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O badaniach nad kształtowaniem się u studentów matematyki podstawowych pojęć analizy matematycznej

100%

Powązka Z.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2013

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tom 5

105-119

PL

This paper is a reflection on the process of forming the basic concepts of mathematical analysis in students. These studies were conducted in the years 2002-2011 and they coincided with the period of major educational reforms in our country. These reforms had an impact on the level of Mathematics teaching in various schools and on anticipatory preparing of young people to study this subject. Thus, this work is an attempt to summarize the results of the already described various stages of the research. The first part contains the answer to the question about the role of the content of this subject in the mathematical preparation of future teachers of Mathematics. In next parts, examples of introducing selected concepts of mathematical analysis are presented and thereafter conclusions from the carried-out research are formulated.

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Simple fractions and linear decomposition of some convolutions of measures

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Misiewicz J., Cooke R.

Discussiones Mathematicae Probability and Statistics

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2001

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tom 21

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nr 2

149-157

EN

Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a}(ξ) = [a/(h(ξ) + a)]$, where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain that convolutions of measures $μ_{a}$ with $μ̂_{a}(ξ) = φ_{a}(ξ)$ are linear combinations of powers of such measures. This can simplify calculations. It is interesting that this simplification uses signed measures since coefficients of linear combinations can be negative numbers. All the results of this paper except Proposition 1 remain true if we replace probability measures with complex valued measures with finite variation, and replace the characteristic function with Fourier transform.

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Z badań nad trudnościami studentów w rozumieniu pojęcia miary i całki

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Powązka Z.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2012

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tom 4

139-154

PL

Measurement is an important form of human activity. It appears at all levels of secondary school education, while discussing the length of a segment, the area of a plane figure and the volume of a solid. The definition of a measure, understood as a non-negative real function defined on a σ-field of set of a space, is taught during the classes on mathematical analysis. It is observed that the concept of measurement causes great difficulties to students. This paper presents the results of research on students' difficulties in understanding and using the basic concept of measure theory and integral. The research conducted in years 2006-2009 shed some light on the causes of those difficulties. The study involved students of the third year of the Mathematics Department at the Pedagogical University of Krakow.

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Przykłady zagadnień z różnych działów matematyki niezbędnych do studiowania teorii miary

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Chronowski A., Powązka Z.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2012

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tom 4

31-60

PL

In the education of mathematicians, including teachers of mathematics, the measure theory plays an important role. From the experience in and research on teaching the measure theory it follows that on of the important reasons why students encounter difficulties in the subject is their insufficient ability to apply their knowledge of other branches of mathematics, especially of the set theory and topology. In this article we propose a series of problem analyses and exercises aimed at preparing students to study the measure theory, especially to understand the proofs of theorems on properties of the measure, including the Lebesgue measure. Most of these problems and exercises are presented with solutions, outlines of solutions, hints, didactic remarks and comments. In this paper we point out, in a practical way, the significance of active reading of mathematical texts and skilful use of mathematical literature. This article is dedicated to both students of mathematics and their academic teachers.

Ograniczanie wyników

3 Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

1 Discussiones Mathematicae Probability and Statistics

3 Powązka Z.

1 Chronowski A.

1 Cooke R.

1 Misiewicz J.

1 2013

2 2012

1 2001

O badaniach nad kształtowaniem się u studentów matematyki podstawowych pojęć analizy matematycznej

Simple fractions and linear decomposition of some convolutions of measures

Z badań nad trudnościami studentów w rozumieniu pojęcia miary i całki

Przykłady zagadnień z różnych działów matematyki niezbędnych do studiowania teorii miary