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

ZDM Mathematics Education (2007) 39:165–171
DOI 10.1007/s11858-006-0016-x
ORIGINAL ARTICLE
Didactics of mathematics: more than mathematics and school!
Rudolf Straesser
Accepted: 22 December 2006 / Published online: 10 February 2007
Ó FIZ Karlsruhe 2007
Abstract In the discipline didactics of mathematics,
the didactical triangle of subject–teacher–learner is
sometimes too narrowly interpreted by shortening the
human side of didactics of mathematics ( fi teacher–
learner) to teaching and learning in institutionalised
settings, especially schools. In his working group of the
‘‘Institute for Didactics of Mathematics’’ (IDM) at Bielefeld
University, Hans-Georg Steiner inserted good
reasons and strong motivations to analyse technical and
vocational mathematics education as well as the vocational
and everyday use of mathematics. In doing so, he
helped developing research on teaching, learning and
use of mathematics outside schools. The text gives a
short description of mathematics in technical and
vocational education and identifies major issues and
problems in this field of mathematics education: realistic
modelling, situated mathematics and the hiding of
mathematics in black boxes as opposed to a utilitarian
reduction of mathematics for the workplace. This implies
widening the horizon of Didactics of Mathematic,
opening a perspective on the concept of ‘‘Bildung’’
much appreciated by Hans-Georg Steiner.
1 A narrow definition of didactics of mathematics:
the didactic triangle
At least in the German or French speaking communities
of didactics of mathematics (or: mathematics
education research—as it is more common in English
speaking communities), the basic research questions
are sometimes identified using the ‘‘didactical triangle’’
of learner–teacher–subject matter, with mathematics as
subject matter taught (for a prominent example see the
introduction to Chevallard 1985/1991). This may look
like Fig. 1.
Such a description has the advantage of clearly
showing that didactics of mathematics has to systematically
take into account the human side of the processes
to be analysed, didactics of mathematics comes
out as a human science. This is in sharp contrast to
mathematics as a scientific discipline, which—at least
in its publications tries to purify the text from all personal
remnants (for an excellent analysis of this see
Heintz 2000). Defining didactics of mathematics as a
human science also ‘‘explains’’ a development clearly
and recently identified within this discipline. If we take
the 1960s and 1970s as the ‘‘era of the curriculum’’ (see
Sfard 2005, who in her report described the ‘‘last two
decades of the twentieth century’’ as ‘‘almost exclusively
the era of the learner’’ and ‘‘we may now be
living in the era of the teacher as the almost uncontested
focus of researcher’s attention’’), we can see the
ongoing importance of the didactical triangle as
structuring didactics of mathematics. We can also see
that didactics of mathematics first somehow neglected
R. Straesser (&)
Institut für Didaktik der Mathematik,
Justus-Liebig-Universität Giessen
and Luleå University of Technology,
Gießen, Germany
e-mail: rudolf.straesser@uni-giessen.de
Fig. 1 Didactic triangle
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166 R. Straesser
the human parts of its discipline by concentrating on
the subject matter side via the focus on the curriculum.
The focus on subject matter, the focus on mathematics
had different faces in different countries. In France for
instance, the IREM movement clearly defined itself as
coming from an epistemological fundament. Some
French colleagues even suggested to define the
fundamental ‘‘Didactique des Mathématiques’’ as
‘‘experimental epistemology’’ (according to Brousseau
(1986, p. 103): ‘‘épistémologie expérimentale’’). For
Germany, we can exemplify the focus on subject matter
with the then widely accepted definition ‘‘didactics
of mathematics is the scientific development of courses
to be effectively used to learn in the area of mathematics.
