Murrell, Peter . The Journal of Negro Education 63. 4 (Fall 1994): 556-569.
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Murrell, Peter . The Journal of Negro Education 63. 4 (Fall 1994): 556-569.
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Murrell synthesizes the results of a critical ethnography of classroom practices of mathematics instruction. Specifically, he analyzes the discourse patterns and speech events that evoke qualitatively different learning experiences for African American male students in urban middle schools. The purpose of this synthesis is to systematically account for the ways that particular teaching practices--practices that are intended to promote deeper understanding of mathematics for all children--actually diminish African American students' opportunities to understand, communicate, and apply mathematical ideas.
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INTRODUCTION
One of the most troubling problems in urban education in the United States today is that African
American children, particularly males, have been categorically underserved by public schools.
Disproportionately large numbers of African American boys in our nation's inner-city schools are expelled, suspended, relegated to special education programs, and subsequently left with fewer personal resources than their European American peers. Clearly, a combination of political, economic, and sociological factors contributes to the inability of teachers, schools, and schools systems to uniformly promote educational success among urban African American children.
Part of the problem stems from an insufficient and incomplete knowledge base about these students' development and socialization. Many factors limit the creation of this knowledge base, including the visceral fear of African American males that is fed by demonized images of
African American maleness in the popular media, and the general lack of access teacher preparation programs have to pedagogical expertise drawn from the culture, language, and history of African American people. Educators are not likely to develop a pedagogical knowledge base of the critical aspects of class and culture for non-mainstream minority-group learners unless a theory is developed that addresses how these students make sense of the curriculum in the context of their unique racial, ethnic, cultural, and political identities. More specifically, teachers cannot fully interpret the developmental learning of these students without an analysis and synthesis of the students' experiences with the curriculum and knowledge of how they position themselves in the culture of the classroom. This necessitates that teachers acquire a deep understanding of the discourse routines and dynamics of the educational settings these students find themselves in.

Developing an understanding of these issues as they relate to African American male students' academic achievement in mathematics is complicated by recent developments in mathematics curricula. A National Council of Teachers of Mathematics' (NCTM) (1989) document entitled
Curriculum and Evaluation Standards for School Mathematics presently is transforming the instructional practices and classroom dynamics of mathematics learning in significant and positive ways. These standards emphasize developing learners' abilities to use mathematics in problem solving, reasoning, and communicating by engendering a greater emphasis on understanding mathematics concepts than on achieving computational competence. They explicitly promote educational outcomes that include dispositions such as self-confidence in doing mathematics and valuing it as a discipline. They further call for instruction that encourages students to:
* articulate their reasons for using a particular mathematics representation or solution,
* summarize the meaning of the data they have collected,
* describe how mathematical concepts are related to physical or pictorial models, and
* justify arguments using deductive or inductive reasoning.
Thus, the NCTM standards influence an important dimension of the classroom culture: the discourse of learning or "math talk." Gee (1991) defines a discourse as "a socially accepted association among ways of using language, of thinking, and of acting that can be used to identify oneself as a member of a socially meaningful group or 'social network'" (p. 3). However, to the degree that many urban African American students do not share mainstream, middle-class perspectives or assumptions about learning and teaching, these students may construct profoundly different subjective worlds than those anticipated by the teachers who teach to these standards (Kochman, 1981).
The recent curricular innovations calling for greater emphasis on communication in mathematical reasoning (as articulated in the NCTM standards, for example), together with the fact that most instructional time is based on teacher-initiated talk (Goodlad, 1981), underscore the importance of classroom discourse as the foundation of children's classroom learning. As the uses of language and forms of discourse are critical determinants of the degree to which children can participate in the social-interactional dynamics of learning, it is important to understand how these new forms of classroom discourse may marginalize those who do not already have access to them (Bowers & Flinders, 1990).
This article synthesizes the results of a critical ethnography of classroom practices of mathematics instruction. Specifically, it analyzes the discourse patterns and speech events that evoke qualitatively different learning experiences for African American male students in urban middle schools. The purpose of this synthesis is to systematically account for the ways that particular teaching practices-practices that are intended to promote deeper understanding of mathematics for all children-actually diminish African American students' opportunities to understand, communicate, and apply mathematical ideas. From this account, a pedagogical framework of responsive mathematics instruction for African American males will be described.

