Displaying similar documents to “O różnych sposobach definiowania wartość bezwzględnej”

Uwagi dotyczące pojęcia wartości bezwzględnej liczby rzeczywistej

Joanna Major, Zbigniew Powązka (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This paper deals with some studies of understanding the absolute value of real numbers. This article is divided into two parts. The first one contains the conclusions drawn from the analysis of the solutions of the mathematical problems with the absolute value which were solved by the secondary school pupils. The second part of the article contains the didactic proposition of teaching the concept of absolute value at various levels of mathematical education.

O ciałach uporządkowanych

Piotr Błaszczyk (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper, we present some basic facts concerning ordered fields. We review definitions of an ordered field, give an example of a field that admits many orderings, and present equivalent definitions of the axiom of Archimedes and the continuity axiom. We show how to extend an ordered field by means of an ultrapower construction and formal power series.

Prime counting function π

Jan Górowski, Adam Łomnicki (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The aim of this paper is to derive new explicit formulas for thefunction π, where π(x) denotes the number of primes not exceeding x. Some justifications and generalisations of the formulas obtained by Willans (1964),Minac (1991) and Kaddoura and Abdul-Nabi (2012) are also obtained.

Uporządkowane struktury liczbowe

Antoni Chronowski (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this article we consider the ordered algebraic structures of thesystems of natural numbers, integers, rational and real numbers. We presentthe ordered algebra of natural numbers, the ordered ring of integers, and theordered fields of rational and real numbers. The main problem considered forordered number structures is the categoricity of these systems determined bya suitable isomorphism. First of all, this article is addressed at Mathematicsstudents of pedagogical studies and at...

Tourism development in small cities on the example of Pistoia in Tuscany

Małgorzata Bajgier-Kowalska, Renata Rettinger (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The dynamic development of tourism in many regions of the world causes competition boost which, in turn, leads to a variety of changes in tourist offers. Tourism is an alternative for other types of economic activity, this relates to spatial units of diverse area, including cities. The tourist function can become the mechanism of local development for cities and the protection of the cultural and natural heritage. The main goal of the article is to present the tourist potential of a...

Przekształcenia wykresów funkcji

Antoni Chronowski (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This article is concerned with some topics of the transformationsof the graphs of functions. It is designated to serve as a text for thestudents of mathematics (prospective teachers) and the teachers of ma-thematics.In this paper we give the answer to the following problem:Given the graph of the function f:R Df

Fixed-point theorems for multi-valued functions and their applications to functional equations

Roman Węgrzyk

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IntroductionIn this paper we present some fixed point theorems for multi-valued functions (m.v. functions for short), generalizing well-known results (Chapter 2).Next, in Chapter 3, we consider continuous solutions of equation (3.1). First we characterize the continuous solutions of this equation, and then we give some theorems on extending continuous solutions.Chapter 4 contains theorems on the existence of continuous solutions when this equation is of order n, i.e. if the set of indices...

Trójkąty średnioboczne o środkowej mającej długość wymierną

Jan Górowski, Adam Łomnicki (2014)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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We use the recurrence relations and the Pell equations to determineall integer triangles whose lengths are consecutive integers and the length ofa fixed median is a rational number.

Housing conditions in selected metropolitan areas in Poland

Piotr Raźniak, Anna Winiarczyk-Raźniak (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Housing conditions are considered an element reflecting the living standard of the population. In addition, it is also an important index illustrating advancement of suburbanisation processes. The article presents diversification of housing conditions in two selected metropolitan areas: Warsaw, Kraków and Potential Rzeszów Metropolitan Area. Two indexes are analysed in the paper: the number of persons per apartment and the average floor space of apartments, in two time periods:1995 and...

Innowacyjność w hotelarstwie z uwzględnieniem adaptacji obiektów zabytkowych na bazę noclegową

Kamila Ziółkowska-Weiss (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Nowadays, tourism is a crucial sector of national economy, and for many countries and regions it is one of the goals of economic activity, at the same time being a means of social and economic revival. Therefore, in order to professionally present the tourist potential and offer, and to attract tourists, tourist products should be organized, promoted and introduced in a relevant way. It is necessary to bear in mind that tourism is a service sector, a sector in which different kinds of...

