Roland Fischer

Sofic systems and graphs

A subclass of the class of the subshifts of finite-state symbolic shifts, which was introduced byB. Weiss under the name “sofic systems”, is characterized and studied by using graphs. It is proved that topologically transitive sofic systems are intrinsically ergodic.

Unterricht als Prozeß der Befreiung vom Gegenstand — Visionen eines neuen Mathematikunterrichts

The central thesis is that one of the functions of mathematics education within the official educational system should be to make a contribution for the liberation from mathematics. It is argued that there exist many possibilities for gaining (scientific) knowledge outside schools, that there exists a lot of (scientific) knowledge, which is sometimes perceived as a burden and has generated phenomena such as “mathophobia”. Mathematics itself has been a means for the liberation of men from (his first) nature, but now it has become a “second nature”, from which delivering is necessary. Mathematics education can fulfil this new function mainly by emphasizing questions of sense in the classroom and, thus, questions of the relation between men and knowledge. Some consequences of these claims are discussed, concerning teacher’s role in the classroom, the use of media (especially textbooks), the use of computers, the dealing with errors in mathematics learning and the connection with social learning.

Mathematik und gesellschaftlicher Wandel

Society today needs more collective self-reflection. Mathematics could contribute to this goal if it undergoes substantial changes (in teaching, research and application). The main role of mathematics is to be an instrument and instrumentalism impedes self-reflection. There are two orientations for mathematics which could broaden the possibilities of mathematics for collective self-reflection: analysis of basic assumptions (mathematics as a mirror of mankind) and problem description instead of problem solving (mathematics as a means of presentation and communication). This orientations can be illustrated by examples from applied mathematics, in special consideration of social sciences and economics.

Sinn mathematischer Inhalte und Begriffsentwicklungen im Analysisunterricht

The question of the “sense” of mathematical contents in classroom cannot be answered in an absolute way. The constitution of sense depends on the context, the social circumstances in general, the classroom situation etc. What can be done are “local” sense-argumentations. Such argumentations are given concerning some concepts of calculus: functions, derivative, limit and continuity, integral and real number. Thereby alternative views on these concepts are emphasized. Finally some general considerations about sense-argumentations and paedagogical motivations are presented.

Materialization and organization

This summary of six articles which have been written in the past fifteen years focus on the question of the social relevance of mathematics on a principal level. The main theses are: Mathematics provides materializations of abstract issues, thereby it supports mass communication. the principles of mathematics are basic for our social organization. The limits of mathematics are limits of organization. But they can be overcome by emphasizing the reflexive potential of mathematics.

Zur Einführung des Vektorgegriffes: Arithmetische Vektoren mit geometrischer Deutung

In [BÜRGER — FISCHER 1978] vectors are introduced in the following way: Vectors are elements of ℝn (n = 2,3), the algebraic operations are defined arithmetically. Vectors and operations can be interpreted geometrically in different ways: vectors as points or arrows, addition for arrows, for point and arrow etc. Flexibility between different interpretations, transfer of concepts and lingual identifications are aspects of working with this vector-concept. In the first part of the paper the concept and its aspects are explained; this is the basis for arguments for evaluation, which follow in the second part.

The number of steps in a finite JACOBI algorithm

Let x=(x1,...,xn) be a rational point and denote by L(x) the length of thefinite JACOBI algorithm associated with x. Let b(x) be the minimal common denominator of x. Then L(x)=const, log b(x) is true in a certain sense for almost all x.

Entscheidungs-Bildung und Mathematik

Ich vertrete ein Allgemeinbildungskonzept, das auf Entscheidungsfahigkeit und Kommunikationsfahigkeit mit ExpertInnen abzielt. Dieses Konzept soll hier kurz dargestellt und es sollen Konsequenzen fur den Mathematikunterricht gezogen werden. Schlieslich soll die Rolle von Mathematik in Entscheidungsprozessen grundsatzlich erortert werden.

The Formal, The Social and the Subjective

With strong reference to his book “Das Formale, das Soziale und das Subjektive” ideas of M. Otte are put into relation with some deliberations of the present author. These concern the role of science in society, general education, the role of mathematics in education, the limits of mathematics and their social relevance.

Technologie, Mathematik und Bewußtsein der Gesellschaft

Mein Thema ist die Rolle der Mathematik in Bezug auf den Zusammenhalt von Gesellschaft und in Hinblick auf das, was ich Bewustsein von Gesellschaft nenne. Technologie spielt dabei insofern eine Rolle, als sie erstens fur den Zusammenhalt der heutigen Gesellschaft eine besondere Bedeutung hat und sie zweitens mit Mathematik immer schon eng verbunden war. “Neue Technologien” haben keine Sonderrolle, sie stellen blos die jungste Ausfonmung bzw. Zuspitzung eines Phanomens dar.