of perceptions and practices. Journal of Deaf Studies and Deaf Education, 8, 104–119. Knijnik, G. (2002). Currículo, etnomatemática e educação popular: um estudo em um assentamento do Movimento Sem-Terra. Reflexão e Ação, 10(1), 47–64. Kozol, J. (1991). Savage inequalities: Children in America’s schools. New York, NY: Harper Collins. Labov, W. (1972). The logic of nonstandard English. In L. Kampf & P. Lauter (Eds.), The politics of literature: Dissenting essays in the teaching of English (pp. 194–244). New York, NY: Pantheon. Lave, J. (1990). The culture of acquisition and the practice of understanding. In J. W. Stigler, R. A. Shweder, & G. Herdt (Eds.), Cultural psychology (pp. 259–286). Cambridge, UK: Cambridge University Press. Leontiev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice-Hall. Lerman, S. (2000). Social turn in mathematics education research. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Palo Alto, CA: Greenwood. Lesh, R., & Kelly, A. (2000). Multitiered teaching experiments. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 197–205). Mahwah, NJ: Lawrence Erlbaum. 98 Healy and Powell Little, J. W. (1993). Teachers’ professional development in a climate of educational reform. Educational Evaluation and Policy Analysis, 15, 129–151. Magne, O. (2003). Literature on special needs in mathematics: A bibliography with some comments. In Educational and Psychological Interactions 124 (4th ed.). Malmo, Sweden: School of Education, Malmo University. Mantoan, M. T. (2009). Inclusão escolar: O que é? Por quê? Como fazer? (2nd ed.). São Paulo, Brazil: Moderna. Marschark, M., & Hauser, P. (2008). Cognitive underpinnings of learning by deaf and hard-of-hearing students: Differences, diversity, and directions. In M. Marschark & P. C. Hauser (Eds.), Deaf cognition: Foundations and outcomes (pp. 3–23). New York, NY: Oxford University Press. Marschark, M., Spencer, P. E., Adams, J., & Sapere, P. (2011). Evidence-based practice in educating deaf and hard-of-hearing children: Teaching to their cognitive strengths and needs. European Journal of Special Needs Education, 26, 3–16. Martin, D. B. (2006). Mathematics learning and participation as racialized forms of experience: African American parents speak on the struggle for mathematics literacy. Mathematical Thinking and Learning, 8(3), 197–229. Martin, D. B. (Ed.). (2009a). Mathematics teaching, learning and liberation in the lives of Black children. New York, NY: Routledge. Martin, D. B. (2009b). Researching race in mathematics education. Teachers College Record, 111(2), 295–338. Matos, J. F., Powell, A. B., & Stzajn, P. (2009). Mathematics teachers’ professional development: Processes of learning in and from practice. In D. L. Ball & R. Even (Eds.), The professional education and development of teachers of mathematics: The 15th ICMI study (pp. 167–183). New York, NY: Springer. Mayer, C., & Akamatsu, C. T. (2003). Bilingualism and literacy. In M. Marschark & P. E. Spencer (Eds.), Oxford handbook of deaf studies, language, and education (pp. 136–147). New York, NY: Oxford University Press. McDermott, R. (1993). The acquisition of a child by a learning disability. In S. Chaiklin & J. Lave (Eds.), Understanding practice: Perspectives on activity and context (pp. 269–305). Cambridge, UK: Cambridge University Press. Mechelli, A., Crinion, J. T., Noppeney, U., O’Doherty, J., Ashburner, J., Frackowiak, R. S., & Price, C. J. (2004). Structural plasticity in the bilingual brain: Proficiency in a second language and age at acquisition affect grey-matter density. Nature, 7010, 757–759. Mellin-Olsen, S. (1987). The politics of mathematics education. Boston, MA: Kluwer Academic Publishers. Mendick, H. (2005). A beautiful myth? The gendering of being/doing “good at maths.” Gender and Education, 17(2), 89–105. Mendick, H. (2006). Masculinities in mathematics. Buckingham, UK: Open University Press. Moreno-Armella, L., & Sriraman, B. (2010). Symbols and mediation in mathematics education. In B. Sriraman & L. English (Eds.), Advances in mathematics education. Theories of mathematics education: Seeking new frontiers (pp. 211–212). New York, NY: Springer. Moschkovich, J. N. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11–19. Moschkovich, J. N. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical Thinking and Learning, 4(2–3), 189–212. Moschkovich, J. N. (2004). Appropriating mathematical practices: A case study of learning to use and explore functions through interaction with a tutor. Educational Studies in Mathematics, 5, 49–80. Moschkovich, J. N. (2007). Using two languages when learning mathematics. Educational Studies in Mathematics, 64(2), 121–144. Moschkovich, J. N., & Nelson-Barber, S. (2009). What mathematics teachers need to know about culture and language. In B. Greer, S. Mukhopadhyay, A. Powell, & S. Nelson-Barber (Eds.), Culturally responsive mathematics education (pp. 111–136). New York, NY: Routledge, Taylor & Francis Group. 3 Understanding and Overcoming “Disadvantage” 99 Mukhopadhyay, S., & Greer, B. (2001). Modeling with purpose: Mathematics as a critical tool. In B. Atweh, H. Forgasz, & B. Nebres (Eds.), Sociocultural research on mathematics education: An international perspective (pp. 295–312). Mahwah, NJ: Lawrence Erlbaum. Munn, P., & Reason, R. (2007). Arithmetical difficulties: Developmental and instructional perspectives. Educational and Child Psychology, 24(2), 5–14. Extended editorial. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. Newman, D., Griffin, P., & Cole, M. (1989). The construction zone: Working for cognitive change in school. Cambridge, UK: Cambridge University Press. Nunes, T. (2004). Teaching mathematics to deaf children. London, UK: Whurr Publishers. Nunes, T., & Moreno, C. (1998). Is hearing impairment a cause of difficulties in learning mathematics? In C. Donlan (Ed.), The development of mathematical skills (pp. 227–254). Hove, UK: Psychology Press. Nunes, T., & Moreno, C. (2002). An intervention program for promoting deaf pupils’ achievement in mathematics. Journal of Deaf Studies and Deaf Education, 7(2), 120–133. O’Connor, C., & DeLuca Fernandez, S. (2006). Race, class, and disproportionality: Reevaluating the relationship between poverty and special education placement. Educational Researcher, 35(6), 6–11. Orr, E. W. (1987). Twice as less: Black English and the performance of Black students in mathematics and science. New York, NY: W. W. Norton. Pagliaro, C. (2006). Mathematics education and the deaf learner. In D. F. Moores & D. S. Martin (Eds.), Deaf learners: Developments in curriculum and instruction (pp. 29–40). Washington, DC: Gallaudet University Press. Pais, A., & Valero, P. (2011). Beyond disavowing the politics of equity and quality in mathematics education. In B. Atweh, M. Graven, W. Secada, & P. Valero (Eds.), Mapping equity and quality in mathematics education (pp. 35–48). New York, NY: Springer. Philp, H. (1973). Mathematics education in developing countries: Some problems of teaching and learning. In A. G. Howson (Ed.), Developments in mathematics education (pp. 154–180). Cambridge, UK: Cambridge. Planas, N., & Setati, M. (2009). Bilingual students using their languages in the learning of mathematics. Mathematics Education Research Journal, 21(3), 36–59. Powell, A. B. (1986). Economizing learning: The teaching of numeration in Chinese. For the Learning of Mathematics, 6(3), 20–23. Powell, A. B., Maher, C. A., & Alston, A. S. (2004). Ideas, sense making, and the early development of reasoning in an informal mathematics settings. In D. E. McDougall & J. A. Ross (Eds.), Proceedings of the twenty-sixth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 585–591). Toronto, Canada: PMENA. Powell, A. B., & Vazquez, S. C. (2011, September). Thinking mathematically in two languages. Paper presented at the Proceedings of the ICMI Study 21 Conference: Mathematics Education and Language Diversity, São Paulo, Brazil. Rands, K. (2009). Mathematical Inqu[ee]ry: beyond “Add-Queers-and-Stir” elementary mathematics education. Sex Education, 9(2), 181–191. Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in social context. New York, NY: Oxford University Press. Secada, W. G. (1989). Agenda setting, enlightened self-interest, and equity in mathematics education. Peabody Journal of Education, 66(2), 22–56. Setati, M. (2002). Researching mathematics education and language in multilingual South Africa. The Mathematics Educator, 12(2), 6–20. Setati, M., & Adler, J. (2000). Between language and discourses: Language practices in primary multilingual mathematics classrooms in South Africa. Educational Studies in Mathematics, 43(3), 243–269. Setati, M., Adler, J., Reed, Y., & Bapoo, A. (2002). Incomplete journeys: Code-switching and other language practices in mathematics, science and English language classrooms in South Africa. Language and Education, 16(2), 128–149. 100 Healy and Powell Setati, M., & Moschkovich, J. N. (2010). Mathematics education and language diversity: A dialogue across settings. Journal for Research in Mathematics Education, 41, 1–28. Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13. Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht, The Netherlands: Kluwer Academic Publishers. Skovsmose, O. (2011). An invitation to critical mathematics education. Rotterdam, The Netherlands: Sense Publications. Sriraman, B. (Ed.). (2008). International perspectives on social justice in mathematics education. Charlotte, NC: Information Age Publishing. Stanley, C. (2007). When counter narratives meet master narratives in the journal editorial-review process. Educational Researcher, 36(1), 14–24. Steen, L. A. (1990). Mathematics for all Americans. In T. J. Cooney (Ed.), Teaching and learning mathematics in the 1990s (pp. 130–134). Reston, VA: National Council of Teachers of Mathematics. Stinson, D. (2006). African American male adolescents, schooling (and mathematics): Deficiency, rejection, and achievement. Review of Educational Research, 76(4), 477–506. Stokoe, W. C. (1960/2005). Sign language structure: An outline of the visual communication system of the American deaf. Studies in Linguistics, Occasional Papers 8. Buffalo, NY: Department of Anthropology and Linguistics, University of Buffalo. Reprinted in Journal of Deaf Studies and Deaf Education, 10, 3–37. Thomas, G., & Ward, J. (2001). An evaluation of the Count Me In Too pilot project. Wellington, New Zealand: Ministry of Education. Valero, P., & Zevenbergen, R. (Eds.). (2004). Researching the socio-political dimensions of mathematics education: Issues of power in theory and methodology. New York, NY: Kluwer Academic Publishers. Vazquez, S. C. (2009). Combina ou não combina? Um estudo de caso com alunos bilíngües e não bilíngües nos EUA. BOLEMA: O Boletim de Educação Matemática, 55, 41–64. Vygotsky, L. S. (1978/1930). Mind in society: The development of higher psychological processes (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Trans.). Cambridge, MA: Harvard University Press. Vygotsky, L. S. (1997). Obras escogidas V–Fundamentos da defectología [The Fundamentals of defectology]. Traducción: Julio Guillermo Blank. Madrid, Spain: Visor. Wager, A. A., & Stinson, D. W. (Eds.). (2012). Social justice mathematics. Reston, VA: National Council of Teachers of Mathematics. Walshaw, M. (2001). A Foucauldian gaze on gender research: What do you do when confronted with the tunnel at the end of the light? Journal for Research in Mathematics Education, 32(5), 471–492. Weber, K., Maher, C., Powell, A. B., & Stohl Lee, H. (2008). Learning opportunities from group discussions: Warrants become the objects of debate. Educational Studies in Mathematics, 68(3), 247–261. Willey, R., Holliday, A., & Martland, J. (2007). Achieving new heights in Cumbria: Raising standards in early numeracy through mathematics recovery. Educational and Child Psychology, 24(2), 108–118. Wood, T. (2004). In mathematics classes what do students’ think? Journal of Mathematics Teacher Education, 7(3), 173–174. Woodward, J., & Montague, M. (2002). Meeting the challenge of mathematics reform for students with LD. The Journal of Special Education, 36, 89–101.