How Teacher Subjectivity In Teaching Mathematics-As-Usual Disenfranchises Students

Mary Klein
James Cook University, Cairns, Australia
 

Abstract

In this article I argue that a teacher's life history, her/his constituted subjectivity, is relevant to what s/he does and sees in classrooms and society generally. Many classroom practices in mathematics are largely determined by the teacher and imposed on students, like a template, leaving little room for them to move in creative or investigatory ways. Since students' enablement depends upon their being positioned as authoritative speakers of established mathematical "truths" it is imperative that teachers recognise coercion where previously they had seen none, and modify, practices that disenfranchise, often according to patterns of class/gender/ethnicity.
 

Introduction

Policy documents in mathematics education stress the importance of investigatory or problem solving processes in students' making sense of mathematics. A National Statement on Mathematics for Australian Schools (1990, p. 4) states "Mathematics involves observing, representing, and investigating patterns and relationships in social and physical phenomena and between mathematical objects themselves". From the early 1980s in the United States of America problem solving has been constructed as "the focus of school mathematics" (NCTM, 1980, p. l). Similarly in the United Kingdom an inquiry approach, inviting pupils to question, challenge, discuss, interpret and explore, has been seen to be relevant since the Cockcroft Report (1982). The reason for the emphasis on inquiry processes in teaching mathematics appears to be an underlying belief that active involvement in constructing mathematical ideas, patterns and connections will lead to mathematically empowered students able to understand and apply these ideas in contexts outside the school (NCTM, 1989).

Underlying my argument in this paper is the perception that if we look beyond the rhetoric we will find that little has changed in classrooms to support students in taking an investigative stance. It appears that students continue to experience mathematics as remembering formulae and procedures where the teacher talks and students listen and try to remember. Price and Loewenberg-Ball (1997, pps 637, 8) believe that any changes have been mainly cosmetic :

Students may use manipulatives instead of just paper and pencil, they -may talk and work in groups instead of working alone, but they still continue with a steady diet of computation and number work. Even when the calculations are dressed up in word problems, the fundamental curriculum - what students learn and how they learn it - remains largely unaltered. Despite all the talk of empowerment, of students and teachers together searching for mathematical patterns and connections and communicating these with confidence to enhance mathematical ideas, teachers continue to teach largely as they were taught (Foss and Kleinsasser, 1996). It appears that they hold on tenaciously, even though they may not admit it (Schuck, 1996), to a view of mathematical knowledge as facts, skills, rules and procedures to be transmitted and the absolute authority of teacher and text.. I argue that even problem solving and sense-making are constructed within schooling sites as abilities or skills that can be transmitted or given to students by the teacher.

A major problem as I theorise it here is that this is an impoverished view of knowledge which does not recognise all that persons learn from the discourses through which they have been constituted. This has implications for teacher change, because teachers' knowledge or knowing comprises much more than factual cognitive skills or constructions; their practice is premised on something other than the explicitly taught knowledges at college or university. Students in turn know much about the world and their lives which is not reflected in "problem solving" and "sense-making" activities in classrooms. Where we accept knowledge as a cognitive construction alone, we take for granted that teachers will be empowered to adopt new approaches to teaching merely from being told about them and their advantages, and that students will be likewise empowered with regard to mathematical knowledge. I use a poststructuralist view of knowledge or knowing, as subjectivity, to reflect on how teaching-mathematics - as-usual is so resistant to policy imperatives of investigation and inquiry.
 

Subjectivity

To talk of subjectivity and processes of subjectification is to muddy the waters of the environment of learning. It is to question the possibilities of student empowerment given the notion that power is not something that can be given and received but something in capillary form that weaves its way throughout the social context, now giving voice to one person and then another. It is not something that one can grasp and hold on forever, it is ephemeral, moving, dynamic. Mathematics can be taken to be a discourse which comprises "socially organised frameworks of meaning that define categories and specify domains of what can be said and done" (Burman, 1994, p. 2). Relationships of power/knowledge/subjectivity constantly circulate throughout the discourse producing or suppressing numerate behaviour in students. The environment or context for learning is never neutral; some participants are able to take themselves up, as agentic numerate subjects while others are not so constituted.

