Blomhøj (2001, p. 223-224) points to the fact that the problem of justification and relevance of mathematics education has both an objective and a subjective side. The objective side concerns the reasons for mathematics education placed in a societal context. The subjective side, on the other hand, concerns the single individual’s (e.g.
teacher or student) sense making of participating in mathematics education. Of course, such a distinction between the objective and subjective may be made concerning the justification of any subject. Nevertheless, the situation for mathematics is a special one in terms of objectivity and subjectivity, a situation sometimes referred to as the relevance paradox (Niss, 1994, p. 371).
3.4 Justification and Relevance of Mathematics Education 47
The Relevance Paradox
The objective side of the paradox has to do with “the unreasonable effectiveness of mathematics” (Wigner, 1960; Hamming, 1980). That is to say mathematics applies to – andis applied in – a wide range of extra-mathematical subjects and practice areas.
(The nature of mathematics both as a pure and an applied science will be addressed in section 3.6.) This use of mathematics permeates society at large, in its past and present functioning and evolution as well as in its future development. The use of mathematics in technology constitutes one example of this, both the material technology (physical objects and systems, e.g. computers and other microelectronic devises) and the immaterial technology (computer software, codes, geographical coordinates, calenders, money transactions, graphical representations, measurements of time, space, weight, currency, etc.) (Niss, 1994, p. 369-370). Other examples, also of an immaterial nature, are the various decision-making and controlling processes which take place as part of the infrastructure of a society and various forms of descriptions and predictions, e.g. about the weather and climate. The application of mathematics to these extra-mathematical areas is brought about by mathematical modeling, including the building, usage, and validation of such models. However, the embeddedness of mathematics into mathematical models and (other) immaterial technologies, as well as the further potentiel embedding of these into material technologies, brings about the subjective side of the paradox – namely that the mathematics in society becomesinvisible, hence irrelevant, to us. Or as Philip J. Davis, co-author ofThe Mathematical Experience, sees it:
... it’s invisible to people because it’s in programs, it’s in chips, it’s in laws...
So you don’t see it, and if you don’t see it, you don’t think it’s there. (Davis, 2005 in Jankvist and Toldbod, 2005c, p. 17)
And not only is the mathematics hidden to the layman, it may also be hidden to the specialists, e.g. engineers and physicists. For example, when interviewed, several of the applied scientists working with the Mars Exploration Rover mission at JPL would, on first hand, reply that they did not use much mathematics in their work. But of course they did, only the mathematics was embedded in commercial software packages – the mathematics had to some extent been outsourced and therefore also hidden (Jankvist and Toldbod, 2007a, p. 14). Scenarios like these are exactly what constitutes the relevance paradox: “the simultaneous objective relevance and subjective irrelevance of mathematics” (Niss, 1994, p. 371). The invisibility of mathematics in society does of course not make it easier to justify mathematics education. We can continue Davis’ line of thought by saying: If you don’t think it is there then why should you care to know about it and study it?
But if the role of mathematics in society is not recognized by the public, what role does mathematics then serve, and why is mathematics then still a major subject in school? The reason is that mathematics education serves several purposes, and that various different arguments for having it there will be put forward by politicians, mathematics educators, mathematicians, people in the humanistic sciences, teachers, parents, and the students themselves. When dealing with some of these, it may be a good idea to distinguish between two different categories of issues; the descriptive/analytic and the normative (Niss, 1996). The analytic category deals with ‘what actuallyis’ (the case) and ‘why’, whereas the normative deals with ‘whatought to be’ (the case) and
‘why’. I shall address a few of the analytic issues first.
48 Research Questions and Theoretical Constructs
A Few Analytic Issues of Justification
One purpose which mathematics education has served in the educational systems throughout time is that of a ‘critical filter’. That is to say mathematics, e.g. in upper secondary education, serves as a “controlling access to many areas of advanced study and better paid and more fulfilling occupations” (Ernest, 1998, p. 42). Students – and parents – are aware of this ‘sorting mechanism’ of mathematics as a subject, and will therefore often pursue studies in mathematics despite the fact that the use of mathematics in society (mathematical modeling etc.) is hidden to them.
Another argument for the dominant role of mathematics education in the educational system is that the study of mathematics promotes students’ logical thinking. Blomhøj (2001, p. 227) refers to this issue as building on the idea of a “thinking muscle”, which may be trained through mathematics and applied in various other areas as well. However, according to Blomhøj, modern research shows that this is not the case: mastering of mathematical reasoning in one area of mathematics cannot even be transferred to other areas of mathematics, despite the fact that there may be significant structural similarities between the areas.
The last of analytical arguments which I shall mention is the so-called utilitarian argument. Niss (1994, p. 374) describes this as follows: “The dominant interest of society at large in relation to mathematics education is to provide for theutilization, maintainanceanddevelopmentof mathematics as anapplied scienceand as aninstrument for practiceas means for technological and socioeconomic development, with the ultimate purpose of increasing the material wealth in society.” To accomplish this, mathematics as a pure science must be kept alive and well, and in order to do that society must provide a fairly advanced mathematics education, at least to a selected few. For the majority of the developed countries the utilitarian argument is probably the general answer to the justification problem of mathematics education. But it may, of course, also have to do with tradition within an educational system, from primary and all the way up through tertiary levels. The utilitarian argument is subject to critique, at least in its more traditional form. Ernest (1998, pp. 38-39) classifies the argument as a ‘myth’, and rates it as “greatly overestimated” and as only providing “poor justification” since the view that “academic mathematics drives its more commercial, practical or popular applications” ignores “the fact that a two way formative dialectic relationship exists between mathematics as practised within and without the academy.”
