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Diversity and equity are issues that are referenced everyday in education, especially mathematics education. Improving Access to Mathematics: Diversity and Equity in the Classroom (Nasir & Cobb, 2007), provides a collection of research that reveals inequities and means to rectify this problem so that true equity exists in the study of mathematics. If you asked people about equity in terms of education, one would hear statements such as; 'everyone' is treated equally, the system is being fair to everyone, and there is no favoritism shown toward any particular student or group of students. It is very common to hear teachers say that they treat all of their students the same or that they do not see 'color', social status, gender or any differences when they look at the students in their classes. All of these comments appear to meet the definition expressed in the Merriam Webster dictionary which states that equity is "freedom from bias or favoritism." Our attempt to adhere to this definition ends up creating and perpetuating inequities in our classrooms. According to Lubienski, Boaler and other authors cited in this book (Nasir and Cobb, 2007), we are not exhibiting fair treatment toward all students when we make such attempts.
Therefore, in an attempt to discuss these issues and influences upon them, the members of our group examined this book, which is one in a series of books edited by James A. Banks on multicultural education. Each book in the series presents a plethora of perspectives and research findings on issues of diversity and equity. Each chapter in the series is written by the original researcher. The researchers examine various aspects of diversity and equity and ways to achieve them. Therefore, in the spirit of the series, our group will present their reactions and viewpoints on the issues raised in the book in the form of short analyses.
Social Valorization of Mathematical Practices: The Implications for Learners in Multicultural Schools
Abreu and Cline (Nasir & Cobb, 2007) suggest that cultural differences not only impact the way individual members of that culture learn mathematics, but which members of their culture learn as well. The relational nature of social valorization is seen in individuals and groups within societies and influences the way individuals and groups define who they are. Mathematical learning, along with other learning, occurs within social groups, which influence an individual's social development and the ways that an individual will learn.
Abreu (Nasir & Cobb, 2007) studied farmers and schoolchildren in a sugarcane farming community in Brazil where the community of farmers used a specific type of mathematics involving weighing and measuring and the schoolchildren were taught distinctly different mathematics which takes some rereading. The continued use of both types of mathematics for distinct purposes led to Abreu's concept of social valorization. Abreu's concept of social valorization helped to explain social status with social groups and how individuals value differences in the status of mathematical practices. She found that farmers were selective as to whom they passed their mathematical knowledge. Furthermore, a negative or lower status was bestowed to members of society not practicing school mathematics. Similar comparisons can be made with many societies today.
Knijnik (cited in Nasir & Cobb, 2007)) studied the Landless' Language Game of Cubagem of wood. He found that in their struggle with the Empire sovereignty, the groups were not educated as a whole and practicing three different ways, were resistant to change, unwilling to share knowledge. Learning about different mathematics and how it relates to their different practices within their groups helped broaden how they view the mathematical world and the relations within it. Of even more importance to the groups, education was helping the groups to successfully build an educational program that would make them stronger against the Empire.
The impact of social valorization on schoolchildren's mathematical learning was studied in primary schools in England on boys and girls, high and low achievers from each ethnic group, who had parents who had learned about learning in a different culture and a different school system. The findings from this study confirmed the fact that children did not understand categorization of practices and comparisons and their justifications for mathematical comparisons emphasized the status of a person's job, or the social valorization of the practices. One student commented that he picked the office administrator as being the best at mathematics because he was wearing flashy clothes and he looked rich. Another student said that if a taxi driver was good at mathematics he wouldn't be working in that field. Devalorization of practices linked to their communities where they live at a status lower than their school may have contributed to their justifications. This can be seen in schools across America today, where we have students from extremely low economic statuses receiving educations that many of their parents did not receive. Without a reference point to draw upon in the home environment, students are at a significant disadvantage compared to their peers.
