Below and attached is the talk given by Lieven Verschaffel on the
occasion of the 25th anniversary of the elementary mathematics
curriculum project "mathe 2000" at Dortmund University in
Germany. During the turbulent international context of the last 25
years, Prof. Erich Wittmann, Prof. Gerhard Müller and the other
members of the "Mathe 2000" project [see
have worked at the development of their own approach to elementary
school mathematics education, in a way that may be considered quite
unique and exemplary, for three reasons that are briefly elaborated
below under the headings: A view of elementary school mathematics, the
methodological approach, and the role of the teacher. The paper is a
tribute to the project given by Lieven Vershaffel of the Center for
Instructional Psychology and Technology, University of Leuven,
You can learn about the project at
Information is given in German and in English (by checking the flag
above 'Short Info').
From the Symposium "mathe 2000", September 21, 2012,
Dortmund University, Dortmund, Germany. See
Growth and success of "Mathe 2000" - a privileged
By Lieven Verschaffel
[Center for Instructional Psychology and Technology University of
It is a great pleasure and a great honor for me to represent the
international community at the 22nd
symposium "Mathe 2000", which is devoted to the 25th
anniversary of this project.
During the past three decades the international scene of mathematics
education has witnessed, in various parts of the world, serious
debates about the goals, the content and the methods of elementary
school mathematics, which sometimes have evolved into true "math
For instance, in the US there have been, since the launch of the
NCTM Standards in the 1980s (National Council of Teachers of
Mathematics, 1989, 2000), highly emotional debates between opponents
and advocates of the reform-based approach to elementary school
mathematics; between traditionalists, who still believe that the
emphasis of math education should be on the direct teaching of fixed,
step-by-step procedures for solving various types of math problems,
and reformers, who favor a more inquiry-based approach in which pupils
are exposed to real-world problems that help them develop deep
conceptual understanding, number sense, reasoning and problem-solving
skills, and positive affects towards mathematics. Only in 2008, the
National Mathematics Advisory Panel, created by president George Bush
himself, succeeded, at least to some extent, in stopping that national
war (United States Department of Education, 2008).
In the Netherlands, another leading country in the international scene
of mathematics education, we have seen a very similar development.
Growing concern about Dutch children's mathematical proficiency in
national and international assessments has led in recent years to a
hot public debate about the way elementary mathematics should be
taught. There were again two opposing camps: those who advocated
teaching mathematics in the "traditional" manner, and those who
supported realistic mathematics education, the reform- based type of
mathematics education that has been conceived and further developed by
Prof. Freudenthal (1983) and his colleagues and successors at the
University of Utrecht (see, e.g., Van den heuvel, 2001). Because of
the intensity of the debate, the Dutch Royal Academy of Sciences
decided to install a Committee, which wrote a report that also
succeeded in calming down, at least to some extent, the public debate
about the quality and future of elementary school mathematics
(Koninklijke Nederlandse Akademie van Wetenschappen, 2009).
In this turbulent international context, Prof. Wittmann, Prof.
Müller and the other members of the Mathe 2000 project have, during
the past 25 years, worked at the development of their own approach to
elementary school mathematics education, in a way that I consider
quite unique and exemplarily, for three reasons that I will elaborate
in a necessarily brief and superficial way in this short
View on elementary school mathematics
In terms of its view on elementary school mathematics, one of the most
important general characteristics of the "Mathe 2000" project is
that it has, from the very beginning, refused to look at math
education, and at its own position in the international scene, in
extreme or polarized terms. I am aware that there exist more nuanced
and sophisticated categorizations, but, I find it conceptually helpful
to conceive of elementary mathematics education as a field consisting
of roughly three major aspects, each of which has been central in a
historically important tradition of elementary school mathematics: a
mechanistic, a structuralistic, and a realistic aspect (Verschaffel,
First, elementary school math has a lot to do with memorization of
basic facts, automatization of techniques for doing mental and written
arithmetic, routine mastery of rules for solving standard problems
dealing with number and space... Historically, this "mechanistic"
element has been emphasized a lot in traditional elementary school
mathematics, and it is this element that has been re-emphasized in
these anti-reform movements in the US and The Netherlands that I
referred to before.
Second, elementary school mathematics is about structures and
patterns. In the various manifestations in concrete mathematical
statements or problems, there may be a common principle, a common
pattern or structure, an underlying "big idea", that has to be
discovered, explored, understood, expressed, formalized,
generalized..., by the learner, and that should become part of his or
her conceptual toolbox. This aspect was central in the structuralistic
approaches to elementary school math, such as the New Math movement,
that was dominant in the fifties to eighties of the previous century
in various parts of the Western world, but is also emphasized in
current approaches that emphasize, for instance, the role of
pre-algebra in elementary school mathematics.
Third, mathematics is a human problem solving activity; it is about
establishing links between real world situations and mathematics, in
both directions; it is about seeing the mathematics in the real world
and about using mathematics to make sense of this world, to understand
and manipulate it, with a view to efficiently solve problems that
arise in that world. This aspect of "mathematical modeling and
applications" is prominently present in approaches, such as the
Dutch realistic approach to mathematics education (although it would
be too simple to reduce RME to that aspect).
Just as in the world-famous tale of a group of blind men each
touching a part of the elephant to learn what it is like, but every
single man being unable to get a complete picture of what it
essentially is, each of these three aspects point to a truly essential
feature of elementary school mathematics, but does not tell the whole
story of what it is about. The great value of the "Mathe 2000"
approach is that it departs from a view of elementary mathematics
education that integrates in a well- balanced way all three aspects.
