Abstract
A numeracy education which enables people across different socio-cultural groups to develop and participate in more numerate discourses is needed. In articulating this need, and thinking about how such discourses may evolve, the paper will focus on the the pervasiveness of mathematical models in our socio-political spheres. In analysing the nature of the presence of mathematics in people’s lives, the paper views mathematics as a form of technology, and suggests that numeracy ought to be seen as part of a broader critical technological literacy.

 
Introduction
It has been said that "structural engineering" is the art of modelling materials we do not wholly understand into shapes we cannot precisely analyse so as to withstand forces we cannot properly assess in such a way that the public at large has no reason to suspect the extent of our ignorance.

[The "technical approach" assumes that] experts have a predominant role in decision-making and citizens are see as ‘consumers’ who are incapable of exerting ethical or practical concerns about the environment ... [while the political approach] adopts a critical view of industrial market society with its growth imperatives and focuses on alternative economic and social strategies which may involve less exploitative values towards the environment.

When we talk about risks, I give them numbers. They give me sociology. There is NO discourse.

 
The first quote appears in a set of guidelines for construction engineers published by the professional body of Australian engineers (Institution of Engineers, Australia, 1994). The second appears in an article that contrasts the technical approach with a political approach in social impact assessments of technological projects (Craig, 1990 cited in Norrie, 1990, p 31). The third was a statement made by an engineering colleague in a conversation about different discourses about public risk.
 

What these quotes suggest is that approaches to thinking about some of the significant problems which impact on the whole of society are classified as either technical or non-technical, and that these approaches are incompatible. Assumptions are also made about the superiority of one over the other, depending of course, on whether the person making that judgement belongs to the "technical" or the "social/ non-technical" community. There is also a sense of futility in trying to engage people in a shared discourse across the technical/ non-technical divide.

An area where the lack of a numerate discourse is noticeable is in public disputes about health risks associated with technological developments such as cellular phones and high voltage transmission lines. These disputes may be eventually "resolved" at the legal or policy levels, but often are not resolved at the socio-cultural and personal levels. The community is accused of arguing along "emotive" lines; the technologists along reductionist and complex mathematical lines. Another example is the lack of meaningful dialogue between managers who concoct quantitative (and often complex) workload and performance measures, and those whose workplaces are prescribed by these models of "work". Is it not the role of numeracy educators to be concerned about the absence of discourse between "communities of mathematical practitioners" from different backgrounds, and to address the gap in our numeracy education which may be contributing to this.

This paper emerges out of a reflection of my work as a teacher of mathematics in engineering courses, and of numeracy and numeracy education in adult basic education (ABE) teacher training programs at a university in Australia. The unusual combination of student groups in my work has presented me with the challenge of looking at numeracy more broadly than something which resides in ABE alone, and finding ways of theorising numeracy in ways which make real links with critical education.

My purpose in this paper is to examine ways in which mathematics "works" in society, and to consider ways in which discourses can be created around these workings towards inclusive social actions. I will argue that looking at mathematics as a technology is useful in this regard, and that therefore numeracy might be viewed as an aspect of technological literacy, which in turn needs to be conceptualised within the framework of critical education.

 
How does mathematics work in society?

Skovsmose (1994) talks about mathematics having a "formatting" power. He expands on the central role played by mathematical models in "giving not only descriptions of phenomena, but also by giving models for changed behaviour" through technologies, including energy, social and information technologies (p 55). Davis (1991, p 2) talks about the descriptive, predictive and prescriptive functions of mathematical models, and how their "ability .. to provide frameworks of reality and of action, and .. to change what is, is very great". So when we talk about the social power (and sometimes violence) or what Skovsomse calls the "formatting power" of mathematics, we are talking about mathematical models, such as economic models, models of risks, models of work and school performance.

Skovsmose shows how mathematical models underpin technologies, and how those technologies in turn exercise social and political power in society. I would suggest that mathematics itself can be viewed as a technology, although it is clearly not associated with an artefact such as computers are with information technology, solar panels are with power generation technology, and ticket machines are with transport technology. It is, as Postman calls, an "invisible technology" (1993). And in its power to create distances between people with different mathematical experiences, it is as Porter calls, a "technology of distance" (1995). When we examine the ways in which public safety and risks, welfare support and access, workplace performance, definitions of "full-time" employment, intelligence levels are prescribed by "numbers", both of the technological metaphors of mathematics seem appropriate.

 
Mathematics as an invisible technology

Mathematical models, as experienced by the community at large is an invisible technology. It is invisible not only in the actual numbers which the models produce as "solutions", but invisible in its derivation. Figure 1 shows a model of a mathematical modelling process. Risks, pay rises, new tax rates, cut-off scores for university entrance are all prescriptions which mathematical models generate. These models are constructed by people who want an answer to a problem they see, based on their assumptions, using methods that they deem appropriate. What the general public sees is typically the answer only. Rarely do they even see the original question that drove the modelling process. Yet, they become co-opted as consumers of these model generated solutions.

