Zevenbergen, R. (2000). “Cracking the code” of mathematics: School success as a function of linguistic, social and cultural background. In J. Boalezr (Ed.), Multiple perspectives on mathematics teaching and learning. . New York: JAI/Ablex.
Methods:
 Three common communicative strategies found in mathematics classrooms form the basis of this chapter. The first is the type of questions commonly found in texts and tests. These represent the register of mathematics that I argue is very structured and that students must come to learn in order to be able to participate in a pro ductive and effective manner. The second communicative strategy is that of classroom talk, which has its own internal rules that are not made explicit to students but form the basis for communication in the classroom. The third and final example is that of what comes to constitute legitimate knowledge in the classroom, and this is bound to the contexts used to embed mathematical tasks.
Findings:
 Drawing on Bourdieu’s notion of cultural capital, or more particu larly, linguistic capital, it is argued that some students will have greater or lesser access to the modes of communication in a classroom, and hence have more or less access to the mathematics inherent in such communications (201)

Bourdieu’s notions of habitus and field are particularly useful in theorizing how social differences are manifested and legitimated through school mathematics. For Bourdieu, habitus is the embodiment of culture, and it provides the lens through which the world is interpreted. Through his detailed work with patterns of con sumption and work, Bourdieu (1979) has shown how different social classes have distinctive preferences toward food, sport, leisure, housing, and so forth. For stu dents who have been socialized within particular familial contexts, distinctive pat terns are observable of which language use is one of the key differences across diverse groups. As students come to hear and use particular forms of language, this language becomes embodied to constitute a linguistic habitus. When students enter mathematics classrooms, they have accepted the language of their home environ ment, the consistency of which will vary with respect to formal school language. Where there is greater continuity between the home and school, there is greater chance of success in school mathematics (Bourdieu, Passeron, & de saint Martin, 1994). (202)
 BriceHeath (Heath, 1982, 1983) has shown that students from economically disadvantaged backgrounds are more likely to be exposed to declarative statements when they are expected to undertake tasks, whereas students from economically advantaged homes are more likely to receive pseudoquestions from parents or guardians requesting their children to undertake tasks. Similarly, in their studies of mothers and daughters, Walkerdine and Lucey (1989) reported similar differences in interactions between social classes. From these familial interactions, children are more likely to embody different patterns of interaction, which will be differentially used and recognized within the formal school context (202)

This is not to suggest a deterministic reading of social background (as is com monly made of Bourdieu’s works) but rather to recognize that differences exist be tween home and school languages and these have an impact on a students’ performance in the classroom. Harker (1984) argues persuasively that the primary habitus can be reconstituted. In his view, for students whose habitus is different from that of the formal school context, there is potential for it to be brought closer to that which is legitimated through school practices, thus suggesting a transformative component of pedagogy rather than a deterministic reading (203)
 Bourdieu argues that: Linguistic competence is not a simple technical ability, but a statutory ability. . . . what goes in verbal communication, even the content of the message itself, remains unin telligible as long as one does not take into account the totality of the structure of the power positions that is present, yet invisible, in the exchange. (Bourdieu & Wacquant, 1992, p. 146)

When students are able to deconstruct texts for the underlying meaning, they are better positioned within the field. In most instances, this requires a familiarity with the language of representation—in this instance, the mathematics register when considering written texts, and interactional competence when considering oral lan guage (204)
 Just as with other languages, mathematics has a particular form and the newcomer must be able to decipher that language. In much the same way as a tourist can make minimal sense of languages in foreign countries, the learning of mathematics is similar for students. Where a student gains compe tency in the intricacies of the mathematics register, he or she will be better able to decipher the subtle but precise meaning of mathematical expressions. (204)

Learning mathematics is, in part, learning the unique correspondence between the signifiers (words) and signifieds (concepts) within a mathematics con text with some words having different meanings depending on the strand1 of the curriculum. (205)
 The lack of specificity in meaning can be problematic. When dealing with common fractions, it is a common perception that numerator is the top number and denominator is the bottom number. However, this simplistic translation cannot be transferred to subtraction equations where the denominator (the bottom number) can be subtracted from the numerator (the top number). It also refers to the specific use of prepositions in mathematics. McGregor (1991, p. 7) has noted that the prepositions used in mathematics are a cause for difficulty in understanding tasks. She notes the use of prepositions in the following manner: The temperature fell to 10 degrees . . . by 10 degrees . . . from 10 degrees; and the effect of omitting the preposi tion: the temperature fell 10 degrees. (206)

Trigger words, often embedded within word problems, need to be interpreted correctly if students are to perform the task contained within the problem. For ex ample, in many word problems, trigger words such as more, less, got, or took away provide cues for the students as to what operation needs to be performed (Schoenfeld, 1988). In part, this is due to the ways mathematics is most frequently taught. This is a common strategy for students who do not have access to the rich ness and specificity of mathematics language. (206)
 Research focused on changing the semantic structure so that it is more in line with the language used by students rather than the formal expression of the problems, in creases students’ capacity to solve the tasks (Carpenter, 1985, cited in De Corte & Verschaffel, 1991). However, while changing the semantic structure of the question may make the question more accessible and help the students find an answer, it does not help them “crack the code” of the mathematics register (206)