Social valorization impacts the way students participate in mathematical practices. Students are often taught mathematical tasks differently at home than the way they are taught in school. For many children this is difficult. They tend to view their home mathematics as of lesser value and do not openly discuss it in school. In these homes, parents will often try to use strategies to hide their mathematics from their child and do try to do what the student is practicing in an effort to support the child's learning at school. However, some parents do not support the school's mathematics and find it less efficient and less conducive to their mathematical thinking. In these families, children value their home mathematics as much as their school mathematics but they still maintain that the school's way of doing mathematics is better. While students can be successful using both approaches, constraints may limit their opportunities for growth and issues of equity can arise.
Hiding home practices from school can lead to devalorization of home practices and submission to the school practices. On the other hand, valorization of home practices can lead to resistance of the school's practices. It is important to take steps to reduce group differences by understanding the importance of home practices in supporting understanding of mathematical knowledge while at the same time being aware of the importance of school mathematical learning. We agree with Abreu and Cline and again, as is evidenced in our schools today. Not only do some students have the conflict of different mathematics in the home, but many students are faced with the challenges of home environments where the parents did not have the schooling and do not know how to do the mathematics.
How social status is valued and/or determined within social groups is a worldwide issue. Furthermore, how individuals value differences in the status of mathematical practices can be seen across races, generations, and socio-economic statuses. Without acknowledging the differences in school mathematics and home learning in the classroom, the devalorization of home practices will only continue.
Equity: Concepts, Roadblocks and Achievements
The National Council of Teachers of Mathematics (NCTM) states that "equity does not mean that every student should receive identical instruction; instead, it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students" (quoted in Nasir & Cobb, 2007, p. 12). Many people would think that this notion of equity is contrary to the definition found in the dictionary and what is widely practiced in schools. The authors stress the point that any treatment that a teacher may select will have a set of values and a cultural identity attached to it. Therefore, the students who identify with the treatment that has their cultural association have an advantage over students who have a cultural identity contrary to the chosen treatment. This is what is known as 'cultural capital' which has advantages and power in the classroom (Nasir & Cobb, 2007). D'Ambrosio (quoted in Ernest, Greer, & Sriraman, 2009) refers to this as "the trap of the same." (p. 3)
Lubienski, Boaler and Davis (Nasir & Cobb, 2007) believe that treating everyone the same hinders some students while at the same time gives others an advantage. Lubienski showed that instructing everyone in the class in the same manner benefited males and members of the social elite while not allowing females to work toward their potential. Davis and his co-authors examined the effects of how students are positioned in relation to their learning of mathematics. They showed that the current manner in which mathematics instructions is presented and courses sequenced disallowed some group access to higher mathematics and hence certain careers. Boaler discussed the effects of gender stereotypes, that are allowed to continue in the current dynamics of mathematics education and culturally accepted norms, which hinder females in mathematics.
Therefore, in order to understand equity and to impart equitable treatment, one must consider issues of culture and identity which issues encompass the notion of race, gender, and socioeconomic status. We must break the hold that the power elite has on mathematics and make it available for all students. As educators, we must be aware of factors, which may appear as fair and equal that may hinder the mathematical success of all students. These factors, along with groups of students not achieving their mathematical potential, lead to a cultural stigmatism on an entire group for generations. When a group of students does not perform mathematically as well as members of the dominant gender, race, class or culture, the root cause of this performance is always relegated to an innate defect of the group. On the contrary when the poorer performance occurs among the dominate group, researchers scurry to find what external factors are affecting their performance. This phenomenon adds to the idea of cultural-deficit theory which ignores the affects of power.
According to the writings presented in Improving Access to Mathematics (Nasir & Cobb, 2007), there are several reasons why equity does not exist. One reason is that equity is complex. It involves the influences of several factors, the cooperation of many participating institutions and a sharing of power. Equity involves far more than mastery of content and it involves factors beyond simple cognition which is an issue that many researchers have avoided in their studies. Lubienski (Nasir & Cobb, 2007) stated that researchers "ignore relevant social and cultural issues" (p. 11) that affect the attainment of equity in the classroom. We know that mathematics is masculine in its construct (Ernest, Greer, & Sriraman, 2009), but we hold females to the same standard as males without any support. Majority of these issues are ignored in mathematical research for fear of presenting one group as deficient or the research misconstrued and the findings are interpreted as being innate to a particular group or gender. Hence the lack of equity is due to the lack of attention given to the influence of culture and instead the focus is on context and mastery (Jacob, 1998, cited in Nasir & Cobb, 2007).
