MATHEMATICAL ACTIVITIES
10.3 The ‘production of ability’: considerations of educational disadvantage
162 mathematical content and techniques within the practices of different cultural groups (Dowling, 1998, p. 12). In this view, (externally imposed) analysis of the practices of different cultural groups will reveal the existence and usage of mathematical structures and concepts by the practitioners in those practices (Dowling, 1998, pp. 11-12). So the weaving patterns used on baskets constructed by Xhosa women are seen to indicate conceptions of transformation geometry (translations, rotation, reflection, and enlargement). In this view, the universal language of mathematics already exists in these practices, and all that remains is for that mathematics to be extracted and revealed to the practitioners in those practices. Revealing the underlying mathematics elevated the status of the ‘primitive practices’ and, therefore, emancipates the practitioners of these practices from their primitive understanding and/or existence (Dowling, 1998, p. 15).
This view, argues Dowling (1998, p. 33), also presents a mythologised view of mathematics: “the myth of emancipation frequently mythologizes diverse non-industrial cultures.” This myth is revealed in two ways. Firstly, this view is once again simply an extension of the ‘mathematical gaze’ scenario – and, so, is inculcated in and distributed from the Apprentice (and Subject) position. When a practice specific to a certain culture and which takes place in a certain context is analysed and deemed to contain mathematics, it is inevitably European mathematical principles, recognition symbols and participation criteria that are imposed and used to provide a language of description of those practices.
And so, Europeans look at the cultural practices with a ‘mathematical gaze’ and through a distinctly European mathematical lens and then claim that the mathematics was always existent in those practices (Dowling, 1998, p. 15). The myth, then, is that rather than celebrating non-European cultural practices and forms of participation, the imposition of the ‘mathematical gaze’ results in a recontextualisation of the practice according to foreign, European structures (Dowling, 1998, p. 17). Non-European practices are re- described and consequently suppressed using European structures. Added to this, Dowling (1998, p. 17) argues that this view is driven not by an intention to promote the cultural practice itself (for example, basket weaving) but rather by a focus on elevating mathematical structures and principles (for example, the learning of transformation geometry).
The second component to the mythologising of this emancipatory perspective relates to the claim within this view that revealing the mathematics in such practices emancipates the practitioners of those practices. By means of challenging this view: will teaching basket weavers about transformation geometry make them more efficient, capable or successful weavers?; is an understanding of transformation geometry essential for being an effective and successful weaver (the Myth of Participation)?; and, crucially, is it only through the internalisation of the European view of such practices that emancipation is achieved? Perhaps Dowling’s response to these questions would be that mathematics is neither a pre-condition or necessary component for the successful participation in such practices and, rather, that such cultural practices have existed successfully for centuries without imposed European knowledge structures and interference.
10.3 The ‘production of ability’: considerations of educational
163 of the different domains of mathematical practice to different groups of students in the teaching and learning of mathematics.
Dowling (1998, p. 51) coins the term ‘production of ability’ to highlight the perspective that “the curriculum does work in order to recontextualize these essentially non-
educational differences [in class, race and gender] as differences in educational
attributes and performances.” In other words, Dowling argues that the notion of ‘ability’
is a constructed notion that has more to do with social difference than with an actual attribute. As a result, differentiation according to ability and, hence, the preservation of social difference and disadvantage is reinforced through curriculum structure and the types of knowledge made available to students from different racial, class and gendered backgrounds (c.f. Dowling, 1998, pp. 49-69).
Dowling presents his argument through an analysis of two mathematics textbooks, one of which is aimed at supposedly higher ability students (who stem from predominantly middle-class backgrounds) and the other at supposedly lower ability students (who stem from predominantly working-class backgrounds). From the results of this analysis, Dowling (1995b) is able to make the following accusation:
The tendency of the two textbook series to specialise their modes of mythologising constitutes a distributing strategy. ‘Higher ability’ students are apprenticed into descriptive mythologising; ‘lower ability’ students are provided with participative mythologising. (p. 219)
‘Descriptive mythologizing’ in this statement refers to esoteric or academic domain activities comprising vertical discourses and exhibiting high discursive saturation (DS+), of which school mathematics is one such practice. The word ‘descriptive’ is used here to infer that students participating in such activities are given access to the regulative principles and structures of the activity through engagement in Esoteric Domain practices. This process affords apprenticeship into the domain, inculcating participants into the Myth of Reference. This process further enables participants to cast a gaze on the world and to generate descriptions of the world using these ‘privileged’ principles and structures. ‘Participative mythologizing’, on the other hand, refers to activities comprising horizontal discourses and exhibiting low discursive saturation (DS−), a feature characterised by everyday domestic activities (Dowling, 1995b, p. 219).
Students involved in these activities are invited to ‘participate’ in the activities and are indicted into the Myth of Participation, but are never given access to the regulative principles and structures underpinning the activities. Such students are not given access to the tools that enable them to cast mathematical descriptions on or of the world and, as such, are relegated to positions of Dependent or Object in the learning process. In emphasising this distinction, Dowling (1995a, p. 7) then asserts that within the context of the schooling system, curriculum associated with Esoteric Domain knowledge – imbued with various forms of descriptive mythologizing − are commonly made available to high ability students commonly in better resourced schools located in middle and upper class environments. By contrast, curriculum focussing on relevance – imbued with various forms of participative mythologising − are made available
primarily to low ability students commonly in poorer resourced schools located in working-class environments. This sentiment is echoed by Bernstein (1999):
164 When segments of horizontal discourse become resources to facilitate access to vertical discourse, such appropriations are likely to be mediated through the distributive rules of the school. … These insertions [of horizontal discourse in vertical discourse] are subject to distributive rules, which allocates these insertions to marginal knowledge and/or social groups. (p. 169)
In the South African context, this situation would translate into Core Mathematics (abstract scientific mathematics) being offered to higher ability students and Mathematical Literacy (mathematics in the world) offered to lower ability students, which is precisely the trend in the majority of schools throughout the country.
