Most German schools are municipal or state institutions, and
the teachers are employees of the state.
The teaching is regulated by many decrees concerning nearly all aspects
of schooling and school organization.
Concerning mathematics teaching, there are regulations extending
from very detailed syllabi to the number and type of tests
to very special aspects of instructions like the notation for division
with remainder or the type of handheld calculator to be used. Additionally,
the textbooks used in class require the state's permission.
On the other hand, the teachers are free in important aspects of their
teaching. It is quite common that the textbook has more influence than
the official syllabus. Also, the tests are constructed by the individual
teachers themselves to suit the special abilities of their pupils. Even
the written final examinations are constructed by the teachers, except
in the states of Baden-Württemberg, Bayern, Brandenburg, and the Saarland.
Therefore the teachers' position is a very strong one with respect to the
pupils.
Syllabi point out leading ideas and goals for mathematics teaching
concerning:
Syllabi attach mathematical content to grades.
1st to 2nd
Objects and their attributes (sets and their elements, diagrams); arithmetic:
the four basic arithmetical operations with numbers less than 100; magnitudes
(lengths, money, and time); and first geometrical experiences: forms and
patterns.
3rd to 4th
Dealing with sets (subsets, intersection, union); arithmetic with natural
numbers up to 1 million, including written calculations; calculating with
simple magnitudes in concrete situations; simple solids and their grids,
tessellations; and first ideas of axial symmetry.
5th to 6th
Use of variables, simple equations; transforming magnitudes; area of
rectangles, volume of blocks; fundamental geometric concepts (such as point,
line, parallel, geometric forms); digits and number systems; elementary
number theory (prime numbers, g.c.d., l.c.m.); and fractions and decimal
fractions.
7th to 8th
Relations and functions; commercial arithmetic (proportion, percent);
congruence transformations; congruent triangles, quadrilaterals; angle
measurement and associated theorems; area of polygons, volume of prisms;
transformation of terms, sentential forms, linear equations; algebraic
structures; and integers and rational numbers.
9th to 10th
Real numbers; quadratic functions and equations; similarity; theorems
on right triangles and circles; powers, exponential functions; area of
circle, volume of pyramid, cylinder, cone, and sphere; and trigonometry.
11th to 13th
Calculus; vector space, analytic geometry; geometric transformations,
conics; algebraic structures; and probability and statistics.
There are several actual problems in teaching mathematics. Two of them seem to be central.
Today computer algebra systems such as DERIVE and MAPLE are available which can solve nearly all algebra problems taught in school. Even handheld calculators like TI 92 can do this. So, what should be the skills in algebra?
Understanding the procedure and the interpretation of the results should become more important than just to handle the procedure.
These developments raise problems in the assessment in mathematics education. How can abilities in understanding and interpretation be tested?
Micro computers offer better chances in visualising mathematics. The plotting of graphs of functions is rather common in mathematics instruction. They are used for studying the influence of parameters. Dynamic processes are demonstrated with the use of a micro computer. Geometry programs like Cabrie Geometry or GEOLOG have good chances in order to study geometric properties and relationships.
Mathematics is no longer a subject for gifted people only. But when mathematics for all is accepted, then it is necessary to discuss the goals.
Some trends of a new orientation of mathematics education can be identified:
These trends will be supported by reducing the training of mathematical skills and practice in order to accept and to use high tech in the form of handheld calculators or micro computers offering strong mathematical tools.