CULTURAL INCOMPATIBILITY AND CLASSROOM DISCOURSE
Increasingly, teachers whose backgrounds are middle-class and mainstream1 are being called upon to promote both conceptual understanding and communication skills among urban school children of color. Preparing teachers to work effectively across boundaries drawn by cultural, racial/ethnic, and class differences continues to be a problem (Murrell, 1991, 1993). However, as most students in U.S. public schools are increasingly children of color in urban settings, the stakes are high with regard to finding ways of providing these students with teachers who can promote their learning, development, and intellectual growth.
Classroom learning is a social process requiring considerable communication, coordinated action, and common understanding (McDermott, 1977). Recent ethnographic research findings demonstrate how cultural and social-class differences influence educational outcomes. Most of these findings suggest that differences in the way the social context of the classroom is construed by mainstream teachers, in contrast to students from historically marginalized groups (e.g.,
African Americans, Native Americans, Hispanic Americans), often result in diminished academic success for the students. These same studies provide examples of culturally responsive teaching-that is, teaching practice that acknowledges and builds upon the culturally shaped perspectives, behaviors, and abilities of non-mainstream communities including those of African
Americans (Foster, 1989; Heath, 1983; Ladson-Billings, 1989; Lubeck, 1985), Mexican
Americans (Diaz, 1989; Erickson, Cazden, Carrasco, & Meldanado-Guzman, 1983), Native
Americans (Erickson & Mohatt, 1982; Phillips, 1972); and Pacific Islanders (Au & Jordan, 1981;
Kawakami, 1991).
The growing ethnographic research literature revealing how cultural incompatibility diminishes school success of culturally non-mainstream children in mainstream schools has provided the impetus for efforts aimed at transforming current pedagogical practices. However, findings from this research literature do not constitute a basis for formulating a pedagogical theory for improving the academic fortunes of those children.2 As Ogbu (1986, 1992) forcefully argues, cultural and language differences between African American children and mainstream culture is insufficient to account for the diminished quality of these children's schooling experiences. He reminds us that culturally linked perspectives, abilities, and ways of knowing do not exist in a vacuum; and that the historical, political and economic fortunes of an ethnic or cultural group determine the extent to which their cultural and linguistic distinctiveness from mainstream
American culture becomes an educational disadvantage. For African Americans specifically, the utility of the cultural incompatibility concept is voided because multiple conceptions exist regarding what constitutes African American culture. Moreover, even if there were a single, agreed-upon conception, no culture or cultural form exists in unaltered form; rather, they almost always arise out of contact or contestation with other forms. Similarly, many of the cultural differences that create problems for children of particular ethnic or cultural groups arose out of their contact with the dominant culture engendered by public schooling. Among these differences, which are called secondary cultural differences, are those that emerge as opposition and resistance to the dominant culture.
One of these differences, cultural inversion, is a phenomenon that occurs when members of a minority group specifically reject those forms of behavior, events, symbols and meanings

