Didactics of mathematics: more than mathematics and school! - CIMM

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168 R. Straesser developmental project piloted by Steiner and Straesser) stressed the importance of mathematics for this vocational area, but also pointed to the fact that mathematics is used as a ‘strategic’ tool to model the workplace situation, to fulfil the quantification needs of the vocational situation in order to have easy predictions of the application of machines and procedures. Mathematics is not needed as such, it offers a (very versatile) tool to model the workplace situation. Consequently, the vocational theory is not interested in a deep understanding of mathematical concepts or procedures, but in an effective and fast prediction of the vocational situation (loc. cit., p. 118). A comparable description can be found for the workplaces related to (traditional, not computer related) electricity: Rauner (1984, p. 151) denies mathematics an independent role in the training of future qualified worker in electricity surroundings, but mentions the role of mathematics as a help to understand vocational situations—with trigonometric functions as the prototypic example helping to understand alternating current. He even goes as far as saying that understanding electro technical devices is simply impossible without mathematics—but wants these mathematical models preferably presented in diagrams and graphs, not in numbers or equations (loc. cit., p. 152f; for examples see the end of his paper). 2.3 The utilitarian reduction Looking into mathematics in vocational training and education is one efficient way to walk out of the classroom to study mathematics and its use out of school. This nevertheless implies a temptation, which Hans-Georg Steiner always resisted, but should nevertheless be mentioned: Studying workplace use of mathematics often tends to reduce the purpose of mathematics to its importance for aims outside mathematics. Falling into this trap implies the idea, that teaching and learning of mathematics is (mainly, if not only) done for utilitarian reasons, for the sake of the role of mathematics’ in its applications. From the description of the developmental project mentioned above, one can see that Steiner (and his team) did not reduce vocational training to utilitarian purposes. Apart from three other problems of teaching mathematics in this type of vocational training, the project wanted to take into account the tension between vocational and general education (in German: ‘‘beruflicher und allgemeiner Bildung’’, see Steiner and Straesser 1982, p.16). The modelling approach of Sect. 2.2, especially the comments from the metalwork and electricity area, already show that a simplistic utilitarian approach to mathematics is inappropriate for vocational training—last not least because the sheer practice of the workplace is not interested in the workers understanding the processes of production and distribution, but will only look into a continuous functioning of machines and the labour force. Research into mathematics at the workplace has shown the limitations of such an approach in more detail: as long as the production/distribution process goes uninterrupted by unexpected events, goes without (unforeseen) problems, understanding the process (maybe with the help of mathematics) is not necessary, sometimes even disturbing. The worker may even override the information produced by a mathematical model. A nice confirmation of this statement from the banking sector may be the comment of a person ‘‘in charge of support and maintenance of computer equipment’’ in an investment bank (see Noss and Hoyles 1996, p. 7f). He commented his evaluation of a piece of equipment by using a pre-programmed spreadsheet by ‘‘I press the button and see what it says. .... I look at the answer. If it seems to indicate what I think we should do, I use the number to justify my decision. If not, I ignore it, or put in figures which will support my hunch.’’ It its all too obvious that mathematics cannot be used as an unquestioned, simple model of the vocational situation. A quote from an experienced teacher trainer in the metalworking area shows a different limitation of a too simplistic utilitarian reduction of the use of mathematics in vocational situations: ‘‘Mathematics in vocational education is serving more as a background knowledge for explaining and avoiding mistakes, recognising safety risks, judicious measurement and various forms of estimation. ... Not practice at the workplace but deepening of the professional knowledge, education to a responsible use of tools and machines and the understanding of and coping with everyday mathematical problems legitimise mathematics in vocational education’’ (quoted from Appelrath 1985, pp. 133–139; translation R.S.). In the statement from metalwork, mathematics takes a specific role to further an understanding of the workplace situation. It is not directly operative in the situation on the shop floor. 2.4 Black boxes As a consequence, mathematics is far from obvious in vocational situations—somehow confirming the difficulties to identify vocational mathematics already mentioned in Sect. 2.1. Difficulties to find mathematics 123
