Hans-Christian Reichel

Lakatos and Aspects of Mathematics Education

Amongst mathematicians and philosophers, Imre Lakatos is well known as a scientist who linked mathematics, philosophy and epistemology perhaps more closely than did any other mathematician during the second half of our century. But Lakatos also had a strong influence on mathematics education. This influence regards everyday mathematics education at school, as well as the scientific background theories. Particularly his well known book Proofs and Refutations (which has been published in German, as well), influenced the theory and practice of mathematics education (or “didactics” as German speakers prefer to say).

Topological characterizations of linearly uniformizable spaces

Abstract The spaces having uniformities with a totally ordered base are characterized in bigger classes of non-archimedean spaces and suborderable spaces. Consequently, several new metrization results are obtained. By examples, we show that the conditions used in our main theorem cannot be weakened essentially. Our examples may be interesting elsewhere, too.

Teaching student teachers: Integration of mathematics education into „classical“ mathematics courses. Examples and various aspects.

This paper continues a theme recently accentuated again by a paper of E. Wittmann ([WI]) published in JMD 10 (1989): “The mathematical training of teachers from the point of view of education”. It partly parallels and continues E. Wittmann’s scenario and ideas, but partly differs from those specifically. In any case, it is to be hoped that the discussion of teacher training at universities and other institutions is stimulated again, as well by our papers as by other recent ones (see e.g.[DO], [FI], or others cited in the bibliography).

Identifying and Promoting Mathematically Gifted Pupils and Students (12–20 years)

Although there have been some attempts to define mathematical giftedness, there is no generally accepted definition in a mathematical way. A formal definition, though, would be extremely difficult to formulate since mathematical giftedness becomes apparent in many different ways and at different stages of age and knowledge. But in order to be better able to discuss mathematical giftedness, this paper presents several subjective ideas on the identification and promotion of what we call mathematical giftedness, taken from more than 30 years as a teacher and a trainer of teachers. Hints and references to situations where gifts or special interests in mathematics are detectable are offered. One aim is to present observations and analyses which may provide a better understanding of mathematical gifts which may be helpful in teaching.

Topological characterizations of ωμ-metrizable spaces

Abstract This paper is a detailed elaboration of a talk given by the second author at the Oxford conference in June 1989. It presents necessary and sufficient conditions for a topological space to be ω μ -metrizable (μ> 0), i.e., linearly uniformizable with uncountable uniform weight. In other words, such spaces are exactly those which can be metrized by a distance function taking its values in a totally ordered Abelian group with cofinality ω μ . (For ω μ = ω 0 , we obtain characterizations of strongly zero-dimensional metric spaces, i.e., nonarchimedeanly metrizable spaces.) It turns out that (strong) suorderability and the existence of a σ-discrete (respectively ω μ - discrete) dense subspace are the most interesting properties in this respect, whenever ω μ > ω 0 , or ω μ = ω 0 and dim X = 0. Therefore, a main part of the paper is devoted to the study of GO- spaces having a σ-discrete (ω μ ) dense subspace (Section 3). The last section (Section 4) is concerned with the characterization of ω μ -metrizability in the realm of generalized metric spaces, in particular, by using g -functions. Since all our spaces are zero-dimensional , the paper also contributes results to this important class of spaces, in particular, to the class of nonarchimedean topological spaces.

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.

Über Distanzfunktionen mit Werten in angeordneten Halbgruppen

This paper presents a general investigation of the relations between structural properties of a totally ordered abelian semigroupS and the properties of various “topological” structures, such as topologies, bitopologies and (semi-)uniformities on a spaceX induced byS-valued distance functionsd∶X×X→S satisfyingd(x,y)=0 iffx=y and the triangular inequalityd(x,z)≤d(x,y)+d(y,z), for allx,y,z∈X. Since a linearly ordered abelian semigroupS need not be a topological semigroup with respect to its order topology we have to consider two cases: the case where addition inS is continuous at 0∈S, and the case where it is not. For both cases, we state several metrization theorems, examples and applications. In this connection, we are also concerned with some special basis-properties of topological spaces. Closely connected is the program stated byAlexandroff-Bourbaki (amongst others) to investigate to what extent countability inherent in matrization theory can be replaced by order-theoretic properties.—Distinguishing between symmetric and not necessarily symmetric distancesdS we obtain a theory containing the theory ofωµ-metrics andωµ- quasimetrics.—As far as it concerns not necessarily symmetric distancesd onX, it seems adequate to study the bitopological structure (τe,τr) induced onX byd and the “inverse” distanced−1 respectively. This is done in § 4 where, in this respect, we also generalize a well-known theorem ofSion andZelmer.

Uniforme Räume mit einer linear geordneten Basis

AbstractA. K. Steiner undE. F. Steiner described the socalled natural topology κ on spacesAB of transfinite sequences (aβ), β∈B,aβ∈A [J. Math. Anal. Appl.19, 174–178 (1967)]. These spaces generalize Baires zerodimensional sequence-spaces. Using these spaces (AB, κ), we generalize two well known theorems of F. Hausdorff, W. Hurewicz, C. Kuratowski and K. Morita on metric spaces and their Lebesgue-dimension respectively, both involving Baires sequence spaces. Thus we obtain a topological characterization of uniform spaces

New approaches to and another view of mathematics and science teaching

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