deemed characteristic of the majority culture. Cultural inversion has special significance for the schooling of African American children. For example, many of the most instructionally relevant cultural expressions of African American children, who are the primary participants in contemporary "hip-hop" culture, emerge at least in part as opposition and resistance to the dominant culture (Perry, 1995; Powell, 1991; Rose, 1991). Additionally, Fordham and Ogbu
(1986) found that academically talented African American high school students in Washington,
D.C., either tacitly or explicitly avoided manifesting characteristics they associated with "acting
White" in an effort to remain culturally "Black." Unfortunately for these students' academic progress, many of these characteristics, rather than merely constituting aspects of "White" behavior, were also ways of being associated with academic success such as class participation, compliance with teacher requests, or turning in homework.
Clearly, the oppositional cultural markers adopted by African American students may not represent culture in conventional ways such as ethnic or national heritage, and they often combine aspects of ethnic and youth cultures. Notwithstanding, they play a powerful role in determining the quality of these students' social and educational experience in culturally mainstream school settings, often by diminishing the students' ability to negotiate the social and cultural demands that schools place on them to perform as well as conform. Many African
American students, for instance, are unavoidably bicultural in these contexts because they must learn to negotiate the cultural world of school and reconcile it with that of their homes and communities. Culturally Black students in "culturally White" school settings may be subject to more than the typical interpersonal conflicts around issues of race and belonging but to intrapersonal conflict as well, as they struggle with the choice between not "selling out" their racial identity on the one hand and achieving academic success on the other.
MATH TALK
Math talk denotes the types and amounts of discourse that occur during mathematics instruction, particularly the oral reasoning performances of learners during teacher-led mathematics inquiry in the classroom. It also refers to students' public talk and public display of mathematics reasoning, in both on-task and off-task situations, as well as their sense of the purpose of mathematics discourse.
Math talk can be viewed as a variation of the typical, three-part IRE discourse sequence described by Cazden (1988), who contends that nearly all classroom discourse can be interpreted according to a basic pattern or concatenations of this pattern. According to Cazden, the IRE sequence consists of: (I) teacher initiation, (R) learner response, and (E) teacher evaluation. The first step usually occurs in the form of a teacher's question about a solution to a problem; however, these questions most often are used to elicit from students a specific response. For example, a teacher might ask, "How many values can X take on here?" (I), to which the selected student might answer, "Two" (R), and obtain the teacher's evaluation of "Correct," or "Good"
(E). Teacher initiation queries can take three forms: (1) open questions (e.g., "Can we treat a ratio the same way we treat a fraction?"), (2) requests for information (e.g., "Now, what did we call these?"), and (3) invitation to initiate a math talk sequence (e.g., "Well, let's see what we can figure out about this, shall we?"). Conversely, learners' responses can also take three forms: (1) choral responses (several students responding together in unison), either to open questions or

calls for information; (2) overlapping talk, such as that which takes place in small-group problem-solving teams, or when the teacher purposely delays an evaluation to see what the class as a whole comes up with when students "have the floor" in the dialogue of inquiry; and (3) sequenced talk-that is, student discourse offered concurrent with the activity or task, such as when each student supplies a part of an answer or negotiates the task procedure with a partner.
Because the teacher controls both the development and direction of the inquiry during math talk, there are two important variations in the mathematics IRE patterns that stem from teacher implementation of the NCTM standards' pedagogical emphasis on learner participation, ownership, and communication of reasoning in the learning processes. One of these is the length of the learner's "turn." The NCTM standards are based on the expectation that students will elaborate answers with supporting reasoning, and that their responses will include both a position and a rationale for their positions. In such math talk, learners are encouraged to provide rationales for every solution they generate. Therefore, relative to other kinds of directed discourse in the classroom, students are expected to do more talking. They are encouraged, and often prompted, to provide a rationale for their problem solving (e.g., "How do you know that?") or to evaluate another student's answer/reasoning (e.g., "Do you agree with what X came up with?"). An additional variation of the mathematics instruction IRE pattern wrought by the
NCTM standards is that teachers may suspend their evaluations of student responses until after several students have spoken (e.g., "Who has still another idea?").
Deeper scrutiny of the discourse in mathematics classrooms may provide additional dimensions for interpreting the distinct patterns of math talk among African American male students. The public school classroom is the critical place to begin such scrutiny, with its implications for revising instructional practices, pedagogical decision making, and classroom interactional routines so that African American male students can learn with understanding.
METHOD
The method of investigation employed in the present study was modeled after the interpretative research framework of the QUASAR Project on mathematics run by the Learning Development and Research Center at the University of Pittsburgh (Stein, Grover, & Silver, 1991). The initial observations of one of the classroom sites were performed by the researcher as a member of the
QUASAR field observation team. The aspects of the QUASAR design adopted for this study were the one-week immersion of the researcher into school life at each site, including formal interviews with the principals and with the site coordinators (curriculum directors), and group interviews with the students. Another adopted aspect was the notion of a target group: a selected group of four students upon whom classroom observations were focused. These students' classroom discourse was recorded electronically and their completed class work was analyzed.
Structured group interviews were conducted with members of this group as were informal individual interviews. Three target groups of African American male focus students were formed.
The Setting

