The Urban Review, Vol. 30, No. 3, 1998

Issues of Culture in Mathematics Teaching and Learning Carol E. Malloy and William W. Malloy Public education of America's students has for over a century provided educational opportunities to students using cultural traditions and theories of learning based on the dominant or majority population. Students who were members of or who were acculturated into the dominant culture were generally successful in mathematics. Others were left behind. This article demonstrates through theory and application that educators can teach mathematics to include more of the excluded students, especially African-American students. Educators must consider the culture of our students as we adopt an accommodating cultural pedagogy—one that gives the students the power to be a part of the mathematics culture as they use the familiar knowledge (which becomes power) of their culture.

The United States is moving from a homogeneous to a more heterogeneous society—from a society dominated by a single Western European cultural foundation to a society in which many cultural groups clamor for increased recognition (Dalin and Rust, 1996). This transition to a multicultural focus has serious implications for institutions in general and educational institutions specifically. For it is within the traditional school setting, with its Western European cultural orientation, that children are sorted into prescribed roles based upon their cultural orientation (Bernstein, 1990; Bourdieu, 1984). Previously schools were successful in facilitating students' access to resources through the process of acculturation. Unfortunately this process is becoming ineffective because the increase in students with diverse cultural backgrounds challenges the notion that one culture dominates the learning process. In fact, the multicultural emphasis, now a "given" in the school milieu, should receive high priority in all school reform movements (Banks, 1993). This article uses the area of mathematics education to describe how this multicultural emphasis has played out and to suggest recommendations for a culturally based pedagogy. Carol E. Malloy, Ph.D., and William W. Malloy, Ed.D., are assistant professors in the School of Education, University of North Carolina at Chapel Hill. Address correspondence to Dr. Carol E. Malloy, University of North Carolina at Chapel Hill, School of Education, CB 3500 Peabody Hall, Chapel Hill, NC 27599-3500. 245

0042-0972/98/0900-0245$15.00/0 © 1998 Human Sciences Press. Inc.

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CULTURE AND LEARNING Culture, as seen by educational anthropologists, is revealed through the ideational aspects of life. Culture is the shared meaning, but not necessarily consensus—the taken-for-granted values and beliefs that are seen in what people do, what they know, and the tools they use (Bellah et al., 1985). The culture that students encounter as they learn is multifaceted. Children's first learning occurs in the home culture; then it extends to the community; and then, as children are exposed to the media, their cultures are extended vicariously into numerous different cultures. When children begin school, they experience the cultures of the school, the classroom, and school subjects. The term used to describe children's' ability to adjust to new cultures is adaptation. In our schools it is generally assumed that children can and will adapt to the culture of classrooms and disciplines. The questions for the educational community are: Should children have to adapt to the instructional classroom culture when they enter school, or should the schools adapt to the cultures that children bring to school? At this juncture, a brief description of the adaptation process at the system, school, and mathematics class levels will assist in providing a frame of reference regarding the necessity for asking the question. Most cultural groups in the United States have been assimilated into this country through a process of acculturation. Acculturation is defined as an advanced culture subsuming lesser more archaic cultures (Schmookler, 1995). This acculturation process mainly entails learning a Western European-oriented code of conduct that is driven by capitalism. Individuals who adjust to this code are considered acculturated—a term that takes on the connotation of becoming cognizant of the code of conduct in a capitalistic society. Acculturation does not necessarily mean accepting that code as one's own; however, it does imply acceptance of the code of conduct that provides access to the rewards of capitalism. The educational system has been the most recognized institution for determining the degree to which individuals have become acculturated and thus eligible to reap the benefits of the system. For over a century, public education in America has provided educational opportunities to students using theories of learning based on the culture of the dominant or majority population. Some students understand or acculturate to the dominant culture of learning and achieve. Other students are left behind. What happens to those who are left behind? In the past, traditional schools were structured in a manner that did not promote diversity. When students who generally were not of the dominate culture experienced difficulty functioning within the parameters of the school, they were placed in lower academic classes, where they were labeled as deficient and devalued for their academic capabilities. Whereas many schools contended that diversity was a valuable asset, frequently students from these diverse popu-

