Erich Ch. Wittmann*

Primary Teacher Education in Mathematics in Germany1

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: The task of the teacher is to find and to offer challenging situations, to provide the children with substantial materials und productive ways of practising skills, and, above all, to build up and sustain a form of communication which serves the learning processes of all children."

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 new syllabus carries the hand-writing of Heinrich Winter, one of our leading mathematics educators. In his work Winter has always pleaded for looking at the teaching of mathematics from the primary level to the university level as one whole whereby mathematics is viewed as a process, not as a ready-made product.

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:

"The learner will no longer be expected to receive knowledge, but to acquire it. In future not guidance and receptivity, but organisation and activity will be the special mark of the teaching/learning process". Since the late eighties considerable progress has been made in developing practical approaches and materials for this new conception of mathematics teaching including innovative textbooks. The developmental work of our project ,,mathe 2000" is a typical example. It includes didactical textbooks for teacher education, a textbook series and accompanying materials.

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 method6 . 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:

"We should teach more along problems than along theories. A theory should be developed only to the extent that is necessary to frame a certain class of problems."
(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)

 Accordingly, our mathematical courses are divided in two parts: The first part is devoted to introducing and clarifying a list of selected problems. The emphasis is on the observation of phenomena and attempts to explain them. The second systematic part provides a theoretical framework for these problems, however, based on students’ experiences in writing the A-scripts. The format of the second part does not differ very much from expository lectures.

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

The first one is "Magic squares"

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

 In our textbook for grade 1 there are the following three pages on magic squares: First children learn to calculate the sums of the rows, columns and diagonals of 3x3 squares made up of the numbers 1,...,9. Then children are guided to discover the Lo Shu square dating back to the ancient Chinese: By murmuring an incantation a magician stimulates shuffles of the nine number cards. The last verse reads as: ,Now be clever and exchange two appropriate cards. Then you will have all sums equal".

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?
    • Later on a systematic treatment of 3x3 and 4x4 magic squares is given in the systematic section of the course which is based on the experiences with these activities.


    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:

    (1) To what extent have you been encouraged for active learning?

    (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?

         A 4-step scale was applied to quantify the answers: 1 = very little, 2 = relatively little, 3 = relatively much, 4 = very much.

    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

    1Paper 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.

    2John 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.

    3John 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.

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

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

    6E.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