MATHEMATICAL ACTIVITIES
10.2 The relationship between mathematical and non-mathematical knowledge and/or practices
10.2.2 The ‘Mathematical Gaze’ and associated mathematical myths
The second aspect of Dowling’s argument regarding the incommensurability of mathematical and everyday knowledge, practice and participation involves the conception of a ‘mathematical gaze’ and four associated mathematical myths. The concept of the gaze and the associated myths are now discussed in detail.
Dowling (1998, p. 10) argues that when mathematicians look at the world and when they bring problems involving everyday contexts and practices into the classroom, they see the world and associated practices and forms of participation and communication through a mathematical lens or a ‘mathematical gaze’. In doing so, they structure the world and daily practices in the world according to mathematical principles and structures and participation criteria: they privilege the world according to a mathematical view (Dowling, 2009b, p. 26). They describe the world and the activities that take place in that world using specific mathematical terms and language and argue that mathematics is an effective tool for making sense of day-to-day practices and for describing the structure of and criteria for legitimate participation in the world:
It is as if mathematics were casting a gaze on people’s lives, reorganising them according to its own structures and then handing them back: you see how much better life would be if we were all mathematicians. (Dowling, 1995a, p. 4).
Mixing concrete becomes about using ratios (or possibly about conversions, or possibly about measuring quantities); painting a room turns into a consideration of surface area and paint conversion factors; running a marathon becomes about calculations involving distance, time and speed (with no mention of training, carbo-loading or steroid use); and
157 shopping is transformed into a practice involving percentages, rates, conversions and a whole multitude of other calculations and concepts (most of which are only calculable with the use of a basic calculator, always readily available and strategically located in a pocket, bag or brassiere). All in the world is calculable and all is able to be described using mathematical terms, symbols and models. As suggested by Ensor and Galant (2005):
It has been a matter of controversy whether, and to what extent, we can describe the practices embedded in routines of work and everyday life as ‘mathematics’, or whether these practices – in shopping malls, on building sites, in games, or arrayed in ethnic artefacts – become mathematical only because we cast a mathematical
‘gaze’ upon them and ‘see’ them as mathematical. (p. 293)
But herein lies the problem. As was discussed above, commonly in interactions between the academic and the everyday, it is the structure and regulative principles of the academic that is prioritised over the everyday (Dowling, 1995b, p. 221). The interaction between mathematics and everyday activities is no different: mathematical structures, principles, knowledge, and forms of communication and participation are generally prioritised and fore-grounded ahead of everyday considerations, to the extent that commonly used (and efficient) everyday forms of communication and practice are no longer deemed as sufficient or appropriate for describing real-world practices. Instead, deliberately selected mathematical activities, symbols, knowledge and language take priority and participation in the everyday practices are now legitimated according to mathematically structured criteria:
I do not intend to claim that, in its origins, mathematics has no connection with the empirical; that would be absurd. But, despite all of its referral to the ‘real world’ − which I take to mean the world beyond mathematics per se − the mathematics curriculum, the non-arbitrary esoteric domain, is primarily constituted as self- referential, self-contained. The ‘real world’, wherever it appears in a mathematics lesson or test must be made to conform with abstract mathematical structures.
(Dowling, 2009b, p. 10)
The result is a ‘recontextualisation’104 of the everyday practice and a reconstituting of the practice as a “virtual reality, a mythical domesticity within which all is rational and all is
104 The term ‘recontextualisation’ is used deliberately by Dowling, and others like Ensor and Galant, to emphasise the distinctive transformation that must take place when moving from one domain of practice and/or activity to another. In particular, Dowling uses the term to refer to the process where the practices of a particular activity are subordinated to the principles of another (2009a, p. 19; 2010c, p. 1). This term links directly to Dowling’s concept of the ‘principle of recontextualization’ referred to on page 152 above.