My argument is that everyday, or horizontal, practices constitute us all and that this is unavoidable and inevitable. Academic, or vertical, practices have been
systematically distributed on class and racial lines, however. This has entailed the effective exclusion of the majority of the populations of both South Africa and Europe from the academic. This is variously achieved via the non-existence or inadequacy of schooling provision or, more subtly, by the insistence of the
inclusion of the everyday and the relevant in terms of participative mythologising.
(Dowling, 1995b, p. 222).
The insistence on ‘relevance’ for optimising participation in everyday (DS−) practices, then, is only seen as a fundamental criterion for lower ability and/or working-class students: “‘Low ability’, by contrast, tends to be constructed as demanding residence in the public domain.” (Dowling, 2009a, p. 31). Dowling (1995b, p. 219) cites the dilemma with this differentiation of curriculum as comprising two components. Firstly, the
‘relevance’ is actually not real; rather, it is a mythical relevance that has been constructed through the imposed lens of the ‘mathematical gaze’ and the associated Myth of Participation. Secondly, the regulative or evaluative mathematical principles underpinning these everyday ‘relevant’ activities are generally hidden or rendered invisible to the students who, instead, are positioned as Dependents that are reliant on the teacher to make these principles explicit. For example, in a question that asks students to calculate the quantity of concrete that a builder needs to make to fill a certain portion of the foundation trench of a house, the underlying mathematical concept that students are expected to make use of is volume. This concept, however, is not explicit and, rather, students have to make the transfer between ‘quantity of concrete’ and volume. The result of this two-fold dilemma is that students come to experience ‘relevant’ problems with a skewed, mythologised, mathematised impression of the problem and without developing a realistic or authentic understanding of either the everyday practice or of the mathematical content inherent in the practice. As Dowling (1995b, p. 219) suggests,
“‘Lower ability’/working class students are, thus, provided with ‘relevance’ at the expense of either mathematical or everyday use-value.”. In short, while drowning in mythical relevance and distorted reality, at no point are the working-class students, supposedly of lesser ability, afforded the opportunity to explore the mathematical concepts that provide access to top-end careers.
By contrast, students engaging with academic or esoteric mathematical content and texts (DS+ practices), comprising primarily higher ability students drawn primarily from middle-class environments, are given direct access to the regulative principles underpinning mathematical activities: “‘High ability’ is therefore constructed as meriting entry into mathematical discourse.” (Dowling, 2009a, p. 31). For example, consider a question relating to factorisation: students are told the type of category of factorisation into which the specific expression falls and are shown appropriate methods for
165 determining the factors of the expression, and at no stage in working with the problem it is unclear or hidden as to what is required to make sense of the problem. Furthermore, although everyday contexts may be incorporated into lessons involving academic or esoteric content, it is always evident to the student that the context is secondary to the mathematical principle, and at no point are students led to believe that the everyday is something other than imaginary or contrived. Students involved in such academic mathematical practices, then, have both access to the mathematical content and control over the mathematical gaze: they understand that the reality of the everyday is different to the mathematical calculations and contexts that they are concerned with. The result is that “‘High ability’/middle class students are thus to be apprenticed into academic mathematics and into the principles of the descriptive gaze.” (Dowling, 1995b, p. 219), all of which are seen as essential traits of high-end professions comprising, amongst other, engineering, medicine, architecture, and economics.
This sentiment is shared by Hoadley (2007) who, through analysis of classroom data from four schools situated in middle-class environments and four schools in working-class contexts in Cape Town, makes the following observation:
This study shows how students in different social-class contexts are given access to different forms of knowledge, that context-dependent meanings and everyday knowledge are privileged in working-class context, and context-independent meanings and school knowledge predominate in the middle-class schooling contexts. (p. 682)
The structure and distributive rules of the schooling system, thus, translate and provide (and limit) access to different types of knowledge, participation, communication, and mythologizing to students from differing class and social backgrounds, hereby producing and reinforcing educational disadvantage and inequality. And all of this is achieved under the guise of differential ‘ability’.
In short and in summary, the emphasis on relevance in mathematics, then, has created a differentiated curriculum that reinforces social division and difference by only making certain aspects of the curriculum available to different social groups. And so, according to Dowling (2010a), an emphasis on relevance in mathematics serves to reproduce social class divisions:
In my analysis of UK junior high school texts, I found that texts directed at ‘high ability’ students moved from the public domain to the esoteric and back (via the descriptive), apprenticing them into new knowledge. Texts directed at ‘low ability’ students remained in the public domain, so that these students were confined to a culture that comprised recontextualised versions of what they already knew; the text mythologised their lives. Furthermore, it was apparent, by an analysis of both the content of the public domains in the different books and of the physical form of the books, that social class was a key indicator of ‘ability’.
We might say, then, that school mathematics functions as a device translating social class into ‘ability’. (Slide 2)
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