Furthermore, regarding mathematics’ role in society as an (utilitarian) argument for justifying mathematics education, Ernest (1998, p. 40) claims that the overt role of academic mathematics in many of the more basic material and immaterial technologies is minimal, at least when it comes to operating them. Therefore “mathematics skills beyond the basic are not needed by most of the general populace in industrial societies”
to ‘cope’ with potential changes as a result of the present information revolution, “if to
‘cope’ means to serve, as here, rather than to critically master”. Thus, the discussion of the role of mathematics in society indeed also touches uponwhat mathematics should be part of the mathematics education (curricula etc.) provided by the society as well as whom should receive it (the general populace or the selected few).
3.4 Justification and Relevance of Mathematics Education 49
A Few Normative Issues of Justification
This brings me to the more normative issues, because what does it mean to “critically master”? Ernest is here referring to empowerment and social justice concerns, i.e. the empowerment of a given student as a highly numerate critical citizen in society. That is a citizen who is, as an example, capable of seeing through misuses of mathematics, e.g.
in graphical representations, statistics, etc. and critically questioning the mathematical models which he or she is subjected to as a citizen. Such intelligent and concerned citizenship is important if society is to be developed in a democratic way (Niss, 1994, p.
376). Skovsmose (1990) talks about ademocratic competenceas “a socially developed characteristic [...] which people to be ruled must possess so they can be able to judge the acts of the people in charge”.1 As indicated, mathematics education has an important role to play here, due to the use of mathematics throughout society. But how may mathematics education equip people with a democratic competence and the capabilities to exercise this kind of citizenship? Surely not every citizen need become an expert in mathematics. Instead it is suggested that the citizens should be provided with an insight into the experts’ expertise, and that this be done through an element of ‘general education’ orAllgemeinbildung (Niss, 1994, pp. 376-377; Blomhøj, 2001, pp. 241-242).
Niss elaborates this view of “mathematics education for democracy”:
Mathematics education should be provided to everyone in order to give insight into ‘the general’, by which I mean: the constitutive features of and the essential driving forces behind the development of nature, society, and the lives of human beings. Insights into the general does not consists in facts and skills alone and for their own sake, but serves the acquisition of overview, knowledge and judgement of main patterns, connections and mechanisms in the world; the ultimate end being to create prerequisites for taking positions on and acting towards processes of significance to society and the individual.
(Niss, 1994, p. 377)
Thus, in the terminology of the framework developed in chapter 2, such an “overview, knowledge and judgement” surely requires more than just an insight into the mathemat-ical in-issues, it requires an insight into the meta-issues of mathematics as well.
Ernest provides a list which may be seen as an exemplification of the “insight into the general”. Ernest (1998, p. 49) adapts a distinction between (1) “developing technological capabilities” and (2) promoting “appreciation and awareness”. The first refers to teaching and having the students learn the mathematical in-issues (of a given curriculum) and developing their skills in relation to these. The second, however, has more to do with the meta-issues of mathematics. Ernest mentions the following seven elements of awareness to illustrate what he believes should be understood by mathematical appreciation:
1. Having a qualitative understanding [of] some of the big ideas of mathe-matics such as infinity, symmetry, structure, proof, chaos, randomness, etc.;
2. Being able to understand the main branches and concepts of mathematics and having a sense of their interconnections, interdependencies, and the overall unity of mathematics;
3. Understanding that there are multiple views of the nature of mathematics and that there is controversy over its philosophical foundations;
4. Being aware of how and the extent to which mathematical thinking
1 Other sources addressing this are: Skovsmose (1994a); Skovsmose (1994b); and Ensor (2008).
50 Research Questions and Theoretical Constructs
permeates every day and shopfloor life and current affairs, even if it is not called mathematics;
5. Critically understanding the uses of mathematics in society: to identify, interpret, evaluate and critique the mathematics embedded in social and political systems and claims, from advertisement to government and interest-group pronouncements;
6. Being aware of the historical development of mathematics, the social contexts of the origins of mathematical concepts, symbolism, theories and problems.
7. Having a sense of mathematics as a central element of culture, art and life, present and past, which permeates and underpins science, technology and all aspects of human culture. (Ernest, 1998, p. 50)
Worth noticing in Ernest’s list is that several of the mathematical awareness elements, or meta-issues, which he mentions are presented in such a way that dealing with them actually requires a certain amount of anchoring in the related mathematical in-issues.
Niss (1996, p. 43) elaborates a bit further on his views of ‘insights into the general’
in a ‘list’ of goals for mathematics education at higher levels, involving aspects of mathematical modeling as well as “the scientific and philosophical nature and status of mathematics and of its position in society and culture, and into the history and development of mathematics as a subject which is a result of human activity”. However, for the purpose in this dissertation, these views are better accounted for through a description of the mathematics program of Danish upper secondary school and the Danish report on competencies and learning of mathematics, the so-called KOM-report (Niss and Jensen, 2002).