Another reason why we do not have equity is that it disturbs the current social order of things. Robert Moses (Nasir & Cobb, 2007), founder of The Algebra Project, states that the reason we do not have equity in education is because of the current system of "serf or sharecropper education" (p. 70) is designed to promote a caste system. In an educational system where equity exist, each student is given an opportunity to have access to knowledge that will allow them to be treated as equal in the experience of learning and they are allowed to be possessors of knowledge. We know that mathematics is the "essential instrument for exercising technological power" (Ernest, Greer, & Sriraman, 2009, p. 22). In the current system, information from one group is valued over another. This cannot be equitable, because there is not a sharing of knowledge or power. In this situation, the thoughts, ideas and culture of one group are placed above that of the other group.
Then, how can we create equity? The authors state that we must move away from a dichotomous debate that positions one teaching method against another. It appears that a combination of methods and approaches are needed to assist all students. Also, we need to move away from the concept that the teacher is the sole dispenser and possessor of knowledge. Instead, a classroom should be about a community of learners who are seeking knowledge from a multitude of sources such as others who may be from a different cultural background, gender or social status. The Algebra Project, created by Robert Moses, is one of many attempts to create an equable classroom for all students. Research in mathematics education must change as well. Boaler and Lubienski (Nasir & Cobb, 2007) identified that a small percentage of the research in mathematics education discuss the effects of culture and gender on mathematical performance. Researchers need to examine mathematics education through a sociohistorical lens due to issues of power and society.
Culture, Identity and Equity in the Mathematics Classroom
In their chapter entitled Culture, Identity, and Equity in the Mathematics Classroom, Cobb and Hodge (Nasir & Cobb, 2007) provide a provisional definition of equity. The authors construe equity as encompassing "students' development of a sense of efficacy (empowerment) in mathematics together with the desire and capability to learn more about mathematics when the opportunity arises" (Nasir & Cobb, 2007, p. 160). They contend that as one considers issues of equity that one of the primary concerns to attend to students' access to opportunities to develop mathematical reasoning that is recognized and valued or-as they put it-reasoning that has clout. Their contention highlights the predominant issues of power as they are represented in the epistemological privilege associated with certain ways of knowing, in the affordances extended or withheld from students, and in the sociomathematical norms-ones specific to mathematic classrooms-that are constructed by students and teachers.
The goals of instructional design, analyses of teaching and learning, and the valuation of student and community knowledge are taken up in Civil's chapter (Nasir & Cobb, 2007), Building on Community Knowledge: An Avenue to Equity in Mathematics Education. Civil draws on collaborative research efforts where she investigates incorporating funds of knowledge into classroom instruction. Funds of knowledge are those historically developed and accumulated ideas and strategies (e.g. skills, abilities, practices, ideas) or bodies of knowledge that are essential to a household's functioning and well-being.
Civil expresses a degree of apprehension regarding some funds of knowledge projects. She is concerned about how to negotiate the tension between activities that are mathematically-rich while at the same time are community-based. She gives considerable thought to what kind of tasks best preserves the purity of funds of knowledge. This apprehension seems to be in response to consequences analogous to the appropriation effect. In the appropriation effect, locally gathered knowledge in less-developed countries is appropriated for use in more-developed nations. Similarly, Civil and other researchers who implement must be mindful of [potential abuses of] the appropriation effect.
