**Primary Teacher Education in Mathematics in Germany ^{1}**

As Jeremy Kilpatrick once remarked: The problems of and the approaches to the teaching of mathematics in the North of most countries are different from those in the South. Germany is no exception from this rule. Our situation is even more varied: We have also a clear distinction between the East and the West. The German Constitution secures the socalled "Kulturhoheit der Länder", that is the independence of the 16 States in all affairs of education, culture and art. You can be assured that all States use this freedom. For this reason it is not possible for me to give an account of primary teacher education in all 16 States of the Federal Republic. 1 will restrict myself to the State Northrhine-Westfalia, with 17 Mill. Inhabitants, by far the largest of all States. As far as the theme of our conference is concerned there is a substantial reason for this restriction: Since the beginning of the eighties Northrhine-Westfalia has played a leading role in the development of primary mathematics teaching and of primary teacher education in mathematics. How strong this influence still is can be derived, for example, from the fact that at present Bavaria is changing its syllabus for primary mathematics according to the principles of the syllabus in Northrhine-Westfalia. This really means something as the Bavarians are usually very reluctant to import things from the North of Germany.

1. **Some Remarks on the German School System and on
Teacher Education**

I would like to start with a few remarks on the German school system: Whereas we have four different types of secondary schools: Gymnasium, Realschule, Hauptschule, Gesamtschule (in some Northern states), our primary school is uniform: All children have to attend it, in most States for 4 years, in some States for 6 years. This special status of primary school was an achievement of the first German republic founded after world war 1. One of our greatest minds in mathematics education of this century, Johannes Kühnel, played a decisive role in getting the new law passed in 1920.

As far as teacher education is concerned Northrhine-Westfalia is the only State which has introduced a special teacher education program for primary teachers. All other States stick more or less to traditional approaches of teacher education and offer combined programs qualifying student teachers for teaching at the primary and secondary level except, however, for teaching at the gymnasium. The regulations for teacher education in the 16 States are as diverse as Germany’s landscapes. It is impossible even to sketch them in this conference.

**2.Subjects in Primary Education and in Primary Teacher
Education**

The subjects in primary education (grades 1 to 4) and
the corresponding school hours per week do not differ very much in the
various States. Approximately we have the following distribution:

German Language | Environmental Education | Mathematics | Art | Music | Physical Education | Religion |

5-6 | 3-4 | 4-5 | 3 | 1 | 3 | 2 |

Language and Mathematics are considered as the major subjects and are accepted as such also in the public. Many primary children like mathematics. In most surveys mathematics ranks second, only slightly behind physical education.

In Germany it is also widely accepted that primary teachers should have special knowledge in socializing and educating children and should be able to teach all subjects, at least in principle. As a rule a primary teacher is responsible for one class of children from grade 1 to grade 4.

However, the approaches to qualifying student teachers for working as a class teacher are quite different in different States. On the one hand we have teacher education programs which emphasize general education and the didactics of some subjects and which introduce student teachers only superficially to the subjects themselves. In particular, there are States where mathematical studies are not required at all. On the other hand there are programs which emphasize the scientific background of school subjects.

There is another point which is important here: Teacher education in Germany is split up in two phases: a more theoretical first phase (in most States 3 years) at the university, and a more practical second phase (2 years) in special seminars independent of the university and in close connection with schools. Against this background there is also the question to what extent practical elements of teaching should be included into the first phase.

In our State Northrhine-Westfalia the emphasis of teacher
education is clearly on the scientific background of teaching: All primary
student teachers have to study mathematics, language, a third subject (music,
environmental education, art, religion, physical education) at free choice
and in addition general education. One of the three subjects must be chosen
as a major subject. The 120 credit hours of the complete program for 3
years are approximately distributed as follows:

Major Subject | Minor Subject | Minor Subject | General Education |

45 | 25 | 25 | 25 |

Mathematics must be chosen by all student teachers either as a minor subject with 25 credit hours or as a major subject with 45 credit hours. Only 10% or less opt for mathematics as a major subject.

