Date: Oct 22, 2012 9:43 AM
Author: Jerry P. Becker
Subject: Symposium - Growth and Success of "mathe 2000"
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, Belgium.
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 observer's view
By Lieven Verschaffel
[Center for Instructional Psychology and
Technology University of Leuven, Belgium]
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 wars".
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
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 tribute.
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, 1995).
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
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
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
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
Baumert, J., Kunter, M., Blum, W., Brunner, M.,
Voss, T., Jordan, A., Klusmann, U., Krauss, S.,
Neubrand, M., & Tsai, Y.-M. (2010). Teachers'
mathematical knowledge, cognitive activation in
the classroom, and student progress. American
Educational Research Journal, 47, 133-180.
De Corte, E. & Verschaffel, L. (2006).
Mathematical thinking and learning. In Damon, W.,
Lerner, R., Sigel, I & Renninger, A. (eds.)
Handbook of child psychology. V. 4: Child
psychology in practice, pp. 103-152. New York:
Education Committee of the European Mathematical
Society (2012). It is necessary that teachers are
mathematically proficient, but is it sufficient?
Solid findings in mathematics education on
teacher knowledge. Newsletter of the European
Mathematical Society, March 2012, 46-50.
Freudenthal, H. (1983). Didactical phenomenology
of mathematical structures. Dordrecht, The
Hess, K. (2003). Lehren - zwischen Belehrung und
Lernbegleitung. Einstellungen, Umsetzungen und
Wirkungen im mathematischen Anfangsunterricht.
Bern: h.e.p. Verlag.
Koninklijke Nederlandse Akademie van
Wetenschappen (2009). Rekenonderwijs op de
basisschool. Analyse en sleutels tot verbetering.
Moser Opitz, E. (2002). Zählen, Zahlbegriff,
Rechnen. Theoretische Grundlagen und eine
empirische Untersuchung zum mathematischen
Erstunterricht in Sonderschulklassen. 2. Auflage.
Bern: Verlag Paul Haupt.
National Council of Teachers of Mathematics.
(1989). Curriculum and evaluation standards for
school mathematics. Reston, VA: National Council
of Teachers of Mathemetics.
National Council of Teachers of Mathematics.
(2000). Principles and standards for school
mathematics. Reston, VA: National Council of
Teachers of Mathematics.
Schmidt, W. (2012). Seizing the Moment for
Mathematics. Education Week [American Education's
Newspaper of Record], Wednesday, July 18, 2012,
Volume 31, Issue 36, pp 24-25. See
Selter, C. (1998). Building on children's
mathematics - A teaching experiment in grade 3.
Educational Studies in Mathematics, 36, 1-27.
Shulman, L. S. (1986). Those who understand:
Knowledge growth in teaching. Educational
Researcher, 15, 4-14.
Shulman, L. S. (1987). Knowledge and teaching:
Foundations of the new reform. Harvard
Educational Review, 57, 1-21.
United States Department of Education (2008).
Foundations for success. The final report of the
National Mathematics Advisory Panel. (Retrieved
January 17 2009 from
M. van den Heuvel-Panhuizen (Ed.) (2001).
Children learn mathematics. Utrecht, The
Netherlands: Freudenthal Institute, University of
Verschaffel, L. (1995). Ontwikkelingen in de
opvattingen over en de praktijk van het
reken/wiskundeonderwijs op de basisschool. In: L.
Verschaffel & E. De Corte (Red.), Naar een nieuwe
reken/wiskundedidactiek voor de basisschool en de
basiseducatie. Deel 1. Achtergronden (pp.
95-128). Brussel: Studiecentrum voor Open Hoger
Verschaffel, L.. (2009). ''Over het muurtje
kijken': Achtergrond, inhoud en receptie van het
Final Report van het 'National Mathematics
Advisory Panel' in de U.S. Panama-Post - Reken-
wiskundeonderwijs: Onderzoek, ontwikkeling,
praktijk, 28(1), 3- 20).
Verschaffel, L., & Greer, B. (in press).
Domain-specific strategies and models:
Mathematics education. In Spector, J. M.,
Merrill, M. D., Elen, J. & Bishop, M. J. (eds.)
Handbook of research on educational
communications and technology. 4th ed. New York:
Wittmann, E. Ch. (1995). Mathematics education as
a design science. Educational Studies in
Mathematics, 29, 355-374.
Wittmann, E. Ch., & Müller, G. R. (2000-2002).
Handbuch produktiver Rechenübungen (Bd.1, Vom
Einspluseins zum Einmaleins, und, Bd.2, Vom
halbschriftlichen und schriftlichen Rechnen: Vom
halbschriftlichen zum schriftlichen Rechnen).
Stuttgart: Ernst Klett Schulbuchverlag.
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]
(618) 457-8903 [H]
Fax: (618) 453-4244