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lations (i.e., minority groups, poor, linguistically different, and neglected and abused) were educated in lower academic tracks or special education (Zola, 1993). Through the use of scholarly research, efforts have been made to explain why diverse populations of students did not fare well in public schools and ultimately society (Cuban, 1989). Explanations range from the cognitive abilities to the social conditions. Whatever the explanation, the traditional educational options provided to meet the challenges of diversity have not met with notable success. The most prevalent solution in mathematics classrooms has been tracking. Research has shown that the stratification of tracking reduces many students' opportunities for learning; that is, students in lower tracks actually learned less than students in the higher tracked classes (Ogbu, 1979; Slavin, 1990; Kulik, 1992). The results of these findings are even more startling when the data are disaggregated to reveal that minority students have not met parity in participation or achievement with majority students (National Assessment of Educational Progress, 1992; Quality Education for Minorities, 1994). With these less than successful results, specifically in mathematics achievement, it is quite clear that past practices had a deleterious effect on the academic achievement of diverse populations of students. Traditionally mathematics, as an academic discipline and school subject, has used acculturation—or the lack thereof—as a reason to further separate students by demonstrating that membership within the exclusive and elusive mathematics culture is dependent upon demonstrated knowledge and skills. Academic mathematics courses always have been gatekeeper courses. Many mathematics teachers, at all levels, have excluded students from the mathematics culture when they thought students did not exhibit a sufficient understanding of mathematics. These teachers understand the mathematics culture(s) and its nuances, but they are slow to share the code of conduct with all students. Sharing the culture is the enculturation process, that is, providing neophytes with the necessary codes for membership in a culture. Using a culturally based process, teachers invite selected students who demonstrate superior understanding of mathematics to become neophytes of the culture. They share with the neophyte students—the gifted and talented—the code of conduct for mathematical superior achievement via mentoring, conceptual development, enrichment activities, explorations, and advanced mathematical problem solving. Those students who do not qualify for neophyte status are relegated to lower tracked mathematics classes, where they experience mathematics learning on algorithmic and procedural levels. With the advent of the reform in mathematics teaching and learning, the culture of learning in the mathematics classroom has begun to change. Many teachers are trying to actively enculturate, rather than acculturate, every student into the mathematics community (Jaworski, 1994). Educational researchers are providing empirical data on how students learn mathematics, and teachers are

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implementing suggested researched-based approaches in their instruction. Teachers are using inquiry-based approaches, real-world problems, varied forms of grouping, alternative assessment, and multicultural and gender-bias-free materials. But educators have not fully examined the role of culture on cognition and thus the use of a culturally based pedagogy in mathematics instruction— they have not contextualized mathematics instruction to their students' learning preferences. Harris (1994) states, The context is the one which results when the culture of the students—all the students—interacts with the teacher's culture. ... It is neither assimilation nor acculturation but accommodation. This does not mean that allegiance to or identification with anyone's individual culture is denied or denigrated. It does mean, however, that a common ground is created wherein interaction can occur that is meaningful to those involved, (p. 78)