Notice that Dowling uses the term differently to the conceptualisation of ‘recontextualisation’ used by Bernstein. For Bernstein, recontextualisation refers to the rules or procedures by which a form of educational knowledge is moved from one site to another (Singh, 1997, p. 7): “Through
recontextualisation, a discourse is moved from its original site of production to another site, where it is altered as it is related to other discourses.” (Singh, 2002, p. 573). For example, a teacher in a classroom will decide on how a particular section of the curriculum must be sequenced, what must be emphasised, and what must be evaluated at the end of the learning process. The teacher has recontextualised the original curriculum according to a particular set of structures to facilitate the teaching and learning process. For Bernstein, recontextualisation of knowledge from one site to another brings with it altered relations of power (classification) and control (framing), which in turn affects the ideological meaning that is attached to the knowledge in that site (Singh, 1997, p. 7).
For a more detailed discussion on the distinction between Dowling and Bernstein’s usage of the term
‘recontextualisation’, refer to Dowling (2009a, pp. 14-16).
158 calculable.” (Dowling, 1998, p. 33). Dowling refers to this virtual or mythical reality as the Public Domain of school mathematics105, a domain that contains a collection of everyday sites and activities that have been transformed by mathematics (Dowling, 2010c, p. 1). The construction of this Public Domain of mathematics entails the casting of a gaze from the Esoteric Domain of mathematics over an aspect of the (non- mathematical) everyday world and the recontextualisation of that world according to mathematical structures, hierarchies, principles, forms of communication, and participation criteria (Dowling, 2009b, p. 26). As Ensor and Galant (2005, p. 293) suggest, “This process of recontextualising denatures these everyday activities, subordinating them to the pedagogic imperatives and internal structuring of school mathematics.” Similarly for Skovsmose (1994a):
If that thesis is acceptable [regarding the central role of reflective knowing in the development of critical mathemacy106] it means that most of the epistemological approaches used in interpreting phenomenon in mathematics education are misguiding or at least biases in concentrating on mathematics, ignoring the conditions for the genesis of reflective knowing. (p. 48)
The dilemma with this imposed mathematical gaze is that the image of the everyday practice that is presented in the Public Domain of the mathematics classroom is not real;
it is a mythical view of what would actually happen in that situation and of how people would actually think, behave, participate and communicate in the situation. As Dowling (1998, p. 33) suggests, “But it wouldn’t be better, because mathematised solutions always fail to grasp the immediacies of the concrete setting within which …, problems and solutions develop dialectically.” Mathematics is not shopping and the techniques, considerations, resources, and forms of communication used and needed to make sense of problems in the classroom compared to in the shops and other everyday activities are often different and unrelated. In short, the criteria according to which successful and endorsed participation in problem situations encountered in the mathematics classroom is legitimised are completely different to the legitimisation criteria for successful participation in everyday practices. Furthermore, while mathematicians may label certain everyday practices as involving mathematics, it is questionable whether the people engaged in everyday practices would constitute their activity in mathematical terms and whether they will make use of formal mathematical techniques and knowledge in solving problems related to those activities. As P. Dowling (2008, p. 4) suggests, “Whilst the esoteric domain objective is mathematically legitimate, the public domain message is suspect, to say the least; … . You might learn mathematics like this, but you’re going to get a naïve view of the nonmathematical world that it recontextualised as its public domain.” What we are left with, according to Dowling, is a powerful mythologising:
namely, that mathematics can be used to generate an accurate and realistic understanding of everyday practices and forms of legitimate participation in such practices. The reality, however, is the reverse: the image of the everyday practice that is presented in the mathematics classroom is not an accurate reflection of how that practice is experienced or engaged in real life or of the knowledge, techniques and considerations that influence the structure of participation in that practice.
105 And, hence, we have now travelled full circle. Namely, from the discussion at the beginning of this chapter of the different Domains of Mathematical Practice that characterise the structural level of Dowling’s language of description, to this discussion of how the Public Domain of mathematical practice represents a particular relationship between mathematical and everyday practices and, particularly, the mythologising of the everyday practices.
106 See Part 4, Chapter 14 and 14.4.5.1 below (starting on page 230) for a detailed description of the notion of ‘reflective knowing’.
