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

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170 R. Straesser 3 Utilitarian mathematics and ‘‘Bildung’’ In Sect. 2, the consequences of looking into vocational mathematics education and some results of research into this type of teaching and learning and on research into mathematics at the workplace have been described. One could see how this field of research throws a new light onto the human struggle with mathematics, how this breaking out of the narrow confines of the institution classroom/school can add to didactics of mathematics. Looking into mathematics in the workplace and in vocational education obviously has something to offer to didactics of mathematics as a scientific discipline. One could also see that current research in this field is still mainly concerned with the subject matter mathematics and its role in the workplace and vocational education. The role and problems of the human learner are still somehow neglected in mathematics in vocational education. The typical German concept of ‘‘Bildung’’ may be an appropriate way to overcome this limitation. Even if the German word cannot be easily translated into English without loosing its distinct meaning, it may be a way to conceptually bridge the divide between the subject matter (to be) taught and the human being struggling with this subject matter. Steiner simply cited the well-known formula defining ‘‘Bildung’’ as what is left when everything that has been learned is forgotten (‘‘Wenn Bildung ... das ist, was übrig bleibt, wenn man vergessen hat, was man gelernt hat’’; Steiner 1972, p. 334f). The afore-mentioned German educationalist Herwig Blankertz especially looked into Bildung from the perspective of vocational education (see Blankertz 1969). In this paper, it is simply impossible to present the long-standing debate on ‘‘Bildung’’ and its implications for mathematics education in general. But it was the concept of ‘‘Bildung’’, which made Steiner aware of the fundamental role of the learner (see the citation in Sect. 1) and opened the way to his activities in vocational education, implying that Steiner inserted the question of ‘‘Bildung’’ into the vocational project he took part in. Mathematics education in and for the workplace can profit from taking into account the human side of the teaching/learning process—possibly with the help of the concept of ‘‘Bildung’’. There is also a lesson to be learnt for general mathematics education. Some ten years later in the 1980s, the human part of didactics of mathematics is so evident, that Steiner makes ‘‘Bildung’’ a point of defence of the reform of the upper secondary mathematics education bound for academic studies (the ‘‘neugestaltete gymnasiale Oberstufe’’). According to Steiner, the pre-defined curriculum for everybody (with academic aspirations) should be substituted by an individually defined ‘‘diet’’ of learning objects and experiences in order to have an adequate upper secondary education including mathematics (‘‘... die Beratungen und ... Betreuungen der Schüler ... unter dem Aspekt der Wahl eines individuellen Bildungsganges und damit unter bestimmte mit dem Bildungsgang verbundene Kompetenzentwicklungen zu stellen’’, see Steiner 1984, p. 21—explicitly referring to Blankertz and his reform work). Here again—and for general education, ‘‘Bildung’’ is the keyword to fight a utilitarian reduction of teaching and learning mathematics in favour of a mathematics created for the individual learner. References Appelrath, K.-H. (1985). Zur Verwendung von Mathematik und zur Situation des Fachrechnens im Berufsfeld Metalltechnik (dargestellt an zwei Unterrichtsbeispielen). In P. Bardy, W. Blum, & H. G. Braun (Eds.), Mathematik in der Berufsschule - Analysen und Vorschläge zum Fachrechenunterricht. (pp. 127–139). Essen: Girardet. Bessot, A., & Ridgway, J. (2000). Education for mathematics in the workplace (Vol. 24). Dordrecht: Kluwer. Blankertz, H. (1969). Bildung im Zeitalter der grossen Industrie. Pädagogik, Schule und Berufsbildung im 19. Jahrhundert. Berlin: Hermann Schroedel. Blum, W., & Straesser, R. (1992). Mathematikunterricht in beruflichen Schulen zwischen Berufskunde und Allgemeinbildung. Zentralblatt für Didaktik der Mathematik 24(7), 242–247. Brousseau, G. (1986). Forschungstendenzen der Mathematikdidaktik in Frankreich. Journal für Mathematikdidaktik 7(2/3), 95–120. Brousseau, G. (1994). Perspectives pour la didactique des mathématiques. In M. Artigue (Eds.), Vingt Ans de Didactique des Mathématiques