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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Strong sequences and partition relations

100%
Jureczko J.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2017
|
tom 16
EN
The first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdös-Rado theorem (1956). On the other hand in 60’s of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdös-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.
2
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Using the graphic display calculator to motivate derivative formulas

100%
Jureczko J.
Didactica Mathematicae
|
2014
|
tom 36
EN
Nowadays using graphic display calculators (GDCs) is becoming more and more popular in learning and teaching mathematics. One of the main usages of this device is to obtain a graph of almost all functions with one variable. If we have a graph of a function we can observe some special properties of the function. Graphic display calculators (even without computer algebra system (CAS)) can do even more - calculate derivative at any point and draw graphs of derivatives of the first and the second degree, find other properties of functions like: minimum, maximum, x- and y-intercepts and so on. Traditionally students are taught differentiation while using limits of functions. Hence the natural question is whether there exists another method of teaching about differentiation? The aim of this paper is to analyze attempts of teaching some aspects of differentiation with using GDC.
3
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κ-strong sequences and the existence of generalized independent families

100%
Jureczko J.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1277-1282
EN
In this paper we will show some relations between generalized versions of strong sequences introduced by Efimov in 1965 and independent families. We also show some inequalities between cardinal invariants associated with these both notions.
4
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Using the graphic display calculator in solving some problems concerning limits of sequences and functions

100%
Jureczko J.
Didactica Mathematicae
|
2015
|
tom 37
EN
Limits are one of the most fundamental and important concepts of calculus, because having a proper understanding of this concept is a prerequisite for understanding such concepts as continuity, differentiation, and integration. Thus, researchers and teachers seem to outdo one another in looking for the best methods of introducing limits. One of the methods is using IT and graphic software to illustrate the idea of limits. It Seems that such a method, while useful, is overflowing with the risk of making mistakes and strengthening the improper understanding of these basic concepts. The aim of this paper is to present how students who use graphic display calculators understand the limits of sequences and functions, before introducing the theoretic and algebraic methods required for solving limits, and what dangers can be caused by using a GDC in this process.
5
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A task about a cube; or, on generalization in 3D

100%
Jureczko J.
Didactica Mathematicae
|
2016
|
tom 38
EN
The necessity of teaching such an activity as generalization was considered by Z. Krygowska in the ‘70s (Krygowska, 1977). The formation of this ability requires an adequate selection of non-stereotypical tasks, for which the algorithm is unknown to the person solving the task, ones in which a student is forced to search for their own method of solving the task on the basis of their knowledge. In literature, there are known studies concerning the process of generalization of students of different ages which use, at most, 2D visual patterns. However, the author still did not find any research based on tasks which examine the process of generalization in 3D. In this paper, results will be shown of using a task concerning a cube carried out in a diverse group of students from middle school, high school, and university.
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