Goodenough (1976) asked the question "What do people need to know in order to operate in a manner that is acceptable to others in a society?" This question provides the stimulus for this paper where the question is rephrased to: "What do students need to know in order to operate in a manner which is acceptable in the mathematics classroom?" Such a question is not without political implications and so needs to be extended to include questions about the consequences of participation in mathematics classroom. To answer this question, I appropriate Gore's (1990) notion of "pedagogy as text" and develop the argument that mathematics pedagogy is a text which students must be able to read in order to be constructed as effective learners of mathematics. These texts, however, are not apolitical and as Bourdieu (in (Wacquant, 1989) has argued persuasively, language is a form of capital which can be exchanged for other forms of capital - social, economic or cultural. Combining these two frameworks, I argue that students enter the mathematics classroom from a range of socio-cultural backgrounds whereby students whose socio-cultural background is congruous with that of the culture represented in and through the practices embedded within the mathematics classroom - including linguistic practices - are more likely to be constructed as successful students.
Bourdieu (in Bourdieu & Wacquant, 1992) argues that access to legitimate language, in this case mathematics, is not equal and that linguistic competence is monopolised by some. In considering the case of mathematics, this suggests that access to the discourses and discursive practices of mathematics is differentially accessible. For those students who enter the mathematics classroom with a competence in the discursive practices, access to mathematics is made more easily. Simultaneously, such students are more likely to be constructed as successful students based on the teacher's judgement of their ability. Within this context, language background is a form of capital which can be converted to academic reward.
Linguistic competence – or incompetence – reveals itself through daily interactions. Within the mathematics classrooms, legitimate participation is acquired and achieved through a competence in the classroom dialogic interactions. Students must be able to display a discursive competence which incorporates a linguistic competence, an interactional competence along with a discursive competence if they are to be seen as competent learners of mathematics. Classroom interactions are imbued with cultural components which facilitate or inhibit access to the mathematical content. To gain access to this knowledge, students must be able to render visible the cultural and political aspects of the interactions.
Bourdieu (in Bourdieu & Wacquant, 1992) argues that
…what goes in verbal communication, even the content of the message itself, remains unintelligible as long as one doe not take into account the totality of the structure of the power positions that is present, yet invisible, in the exchange. (p. 146)
From this perspective,
language must be understood as the linguistic component of a universe of
practices which are composted within a class habitus. Hence language should
be seen to be considered as another cultural product - in much the same
ways as patterns of consumption, housing, marriage and so forth. When considered
in this way, Bourdieu proposed that language is the expression of the class
habitus which is realised through the linguistic habitus.
Gore (1990) has argued that pedagogy can be seen as a text which can be read and interpreted by the reader. Texts can be read in a multiple of ways, so that the student entering the mathematics classroom will be required to read, interpret and make sense of what transpires in the mathematics classroom - not only of the mathematical content, but also the pedagogical approaches within which the content is relayed. To be able to read these texts, the students must have some linguistic competence in the reading of social texts and discursive practices.
The interactions that
occur within the classroom have been subjected to ethnomethodological approaches
and have been found to have highly ritualised components with clearly identifiable
discursive practices (Lemke, 1990; Mehan, 1982a). They argue that these
components are not explicitly taught but are embedded within the culture
of the classroom. The highly ritualised practices of classroom interactions
can be seen in the types of interactions which occur across the various
phases of the lesson. For example, the most common form of interaction
consists of a practice in which the teacher initiates a question, the students
respond and the teacher evaluates that response which Lemke (1990) refers
to as "triadic dialogue". This interactional practice can be observed in
S: The outside of the square
T: Not quite, someone else? Tom?
S: When cover the whole surface, that's area.
T: That's good
Furthering the work of Lemke, Mehan (1982b) has identified three key phases of a lesson - the introduction, the work phase and the concluding/revision phase. In each of these phases there is a shift in the power relations between the students and teacher which permits different forms of interactions to occur (Mehan, 1982b; Schultz, Florio, & Erickson, 1982). For the purposes of this paper, it is my intention to discuss the introductory phase only.
Mehan (1982) argues that during the intoductory phase of the lesson, the teacher maintains tight control over the students, initially to ensure that the students are ready for the content of the lesson. Once control has been established and attention gained, the lesson can then proceed. Triadic dialogue is commonly observed in this phase in order to keep control of the academic content of the lesson and the control of the students. Dialogue between students and between teacher and students is not generally part of this phase. If the teacher initiates a question but the student is not able to respond, it is not appropriate for students to express their lack of understanding since this will interrupt the flow of the phase. If there is a misunderstanding or lack of understanding, it is more appropriate for this to be voiced in the work phase of the lesson.
What is lacking from
this corpus of knowledge of classroom interaction is the failure to recognise
that these interactions recognise a particular linguistic form which will
be more accessible to some students than others. In this sense, the interactions
within the classroom can be considered as another cultural product which
is more familiar to some students and not others. The linguistic habitus
of the students will or hinder a students capacity to render visible the
mathematical content embedded in the pedagogic action. The
This school serves
a middle- to upper-class client group. The mathematics teaching, learning,
assessment and curriculum are relatively conservative with a strong emphasis
on rote learning, preparation for examinations and teacher-directed pedagogy.
The class sizes are small with only 12 students in classroom observed.
In the lesson presented here, the students were undertaking an activity
from the Mathematics Curriculum Teaching Package. Prior to the extract
shown, the teacher (Helen) has used a number of short mental arithmetic
tasks. The following is the introduction to the lesson.