Four urban middle schools-Easton, Norton, Alston, and Sutton-were selected for this study.3 At the time of the study, all of the schools had African American populations exceeding 50%:
Alston (58%), Easton (49%), Norton (56%), and Sutton (59%). All four schools had total enrollments exceeding 800 students. Table I compares mathematics achievement at the schools as indicated by the percentage of students who scored at or above the national average on the
Iowa Test of Basic Skills. As depicted in this table, the mathematics achievement test results of seventh graders in each school reveal a discrepancy in the performance of White and African
American students. In all four cases, the academic mathematics achievement for Hispanics and
African Americans was significantly lower than that of Whites.
Four mathematics classrooms, one in each school, were selected for observation and its occupants selected for periodic, in-depth interviews exploring the focus students' emic experience of the mathematics curriculum. Each class was taught by a regular classroom teacher with the assistance of a student teacher for whom the researcher served as the college supervisor.
Sample
Twelve African American male students (6 sixth graders and 6 seventh graders) were selected as the focus students for the study. The students were selected on the basis of their being designated as "low-ability" in mathematics by the classroom and student teachers.
Procedures
As part of an assignment routinely given to student teachers by the researcher, the four student teachers participating in the present study were required to (1) select from each of their classrooms three students who seemed to be having difficulty learning mathematics, (2) frame the nature of the students' learning problem or difficulty, (3) reflect on the contributing causes, and (4) generate an appropriate intervention. In the tradition of micro-ethnography, the student teachers were to record in their teaching journals observational data consisting of continuous handwritten accounts of classroom interactions and activities during instruction. They were to note the frequency and types of IRE interactions that occurred in the classroom, the kinds of inquiries and responses that took place in group work, and the tropes of teacher-talk and studenttalk (e.g., giving direction, providing information, displaying knowledge, etc.). Additionally, classroom artifacts, curricular materials, teaching materials, student classwork and homework in portfolios were analyzed in order to construct the conceptual landscape of the enacted curriculum. Observations were also made of informal interactions with students and school personnel outside the focus of the class period during which mathematics instruction took place.
Data gathering consisted of classroom observation and interviews, both formal and informal. The focus students in each classroom were interviewed as a group. Interviews with the principal and the curriculum coordinator were conducted individually. Interviews with teachers were informal, impromptu, and unscheduled. The bulk of the field observations were conducted during biweekly visits to the school sites. These visits included classroom visitation sessions as well as meetings with the participating teachers.