It is also made up of the practical teaching of
such courses and their empirical evaluation. This includes
reflections on the aims and subject matter
selection of such courses’’ (‘‘Didaktik der Mathematik
ist die Wissenschaft von der Entwicklung praktikabler
Kurse für das Lernen im Bereich Mathematik sowie
der praktischen Durchführung und empirischen
Überprüfung der Kurse einschliesslich der Überlegungen
zur Zielsetzung der Kurse und der Stoffauswahl’’,
see Griesel (1971, p. 296), nearly identical in
Griesel (1974); a more recent, internationally read text
is Wittmann (1992, English version 1995). In the
beginning of the 1970s, Hans-Georg Steiner also took
didactics of mathematics as fundamentally defined by
the knowledge to be taught, by mathematics—as can
be seen from his publication on ‘‘Bildung’’ (see Steiner
1972). He described didactics of mathematics as
‘‘thinking through mathematics under the perspective
of the elementary and the motivation for the learner’’
(loc. cit., free translation from p. 326; ‘‘Das Durchdenken
des Gebäudes der Mathematik unter dem
Gesichtspunkt des Elementaren und im Hinblick auf
die Motivation des Lernenden gibt dieser Wissenschaft
ein lebensnahes Fundament ...’’). Until around the end
of the 1970s, the so-called Stoffdidaktik was the predominant
approach to didactics of mathematics in
Germany, sometimes even reducing didactics of
mathematics to a mere ‘‘elementarisation’’ of mathematics
for the purpose of its teaching.
At the end of the 1970s, Steiner had a broader definition
of didactics of mathematics. He stated that the
constraints and factors, which determine the classroom
of mathematics and its changeability, belong to didactics
of mathematics. This includes the complex processes of
social interaction related to the teaching and learning of
content matter and ways of behaviour (free translation
by RS of ‘‘Mathematikdidaktik wird dabei in einem
erweiterten Sinne verstanden -als vorwiegend materialorientierte
Entwicklungsarbeit, Einfügung RS, wonach
sowohl die den mathematischen Unterricht und
seine Veränderbarkeit bestimmenden Bedingungen
und Faktoren als auch die komplexen sozialen Interaktionsprozesse
im Unterricht in ihrem Zusammenhang
mit der Vermittlung und dem Erwerb von Inhalten und
Verhaltensweisen wesentliche Bestandteile bzw. Dimensionen
der Untersuchungsgegenstände sind’’; see
Steiner 1978, p. XL). Obviously, and probably in relation
with the creation of the national research institute
for didactics of mathematics (Institut für Didaktik der
Mathematik—IDM) in Bielefeld, Steiner looked further
than the curriculum and course developers abundant
at that time—at least in Germany. Nevertheless,
this perspective was still concentrating on the process of
teaching and learning in schools, institutions to offer (an
understanding of) mathematics to everybody.
2 An opening: mathematics in technical
and vocational education
About 10 years later and in France, we find a slightly
broader definition: didactics of mathematics (is) the
science of the specific conditions of the diffusion of
mathematical knowledge useful for the functioning of
the human institutions (‘‘La didactique des mathématiques
se place ... dans le cadre des sciences cognitives
comme la science des conditions spécifiques de la diffusion
des connaissances mathématiques utiles au
fonctionnement des institutions humaines’’, see
Brousseau 1994, p. 52). Here, didactics of mathematics
has left the narrow area of schools and looks into the
diffusion of mathematics in society at large.
Even if nearly invisible in his list of publications,
Steiner too had crossed this borderline of schools in his
activities: Because of his interest in the pedagogical
concept of ‘‘Bildung’’ (see the text entitled ‘‘Mathematik
und Bildung’’; Steiner 1972) and an ongoing
German debate on a reform of upper secondary
teaching in general, Steiner had taken the job of a
scientific counsellor of a major German reform project—the
‘‘Kollegstufe’’ piloted by Herwig Blankertz.
Kollegstufe aimed at offering upper secondary education
for virtually everybody in need of this—from the
future unskilled worker to the young adult planning an
academic career. This reform project especially fitted
with the intentions of the Bielefeld institute IDM and
Hans-Georg Steiner, because Blankertz wanted to
build his project on the joint efforts of the different
subject matter didactics (‘‘Fachdidaktiken’’) available
at that time. Steiner integrated a mathematics teacher
from the ‘‘Kollegstufe’’-project in his working group
in Bielefeld after having made upper secondary
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Didactics of mathematics: more than mathematics and school! 167
mathematics teaching and learning the field of research
of his working group in the mid 1970s. With a first
academic training as a mathematics teacher for Gymnasium,
Steiner at that time broadened his interests to
all types of education and training at the upper secondary
level. As a consequence and in the German
organisation of the educational system, this implied
that mathematics also in technical and vocational
education had to be studied in his working group. In
addition to this and with the apprenticeship system as a
major part of technical and vocational education and
training, the German upper secondary educational
system offered an opportunity not to study only classroom
environments, but also look into mathematics in
commercial and industrial settings.