Civil proposes an apprenticeship model in which students are engaged in activities and tasks that are challenging, yet meaningful to them-not merely to teachers or even solely, the parents/community. This model of mathematical apprenticeship emphasizes inquiry-based learning and participatory research. The participatory element in such mathematics instruction may serve to broaden characterizations of mathematics competence. This becomes the case as competence becomes more of an interaction between the opportunities that a student has to participate and how the student responds to the opportunities.
Civil articulates a legitimate concern for mathematics educators. She recalls the Millroy paradox which inquires: "How can anyone who is schooled in conventional Western mathematics 'see' any form of mathematics other than that which resembles the conventional mathematics with which she is familiar?" This concern has implications for what gets constructed as mathematical competence and, more fundamentally, what gets constructed as doing mathematics. These constructions are authored by both teachers and students. Therefore, how students are positioned as learners and doers of mathematics is dependent on the particular lenses through which we view these constructs. Although the work of Civil and others referenced focused on particular cultural communities, FKT research has implications for many working-class neighborhoods with a large concentration of students are of ethnic and/or linguistic minorities.
In his chapter, Martin (Nasir & Cobb, 2007) decries the master narrative, a dominant discourse which has portrayed African Americans in a light that is not particularly flattering. This discourse, Martin argues, is one that depicts African Americans as "passive in the face of differential treatment and denied opportunity in mathematics and as lacking both agency and voice" (Nasir & Cobb, 2007, p. 149). Martin's work serves the essential role of providing counternarratives in the tradition of critical race theory. Martin's analysis also provides details about African American identities of participation and nonparticipation. Martin's work is powerful in that he merges conceptions of culture-particularly African American culture-with aspects of identity, both culturally and as doers of mathematics.
Cobb and Hodge (Nasir & Cobb, 2007) suggest an interpretive framework for exploring identity. They delineate three identity types: normative identity, core identity, and personal identity. They also articulate two views of culture. In one view, culture is a characteristic of identified, circumscribable communities. That is, it is a way of life of a particular bounded community. The more recent view of culture they present is that it describes a set of locally instantiated practices. Thus culture, like the practices it subsumes, is both dynamic and improvisational. It is this second view of culture to which Cobb and Hodge subscribe. This perspective emphasizes people's participation in multiple communities. Also, it considers the boundaries that exist between groups or communities to be permeable. This reconceptualization is more suitable than the more static view that preceded it in accounting for the interplay between the individual and the sociocultural processes that shape mathematics education and interactions with it.
So One Question Leads to Another: Using Mathematics to Develop a Pedagogy of Questioning
We felt like we were well educated by this article on teaching for social justice and Eric Gutstein (Nasir & Cobb, 2007) has indeed done justice to the title of this book through this article. The book aims to investigate the reasons behind the mathematics achievement inequities existing between white students, students of color, and students from low socio economic status and to find interventions for eliminating these inequities. However considering the fact that the book and the article are targeted at mathematics teachers and educators, we conjecture that the article is incomplete in some sense. We will examine the strengths and incompleteness of the article in the following paragraph
Gutstein begins the article supporting the views of Freire (1970/1998, cited in Nasir & Cobb, 2007) in condoning the act of passing mathematics knowledge by teachers to students as if mathematical knowledge were bank deposits to be safeguarded and utilized when necessary. The passive role taken by students in banking the knowledge allows students to be passive citizens, accepting the world of injustice without questioning. On the other hand, a problem posing environment created in a classroom allows students to be critical thinkers capable of questioning authority (Freire cited in Nasir & Cobb, 2007). Gutstein (Nasir & Cobb, 2007) through his empirical study advocates 'pedagogy of questioning' as a pedagogy for teaching social justice that empowers students to be active participants in questioning and changing the injustices in society. According to Gutstein, academic success, sociopolitical consciousness, sense of social agency and positive cultural and social identities constitute the components of the 'pedagogy of questioning' along with classroom norms and culture that is developed through student teacher relationship.