Didactics of mathematics: more than mathematics and school! 169 in the workplace seem to even grow with the increasing use of (modern) technology. Some people even speak of the ,,disappearance of mathematics from society’s perception’’ (title of Straesser 2002). The use of mathematics (in a simple utilitarian sense) as backbone of models of a vocational situation is often implemented in graphs and algorithms, maybe even machines, hides the inherent mathematical procedures and concepts from the direct perception of the user. S/he may see the situation as not containing any mathematics, because the mathematics parts is ‘‘crystallised’’ in artefacts like graphs, tables or machines like automatic wages, control devices and protocols. The user might only be confronted with some pointer, which has to be kept within a given interval (in a steering/control situation) or answered by a certain activity (for instance in a buying situation with the price automatically given by a digital balance; for details of the development of weighing, see Straesser 2002). The highly integrated automatic control systems of the flow of cash and goods are another illustration of this tendency to hide mathematics from being noticed. The whole development can be described as the gradual, but steady introduction of ‘‘black boxes’’ into the production and distribution of goods and services in a modern economy. Consequences for the teaching and learning of mathematics are far from obvious. In accordance with the proclaimed habit of mathematicians to always go back to the source of knowledge and leave no statement unproved, some educators explicitly opt for an opening of these boxes, which should go as far as possible—even avoiding the use of such algorithms and automatisms as long as they have not been explained and understood (for an example see Buchberger 1989). In sharp contrast, Peschek for instance clearly stated the impossibility of such a procedure, advocating an intelligent and efficient use of black boxes (see Peschek 1999, p. 270). Straesser (2002) even goes further in pointing to the fact that the role of technology in itself can be quite ambivalent. On the one hand, (modern computer) technology may be a very effective way to hide the use of mathematics from societal perception. ‘‘On the other hand, ... the same technology can be used to demystify the black boxes, to show the mathematical relations and offer an opportunity to explore the inherent, implemented relations in a way workplace reality would never allow because of the risk of material, financial and time losses. The practice of using sophisticated mathematics can be brought to the foreground and hence to the consciousness of the user by appropriate software and vocational training/education. It is modern computer technology and appropriate software which can be successfully used to really explore and understand the underlying ... mathematics’’ (Straesser 2002, p. 130). 2.5 Situated mathematics What was reported about mathematics at the workplace and vocational training/education related to this all starts from a common, normally not contested assumption: mathematics is a special type of knowledge with a specific structure, special ways of proving and a clear borderline to separate mathematics from the ‘rest of the world’ (as Pollak described it at ICME 3, see Pollak 1979). A modelling approach to applications of mathematics as well as the idea of black boxes hiding mathematics heavily rely on this separation of mathematics from the rest of the world. Workplace mathematics somehow questions this idea—last not least because the difficulties of identifying mathematics used in the workplace. Inspired by research into the use of mathematical ideas in workplace situations like banking, nursing, piloting an airplane and engineering, Hoyles and Noss (2004) detailed the concept of ‘situated abstractions’ in mathematics they had already brought forward earlier. • ‘‘mathematical meanings that emerge through webbing with tools that ‘embed’ mathematics • in use or communication display ‘correct’ mathematical invariants • are expressed with the language of the tools and community • are abstracted within, not away from (the workplace, insert RS) community’’. It is obvious that these situated abstractions related to mathematics are no longer part of mathematics or ‘the rest of the world’ alone, but are a new type of knowledge bridging the divide between mathematics and the rest of the world. To make this idea functional for teaching and learning, Hoyles added the concept of ‘techno-mathematical literacies’ (see Hoyles and Noss 2004). Even if these concepts are developed from research into mathematics in the workplace, these concepts can obviously be used in a more general sense, in the general scientific debate about teaching and learning mathematics—not isolating workplace mathematics from the vast majority of a didactics of mathematics research community not necessarily interested in workplace mathematics. 123
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Didactics of mathematics: more than mathematics and school! - CIMM

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