The public’s relationships with mathematical models is akin to the public’s relationships with many other technological developments. Many of us find ourselves in the position of having to "upgrade" our perfectly functional computers to more powerful, faster and bigger machines to accommodate the software which will replace the perfectly functional software that we have been using. We may not have seen or felt any problems with our software or the computer, but a "solution" to someone else’s problem is imposed upon us, and we have little choice but to upgrade our(?) technology.

There are some mathematical models which produce solutions which the public will reject. An example of an area in which this occurs is in decisions about personal and environmental risks. In many of these cases, the community is also ignorant of the mathematics which has gone into measuring the risk factor of, say, a technological development. The question may be assumed to be whether or not use of cellular phones causes brain cancer, or whether or not living under high voltage transmission lines causes leukaemia. But typically, the specific questions which drive the decision making processes for the different interest groups are quite different. What is rejected by the community may not really be what they thought they were rejecting.

The community is typically looking at risk versus no risk; safety versus danger. They are searching for "answers" which would give them some definitive assurance or a disaster signal. The technical experts are looking at minimising risks within a complex "systems framework", not focused on the particular individual incidences, but on statistical information and subjective probabilities based on their assumptions and knowledge. But within that systems framework, there is never the possibility of achieving a zero risk solution (unless the project is brought to a halt altogether). The critical issue here is not whether one approach is more valid than the other; rather it is the invisibility of the magnitude of differences in motivation behind the questions being asked by the two groups.

To resolve the conflict between the two groups, it is not sufficient, nor is it appropriate to focus only on the "solution" that the expert’s model produce. Ideally, there would be a process of developing a shared understanding of the question, engaging in a shared process of modelling the problem - mathematically or otherwise, which includes an agreed set of assumptions, using a methodology which takes into account information which all parties grow to accept as relevant to the problem. An answer, or a set of answers thus derived would not only be better understood by all concerned, but also have shared ownership. So long as the process of modelling employed by the technical experts are invisible, especially in the early stages where the question itself is being specified, and assumptions are made, this shared understanding and ownership (and with it responsibility) of the decisions based on the models would not be possible.
 

Mathematics as a technology of distance

How does the technology of mathematics create distances within social groups? It is often said that the perception of objectivity of mathematics also contributes to the lack of contestation against mathematically prescribed decisions. Porter puts forth the thesis that objectivity has "to do with the exclusion of judgement, the struggle against subjectivity. ... this, more than anything else accounts for the authority of scientific pronouncements in contemporary political affairs. ... In science, as in political and administrative affairs, objectivity names a set of strategies for dealing with distance and distrust" (p ix). He presents a thesis that the high level of discipline in the discourse of (formal) mathematics helps to make it something that is well suited for communication that "goes beyond the boundaries of locality and community" (p ix). But where does it go?

The authority of mathematics, based on the perceived objectivity of what "truths" it can convey, is at one level, a critical part of explaining how mathematics creates a distance (and in some cases gulfs) between the "haves" and "have nots" of mathematical knowledge. But it doesn’t fully explain how mathematical models work as a technology in the wider community. More maths will not necessarily enable people to challenge expert claims on risk and safety, economic policies, or workload formulae. In order to understand this better, it helps to look at some of the theories of technology.

Kranzberg (1997) illustrates a constructivist view of technology with a set of six "laws" which he (unashamedly) calls "Kranzberg’s Laws". Two of these laws are helpful in explaining how mathematics can be seen as a technology of distance. They are his -

Third Law: Technology comes in packages big and small. (p 10)

and

Fourth Law: Although technology might be a prime element in many public issues, nontechnical factors take precedence in technology-policy decisions. (p 11)

The Third Law seeks to explain how a technological innovation is embedded within complex systems, which in turn interact with other systems. He presents as an example, the Ford assembly line. One could view the assembly line as composed of different technological elements, such as conveyor lines, which are integrated within a comprehensive system (p11). The assembly line itself can also be seen as an integral part of the "technology" of manufacturing systems, which exists in particular ways within a larger socio-economic system. This then leads to his Fourth Law which explains how technological capability is only one of many factors which determine the ways in which technological infrastructures and policies are realised in society. These include the nature of the economic systems, dominant social values, and environmental contexts which determine the location of the technological system within the society. In terms of how mathematical models are embedded within a tight and socially non-inclusive system, the following quote about a dominant ethos in engineering is illustrative -

Systems engineering is a choreographed "dance" between the client and the project team. The client is the person/organisation who "owns" the project. It is the client that has the need for the results of the project, who will pay, and who will put the results of the project into operation. The project team is the technical group that will actually develop the system. The client knows what they want (or at least would like to think that they do), while the project team knows how to build it (or at least claims competence in building such systems). Systems engineering is the process of matching the client’s understanding of what is needed with the technical competency necessary to build a complex system. (Drane & Rizos, 1998, pp 17-18)