A poststructuralist view of knowledge can be used as an analytic tool which supports reflection on why it is that teachers experience such difficulties in attempting to implement inquiry or investigative approaches when teaching mathematics. My aim is not to reject or replace prior views of knowledge, but to overlay and deepen these with a view of knowledge which explains how teachers have been made subjects such that they know at an intuitive, or visceral level, what mathematics is and how it should be taught. This has implications for teaching practice because the poststructuralist conception of knowledge holds that "in our action is our knowing" (Lather, 1991, p. xv). Classroom practice is based on teachers’ knowing about themselves (their subjectivities) and about mathematics, neither of which may include the investigative processes encouraged in policy documents.

Although teachers often express the desire to teach in more investigatory ways (Foss and Kleinsasser, 1996), processes of subjectification have taught them the absolute authority of teacher and text (Klein, 1998). Many, if not most of the teachers in schools around the world today have lived experiences of classrooms where the teacher and text were authoritative holders of mathematical "truths" which had to be remembered, or, more recently, constructed. Despite von Glasersfeld's (1989, p.124) dictum that knowledge "refers to conceptual structures that epistemic agents, given the range of present experience within their tradition of thought and language consider viable" (emphasis in original) and that there are no absolute mathematical truths, only socially agreed upon consensus, classroom practice continues to be premised on the binaries of right/wrong answers, appropriate/inappropriate behaviour, and competent/incompetent students. It is the processes of inclusion/exclusion which support these constructed binaries in classrooms which are difficult to interrupt.

Teachers have also been constituted to know that where students fail it is their own fault: that their parents have not encouraged them enough, or they have been lazy or unmotivated. This is the liberal humanistic framework under which they labour and have lived out their prior experiences of classrooms. Liberal humanists place great faith in the naturally "supportive" classroom environment and presume that it is equally supportive of all participants. Failure is taken to be a personal, irrational, choice. The liberal humanist lens is one which reveals no coercion, no classification or marginalisation from learning essential mathematical "truths" and blames the victim for failure. Unfortunately many teachers have been constituted through liberal humanistic discourses and do not question processes of marginalisation in the classroom. Ladson-Billings (1994, cited in Davidson and Kramer, 1997, p.139) elaborates on this point:

My own experiences with white teachers, both pre-service and veteran, indicate that many are uncomfortable acknowledging any student differences and particularly racial differences. Thus some teachers make statements such as 'I don't really see colour, I just see children' or 'I don't care if they're red, green, or polka dot, I just treat them all like children'. However these attempts at colour blindness mask a 'dysconscious racism', and 'uncritical habit of mind that justifies inequity and exploitation by accepting the existing order of things as given'...by claiming not to notice, the teacher is saying that she is dismissing one of the most salient features of the child's identity and that she does not account for it in her curricular planning and instruction. Such practices are based on liberal humanistic understandings of the universal child scientist "naturally" able to choose to be competent.

Students, too, have constituted subjectivities which are further constructed in mathematics classrooms, it is imperative, as we move into the twenty first century, that all students are able to engage themselves appropriately in the social practice that we know as mathematics and to come to know and speak its socially constructed "truths". Unfortunately, inappropriate teacher practices often work against students' solving problems and making sense of mathematics as they continue to control and circumvent investigation and inquiry. In the following section of this article I examine how teachers' subjectivities influence the ways problem solving and sense- making are realised in classrooms and how this can affect student enablement in the discourse.
 

Problem Solving

"Problem solving" as it is constructed in policy documents is closely allied to notions of free and competent individuals "naturally" inclined to explore mathematical ideas and construct knowledge. A further notion is that this is an empowering process for learners who will also naturally be able to solve problems in wider societal contexts. The Australian Mathematics Education Program (1982, p.3) states: Problem solving is the process of applying previously acquired knowledge in new and unfamiliar situations. Being able to use mathematics to solve problems is a major reason for studying mathematics at school. However, in examining more closely how problem solving is inserted into schooling sites, it becomes clear that many students do not find the experience enabling. The teacher's constituted subjectivity insinuates itself into classroom practice in ways that dampen the investigative and sense making impulses in students. Take, for example, the following problems. The first is a problem taken from Zolkower (1996, p.67) and given to a "remedial" group of bilingual students in El Barrio (East Harlem, New York City). Ten fourth grade boys and girls sit around the table with a teacher's aide to help them. Among the students are Kayla, who was horn in New York City, and Asia, who came from Jamaica six years ago. The teacher has just given workbooks to the students:

THE PROBLEM: What number belongs at the start?