T So we have to talk about degrees of difficulty. What do you think that means? What does that actually mean? Robert?
Robert You have to add a bit more to the score because of the degrees of difficulty.
T Good boy. Yes, good. Daniel?
Daniel Well the performance of their dive, how they dive and well like they might have a very good dive and make a very big splash and may even get off
T Right, good. OK you are on the right track. What do you want to say about degree of difficulty Cate?
Cate How hard it is?
T How hard it is. Tom what would you like to say about degree of difficulty? That's not a word we use much in our everyday language..... degree of difficulty.
Tom The percentage of how hard it is
T Good. Because you're focussing on the word degree though aren't you. So a really hard dive. Now you can see on this sheet they're talking about DD which is short for degree of difficulty and a really hard dive. What would be a really hard dive? What would be the highest number for a degree of difficulty be? Have a look at your sheet. Try and work out the degree of difficulty. Vicky?
Examining the flow of the interactions indicates that there is a complicit agreement between the teacher and students to participate in the interactions. There are no transgressions or challenges to the teacher's authority. This allows for substantive content to be covered.
The teacher is able to maintain control over both the form and content of the lesson and the students through a mutual compliance with the implicit rules by both the students and the teacher. She has used Triadic Dialogue to structure the interactions and students infrequently transgress the rules. This allows her to retain the focus of the lesson and in so doing, the students are exposed to a significant amount of mathematical knowledge that is embedded in that dialogue. The teacher's capacity to deliver the lesson in this way allows for her to use a very rich mathematical language as she discusses the mathematical content. In other words, the students are exposed to mathematical language and concepts in a style which takes for granted their linguistic background. The work of Brice-Heath (1982) has shown that middle-class students are more likely to be familiar with these forms of school interactions due to their similarity with the linguistic patterns of the home environment. This familiarity has facilitated a linguistic habitus which is similar to that of the formal mathematics classroom and hence permits access to the codes and signifiers of school mathematics.
Connewarre is a large
government school which is located within a large housing commission estate.
The clientele of the school is predominantly working class with many of
the parents receiving government support. The classrooms are smaller than
Angahook with approx 25-30 students in each class. The teacher introduces
the mathematics lessons with problem solving activities which the students
undertake as small groups. They are able to be physically involved in the
activities and it is not uncommon for the students to draw on the carpet
with chalk to represent the task or physically construct the problem. The
mathematics has a strong emphasis on real life situations. The following
extract is the introduction to a lesson in which the teacher has drawn
a net on the board which the students will have to draw onto card and then
construct. Students are then required to develop a number of nets for nominated
C A cube
T He said a cube. Don't call out please.
C A rectangular rectangle
T You're on the right track
C A 3D rectangle
T Three dimensions, technically I suppose you're right.
C A rectangular
It's a rectangular something. Does anyone know what it is called?
C A parallelogram
T Put your hand up please.
More calling out
T I guess you could have a rectangular parallelogram, but no. A rectangle is a special parallelogram.,
C A rectangular oblong
T The word we are looking for is prism
C Yeah that's what I said
T Say the word please
T Not like you go to jail "prison", that's prison. Excuse me, could you return those please.
T So one thing that we think about with rectangular prisms and that this shape on here is, excuse me...Now you can leave them down please. You need a little bit of practice at lunch because you can;t stop fiddling. This shape here is drawn out on the graph, this grid here [net for a rectangular prism]. We 're going to try and do the same thing. Draw the shape and then cut it out. If you look at the shape, it's made up of rectangles and squares.
There are many transgressions of the implicit rules of classroom interactions. The flow of the lesson is fragmented as students challenge the teacher's control for the floor and content of the lesson.
The linguistic habitus of the student implies a propensity to speak in particular ways which, as can be observed in the case of the interactions in this extract, works to exclude students from the mathematical content. The students are not as competent in the linguistic exchanges of the mathematical interactions as their middle-class peers thereby marginalising them in the process of learning. The teaching of mathematics in this way tacitly presupposes that the students will have the discursive knowledge and dispositions of particular social groups, namely the middle-class. The students are not as complicit in the classroom practices and in so doing are being excluded from active and full participation in the mathematics of the interactions. In this way, students have been exposed to the symbolic violence of formal education.
Using data from mathematics classrooms, Voigt (1985, p. 81) has argued, "The hidden regularities, the interaction patterns and routines allow the participants to behave in an orderly fashion without having to keep up visible order" so the idea is far from new. However, what I have sought to uncover using an interactionist approach is the ways in which some students are able to gain access to mathematical content and processes more readily than others. I have proposed that one subtle and coercive way is through the linguistic habitus of the students and the practices of classroom interactions whereby some students enter the formal mathematics classrooms with a habitus that is akin to that which is valorised within that context. These students will be able to participate more effectively and efficiently than their peers for whom the patterns of interaction are foreign to their habitus, thereby making the habitus a form of capital which can be exchanged for academic success within this context.
The predominantly implicit
codes of curriculum and classroom interactions take as a given that students
will have a familiarity with the legitimate linguistic practices of the
mathematics classroom, but neither curriculum nor pedagogy render that
language visible. gaining access to mathematical knowledge is facilitated,
or hindered, but a match or mismatch of codes. Rather than perceive this
a function of language deficiency, but as systemic through which the dominant
classes are able to maintain control:
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