What this definition of systems engineering shows is that it is not the lack of mathematical knowledge which makes technological decisions impenetrable for the general public. It is the way in which problems/projects are conceived within a client/patron relationship which has little visible accountability to the society at large. Gilchrist (1995) reveals another example where an Australian Government bureau’s mathematical model of greenhouse gas emissions led to misguided policy recommendations, because it was not immediately revealed that the model was funded by major Australian coal producers whose immediate interests clearly would not have been served by tighter greenhouse policies. While a reasonably sophisticated level of mathematical knowledge would most likely be needed to decode the workings of the model, both technical and non-technical details of the model were unavailable for scrutiny. The public in these cases are therefore distanced firstly and more significantly by the boundaries of the "system" agreed upon by the technical/economic interest groups, than by the possible lack of mathematical knowledge. In these cases, "more mathematics" would not help to penetrate the boundaries set by the patron/client systems.
 

Penetrating the impenetrable - the role of numeracy education?

[Numeracy] is a social consciousness reflected in one’s social practices which bridges the gap between the world of academic maths and the real world, in all its diversity. The consciousness enables one to challenge the boundaries and the role of mathematics in social contexts. For this reason, numeracy is not linked to any level of knowledge in the hierarchy of academic mathematics; there is a need for numeracy associated with all levels of mathematical knowledge. Further, being an ingredient in the expansion of social justice, if it isn’t political, it’s not numeracy and if it’s not in context, it’s not numeracy.(Yasukawa & Johnston, 1994, p 198)

 
A "noble" definition of numeracy. But how can numeracy education tackle an invisible technology which excludes and distances the community from decisions and policies which affect all of us? What does it mean for numeracy education to expand social justice? What pedagogical frameworks are available to help us find direction in critical numeracy education?

For some years now, there has been effective "translation" of constructivist learning theories into effective mathematics pedagogy by applying what has been observed in socio-cultural studies of mathematics and mathematics education (Lave, 1988; Nunes, et al, 1993; D’Ambrosio, 1995; Walkerdine, 1990). Teaching in contexts relevant to the learner, celebrating different ways of doing maths, and negotiating the curriculum are some of the ways in which as teachers, we have tried to make mathematics more meaningful to our learners. But is there a danger that teaching only in the learner’s contexts may leave the maths "trapped" in the learner’s immediate and personal contexts, and thereby also the learner? Is there a bigger danger in learners defining their contexts through the maths they perceive themselves capable (and incapable) of doing?

Many teachers have adopted the approach of teaching "in context". For teachers who work in the ABE sector, this may mean using contexts such as shopping, utility bills, and social statistics in their teaching (Helme, 1995). For teaching engineering students, it can mean, as I have done, engaging students in project work which makes explicit the role of mathematics in a great variety of engineering processes and systems such as energy demand forecasting, traffic control, product reliability analysis, and so on (Yasukawa, 1995).

But where in our education system do we educate students who will become engineers, economists, business managers who respect the ethics and practical concerns of the wider community? Where do we give students in the non-technical areas the strategies to critique the technical approach effectively, rather than simply criticise its "narrowness" and "reductionist approach"? How can we ensure that "critical numeracy" education for the wider community does not end with "a bit more maths in context", resistance to learning statistics because "statistics lie", and perhaps a letter to the editor about media’s selected use of data?
 

Numeracy, technological literacy and critical education

The ramifications of decisions based on models can be too broad and far-reaching, both in terms of decisions based on public policy and private investment, for society not to take an active role. (Leet and Wallace, 1994, p 243)

If the subject labelled Technology is to be largely focused on practical aspects of designing and making, then it cannot possibly bear the sole weight of responsibility for enabling students to make sense of technology. To achieve the latter aim, other subject areas must take technology seriously. However, an arrangement by which responsibility for practical capability rested with Technology, and for critical awareness with subjects ... where values had been driven into exile from out of Technology, would be undesirable. This would tend to confirm Technology as a ghetto for ingenious, specialist tinkerers, and the Humanities as the natural home for anti-technologists. (Barnett, 1994, pp 62-63)

[Critical education] must assume an active role in identifying inequalities in society, in identifying causes for the emergent sociological and ecological crises and in explaining and outlining ways of dealing with such problems .... [as Giroux states] schools must be defended as an important public service that educates students to be critical citizens who can think, challenge, take risks, and believe that their actions will make a difference in the larger society. (Skovsmose, pp 40-41)

Leet and Wallace advocate that society has a role in developing an ethics of modelling. They recommend that "the discipline of Applied Ethics be employed to provide the knowledge upon which we can begin to prescribe ethical behaviour for model builders. ... to sensitize practitioners to where ethical dilemmas arise in practice, ... and to encourage reflective decision making in order to arrive at a consensus that satisfies a ‘greater moral good’" (pp 243-244). But how can this be realised in a way that is a meaningful and practical part of the education for model builders and users, rather than an ethical "prescription"?