Interpretation of the intended instructional goals in each of the four classrooms was obtained through three-way conferences among the cooperating classroom teachers, the student teachers, and the researcher. This permitted a triangulation of perspectives regarding what constituted the learning goals in each classroom, how well instruction brought about these goals, and how well the instruction reflected the NCTM mathematics standards. In addition to conferences evaluating the student teachers' performances, the researcher also assessed their teaching practice through an evaluation of their lesson plan designs and analytical teaching journals.
Structured post-observation group interviews were conducted with the three focus students nominated from each classroom. The group interview focused on the students' perceptions of (1) the classroom experience in general, as derived from answers to interview "brainstorming" questions (e.g., "What does it take to get a good grade in Ms./Mr. ________'s class?"); and (2) specific elements of mathematics instruction (e.g., "What do you think was the main thing you were supposed to learn during this activity?"). As the three students brainstormed, their responses were noted by the researcher on a large drawing pad mounted upright on a table in front of the group.
RESULTS
All 12 of the focus students most frequently entered the mathematics discourse in their respective classrooms following request-for-information types of initiating questions. Only rarely did they respond to open questions, unless their responses were part of a unison response or responses to teachers' invitations to initiate math talk. The infrequency with which the focus students participated in the class inquiry following these latter types of initiators suggests that the participation of the African American focus students was limited to the classic IRE role of supplying information. Table II summarizes the distinctions in the inquiry discourse of the present study's focus students contrasted with those intended by the NCTM curriculum standards. As this table shows, the contrast between the math talk roles intended by the instructional outcomes of the teacher and those assumed by this sample of African American male students differs markedly.
All students (African American and White, male and female, sixth and seventh graders, etc.) tend to limit their explanations to the point at which they sense other members of the class may conclude they are saying too much, showing off, or otherwise "acting like a brain." In public math talk during whole class instruction, students almost always can supply information to supplement their initial response to the teacher's question. Typically, they merely await the prompts offered by the teacher's probe-type questions. In this manner, the IRE pattern is a familiar and frequently enacted discourse frame. In situations when a student continues talking and does not await the teacher's prompt of either an evaluation (E) or new initiating question (I), other students will exchange looks or make subtle comments.
In the present study, however, the African American male focus students were not at all influenced by such cues from peers. For example, in each of the classrooms it was customary for the teacher to require a representative from a group to come to the front and illustrate the set of solutions they had generated for an activity or problem. However, in such situations, the focus students frequently tended not only to hold the floor in the front of the class but also to insert

ideas of their own that were not necessarily those of the group. They proceeded to "hold the floor" in front of the class for as long as they could, even after they had exhausted their set of meaningful things to say. Thus, the identifier "controller" was selected to designate the prominent features of the focus students' math talk participation in whole group settings.
Moreover, in many instances it was obvious that the focus students were attempting through their discourse merely to "get over"-that is, to respond in such a way, particularly when called upon, that they appeared to have a grasp of the subject matter when, in fact, they did not. To disguise their inability to answer a request for information, the students would often engage in superficial aspects of math talk. While their responses momentarily satisfied their teachers, assessments of the students' mathematics performance reveal that the students did not attain the conceptual understanding math talk is presumed to engender. What seemed to the students of greater importance than the inquiry and the need to understand mathematics concepts and ideas was their participation as a talker. The focus students seemed to regard verbal adroitness, whether or not it was substantive in terms of mathematics learning goals, as a criteria for doing well in mathematics class.
The focus students also frequently attempted to reframe the nature of the interactions with the teachers during small group interactions. They seemed to relish discovering small mistakes in the teachers' discourse, and often became obstinate just when they appeared to be on the verge of a critical point of understanding. Some discussion of the basis for this resistance and obstinacy is merited here. The data suggest that the teachers and focus students in this study operated from two different frames in their math talk interactions. Kochman (1981) would describe this as a classic example of a contestation of interactional styles: The teachers sought to engage in the kinds of discourse called for by the NCTM standards. The focus students, on the other hand, drew from their cache of verbal verve and adroitness to engage in discourse that allowed them to disguise the fact that they did not know the right answers and avoid appearing dumb. The students were operating from the frame of maintaining face, or, as Majors and Billson (1992) would contend, they were more concerned about not losing their "cool pose," or sense of masculine identity. Thus, at one level, the focus students' resistance and obstinacy was a result of interpersonal conflict. That is, to admit that the teachers helped them get the answers or that they even needed help was, for these students, an admission of inadequacy. As though their identities as mathematics learners contradicted their racial, personal, and masculine identities, participating in math talk constituted an approach-avoidance conflict of the motive to be assisted versus the motive to maintain a cool pose.
In general, the African American male students in this sample did not construe discourse-laden, inquiry-based mathematics instruction to represent a greater emphasis on understanding mathematics ideas. As teachers engaged in more math talk as a means of exploring and elaborating mathematics principles and concepts, these students did not regard the discussion as an increased focus on mathematics learning. Rather, they tended to regard the emphasis on math talk simply as a new regimen to be mastered to meet their teachers' requirements, as a string of operations and concepts with little or no thematic coherence. As they acted on these perceptions, their learning performance differed dramatically from those intended by the curricular innovations. These differences are summarized in Table III.

