Mathematics

3.1 The mathematical culture of classes
Many authors analysed the situation of German mathematics education and
found out what has to be done in a better way. (TIMSS, PISA, Coactiv,
expertises of Winter/Baptist, Borneleit et. al., Kaiser/Henn) In addition to all
those findings I observe that mathematics teachers tend to use the known
algorithms without reflecting critically about the specific task (tendency to
uncritical attitude). The following examples show the uncritical attitude towards
tasks of central examinations.
The first example of the final examinations 2011 gives an impression of what
usually happens in classes: “Given is a box ABCDEFGH (the base is a square
with length 3, height of the box: 13) by two points A(0,0,0) and G(3,3,13). Draw
the image of the solid in a rectangular coordinate system …” Nobody noticed
and complained about the incomplete description of the position of the solid. I
talked with other colleagues after the examination. They did not recognise the
gap. In their opinion this mistake within the task is not worth mentioning.
This attitude is reasoned by the belief system many teachers have.
Mathematics is more and more considered as a set of knowledge that has to
be known and as a set of procedures which have to be followed and executed.
Unfortunately these attitudes are supported by the interpretations of the
educational standards through simple and short tasks in tests and the final
examination.
The second example is of the final examination in 2005. “A function f is given
by
… Draw the graph … There exist exactly three tangents at the
graph which have no further point with the graph in common. Give the
equation of these tangents.” There are 5 tangents of this property. Only four
teacher considered this.
3.2 The mathematical contents which are widely unknown
For many the reasons mathematical contents and problems have to be
selected carefully. But they are mainly given by the curriculum and central
examinations. Therefore the domain of tasks and problems is very restricted.
That shapes the professional knowledge of the teacher.
Example 1 (central examination): “Prove that y=2x-2 is the equation of the
tangent at the graph of the function with y=x^2-2x+2 at P(2,2).” It was unknown
to the teachers who was responsible for the tasks of the final examination that
one could confirm this assertion by transforming x^2-2x+2=2x-2 to (x-2)^2=0. It
is unusual to introduce the concept of tangents as a line that snuggles up to
the graph and that gives the linear approximation at the point P. The
competence which is desired is the use of the differential calculus which
students should always practise.
Example 2: Drawing without replacement: a box with 8 white and 12 black
spheres; drawing 5 times; X = number of white spheres which are drawn. Are
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