This article is a report on an empirical study conducted by Gutstein over a period of 5 months, which he calls as practitioner-research investigation. The participants were 30 students from a 7th grade class belonging to a Latino/a community. The author cautions the reader that the 'pedagogy of questioning' is not a set of rules to be followed and describes the same as a classroom culture to be developed collaboratively by the teacher and the students where students are able to meaningfully understand their sociopolitical reality and are encouraged to ask and answer questions to one another and the teacher.
The author describes his use of the mortgage project and B2 bomber project in his 7th grade classroom to critically analyze racism and accountability of tax money in this country. A detailed description of the project along with the artifacts (tables and figures) given to students to complete the project will illuminate the reader and benefit teachers. The author has provided rich qualitative data in terms of journal entry by students and responses by the researcher/ author. While the reader can highly appreciate the deep understanding of context of racism and the quality of sincere straightforward writing by 7th graders, one wonders how much the author had to compromise in terms of missing mathematics standards. The author has not discussed time factor while discussing the complexities involved in terms of implementation of the project.
Gutstein (Nasir & Cobb, 2007) opines that reform mathematics gives authority to students to read (investigate and analyze) the world with mathematics, but diminishes the power of mathematics by saying much more than mathematical power is needed to empower students as critical thinkers. Looking at this, Gutstein is not giving importance to mathematics, but the ability of mathematics to act as a tool to question authority all the time.
Theories/reforms do not come into practice on their own and especially when they do not serve the interests of the oppressor. Pedagogy of questioning is not going to be adapted by teachers in the classroom unless it is introduced through teacher education programs. The author has not mentioned any implications to the teacher education program nor to the classroom teacher. The author also goes to extent of cautioning the reader not to consider his study as a prescribed method to practice pedagogy of questioning. In this aspect, the article is incomplete.
Identity, Goals and Learning: The Case of Basketball Mathematics
I started reading this article with enthusiasm to find out some real connections between mathematics and the basketball game only to find out the article is not as outstanding as the title is. Nasir (Nasir & Cobb, 2007) claims that learning goals in mathematics are not necessarily constructed in the mathematics classroom and many a times are influenced by practices in the community. The article is based on an empirical study of the construct of goals and identities as related to the process of learning based on a study on the cultural practices of African American basketball players.
The author defines identities as dynamic constructs that students build based on the influences from the society (Nasir & Cobb, 2007). The theoretical framework used in the study has been explained in detail. The author uses the 'three modes of belonging namely engagement, imagination and alignment' to discuss how students frame their identity and the case of basketball mathematics to propose his theory on identity, goals and learning. High school basketball players engage themselves at a higher degree than the middle school players and make strong connections and rapport with their teammates. High school basketball players establish their imagination mode of their identity by relating themselves to the veterans in the field and compare their scores to match to the player they want to be in the next level. This identity triggers them to elevate their goals thus influencing their learning. On the other hand, the knowledge gained through learning creates a new identity and influence them to set up new goals. Thus, there exists a two way association between identity and goals, goals and learning, and learning and identity.
Nasir (Nasir & Cobb, 2007) concludes the article with a set of implications for schooling and mathematics learning and for African American children. However, there is no mention of description of participants of the study except for the fact that they are African American basketball players. The way the basketball players are tested on their mathematical skill, the kind of questions and the number of questions has not been described in detail. Also, the reader is not able to make a connection between data (there is no organized data) and the claims made. Having discussed the theory behind identities, goals and learning, the author could have taken responsibility in describing the study in detail as well. Reporting the data and making a connection between research and theory would have illuminated the reader and would have impacted the teacher in the classroom.
From the group's reactions to the works presented in this book, one can see that diversity and equity are very complex issues. They cannot be achieved by giving every student the same curriculum and instruction. They require communal efforts whereby all groups must share power and knowledge. Researchers of mathematics education must realize that there are other forces acting upon the learning and access to mathematics. Therefore, mathematics education must be studied in context of these forces. We must look beyond the attainment of mathematical content if we are going to give every student an opportunity for mathematical success.