As a whole this program turns out as a good compromise: The basic school subjects language and mathematics are compulsory for all student teachers. A

third subject complements their subject matter knowledge. So teacher education covers a major part of the syllabus. That less importance is put on general education is to some extent mitigated by the fact that education is fundamentally integrated into the study of the subjects, too. Also general education plays a major role in the second phase of training.

**3. The Syllabus for Primary Mathematics in Northrhine-Westfalia**

The syllabus for primary mathematics in Northrhine-Westfalia describes three major content areas: Arithmetic, geometry, magnitudes and applications. The same areas can be found in the syllabuses of the other States. There is nothing special about them. The arithmetical contents are quite traditional: Grade 1: Development of the number concept, numbers up to 20, addition table. Grade 2: Numbers up to 100, multiplication table. Grade 3: Numbers up to 1000, informal methods of calculation, standard algorithms of addition and subtraction. Grade 4: Numbers up to 1 million, standard algorithms of multiplication and division.

Why this syllabus is nevertheless playing a leading role in the development of mathematics teaching and teacher education in Germany is due to three other features:

1. For the first time in history this syllabus prescribed the principle of learning by guided discovery as the basic principle of teaching and learning:

"The tasks and objectives of mathematics teaching are best served by a conception in which learning mathematics is considered as a constructive and exploring process. Therefore teaching has to be organized such that the children are offered as many opportunities as possible for self-reliant learning in all phases of the learning process:

- starting from challenging situations; stimulating children to observe , to ask questions, to guess
- exposing a problem or a complex of problems for investigation; encouraging individual approaches; offering help for individual solutions
- relating new results to known facts in a diversity of ways; presenting results in a more and more concise way; assisting memory storage; stimulating individual practice of skills
- talking about the value of new knowledge and about the process of acquiring it; suggesting the transfer to new, analogous situations.

2. The section "Tasks and objectives" lists the following four "general objectives of mathematics teaching": Mathematizing, Exploring, Reasoning and Communicating, which reflect basic components of doing mathematics at all levels.

3. The syllabus describes in some detail why mathematical structures on the one hand and applications of mathematics on the other hand are two sides of one medal and how they can be interlocked in teaching. The explicit statement of this complementarity is novel for German primary schools. Traditionally, the main purpose of primary schools was seen exclusively in preparing for solving practical problems arising in real life.

The development of the new syllabus was certainly very much influenced by similar developments in other European countries, in particular the Netherlands where Hans Freudenthal exerted his seminal influence. However, there has also been a strong trend towards active learning within traditional German mathematics education. At the beginning of this century, Johannes Kühnel wrote his famous book "Neubau des Rechenunterrichts" ("Reconstructing Arithmetic") in which he described the "teaching/learning method of the future" as follows:

**4. Primary Mathematics Teacher Education in Northrhine-Westfalia**

The framework for the study of mathematics as a minor
subject in primary teacher education in Northrhine-Westfalia is as follows:

Ground level (3 terms) | Two introductory mathematics courses:
1. Arithmetic (2+2) 2. Geometry (2+2) |
Introductory course on didactics of primary teaching (2+2) | Practical studies (2) |

Advanced level
(3 terms) |
One mathematics course (2+2) | One course on didactics (2+2) | Practical studies (2) |

The numbers "2+2" mean that the course consists of a 2-hour "lecture" and 2 hours of work in groups of approximately 25 students. This group work is conducted by graduate students. I should not forget to mention that at larger universities like Cologne, Münster or Dortmund the courses are attended by hundreds of student teachers - a huge challenge for teacher educators.

What is more important than the structure of the program is of course how the courses are organized. As the implementation of the new primary syllabus depends crucially on the teachers' ability to abandon the deeply rooted traditional guidance/receptivity model of teaching and learning in favour of the organisation/activity model it is not enough for teacher educators just to preach new ways of teaching. Instead they must reform their own courses according to the organisation/activity model.