I have learned from my experience of teaching a subject called Mathematical Modelling to senior engineering students that there is a great dilemma in saying, on the one hand, "be aware that your view of reality is not the only view ... don’t presuppose that a valid solution to the problem you wish to investigate is a mathematical one" and on the other, to say that they must complete a project which meets the subject objectives, one of which is "application of mathematics in a ‘real’ world problem". I can teach them "the right line" about ethics in modelling, but within a program which primarily emphasises technical expertise, it has been difficult to develop strategies which students can use to turn a social critique of their technical approach into a change in their own practice of engineering in the community.

There is also a problem, as Barnett points out in having subjects which isolate the critique from the practice of technology. Society which is divided into a group of tinkerers and another group of critics can never properly challenge the formatting power of mathematical or any other technology. What is needed is a technological literacy education which is strongly grounded in the project of critical education, that is educating those people who as Giroux’s quote by Skovsmose states, can "challenge, take risks, and believe that their actions can make a difference in society".

In order to challenge, people need to understand what they are challenging. As I have attempted to describe earlier, the formatting power of mathematics is not determined by mathematics alone. It is determined by the nature of the "contracts" by and the "systems" within which models are built, and by the specific questions which drive the modelling projects. Before people can meaningfully challenge prescriptions delivered by models, they need to understand what Lemke and others call the "community of practice" (CoP) of these model builders, that is the "ecology..., meanings and things" with which they deal (1997, p42). Equally, the model builders would need to develop an understanding of the different communities of practice who are affected by what they do, and what their "things" prescribe to them.

In considering a pedagogy for such understandings, we can draw from the theories we already have about "meaningful" mathematics learning. We have varieties of constructivist theories, all of which are underpinned by the notion that learners negotiate meanings, and meanings cannot be imposed by transmission. That is why many of us try to teach in ways which engage students in hands-on, interactive learning activities. Is it too optimistic to suggest that meanings can be negotiated across people who identify with different communities of practice?

Situated cognition theory has used ways of researching mathematical and literacy practices, ways of understanding people’s practices in relation to their socio-cultural contexts. But Lemke also suggests that we must "look at networks of interdependencies among practices, activities and CoPs to understand the dynamics of ecosocial systems" (p49). This, he says, is because each individual changes through participation in a CoP, and is a member of more than one CoP at any one time, and through life. Could we not make the research methodology available to our students, so they may understand not only the CoPs with which they themselves identify, but also the CoP of other groups in society? Is it not possible that through that, people of different CoPs can evolve a new CoP in which they all belong, and which can engage in a common discourse about technologies?

What is a hopeful ingredient for this vision of technological literacy, is the recognition that every individual identifies with different CoPs at different times. For example, an engineering student may identify her/himself as an engineer when s/he is tackling a problem about traffic control technology in say, a Computer Systems Analysis class. But the same student also experiences aspects of the transport system in her/his everyday life, and most likely without applying formal analytical methods to negotiate them.

A pre-requisite for a technological literacy program which can educate active citizens is an educational framework which makes explicit the holistic influence that education has on individual learners, that is not only the technical/ discipline specific expertise which the students are expected to develop. It needs to make visible the different communities of practice which will interact with students in their professional and private lives, and promote critical reflection on how they can interact effectively in these networks of CoPs for the greater social good. It needs to engage students in a discourse "within themselves" as members of different CoPs and challenge them to reconcile their technical approaches and their needs and issues in their personal lives. We need to build on this reflection and engage students in discourses with other student groups and wider community groups on real social issues so that they may evolve a new CoP which represents, respects, and negotiates, rather than build walls against different interest groups. We can engage students from non-technical areas in "tinkering" with design projects of their choice, so make visible that even in their own lives, they use analytic methods to deal with uncertainties and risks. This may be less visible, but similar to the formal methods used by "professional" risk practitioners. If we as professional members of society cannot bridge gaps between different CoPs, we have to ensure that at least our students become agents for developing new practices and discourses.

It is one thing to acknowledge and respect people’s different realities. However, we all belong to the one planet, and while the "degree" of social injustice and environmental degradation on the planet may be debatable, social injustice and environmental degradation are real. We as educators must recognise this, and be actively seeking ways to educate people who are going to help us move towards a more socially equitable and environmentally sustainable future. Numeracy for a sustainable society has to be much more than maths in context, and criticism of other people’s maths. It must ultimately be about building numerate discourses across different communities of practice. We as numeracy educators need to extend beyond our traditional communities of numeracy educators, and be active participants in this project.

 
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