He sums up the notion of lexical density as being “the number of lexical items as a ratio of the number of clauses” (1988, p. 67). Halliday suggests that lexi cal density contributes to the complexity of written problems in mathematics and may be a further barrier to learning. Mathematical tasks are often characterized by their conciseness and preciseness, where there are few redundant words and where all words have highly specific meaning. As noted previously, that specificity of meaning may not be the same as in the nonmathematics contexts. To translate a mathematical task into a more accessible form would require, in most cases, a more convoluted and lengthy description. As a consequence, the lexical density results in a high level of complexity in the translation of the problem (207)
 Bernstein (1990) has developed the notion of “relay” to describe the elusive ways in which cultural norms and knowledge are transmitted. Successful interac tion patterns are rarely taught explicitly to our students; they must come to learn them covertly. In the following sections, I draw on the work of ethnomethodology that seeks to identify the micro interactions of classrooms that become a compo nent of the culture of classrooms. (212)

One of the most documented patterns of interaction in the classroom is that of “triadic dialogue” (Lemke, 1990). It has been found across all curriculum areas and all sectors of formal schooling. The phrase “triadic dialogue” as coined by Lemke is the one I use in this chapter, al though others have described the same interaction patterns in different terms. Triadic dialogue consists of three key parts: the teacher initiates a question to which the students usually know the answer; a student responds; and the teacher then eval uates the student’s response (Mehan, 1982; Sinclair & Coulthard, 1975). (212)
 Triadic interactions serve the purpose of controlling student behavior while also prescribing the content of lessons. Lemke (1990) argues that rules for interacting are not explicitly taught and students have to learn them through participation in the interactions (213)

Triadic dialogue is common in the intro ductory phase of a lesson where the teacher attempts to keep tight control of the content and students. Hence, a significant amount of power resides with the teacher. Similar observations are made of the concluding phase of the lesson. However, dur ing the “work” phase of the lesson, the patterns of power are somewhat more equal and students can express their lack of understanding. The role of teachers’ ques tions are critical in controlling the interactions with classrooms. As Lemke (1990) has shown through triadic dialogue, questions are used to control the flow of the lesson, the content to be covered, and the behavior of students, and to provide pro gressive evaluation of student learning and lesson implementation. (213)
 In a 1–year, ethnographic study of three schools (an independent elite school serving a middle to upperclass clientele, a government school serving a predominantly middleclass clientele, and a government school serving a predominantly workingclass clientele), two classrooms were observed and mathematics lessons video recorded throughout the year. Interviews were conducted with students and participating teachers. These data were used to examine the practices within the schools in order to identify the ways in which social differences were being realized in and through the practices of mathematics. From the analysis of classroom inter actions in middleclass settings, it was noted that there was a strong compliance with triadic dialogue in mathematics lessons. Students and teachers used this model of interaction effectively and efficiently to convey information and maintain control of the lesson and students. In contrast, in both classrooms at the school where the students were predominantly from workingclass backgrounds, there were many challenges to the triadic dialogue and, hence, its use by the teachers was thwarted by the actions of the students. This made control of the content and students less effective (Zevenbergen, 1994, 1998). Freebody and associates’ (Freebody, Ludwig, & Gunn, 1995) research in literacy classrooms has found simi lar patterns of interactions with students from disadvantaged backgrounds. (214)

As Cooper and Dunne (1998) have demonstrated, workingclass students are considerably disadvantaged by the embedding of tasks in a pseudomathematical context. In part, this is owing to the increasing complexity of demands of the task. Newman (Ellerton & Clements, 1992) has noted that several cognitive steps are needed in deconstructing and responding appropriately to a word problem: reading the problem, compre hending the problem, translating the problem into a mathematical task, undertaking the mathematics necessary for the task, and, finally, interpreting what the answer means. Students are able to make mistakes at any of these steps and hence produce incorrect responses. This is then further compounded by the language differences between that which is represented in and through the mathematics curriculum—as evident in the mathematics register—and the language of the students. Where there is greater synergy between the language of the mathematics problem and that of the student, there is greater potential for success. In contrast, where there is significant difference between the two registers, the chances of success are reduced. (216)
 When the culture, and hence in many cases, knowledge, of the stu dents in many classrooms is different from that represented in the curriculum, then there is likely to be a greater mismatch between what is seen as relevant and mean ingful. In many cases, this is blatantly obvious; for instance, where there is a big mismatch in cultures. Where there is less of a mismatch between the classroom and school, there is greater opportunity for seeing the activities as being relevant to all students (217)

In many cases, the activities work on a mathematical assumption that students will need to be able to apply the mathematics in contexts that they are likely to encounter in their lives beyond school. It is seen to be the task of school mathematics to empower students to make the “right” choices in their adult lives, with mathematics being regarded as a key tool for making such informed decisions. (218)
 For students coming into mathematics classrooms, there are particular language games they come to learn, with which they have some familiarity because of simi larities with their home environments. For these students, the language games of the classroom—the registers, the interactions, the contexts—are familiar enough that they are able to participate more effectively and more efficiently and be seen as mathematically able students. For these students, their familiarity with the language games becomes a quality they can use or trade for success in classrooms. Ac cordingly, such familiarity is a form of capital that can be traded for educational rewards. If the language games of students are not part of their social or cultural backgrounds, then subsequent constructions of their success are far more elusive. (220)
Posted on September 22, 2012 by B.J.D.Armas
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