Contextualization occurs when mathematics educators consider cultural influences on learning and thus restructure their pedagogy—accommodate for their students. Questions may arise about the nature of cultural influences and how to change pedagogy to acknowledge these influences. We contend that mathematics instruction needs to address the cultural heritage of all students. This article uses learning preferences of African-American students to discuss the role of culture on learning and offer pedagogical suggestions that accommodate their learning. We do not make the assumption that all African-American students fall within prescribed categories or preferences because students are individuals and participate in cultures that overlap ethnic and racial lines. Cultural preference theory is a beginning point to develop pedagogy for culturally diverse populations. EXISTING CULTURAL CONFLICTS IN TEACHING AND LEARNING The social, cultural, and historical context in which students live defines and shapes students and their experiences. Culture, related to mathematics learning, impacts students' perceptions of themselves as members of the mathematics community. Banks (1993) contends that "the assumptions, perspectives, and insights that students derive from their experiences in their homes and community cultures are used as screens to view and interpret the knowledge and experiences that they encounter in the school and in other institutions within the larger society" (p.7). Personal and cultural knowledge can be problematic in schools because they conflict with scientific ways of validating knowledge, often are oppositional to the culture of the school, and challenge the main tenets and assumptions of mainstream academic knowledge (Banks, 1993). The practice of schools disregarding the personal and cultural knowledge of students and concentrating instead on teaching them school knowledge is encouraged by

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three conditions in the schools: the sorting function, impersonal conditions, and language. These conditions support the standard operation procedures in schools (standardized testing, competition, tracking), which are designed to exclude rather than include. Schools, as institutions, reproduce the culture of the broader society in structures, rules, and hegemony (Delpit, 1988). Regardless of the subject matter, the classroom culture exemplifies power of one person and of the institution itself. The success of students in a class depends upon the ability of the students to understand and participate in the culture of those who are in power. Delpit (1988) describes the culture of power in our schools as coming from the power of the teacher, textbooks, curriculum developers, school districts, and society. Children of the middle and upper class come to school with tools to participate in this culture, whereas children from other families, minority and poor, operate within viable cultures that do not carry the same power codes or rules. The learning of African-American students and the instruction in schools are oppositional both cognitively and affectively (Hale-Benson, 1986; Milliard, 1976; Shade, 1989; Willis, 1992). The African-American community values and encourages the acquisition of unique verbal expressiveness, but schools place highest value on the written demonstration of verbal knowledge. AfricanAmerican students' view of the world is that of a unified environment (Shade, 1989; Stiff and Harvey, 1988); thus they use a mixture of holistic and analytical reasoning, but schools concentrate on analytical reasoning. African-American students are taught that an interdependence of people and environment is respected and encouraged, but schools teach individualism (Hale-Benson, 1986; Hilliard, 1976; Willis, 1992). African-American students rely on personalistic stimulation in learning; schools focus on inanimate or object stimulation (Shade, 1989). Students' cultural knowledge and school knowledge often conflict with variables related to the ways that the individual should relate and interact within the group, normative communication, and styles and interactions. These preferences have underlying assumptions of strong influences from culture; are preferences for student interaction with the environment; are influences on cognition, attitude, behavior and personality; and may be different from the majority population but are not deficient. THE PROCESS OF ACCOMMODATION IN TEACHING MATHEMATICS With the given cultural conflicts in teaching and learning mathematics, there are three areas that must be considered in order for pedagogy and classroom interaction to be modified to accommodate African-American student learning. Teachers must have knowledge of and establish the classroom culture that promotes mathematics learning; they must understand the importance of multi-

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cultural education and integrate multicultural educational materials and pedagogy into the teaching of mathematics, and they must be aware of AfricanAmerican learning preferences and adjust their pedagogy to capitalize on these preferences. Classroom Culture

Mathematics educators have a firm set of beliefs about classroom culture that optimizes student learning. The classroom atmosphere should provide students with opportunities to learn, appropriate levels of challenge, meaningful objectives, opportunities to take risks, and a variety of strategies (Grouws and Lembke, 1995). The inquiry-based approach to learning places knowledge in the hands of learners, the use of communication in mathematics increases students' understanding of mathematical concepts, and variation in classroom organization permits students to experience mathematics socially (National Council of Teachers of Mathematics, 1989, 1991). These suggested practices promote a positive learning culture in the mathematics classroom, but they do not address the interaction between the culture of the student and the culture of the classroom. Multicultural Education