en France - Hommage à Guy Brousseau et Gerard Vergnaud (pp. 51–66). Grenoble: La Pensée Sauvage. Buchberger, B. (1989). Should students learn integration rules? (RISC-Linz-Series No. 89-07.0). Linz: RISC. Chevallard, Y. (1985/1991). La transposition didactique. Grenoble: Pensées sauvages. Griesel, H. (1971). Die Neue Mathematik für Lehrer und Studenten - Band 1: Mengen, Zahlen, Relationen, Topologie (Vol. 1). Hannover: Schroedel. Griesel, H. (1974). Überlegungen zur Didaktik der Mathematik als Wissenschaft. Zentralblatt für Didaktik der Mathematik 6(3), 115–119. Heintz, B. (2000). Die Innenwelt der Mathematik. Zur Kultur und Praxis einer beweisenden Disziplin. Heidelberg: Springer. Howson, A. G., Kahane, J. P., Lauginie, P. & Turckheim, E. (Eds.). (1988). Mathematics as a service subject. Cambridge: Cambridge University Press (additional selected papers published by Springer (1988): Clements, R. B., Lauginie, P. & Turckheim, E. (Eds.) as CISM Courses and Lectures No. 305). Hoyles, C. & Noss, R. (2004). Abstraction in workplace expertise. Copenhagen: ICME 10 conference website (paper from TSG 7 at www.ICME10.dk). 123
Didactics of mathematics: more than mathematics and school! 171 Noss, R., & Hoyles, C. (1996). The visibility of meanings: modelling the mathematics of banking. International Journal of Computers for Mathematical Learning 1(1), 3–31. Peschek, W. (1999). Mathematische Bildung meint auch Verzicht auf Wissen. In G. Kadunz, G. Ossimitz, W. Peschek, E. Schneider, &B. Winkelmann (Eds.), Mathematische Bildung und neue Technologien (pp. 263–270). Stuttgart: Teubner. Pollak, H. O. (1979). The interaction between mathematics and other school subjects. In I. C. o. M. I. (ICMI) (Ed.), New trends in mathematics teaching (Vol. 4, pp. 232–248). Paris: United Nations Educational, Scientific and Cultural Organisation (UNESCO). Rauner, F. (1984). Die Lehre von der Elektrotechnik in der Berufsschule. In R. Straesser (Ed.), Bausteine zu einer Didaktik des mathematischen Unterrichts an Berufsschulen (Vol. 34, pp. 129–160). Bielefeld: Institut für Didaktik der Mathematik (IDM). Schilling, E.-G. (1984). Didaktischer Bezugsrahmen für die Beurteilung von Funktion und Rolle der Mathematik im Unterricht der Berufsschule (Berufsfeld Metalltechnik). In R. Straesser (Ed.), Bausteine zu einer Didaktik des mathematischen Unterrichts an Berufsschulen (Vol. 34, pp. 97– 128). Bielefeld: Institut für Didaktik der Mathematik (IDM). Sfard, A. (2005). What could be more practical than a good research? On mutual relations between research and practice of mathematics eduaction (Survey Team 1 Report on Relations between research and Practice of Mathematics Education). Educational Studies in Mathematics 58, 393–413. Steiner, H.-G. (1972). Mathematik und Bildung. In H. Meschkowski (Ed.), Grundlagen der modernen Mathematik (pp. 310–340). Darmstadt: Wissenschaftliche Buchgesellschaft. Steiner, H.-G. (1978). Didaktik der Mathematik – Einleitung. In H.-G. Steiner (Ed.), Didaktik der Mathematik (Vol. CCCLXI, pp. IX–XLVIII). Darmstadt: Wissenschaftliche Buchgesellschaft. Steiner, H.-G. (1984). Mathematisch-naturwissenschaftliche Bildung. Kritisch-konstruktive Fragen und Bemerkungen zum Aufruf einiger Fachverbände. In M. Reiss & H.-G. Steiner (Eds.), Mathematikkenntnisse – Leistungsmessung – Studierfähigkeit (Vol. 9, pp. 5–59). Köln: Aulis Verlag Deubner. Steiner, H.-G., & Straesser, R. (1982). Mathematik in der Berufsschule – gegenwärtiger Stand, Entwicklungstendenzen, drängende Probleme. In R. Straesser (Ed.), Mathematik in der Berufsschule (Vol. 28, pp. 9–51). Bielefeld: Institut für Didaktik der Mathematik (IDM). Straesser, R. (2002). On the disappearance of mathematics from society’s perception. In H.-G. Weigand (Eds.), Developments in mathematics education in German-speaking countries. selected papers from the annual conference on didactics of mathematics, Bern, 1999 (pp. 124–133). Hildesheim: Franzbecker. Straesser, R., & Zevenbergen, R. (1996). Further mathematics mducation. In A. Bishop (Eds.), International Handbook on Mathematics Education (Vol. 1, pp. 647–674). Dordrecht: Kluwer. UNESCO. (1978). Terminology of technical and vocational education – Terminologie de l’enseignement technique et professionnel. Paris: UNESCO. Wittmann, E. C. (1992). Mathematikdidaktik als ‘‘design science’’. Journal für Mathematikdidaktik, 13(1), 55–70. English version (1995): Mathematics Education as a ‘‘Design Science’’. Educational Studies in Mathematics (29), 355–374. 123
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