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

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