Comments offered during the group interviews reveal that the sampled African American male students' perception of mathematics instruction did not distinguish behaviors of compliance, procedures, and conduct from the mathematics conceptions and ideas. Table IV presents a composite list of brainstormed items generated by the sixth graders in response to the question,
"What do you think of when I say mathematics?" What is significant about this list is that the volume of items associated with conduct and classroom behaviors nearly matches that of the items associated with mathematics concepts. Similarly prominent were conduct/behavioral items in response to other questions such as "What does it take to get a good grade in this class" (see
Table IV). The focus students generally did not distinguish behaviors associated with compliance, control, and classroom routine (e.g., turn in homework, listen to the teacher, not talk, do your work, etc.) from mathematics concepts.
DISCUSSION
The intended outcome of the NCTM standards was to move students beyond routinized, unreflective, and procedural approaches to mathematics and toward greater conceptual learning in that subject. However, for this study's sample of African American male students, the result was not an increase in conceptual learning, but rather greater attention to discourse performances unrelated to actual conceptual understanding. The students' conception of math talk-the discourse routines of the classroom mathematics inquirywas that of a new competence to be mastered and exhibited in the same way they exhibited other aspects of school performance such as doing one's work, turning in homework, and listening to the teachers. When they participated in math talk, it was not as a discourse of conceptualization and learning, as was intended by the NCTM standards of communication of mathematics ideas. Math talk was not seen as a means for deepening mathematics understanding, but rather as an aspect of showing off what they were expected to learn and be like.
To what extent does math talk represent an unauthentic, idealized relational framework for all children, not only African American males? On one level, engaging in math talk may represent for African American male students a variation of "acting White" (Fordham & Ogbu, 1986). On another level, any child who has not been socialized into the mathematics discourse forms assumed by mainstream teachers may participate in math talk in the same manner as the students sampled in the present study. What may make the experience unique for African American males is the extent to which engaging in math talk conflicts with their personal styles of selfexpression. As Gee (1991) notes, in distinguishing between primary and secondary discourses, the degree of compatibility between a child's primary and secondary discourses shapes his or her experience of academic success.
Responsive Teaching: A Theoretical Framework
Responsive teaching is essentially the systematic and analytic implementation of discourse patterns and speech activities4 that optimally support and sustain an ecology of developmental learning, reasoning ability, and performance for all children. Applying Vygotsky's (1978) conception of internalization, responsive teachers of mathematics promote opportunities for mathematics discourse to become internalized as mathematics reasoning and performance. In short, the purpose of responsive mathematics teaching is to assist children in the internalization