START DIVIDE ADD SUBTRACT MULTIPLY END

?                14      39            18              by 6        216

HINT: Work backwards

THE TEACHER'S AIDE: Asia, they tell you to work backwards. I will give you another hint. What I'm giving you is 'work backwards' but don't do exactly what they're telling you to do. No, if you do exactly what they're telling you to do you're not gonna get the right answer. So, if you don't do that, what else can you do? Try the opposite. That's the only word I will say. Try the opposite of what they say.

ASIA: Miss, I got it!

KAYLA: I don't get it, miss!

THE MATH LAB TEACHER: If you take a minute and give not gonna get it in time...Kayla, why are rather? less than one you writing second backwards? It get it. You're You need some Upside down

 The teacher presents the problem to students. Various cultures are represented in the bilingual school though it is taken for granted that all students will equally become involved in solving the problem. Although the teacher does not remain physically present in the classroom, the activity s/he has chosen from a workbook imposes parameters on the degree to which the various students will engage with the activity. Certainly there appears to be little in the problem that would light the fire of student imagination, as in imploring students to "work backwards" it go against all they have learned about solving problems previously. Suspect gestures of support, the "hints" given by the teacher’s aide, merely serve to position the students as lacking in authoritative knowing as they attempt to uncover the one correct answer, the teacher's and textual "truth". In an attempt to please or perhaps in defiance, Kayla tries "the opposite of what they say to do" and writes upside down! As Zolkower (1996, p.85) states: By bringing out the nonsensical in the teaching machine, the student's view from below may be seen as an immanent critique of the pretensions of realism, suspect gestures of inclusion, and illusions of control which characterise cutting edge mathematical pedagogies. Where teachers are not able to question that part of their subjectivity that has faith in Piaget's child scientist "naturally" seeking equilibrium in a "natural" and supportive environment, they will continue to disadvantage students. This is because there is nothing natural about the environment for learning anything, the context is eminently social and constitutive of a person's subjectivity, not external, as some would suppose. The teacher has chosen a mathematical task s/he considers important; but rather than explicitly teach the reversibility of operations s/he assumes that students will "naturally" construct this knowledge for themselves.

Teachers sometimes attempt to make problems appear more inviting and relevant by "painting on" a context (Boaler, 1993). The context chosen is usually meant to reflect the "real world" though the practices depicted often do not mesh with students' lived experience. For example, Baroody (1993, p. 2-23) uses a home based, flagrantly sexist, scenario to illuminate a mathematical problem:

Sophie had baked a large quantity of cookies for the holidays. Her husband football Jelko ate half the cookies while watching Saturday games. He ate another quarter of the cookies while watching the Sunday pro-football games. Jelko sat down to watch Monday night pro football and asked for some cookies. Sophie, who was concerned about her husband's weight, tactfully noted he had already consumed 54 cookies over the weekend. How many cookies had Sophie baked? In whose "real world" do people talk about the number of cookies eaten as "half the cookies" or "another quarter of the cookies"? Other problems involve farmers counting heads and feet of animals to determine the number of hens and cows in their paddocks; people whose pay doubles day by day and runners who progress at a constant speed over several days. Students have to suspend all that they know, as constituted subjectivities, to engage with the unbelievable situations encountered in mathematical problem solving.

As long as we are to continue with the term problem solving we are going to produce disenfranchised learners. Where classroom practice is premised on reaching a solution, a "truth" it calls into play process which produce right/wrong answers, motivated/unmotivated behaviours and competent/incompetent students. If there is competition for the one correct answer, as in the two examples above, there will always have to be winners and losers. The teacher and text hold the authority and the student is classified as an "incompetent" problem solver where s/he does not arrive at the pre-specified answer. Inappropriate classroom practices are not questioned where teachers continue to defer to liberal humanistic notions of the child as "natural" problem solver in a "natural" environment.

