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 :
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.
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.
THE PROBLEM: What number belongs at the start?
START DIVIDE ADD SUBTRACT MULTIPLY END
? 14 39 18 by 6 216
HINT: Work backwards
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
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:
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:
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....
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.
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,
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
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