of math talk (discourse) so that it becomes "math thought" (reasoning). As such, responsive mathematics teachers are compelled to attain proficiency in framing and reframing the dynamics of discourse in their classrooms to meet the needs of diverse learners.
As responsive teachers conceptualize their pedagogy for teaching mathematics, they must simultaneously conceptualize and routinize speech activities that promote the development of reasoning and thinking abilities. This means that they must develop their instructional plans not merely in terms of content but also in terms of the social and interactional dynamics of the classroom that unfold as speech activities. Thus, in conceptualizing a speech activity, they must make explicit the rules of talk and performance expectancies for all occasions of classroom discourse, including cooperative group discussions and informal off-task talk as well as wholeclass inquiry. They must also diagnose and change those frames of discourse that do not support learning for non-mainstream students.
The following are tenets of the theoretical framework supporting the use of discourse in responsive teaching:
(1) Discourse is the most important mode of learning for understanding. The relationships children form with their teachers, with each other, and with the subject matter are shaped by the discourse patterns emerging from classroom interactions and activity settings.
(2) The discourse routines, patterns, and speech activities that support learning for understanding are determined through careful observation (e.g., Hymes's [1981] notion of "ethnographic monitoring") of children's interactions in light of the learning goals that develop their abilities.
(3) The responsive teacher routinizes discursive practices and approaches that have been determined to optimize the participation, conceptual learning, and development of social interaction abilities among the students in his or her classroom. Responsive teaching combines pursuit of the learning goal with the development of sensibilities that respond to the social ecology of the classroom.
Responsive teaching for any group of students depends on how well the teacher orchestrates a learning experience or event incorporating: (1) an understanding and appropriate use of discourse frames; (2) instructional strategies that create particular discourse frames that optimize conversation for the purpose of understanding; and (3) purposeful, meaningful, and intellectually worthwhile learning goals that develop abilities. Toward this end, responsive teachers use strategies for assisting performance such as modeling, contingency management, productive feedback, direct instruction, assessment questioning, and cognitive restructuring to shape the social and intellectual ecology of classroom activity (Tharp & Gallimore, 1991). They also assist student performance toward significant outcomes such as those indicated by the NCTM standards.
What understandings about responsive teaching for middle-level mathematics instruction can be gleaned from the study described in this article? What makes mathematics teaching responsive to
African American male children at risk of academic failure? Responsive teachers must be continuously aware that the relationship students construct with their teachers, as well as with the

subject matter, is shaped by the degree to which discourse routines and speech events promote interest, social participation, a sense of efficacy and industry, and sense of purpose. As such, responsive teachers recognize and capitalize on the frames of discourse within which African
American male students routinely operate. These include:
(1) a question-posing, teacher-challenging approach,
(2) a preference for request-for-information teacher inquiries,
(3) an eagerness to show off the information they possess,
(4) a penchant for extended explanation, and
(5) a preference for "getting over" rather than admitting ignorance.
The central issue raised in the findings of the present study was that the African American male focus students (and probably most other students as well) construed or framed math talk and the instructional intent of the discourse differently than did their teachers. Arguably, this finding reflects the inability of the teachers to frame the instructional discourse of their classrooms to accommodate the learning and development of these students. To teach more responsively, the teachers needed to make more explicit both the students' frame of discourse and the frame of discourse for understanding. They needed to employ math talk to communicate both an instructional connection and disciplinary control. As Delpit (1988) argues, making explicit the rules, codes, and expected performances of classroom discourse is essential to helping nonmainstream students develop reasoning competence in discourse. She further maintains that it is by these means, among others, that these students are empowered.
CONCLUSION
This article began by framing an educational problem: African American schoolchildren, especially males, are seriously and dramatically underserved by public schools in the United
States. Exploration of this problem involved an ethnographic study of middleschool mathematics instruction directed at African American male students at risk of academic failure. The study exposed the limitations of contemporary instructional practice despite attempts to incorporate the innovative NCTM standards for the teaching of mathematics, particularly in the discourse of mathematics learning or math talk. It concluded by laying out a theoretical framework for responsive mathematics teaching aimed at improving the mathematics understanding and performance of these students.
The focus students in the study described above placed greater emphasis on their ability to: (1) manipulate situations and people as opposed to mathematics ideas and information through their discourse, and (2) meet perceived performance requirements (e.g., rules of classroom participation) as opposed to gaining understanding of the conceptual content. To the extent to which these students viewed math talk as a set of competencies, their teachers should have made learning math talk competencies more explicit. Instructionally, this would have entailed teachers' commenting on the form of the students' presentations as well as the content of their discourse.