159 It is also important to point out that different individuals employ differing generative principles in the construction of a mathematical gaze for a particular recontextualised practice. For example, while a teacher may legitimise participation in calculations involving concrete quantities through engagement with the mathematical structure of ratios, a different teacher or a textbook author may legitimise participation in the same practice through engagement with the mathematical structures of measurement and volume calculations. This is significant in that it highlights that it is the individual who imposes a mathematical gaze on an everyday practice who controls and defines the criteria, principles and structures on which the mathematisation process is based. As a result, any participants who are expected to engage with a recontextualised and mathematised practice are reliant on the person who conducted the mathematisation process to make explicit and visible the generative principles and structures of the practice. And, as has already been discussed, this has implications for participants who engage exclusively in Public Domain practices: such participants are utterly dependent on a Subject in the practice (e.g. a teacher) to make explicit the regulative principles and criteria of these Public Domain practices.
The imposition of a mathematical gaze on everyday non-mathematical practices gives rise to three ‘mathematical myths’ about the practice and also about the relationship of the participant (who is imposing the gaze) to the practice. These myths include the Myth of Reference, the Myth of Participation, and the Myth of Emancipation. Each of these myths are now discussed in detail.
10.2.2.1 Myth of Reference
Mathematics is mythologized as being, at least potentially, about something other than itself. (Dowling, 1998, p. 4).
The Myth of Reference describes the widely held assumption that mathematics can refer to activities and practices other than itself and can be used as a tool for making sense of and describing those activities and the structure of participation in the activities (P.
Dowling, 2008, p. 1). In Dowling’s (1998, p. 6 & 16) terms, mathematics is seen as a set of ‘exchange values’, something that can be used to cast a commentary on non- mathematical activities. The myth encourages us to believe that it is possible to move between two spheres, one of which is always mathematical. As Dowling (1998, pp. 6-7) points out, though, it is always the mathematics casting a commentary on something else and seldom the other way around.
Importantly, when mathematics is employed to make sense of other activities, the recontextualised activity is presented as being real, as an accurate and realistic reflection of the actual practice and of participation in the practice. This process denies that the recontextualised activity is something less than real. For example, in the mathematics classroom, shopping is presented as an activity that really does involve ratio and rates.
And herein lies the myth: when mathematical principles are used to make sense of non- mathematical activities in the classroom, the result is always a “colonising of non- mathematical activities” (Dowling, 1998, p. 33), with the non-mathematical setting consumed by the mathematics and leaving behind only a trace that there is something outside of the mathematics (Dowling, 1998, p. 16). As such, the image of the everyday activity that is recontextualised in the mathematics classroom is not authentic or realistic
160 and does not give accurate consideration to how people actually act, think, behave and communicate in that activity in daily life. Rather, the recontextualised image of the everyday is the image as seen from the perspective of the mathematician, seen through the ‘mathematical gaze’. As Dowling (1995a, p. 9) argues, “However, they [the mathematical/scientific texts] are no more likely to generate plausible solutions to everyday practical problems, because the everyday is not structured according to mathematical principles.” The myth, then, is that “the descriptions resulting from casting a mathematical gaze upon the world are indeed about that which they appear to describe.”
(Christiansen, 2007, p. 98).
As a final comment, it is worth noting that the Myth of Reference operates through an explicit recognition and privileging of mathematical principles and forms of participation over any real-world considerations: for example, the appropriate use of ratio and proportion is privileged over any real-world considerations in a best-buy shopping scenario. As such, although the Myth of Reference views mathematics as a tool that can be employed to make sense of any real-world situation, the movement is always towards the Esoteric Domain of mathematics and away from the real world (Dowling, 2001, p.
22). The Myth of Reference, then, can be associated with a move beyond the Public Domain towards the Esoteric Domain.
Importantly, the “myth of reference is distributed to the apprenticed voice” (Dowling, 1998, p. 295) and, so, is associated with the Apprentice position. Students who are apprenticed into mathematics are given direct access to the Esoteric Domain content and practices of the subject. This Esoteric Domain content is always prioritised over Public Domain content and/or contexts, and the inclusion of Public Domain contexts serves purely to draw apprentices into the activity. In this way, the ‘mathematical gaze’ is inculcated into the apprenticed practice, voice and position such that apprentices are given the means and the control to decide and determine the criteria according to which a mathematical gaze is to be cast and the generative mathematical structures and principles that define the gaze (Dowling, 1998, p. 292; 295). In short, it is the Apprentice and Subject of the activity who are able and who believe it appropriate and justifiable to cast a gaze beyond the domain of mathematics on the extra-mathematical world.