All students, and teachers, are constantly striving to make sense of their lives according to the discourses that have structured their existence so far. However, discourses overlap and affect practices such that gendered discourses, or racist or classist discourses may determine largely how or if a student is likely to be fully involved in the mathematical enterprise. Disadvantaged students, those from culturally diverse backgrounds and girls/women may have distinctive, though not valued in classrooms, ways of making "sense of experience. When a student's attempts at sense making do not correspond to the authoritative ways of the classroom, s/he is excluded from participation in the dominant discourse. Davies and Hunt (1994) tell an interesting story of Lenny, an Aboriginal boy, who attempts to establish himself as a competent student in the Welcome Back Kotter style. Sadly, this reading of competence is not visible as such in the particular classroom in which he finds himself. A short excerpt from Davies and Hunt (1994, pp. 402-403) shows how Lenny's constituted subjectivity clashes with the established ways power/knowledge is realised in the classroom:

LENNY: (Lenny is seated... with his legs casually up on the desk, leaning back in his chair. He calls out loudly). Hey Miss, hey Mister Kotter Mister Kotter! (John reaches across in front of Lenny for some cuisenaire rods. Lenny re-positions himself slightly and looks in·the direction of the teacher'. No eye contact is achieved. The teacher does not acknowledge Lenny). Fuck. Mr Bloody Kotter, ya ( ) (Lenny shifts his legs down and sits in a 'good pupil’ position) Hey man, I want some work over here! Hey (bangs elbow on desk) Mr Kotter (Annoyed He leans back in his chair and puts his legs up on his direction). tone of voice. the desk. the teacher walks past while he is in the middle of this action, touches him lightly on the head as she passes and moves on to attend to Jenny) I want some work down here (he waves his pencil and bangs his paper on the desk).

TEACHER : When you sit quiet! I'll come and see you. (she moves back past him, tapping him slightly on the leg) Sit round so that I can ( ). When you sit nicely I'11 come back and see you....

 Lenny's subjectivity is constituted in such a way that he wants to take himself up as a cool, competent, "sweathog" type of student but such a student can not be heard in this classroom. A way of behaving that seems eminently reasonable or sensible to Lenny is not recognised as such in the schooling context and affects his chances of learning any mathematical knowledge at all. The context of learning in not "natural" or neutral!

The ability to make sense of/in mathematics is much more than a cognitive ability. It is a function the learner's subjectivity and the dynamics of the learning context. Recent policy initiatives suggest that teachers move away from direct teaching methods to approaches that are more inclusive of children's needs and desires. However, a poststructuralist view of knowledge recognises that participants are always actively involved in discourses such as mathematics, even when not reproducing the mathematical truths teachers would wish. The children are learning what it means to be a competent student, and so on. It is important that classroom practices, from which students learn so much, are productive of knowledges that are enabling. Problems can arise with the use of new materials, for example the multi-base arithmetic blocks, where teachers' use of the materials is structured by dated beliefs about mathematical knowledge and teacher authority.

Use of the blocks is often aimed at getting children to understand the base ten system of numeration. The blocks are used to do algorithms and to build numbers, to add or subtract ten and similar examples. However, it sometimes happens that the extent to which teachers orchestrate how the blocks must be used places a straightjacket on students' creative use of the blocks in understanding number in ways which are self generated and related to experience. Indeed, because the blocks have been constructed by adults and inserted into classroom, it is not clear to what extent children actually extract the base ten concept from the use of the blocks. Orton and Frobisher (1996, p.65) cite Boulton-Lewis and Halford that "although children can physically manipulate objects, and allocate appropriate names, they are not recognising the structural correspondence between concrete representations and the mathematical concept it was intended to illustrate..."