While this conclusion is generally accepted there are
controversial positions how it should be realized. One group of mathematics
educators around Heinrich Bauersfeld who have their professional roots
in psychology and general education conceive the principle of learning
by guided discovery more as a psychological and educational principle.
These colleagues are in favour of emphasizing didactics instead of mathematics
and plead for more credit hours for mathematics education. On the opposite
side we have mathematics educators who have their professional background
in mathematics and who follow Heinrich Winter’s holistic view of mathematics
teaching. This group to which the mathe 2000 team belongs believes that
only teachers with first hand experiences in mathematical activity can
be expected to apply active methods in their own teaching. Basic references
for this group are John Dewey’s fundamental papers "The Relationship of
Theory and Practice in Education"^{2}
and "The Child and the Curriculum"^{3}.
At the request of our ministery of education a working group of maths educators
has just written a position paper which describes the more mathematical
approach in some detail. However, the ministry hesitates to adopt this
paper for whatever reasons.

**5. The Primary Teacher Education Program in Mathematics
at the University of Dortmund**

The program we offer to our primary student teachers
is based on the developmental research we are conducting in our project
,mathe 2000"^{4}
. We

consider mathematics education as a "design science"^{5}
and therefore our

research is focused on designing substantial learning
environments and investigating them empirically.

Substantial learning environments are used in teacher
education for multiple purposes:

In didactical courses each theme is introduced by means
of a set of typical learning environments. These environments are described
verbally and illustrated by means of videoclips, students’ documents or
pages from a textbook. As the selection of each set is also made with respect
to some general principle it is possible to explain this principle by refering
to these environments. In this way concrete examples and theoretical principles
support one another. My practical way of making this structure apparent
is to split any lecture into two parts: a first one in which the learning
environments are described, and a second one which is devoted to discussing
the attached theoretical principle.

Student teachers can use these learning environments for conducting teaching experiments during their practical studies in the first and in the second phase of training. So a lively relationship between theory and practice is secured.

Relating mathematical courses to learning environments
is much more difficult to realize for the following reason: The traditional
pattern of introductory mathematics courses at German universities is a
combination of a 2 to 4 hours per week lecture ("Vorlesung") and 2 hours
of training per week taking place with groups of about 30 students ("Übungen").
The problem is that the lecture/training pattern has a strong inherent
tendency towards guidance and receptivity: Often the tasks and exercices
offered to students for elaboration require mainly a reproduction of the
conceptual and technical tools introduced in the lecture. So more or less
students’ individual work and group work tends to be subordinated to the
lecture. Frequently, group work degenerates into a continuation of the
lecture: The graduate student responsible for the group just presents the
correct solutions’ of the tasks and exercises.

In order to get rid of the lecture/training format we
have developed the O-script/A-script method^{6}
. The basic idea, the Alpha and Omega, of this method is very simple: We
take Johannes Kühnel’s postulate seriously also in teacher education
and replace "guidance and receptivity" by "Organisation and Activity",
that is, we use both the lecture and the group work for organizing student
teachers’ activities.

Part of this new teaching/learning format is a clear distinction between the text written down by the lecturer on the blackboard or the overhead projector and the texts elaborated by the individual students. As the lecturer’s main task is to organize students’ learning her or his text is called the "O-script". It is not a closed text, but it contains many fragments, leaves gaps, gives often only hints. Therefore it is a torso which is to be elaborated by the students. As this elaborated text expresses the student’s individual activity it is called the "personal A-script".

How to organize students’ activity in a lecture? In trying to find an answer to this question we got inspired by two quotations:

(Giovanni Prodi, mathematician at the University of Pisa and a brother of the present president of the European union)

"The main goal of all science is first to observe, then
to explain phenomena. In mathematics the explanation is the proof."

(David Gale, geometer at the University of Berkely)

As an illustration of how learning environments are used in our teacher education program let me give two examples.

(see the copies from the textbook "Das Zahlenbuch" in appendix 1).

The third page is devoted to verifying all 8 possible magic squares with the numbers 1, ..., 9. Of course all these are symmetric.

In grade 2 children meet magic triangles, in grade 4 the famous magic square on Dürer’s copper engraving "Melencolia".

Student teachers are introduced to magic squares in the section "Playing with numbers" of the introductory course on arithmetic. The following questions provide a good starting point for mathematical activities which draw fundamentally on the general objectives "Exploring, Reasoning, Communicating":

Why must the magic sum be 15? Why must 5 be placed in the middle? How many magic squares with numbers 1, ..., 9 do exist? How to make magic squares of the first nine even or the first nine odd numbers? If the field in the middle is excluded: What magic sums are possible with numbers 1, ..., 9? What magic sums are possible in magic triangles with 6 or with 9 boxes?