Nelson, Joseph, and Williams (1993) offer suggestions for using culture in the classroom by identify three distinctive concepts that have evolved through multicultural educational practice: education through many cultures, into many cultures, and for many cultures. Education through many cultures is based on the belief that "knowledge of and empathy with several cultures is essential for the mental health of children living in a multicultural neighborhood" (Nelson et al., 1993, p. 2). In this setting, the use of multicultural materials is important and necessary to promote better self-development and intercultural understanding. Education into many cultures is based on the premise that knowledge about cultures is necessary for educators as well as students because knowledge of one's own culture is dependent upon the knowledge of other cultures. This cultural knowledge is a form of affective commitment that enables educators to value the culture that students bring with them into the learning process (Nelson, et al., 1993). Education for many cultures—that is for a multicultural society—is based on the premise that the present education system is unjust and that bringing cultures into the classroom is not enough. The purpose of education for many cultures is the emancipation of students. Educational practice for many cultures requires changes to pedagogy that follow from the realization that education has been a sphere where inequities have existed and must be rectified (Nelson, et al., 1993).

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Preferences Traditionally mathematics instruction has addressed the needs of the analytic, field-independent, individual learner. These students were instructed in ways that encourage them to focus on detail and use sequential/structured thinking, recall abstract ideas and irrelevant detail, engage in inanimate material, respond to intrinsic motivation, focus on the task, learn from formal lecture, achieve individually, and emphasize facts and principles. But this instruction has not benefited all students. Mathematics pedagogy can be modified to give students who have other learning preferences—the holistic, field-dependent, interdependent thinkers— the opportunity to learn mathematics by seeking solution paths that match their learning preferences. The learning preferences and approaches of AfricanAmerican learners (Dance, 1997; Hale-Benson, 1986; Hilliard, 1976; Shade, 1989; Willis, 1992) isolate several areas where teachers can accommodate to students' preferences for learning. Instruction should be given in ways that encourage students to focus on the whole, use improvisational and intuitive thinking, recall relevant verbal ideas, engage in human and social content material, respond to extrinsic motivation, focus on interests, learn from informal class discussion, achieve interdependently, and narrate human concepts. Inclusion of these types of mathematics instruction will offer new learning opportunities for students who have had learning conflicts with the rigid, formalized structure of traditional classrooms and may provide continued success and expanded knowledge for students who always have achieved. Culturally based pedagogy can give all students, regardless of their learning preferences, the opportunity to learn mathematics. Below we synthesize knowledge from established pedagogical practices that promote learning, the cultural concepts of Nelson et al. (1993), and AfricanAmerican learning preferences to suggest further modifications in pedagogy and curriculum that will accommodate student learning.

SUGGESTED PRACTICES Pedagogy Pedagogy is predicated on how the teacher interprets, understands, recognizes, and integrates the students' culture within the learning process; how the teacher allows students to construct knowledge based on their experiences; and effective classroom practice (von Glaserfeld, 1995). Moreover, the teacher must respect and have knowledge about students' lives, culture, and experiences in order to use students' life experiences in instruction. "The teacher must be aware of sensitivity to the needs of her students or she is in danger of assuming

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a teaching style that satisfies her own needs instead of the motivational needs of her students" (Grouws and Lembke 1995, p. 41). A student's culture is a guide to ensure desired academic competence which will produce meaningful learning. Teachers should initiate instruction in the areas where students exhibit strengths and then should stretch students into thinking in ways that are culturally familiar and unfamiliar. For instance, if students are more comfortable with divergent thinking, teachers should use divergence to introduce mathematics concepts. Then they should teach students to use convergence in reaching conclusions. To accomplish this, teachers need sufficient in-depth understanding of their students' backgrounds and the relationship between their culture and their learning. How do you ask a question that will allow students to approach the problem using their preferred way of thinking and then have you teach them to extend their knowledge by using other methods? Consider the different approaches students with different learning preferences might use to solve this problem: Augustus is trying to make chocolate milk. So far he has made a 10% chocolate milk solution and a 25% chocolate milk solution. Unfortunately, the 10% solution is too weak and the 25% solution is too chocolatey. He has a whole lot of the 10% solution, but he only has 30 gallons of the 25% solution. How many gallons of the 10% solution must be added to the 25% solution to make a mixture that is 15% chocolate? Augustus is sure it will be absolutely perfect. (Herr and Johnson, 1994, p. 184)