A second problem concerning students' sensemaking is that many teachers consider concrete experiences to be essential for all children at a certain stage of development. Deferring to Piagetian notions of "natural" development, some teachers consider themselves failures if their students cannot use the blocks efficiently and correctly at what they consider to be the appropriate stage. Zevenbergen (1996) mentions a teacher having problems with a nine-year-old student recently arrived from Papua New Guinea. The boy had no idea of how to use the concrete manipulatives of the Western classroom. The teacher was concerned (an aspect of her/his subjectivity) that even though the student could perform mathematically, he was not able to correctly use the materials. This proved disruptive of the smooth flow of the classroom, as the newcomer was not able to join in many classroom activities. In this case a teacher's constituted subjectivity regarding the necessary use of manipulatives at certain stages of development and her prior belief that a Papua New Guinean would "naturally" rely on concrete experiences threatens to position a most capable student as not able in this classroom.
 

Teacher Agency

A major problem we face is that teachers constituted through past discourses do not readily know how to position themselves as teachers with agency who know how to teach "against the grain". They have learned some content, but it is content that can nowadays be done more efficiently by machines. Their knowledge and the context of schooling-as-usual make it difficult for them to speak the mathematical knowledges considered relevant (Australian Education Council, 1990: Department of Employment, Education and Training, 1989; National Council of Teachers of Mathematics, 1989) to the twenty-first century. They have not experienced conjecture, exploration and enquiry as important elements of knowing mathematics as a social and intellectual practice. The currently established discourses in texts of problem solving approaches to teaching and sense-making on the part of students remains at the level of rhetoric rather than constitutive of practice.. I believe this is because these high ideals are incorporated into teaching- mathematics-as-usual based on competition, the one correct answer and teacher authority. Thus problem solving often becomes finding the correct answer as quickly as possible, sense-making means making the teacher's sense for a tick or good marks and technologies such as games and concrete materials are used m standardised ways to practise or consolidate some skill or procedure.

Teachers need to be competent mathematically and they need to be agentic. I concur with the new pedagogical discourses on the construction of mathematical ideas, connections and relationships and recognise as problematic the fact-that this discourse may have been absent in their schooling. I also take seriously the notion that they must be encouraged and supported in learning to orchestrate an investigatory discourse; they must develop the skills of questioning which keep the mathematical conversation alive and which do not cut off inquiry by asking closed questions. Just as importantly, to be agentic teachers of change, they must know how the discourse of mathematics currently operates to disenfranchise learners, how they themselves have been caught up in its operations, and how classroom uses of language and practices might be changed in ways that prove to be empowering for more students. Davies (1991. p. 51) states: "Agency is never freedom from discursive constitution of self but the capacity to recognise that constitution and to resist, subvert and change the discourses themselves through which one is being constituted". Teacher development programs are implicated here.

Perhaps it is important that in teacher development programs processes of subjectification are made visible. Perhaps, as teachers, we need to recognise how, in interaction, we collectively manage to categorise and classify each other into marginal or authoritative positions within all manner of discourses. In mathematics education, as we work together to construct the mathematical concepts, ideas and connections considered important for the twenty-first century, we might focus some discussion on the following concepts (adapted from Davies, 1994) which together try to encapsulate the kind of context participants actively create for one another: positioning, subject positions made available, and story-lines that are made relevant. For example,

If these teachers are to teach in investigatory ways, they will need to recraft their eyes to recognise the limitations of teaching-as-usual: to see how filling in worksheets, hearing tables, doing irrelevant problems where there is one correct answer, streaming, practising algorithms and formulae, position the student as always unknowing and the teacher as sole authority. Furthermore, because these classroom practices are all premised on the regulatory establishment and maintenance of constructed binaries of right/wrong answers and competent/incompetent students they effectively undermine any investigative impulses on the part of part of students or teachers. Teachers need to work together to think about classroom processes that might better facilitate investigation.

Many would argue that it is the large structures of society and the school that need to change to bring about pedagogical and social change. I have argued, as has Davies (1996) that we ignore subjectivity at our peril and that the larger structures will change only when we have our schools staffed with agentic teachers of vision and voice who, with eyes recrafted to recognise potential marginalisation and oppression and agency at the local level, act to ensure a positive learning experience for as many of their pupils as possible. Subjectivity, I would suggest, as well as constructed cognitive knowledge, significantly influences practice.
 

References

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