250038 VO School mathematics 1 (Arithmetic and Algebra) (2010W)

Arithmetic is the bridge in mathematics education from primary level to the
secondary one. It will be continued to the end of school education changing
its name. Calculating with natural numbers is beside geometry the basic
issue of (school-)mathematics, it won't lose its meaning if the number
range is extended. These steps forward to "new" number sets are remarkable: they represent in a didactical sense real breaks, interruptions in the basic beliefs of the pupils, which could lead to misunderstandings and errors. Thus in this lesson we will focus on these number extensions. Mainly the fraction numbers are different to the numbers "before" (i. e. integers): there exist no antecessor and no successor of a fraction, for instance, and already the basic arithmetics (addition, multiplication and their inverse operations) are quite different to manage in comparision to their pendants in the natural numbers. And the real numbers? What's up with them? Unfortunately the answers to these questions which are given in the calculus lessons are often cloudy presented, not sufficient for a competent teaching afterwards. This lesson will try to close this gap.
Elementary Algebra is in addition to calculating with fractions one of the
most important fields which are content of mathematics education in
arithmetics at secondary level one. The ability of "calculating with
characters" is everywhere needed in mathematics indifferent at which level
it is done. Simultaneously mathematics loses its "innocence", this means
that the step from the concrete to the abstract point of view is done at
this time and it will never be retracted. Far from it one of the
characteristic features of mathematics gets so included in general
education.
This lecture will focus this essential step with all its difficulties which appear in mathematics education, in literature this phenomena is known as "pupil's mistakes in algebra". The cognition of term structures (and the consequent acting) is the key ability to successful manipulating algebraic formulas (besides a certain training but this is not a (primarily) specific point of this theme). In addition the use of computer algebra systems in mathematics education must be discussed. So the treated topics will also touch the contents in school mathematics at the secondary level two including leaving examinations.
The basis of all our considerations will be always the current Austrian curriculum in mathematics, of course.