There are several approaches that students could use to solve this problem. Some students might use the symbolic model to find the answer. Other students might draw pictures, make charts, or guess and check by using a physical model to mix equal and unequal amounts of the solutions. The holistic learner may not use the same method as the analytic learner, just as the convergent thinker may not use the same method as the divergent thinker. Whatever method students use, the teacher has the ability to stretch the students from their cultural base into other ways of thinking about a solution. The most important concept to be learned by the students is what happens when two solutions of different strengths and quantities are mixed. This understanding could be demonstrated through representations that are verbal, symbolic, tabular, or graphic: (1) verbal representations could be oral or written explanations of the problem solution; (2) a symbolic solution plan would use an algebraic expression or equation; (3) a tabular display lists data values; and (4) a graphic representation gives a visual picture of the data. Teachers who help students use their experiences and cultural preferences for learning to think mathematically without incrimination are using a culturally based pedagogy. Ladson-Billings (1994) defines culturally relevant pedagogy as teaching that uses student culture in order to maintain it and to transcend the negative effects

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of the dominant culture. Negative effects could be being openly or privately denigrated for not solving a problem the way that books, other students, or teachers might solve the problem. The negative effects also could be not seeing one's culture in books or curriculum or experiencing a staffing pattern where minorities are janitors and cafeteria workers and teachers and principals are white. "The primary aim of culturally relevant teaching is to assist in the development of a relevant black personality that allows African American students to choose academic excellence yet still identify with their African and African American culture" (Ladson-Billings 1994, p. 17). Teachers who are culturally relevant have many characteristics that Ladson-Billings compares to assimilationist teachers—whose role it is to ensure that students fit into society. Ideologies and actions of assimilationist teachers reinforce their active need to have students assimilate without regard to the students' particular cultural characteristics. Often they reproduce a society that is debilitating instead of liberating. Culturally relevant teachers, on the other hand, seek excellence within the students' culture thereby enabling students to become emancipated lifetime learners. Curriculum

State departments of education, local school systems, the National Council of Teachers of Mathematics, and others have spent nearly a decade refining and revising the mathematics curriculum. The suggested curriculum encourages students to think about, handle, and conjecture as they learn mathematics. The curriculum suggests that students be taught through many cultures by being exposed to role models from their communities, see students like themselves in textbooks and on classroom displays, and talk about traditions and symbols of their ethnic cultures. Teachers are encouraged to learn about cultural issues in the broader society and discuss these issues with their students. The curriculum recommends a multicultural approach both through and into many cultures, and it hints toward the Nelson et al. (1993) "for many cultures" by supporting mathematics in context. Even with this revised curriculum, many students are still taught that mathematics only comprises symbolic manipulation. Some students learn with this implementation of the curriculum, but others—holistic learners—learn through contextual problems that are relevant to their lives. Research has shown that African-American students are successful in mathematics when they can see the utility of the mathematics (Dance, 1997; Malloy, 1997; QEM, 1992). Dance (1997) suggests that real-world context has been avoided in the past because of the messiness of the problems. She points out that technology can enable students to experience real problems with real data and to answer real questions based on the data. The proposed mixed-race classification for the United States