10.2.2.2 Myth of Participation
Nevertheless, school mathematics frequently presents the myth that its ‘real world’
applications constitute a necessary condition for adequate domestic practice and, indeed, for adequacy in a whole range of other non-mathematical activities. I have referred to this myth as the myth of participation. (Dowling, 2009b, p. 26)
The Myth of Participation of mathematics claims that mathematics is a universal tool and that understanding of mathematical concepts and techniques allows us to understand and control the world (P. Dowling, 2008, p. 1). Mathematics is, thus, presented as a necessary pre-condition for effective participation and functioning in everyday domestic practices:
“mathematics is a necessary supplement to what the student already knows if they are to optimise their own lives.” (Dowling, 2010b, p. 8).
This view of mathematics promotes a mythologising in two ways. Firstly, this view is simply an extension of the ‘mathematical gaze’ discussed above. By claiming that mathematics provides a means for making sense of the world and for understanding everyday practices, everyday practices are scrutinised though a mathematical lens,
161 mathematical structures are imposed on those practices, and effective participation in those practices is only deemed possible if the imposed mathematical structure is understood (Dowling, 1998, p. 10). The everyday practice is transformed or recontextualised into a ‘virtual reality’ which, although it may bear some resemblance to what would actually happen in daily life, is still a mathematised and distorted view of reality. Hence, the everyday practice presented is a mythical image of the reality.
Secondly, this view is a myth precisely because many people are able to function perfectly well in the context of everyday or domestic practices without understanding the mathematical concepts or techniques that the mathematical gaze imposes on such practices. As Dowling (1998, p. 10) suggests, “mathematical skill is neither necessary nor sufficient for optimum participation within these [everyday] practices”. You do not need to understand ratios and rates to be able to complete your shopping, nor do you need to have an understanding of surface area to be able to paint a room.
In contrast to the Myth of Reference – where there is an explicit privileging of mathematical principles over real-world considerations and where there is a direct move beyond the Public Domain, the Myth of Participation draws participants into believing that the context is real and they have a direct participatory role in that context (i.e. are
‘objectified’ by the context (Dowling, 1998, p. 144)). As such, the Esoteric Domain origin of the gaze is hidden and there is no deliberate movement beyond the Public Domain (Dowling, 2001, p. 22). The Myth of Participation facilitates and is in turn facilitated by practices associated exclusively with the Public Domain.
Whereas the Myth of Reference is distributed to Apprentices – who are given control over the generative principles and structure of the mathematical gaze, the Myth of Participation is distributed to participants positioned as Dependents (and associated dependent voice in pedagogic texts) in the domain of mathematical practice (Dowling, 1998, p. 295).
Students who are exposed primarily to Public Domain content and contexts and to the fragmenting and localising strategies that characterise practices in this domain are commonly subjugated to the position of Dependent or Object. Students are lead to believe that the problems that they are solving are about real life (rather than mathematics), that the problems provide a realistic map of the everyday world, and that they are active participants in the problem-solving process. They are invited to identify with the problem and the characters in the problem, as though the problems and contexts are their own; and they are invited to provide opinions and to suggest strategies for solving the problems.
Students are further led to believe that understanding of mathematics provides the key to solving these problems and for making sense of the contexts in which the problems are situated. This is precisely the Myth of Participation. Participants in Dependent or Objectified positions, then, are directly inculcated into the Myth of Participation (Dowling, 1998, p. 250; 293; 295).
10.2.2.3 Myth of Emancipation
Revealing the truly mathematical content of what might otherwise be regarded as primitive practices elevates the practices and, ultimately, emancipates the
practitioners. This is the myth of emancipation. (Dowling, 1998, p. 15).
The Myth of Emancipation – considered by Dowling (2001, p. 20) to be a globalized version of the Myth of Participation − is commonly associated with the term
‘ethnomathematics’ and with research that strives to celebrate the already existence of
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.