It does so both in its theoretical foundations and in the concrete
textbook pages and materials of its textbook, Das Zahlenbuch. To the
best of my knowledge, there are few textbooks in the world that have
been so successful in realizing this balance so subtly and so
successfully as Das Zahlenbuch.
Moreover, in realizing that subtle balance between these three major
pillars of elementary school mathematics, it adheres to three
principles that have been found in the curricula of the world's
highest-performing countries, according to a recent study by Houang
and Schmidt (2012) namely (1) coherence (the logical structure that
guides students from basic to more advanced material in a systematic
way); (2) focus (the push for mastery of a few key concepts at each
grade rather than skim over dozens of disconnected topics every year);
and (3) rigor (the level of difficulty at each grade level).
Closely related to the above-mentioned international debate
between advocates and opponents of reform-based approaches to math
education, there is an ongoing methodological fight about the kind of
scientific research that is primarily suited and needed for improving
elementary school mathematics. Stated again somewhat boldly, there
are, on the one hand, researchers who adhere the so-called
"evidence-based approach", which postulates that only effective type
of research is the (quasi-)experiment, whereby one compares the effect
on learners of two or more approaches to teach a given mathematical
topic, with randomly selected classes, in well- controlled conditions,
using only psychometrically adequate standard achievement tests; and,
on the other hand, those who argue that this evidence-based approach
is not and will never be able to capture the rich, complex and
contextual nature of teaching and learning in a real mathematics
classroom, and therefore argue that the only useful kind of research
is of a more qualitative nature, that documents in detail how one
arrived at the design of a new teaching/learning unit, how teachers
and learners reacted to it, and what was learnt from it in view of the
improvement of the design of that unit (Verschaffel, 2009). Also in
this international methodological battlefield, the "Mathe 2000"
project has always taken a nuanced, broad-spectrum view, by pleading,
on the one hand, for the existence of "design experiments" as a
central research method in the domain of mathematics education, but,
on the other hand, also supporting more large-scale and systematic
evaluation studies aimed at unraveling the relative strength and
weaknesses of its newly designed instructional materials and
approaches. As illustrations of the former, I refer to Prof.
Wittman's paper "Mathematics education" as a design science",
published in Educational Studies in Mathematics (1995), which has
become an internationally recognized "classic" in the field of
mathematics education, as well as Prof. Selter's exemplary design
study about building on children's mathematical productions in grade
3, published in 1998 in the same journal. Illustrations of the latter
are the evaluation studies by Moser Opitz (2002) and Hess (2003), both
comparing teaching and learning in classes in which a traditional
textbook was used with teaching and learning in classes which worked
with (an adaptation of) the "Zahlenbuch", and both providing
substantial empirical support for the "Mathe 2000" approach,
particularly for the mathematically weaker children.
Role of the teacher
Referring back to the two reports that tried to stop the math
wars in the US and The Netherlands, it is interesting to see that
according to both reports the key to improving children's
mathematical proficiency does not lie in the textbook in itself, but
in the competencies of the teachers who have to use it. And, by these
competencies, they do not only mean their mathematical content
knowledge, but also, and according to some even primarily, their
"Fachdidaktische Kompetenz", or, in Shulman's (1986, 1987)
terminology, their pedagogical content knowledge (PCK). Many studies
and surveys have indicated the importance of this PCK. In a recent
German study (COACTIV project - see, Baumert et al., 2010), it has
been shown that students taught by teachers with a high PCK showed
better PISA results than those of other students, mainly because
teachers with a high PCK design their teaching so that the students
are more actively cognitively engaged. Further analyses revealed that
PCK has greater predictive power for student progress and is more
decisive for the quality of instruction than their content knowledge
(Baumert et al., 2010, p. 164). Moreover, the available international
research on mathematics teachers' knowledge and professional
development (as nicely summarized in a recent publication by the
Education Committee of the European Mathematical Society (2012) headed
by prof. Konrad Krainer), indicates the positive impact of
"collaboration" among teachers and of teachers' collegial
learning, i.e. of teachers belonging to "communities" consisting
of experts, teachers and researchers and improving their teaching
actions and upgrading their professional theory through unfolding
their learning process in cooperation with the other members of the
community. Clearly, the "Mathe 2000" project has, from the very
beginning, deeply endorsed the idea that the teacher is the critical
factor in the curriculum implementation process, and that, therefore,
a textbook series project without a parallel well-established
supportive system for its teachers, is doomed to fail. This is not
only evidenced by the two excellent volumes of the Handbuch
produktiver Rechenübungen (Wittmann & Müller, 2000-2002) that
accompany the textbook Das Zahlenbuch, and that provide the teachers
with the PCK and the accompanying beliefs needed to implement the
textbook in a proper way; but also by the organization of the annual
meetings of the "Mathe 2000" community allowing intensive
exchanges of ideas, findings and experiences between teachers,
researchers and other kinds of experts.
As a scholar from abroad, it was a great privilege to observe
from close-by, through my long-standing and intensive contacts with
the members from the Dortmund "Institut für Entwicklung und
Erforschung des Mathematikunterrichts" (IEEM), the development of
the "Mathe 2000" project. The project can be really proud of what
it has accomplished during the past 25 years and the impact it has had
on the research on and practice of elementary school mathematics, in
Nordrhein-Westfalen, in Germany, and abroad. I wish you all very nice
and stimulating conference celebrating this 25th anniversary.
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Klusmann, U., Krauss, S., Neubrand, M., & Tsai, Y.-M. (2010).
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De Corte, E. & Verschaffel, L. (2006). Mathematical thinking and
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halbschriftlichen zum schriftlichen Rechnen). Stuttgart: Ernst
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
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