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census is an example of such a problem. Students can investigate the effects of reclassification of mixed-race people in the 2000 census. They can determine how the changes to U.S. congressional district appropriations could change the number of representatives from each party. Students can then speculate on the future existence of funding for current programs developed to increase underrepresented minority participation in mathematics and science. There are numerous other real problems that can be used in elementary, middle, and high school mathematics classes where students are not asked to substitute numbers into a formula to find an answer. Educators help students to be emancipated by using the present to shape the future instead of the future to shape the present. Culturally emancipating curriculum enables students to experience and critique human behavior in a world that does not always value their possible contribution. Emancipation helps students to become aware of social inequities and to understand the motivation for policy decisions and solutions. Educators must be willing to share with students the motivations and the hidden agenda (curriculum) in their world. These agendas are prevalent and support the social structures within all communities. For example, students who investigate the placement of dumps or hazardous waste plants in poor neighborhoods are learning about the social realities of the poor and powerless. They are learning to think critically about their environment. "Moreover, African-American students need forms of education that permit them to choose collective liberation and survival as a goal and to see this as part of a larger struggle for social change. To do this they need the skills and knowledge to help reinvent America as a more just, democratic and culturally diverse society" (King, 1994). The curriculum that promotes all students participating in mathematics learning is problem-based. The problems are real and can be solved using multiple approaches and methodology. Extending learning and emancipation to all populations can be accomplished through instituting a curriculum that is multicultural and critical. This form of multicultural education helps students to understand how knowledge is constructed. It gives students opportunities to investigate and determine how cultural assumptions, frames of references, perspectives, and the biases within a discipline influence the ways the knowledge is constructed (Banks, 1993). Teachers should be sensitive to the motivations, conversations, social preferences, and thought patterns of their students. They should emphasize language development and take care to contextualize instruction in the meaningful experiences of their students (Tharp, 1989). Moreover, teachers must frequently acknowledge, respect, and use students' cultural heritage in classroom instruction. Using the learning strengths of students' cultures in our pedagogy and our curriculum, educators can serve all students. Schools can provide an academic environment that relies on students' cultural backgrounds as the foundation for

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teaching and learning and enlists the students to become responsible for their mathematics learning. Because any student, regardless of culture, religion, ethnicity, or gender, might learn better using multiple instructional methodologies and real curriculum, these suggestions should be useful in most mathematics classrooms.

CONCLUSION Public education of America's students has for over a century provided educational opportunities to students using cultural traditions and theories of learning based on the dominant or majority population. The mathematics instruction was linear, independent, competitive, orderly, direct, unemotional, and analytical. However, many students have cultural learning preferences that are different from the schools' because these students may be circular, interdependent, cooperative, stylistic, associational, emotionally intense, socially oriented, and holistic. These cultural preferences, in many cases, dominate the learning of students from many minority groups. The process of total acculturation in the schools has not been ubiquitous because the ideology has never been accepted by minority groups. At the same time, these communities have exhibited high self-expectations, valued education, and sought to provide educational opportunities to their children, but in reaction to the lack of opportunities and low expectations, the community, particularly the African-American community, has developed distrust and vulnerability to achievement (Anderson, 1988; Ogbu, 1988; Steele, 1992). Educators can alleviate this distrust and thus the vulnerability. We can use the visible, as well as invisible, aspects of culture that influence learning. Visible aspects include customs related to music, dance, diet, dress, and so on. More important to learning are the invisible aspects, including "language and dialect, non-verbal communications, perspectives and personal world views, behavioral styles and nuances, methods of reasoning and validating knowledge, and cultural identification to the teaching-learning-assessing enterprise" (Harris, 1994, p. 87). Pedagogy must be enriched by using the strengths of all cultures to serve all students. Certainly educators must make students familiar and comfortable with the culture of mathematics, but in doing so, they must expand this enculturating process by accommodating the individual in order to make the enculturation dominant rather than selective in our mathematics classrooms. Educators must consider the culture of students as they adopt an accommodating cultural pedagogy—one that gives the students the power to be a part of the mathematics culture as they use the familiar knowledge (which becomes power) of their culture. Implicit is the realization that learning preferences of any group of students are positives and can be used in instruction.

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