2.1 Vocational mathematics: a description
Contrary to the practice of defining mathematics education
at upper secondary by mathematics to prepare
for university studies, looking into mathematics in
technical and vocational education implies a different
approach. The book edited by Bessot and Ridgeway
(2000) gives an excellent overview of research into this
field, showing that—apart from university bound
young adults—one has to also care for the vast
majority of adolescents who more or less directly go
into the labour market and everyday life. In most,
especially industrialised countries, some sort of mathematical
knowledge is part of the vocational training—if
there is an institutionalised training for
vocations at all.
Once upon a time, UNESCO described this training
and/or education as ‘‘the educational process ...
(which) involves, in addition to general education, the
study of technologies and related sciences and the
acquisition of practical skills and knowledge relating
occupations in various sectors of economic and social
life’’ (UNESCO 1978, p. 17). Even if it is methodologically
rather difficult to describe the mathematical
needs of the workplace, the common core of this part
of mathematics was often described as ‘‘basic arithmetic,
percentages and proportion, ... for some vocations,
these techniques already seem to be all the
mathematics’ topics needed’’ (cf. Straesser and
Zevenbergen 1996, p. 653). Apart from this, it seems to
be very difficult to discern the ‘real’ professional use of
mathematics. Straesser and Zevenbergen (loc. cit.) cite
from the 1985-Cockcroft-report: ‘‘It is therefore not
possible to produce definitive lists of the mathematical
topics of which a knowledge will be needed in order to
carry out jobs with a particular title’’. Additional
algebraic techniques seem to be in use in technical and
economical vocations, Geometry use seems to be restricted
to the technical and building sector (loc. cit.).
Little is known about the mathematics ‘‘needed’’ for
the upper part of the qualification spectrum, even if
‘‘mathematics as a service subject’’ is part of a lot of
academic training for future engineers and management
personnel (see the pertinent ICMI-study no. 3 on
this issue with Howson et al. 1988 as its proceedings).
If one looks into the list of publications of Hans-
Georg Steiner, studies into this broader field of upper
secondary (mathematics) education are nearly invisible—with
one major exception: Steiner and Straesser
(1982). Here, the vast range of technical and vocational
education is narrowed down to that part of the training
and education of young adults who try a direct entry to
the world of work after compulsory schooling. At least
in Germany, this is still more than half of the population
and has a special organization called ‘Lehrlingsausbildung’,
internationally known as ‘apprenticeship’,
i.e. joint efforts of vocational colleges and companies
to train a skilled workforce. From this short description,
a major tension may already be obvious: How to
cope with the specific, but ‘‘needs’’ of a whole variety
of different vocations from hairdressing to the building
sector and business administration or electrical and
even computer based information technology? What
about the mathematical ‘‘needs’’ of this diverse occupational
fields and (in Germany: more than 400)
vocations? What has didactics of mathematics to offer
to this training/education linking classrooms of vocational
colleges to workplaces in the different occupational
fields?
2.2 Realistic modelling
Looking into vocational mathematics—in the narrow
sense of a direct move or preparation for workplaces—offers
a chance, which mathematics in compulsory
schools often do not have: with the workplace and
vocation to be trained for clearly identified, there is an
obvious field of application for mathematics in this
type of training/education. To put it in a different
wording: vocational mathematics in principle immediately
offers a field outside mathematics, to which
mathematics can be applied, it does not have to look in
the vague ‘‘rest of the world’’ often mentioned when
applications of mathematics are discussed. If the
vocational field ‘‘asks’’ for mathematics to be used
(and this is not always obvious, see the next subsection),
the vocational field offers a vast array of applications
for mathematics.
If we take the example of technical vocations related
to metalwork, Schilling (1984, p. 116; in a text for a
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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
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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.
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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.
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