Magic figures also appear in the introductory course
on didactics as an example of the section "Practicing skills". Students
are introduced into "Magic Triangles", a very rich material invented by
a German primary teacher. It provides also a good illustration for the
principle of "immanent practice".

My second example is "Regular and semiregular tesselations"(see the copies from ,,Das Zahlenbuch" in appendix 2)

This theme is covered in our textbook for grade 3 in the following way:

The first page introduces the regular polygons which can be drawn by means of a stencil. The second and third page stimulate children to draw regular and semiregular tesselations as well as strings and rings of forms - and to design their own patterns!

Our student teachers meet tesselations in the unit on
"congruent figures" of the introductory course on geometry. First they
try to find the 3 regular and 8 semiregular tesselations by working with
cardboard polygons. They look at the angles and investigate which angles
fit together. Later on the semiregular tesselations are derived systematically.

In the didactical courses tesselations are met again
in connection with fundamental ideas of elementary geometry.

These two examples show that our interest in primary teacher education lies definitely in elementary, not in higher mathematics. Moreover: What counts for us in the first line are not ready-made mathematical theories, but mathematical activities as captured in the general objectives "Mathematizing, Exploring, Reasoning and Communicating". The view we adopt of mathematics is a genetic one: Mathematics is an organism growing out of elementary seeds. Elementary number theory, combinatorics and geometry are so substantial, and they provide such rich contexts for mathematical activities that there can be no better introduction into mathematics as offered by these domains. The use that is made of them in mathematical olympiads confirms this fact.

As there is no systematic mathematical literature which would serve our purposes in primary teacher education a group of 16 authors has started a series of mathematical textbooks for teacher education based on the experiences of the "mathe 2000" project. The first volume entitled "Arithmetic as a Process" is well underway. We hope to get it published next year.

The experiences with our approach are very good as was
found some years ago in a survey made by the centre of teacher education
at our university. Student

teachers in their second phase of training were asked
to evaluate the professional education they had received in German language,
mathematics and general education at their university. The questionaire
contained the following questions:

(2) To what extent were theories introduced as an answer to problems?

(3) How was the didactical training related to the mathematical training?

(4) How was the didactical training related to the practice of teaching?

(5) To what extent were you introduced to innovative approaches
as

described in the syllabes?

The empirical findings (see appendix 3) show that the teacher education programs in mathematics at the Universities of Paderborn and Dortmund which are based on similar principles were by far best evaluated by student teachers. We consider this as a very encouraging feedback.

REFERENCES

^{1}Paper submitted to
the conference on "The training and performance of primery teachers in
mathematics education" organized by the Spanish Royal Academy of Sciences,
Madrid, October 16, 1999.

^{2}John Dewey, The relation
of Theory to Practice in Education. In: John Dewey, The Middle Works 1899-1924,
vol.3, ed. by Jo Ann Boydston, Carbondale/Ill.: Southern Illinois University
Press 1983, 249-272.

^{3}John Dewey, The Child
and the Curriculum. In: John Dewey, The Middle Works 1899-1924, vol.3,
ed. by Jo Ann Boydston, Carbondale/Ill.: Southern Illinois University Press
1983, 271-292.

^{4}E.Ch. Wittmann, Toward
an Activity-Based and Focused Curriculum: The Project "mathe2000"Paper
submitted to the EARLI conference, Gothenburg/Sweden August 1999

^{5}E.Ch. Wittmann, Mathematics
Education as a "Design Science". Educational Studies in Mathematics 29
(1995), 355-374

^{6}E.Ch. Wittmann, The
Alpha and Omega of Teacher Education: Organizing Mathematical Activities.
To appear in the ICMI-Study "Teaching Mathematics at University Level"

* Address of the author:

Prof. Dr.h. Erich Ch. Wittmann

University of Dortmund

Dept. of Mathematics, IEEM,

D-44221 Dortmund

ewittmann@mathematik.uni-dortmund.de