Teachers could have rectified the situation by giving the students credit or positive reinforcement for what they obviously did well-that is, talking and "holding the floor"-while at the same time making clear to the students that other performance criteria related to the quality of mathematics reasoning and thinking-specifically, math talk-were vitally important.
An important principle for responsive teaching for African American male students is that the instructional practices designed to bring about conceptual understanding must be grounded in the organization of activity settings in the classroom, not merely in the classroom discourse.
Learning activity should be structured by action, not merely verbal direction. This means that teachers must (1) realize that learning requires doing, and relinquish the assumption that conceptual understanding can be imparted to youngsters through talk; (2) require that learning achievements be based in the demonstration of understanding rather than merely the display of knowledge through talk; and (3) find ways to align the high achievement motive of African
American children with authentic demonstrations of understanding.
Footnote
1 Though the term "mainstream" has been ill-defined in both popular and scholarly literature, I adopt it herein to coincide with the specific meaning given it by Heath (1983) in her now-classic ethnography, Ways With Words: Language, Life, and Work in Communities and Classrooms. As she states: "Mainstreamers exist in societies around the world that rely on formal educational systems to prepare children for participation in settings involving literacy. Cross-national descriptions characterize these groups as literate, school-oriented, aspiring to upward mobility through success in formal institutions, and looking beyond the primary networks of family and community for behavioral models and value orientations" (p. 391-392). In the United States, mainstream individuals and populations tend to be white, middle-class, and politically moderateto-conservative. see Ogbu (1992) for a discussion of why the historical experience of African
Americans as "involuntary minorities" virtually guarantees them non-mainstream status in
America.
Footnote
2 There are several reasons for this. Some have to do with the limitations of the concept of cultural incompatibility in the formulation of theory. First, use of the concept has the unfortunate effect of dichotomizing cultural background into two categories: mainstream and "other." This bifurcated notion of culture recapitulates the problems of diversity as faced by public schoolsnamely, reductionistic and stereotypical thinking about what constitutes the culture of the other, and the conceptualization of African American culture only according to the ways in which it differs from mainstream cultural sensibilities. Implications for practice are left hanging at that point because "difference" too has become reified without a practical understanding of the meaning and significance of cultural differences. secondly, use of the concept retards the critical understandings teachers need to develop about their own culture relative to those of the students in their classrooms. The cultural incompatibility notion presupposes an understanding of both a school culture and a home culture that rarely exists in practice. Without a deep understanding of which aspects in a student's cultural background supports learning and which do not, the culture of a student becomes a self-fulfilling category in the process of schooling. Without first

providing a means of making commonly sensible the important elements of culture with respect to schooling in both locations, there really is no foundation on which to build a pedagogy of culturally responsive teaching and learning.
Footnote
3 Pseudonyms are used to identify the schools and subjects mentioned in this study.
Footnote
4 Gumperz (1982) defines a speech activity as "a set of social relationships enacted about a set of schemata in relation to some communicative goal" (p. 166).
References
REFERENCES
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AuthorAffiliation
Peter C. Murrell, Jr., Wheelock College
Copyright Howard University Fall 1994
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Cite

Volume
63
Issue
4
Pages
556-569
Number of pages
14
Publication year
1994
Publication date
Fall 1994
Year
1994
Subject
Teaching methods ;
Mathematics education ;
Urban schools ;
Males ;
Black students ;
Middle school students
Title
In Search of Responsive Teaching for African American Males: An Investigation of Students' Experiences of Middle School Mathematics Curriculum
Author
Murrell, Peter C, Jr
Publication title
The Journal of Negro Education

Publisher
Washington
Publisher
Howard University
Place of publication
Washington
Country of publication
United States
Journal subject
Ethnic Interests , Education
ISSN
00222984
CODEN
JNEEAK
Source type
Scholarly Journals
Language of publication
English
Document type
Feature
Document feature
Tables;References
Subfile
Mathematics education, Black students, Males, Middle school students, Urban schools, Teaching methods
ProQuest document ID

222125622
Document URL http://ts.isil.westga.edu/login?url=http://search.proquest.com.proxygsuwgc1.galileo.usg.edu/docview/222125622?accountid=15017
Copyright
Copyright Howard University Fall 1994
Last updated
2010-06-09
Database
3 databases
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