Информационно-коммуникационные технологии в образовании студентов специальности >>5В010900-Математика<< педагогического вуза

Бакыткуль Р. Каскатаева, Назира С. Баймулдина (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the paper the authors discuss educational usefulness of special applied software (Mathematica) in the process of studying Mathematics at the pedagogical university. The authors also present the levels of thecompetences connected with using an educational electronic means, necessaryfor the students of mathematics.

Different ways of solving quadratic equations

Barbara Pieronkiewicz, James Tanton (2019)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper we explore different ways of solving quadratic equations. Our main goal is to review traditional textbooks methods and offer an alternative, often side-stepped method based on the area model. We conclude that whereas traditional methods offer effective algorithms that quickly lead to the desired results, alternative methods may enhance meaningful and joyful learning.

Antropopresja turystyki wysokogórskiej na przykładzie Kilimandżaro

Michał Apollo (2010)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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This article outlines the negative impact on the environment of the Kilimanjaro Mountain which is a result of commercial approach to mountaineering and change of the climber’s profile. Kilimanjaro, which was called “trampled mountain” in the years 1977–2007, has been visited by 627,974 tourists. The National Park of Kilimanjaro pocketed 25.7 millions USD from admission fees in 2006 only. Officials of the Park are more interested in the commercial aspect than the environmental one. A...

Różne reprezentacje liczb rzeczywistych

Barbara Pieronkiewicz (2017)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This article is devoted to the different representations of real numbers. In particular, the following types are distinguished and discussed: (1)representations based on theorems referring to the axiomatic characterization of the field of real numbers, (2) genetic representations – related to the construction of real numbers, (3) visual representations – mainly related to the geometrical way of presenting numbers. The paper addresses different representations of real numbers from a higher...

Актуальные проблемы обучения математике в школе и вузе в свете идей Л. С. Выготского

Людмила Боженкова, Марина Егупова (2016)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The article gives a brief overview of the scientific results of the III  International Conference held in Moscow State Pedagogical University in November 2016. The main directions of the development of ideas of L. S. Vygotsky, presented in the reports of leading Russian and foreign scientists – participants of the conference.

Idea czynnościowego wprowadzania pojęć geometrii elementarnej z pomocą kursu e-learningowego

Magdalena Kucio, Tadeusz Ratusiński (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The present article introduces a didactic proposal of supporting the process of teaching mathematics with blended learning. The authors discuss both benefits and dangers of this idea. Also, the concept of functional method of teaching mathematics is brought closer, as it is naturally inscribed in the concept of blended learning. In the second part of the article, the authors describe the materials processed for employment of the discussed foundation in teaching elementary geometry. Estimate...

Geometria i kształcenie geometryczne we współczesnej szkole średniej i wyższej - praktyka i problemy

Roza A. Utiejewa, Antoni Pardała (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The present article presents the genesis of the international conference: andhighGeometry and geometrical education in the current second high school, which took place on November 22-25, 2012 in Togliatti StateUniversity. This conference was dedicated to the 70th jubilee of professorW.A. Gusiew (В.А. Гусев), world-wide famous professor of Mathematicseducation, author and co-author of Geometry textbooks and methodologicalguidebooks for teachers of Maths, published for the need of students’...

Nowe spojrzenie na gry Penneya

Maciej Major, Barbara Nawolska (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The problem of gaming is an important part of the educational process on all levels of Mathematics teaching. In this work, we present certain stochastic problems, based on the so-called Penney games. It provides the opportunity to formulate new problems and to construct the tools needed to solve them. This process is accompanied by a variety of mathematical activities.

Iterations of homographic functions and recurrence equations involving a homographic function

Jan Górowski, Adam Łomnicki (2015)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The formulas for the m-th iterate ( m ∈ N ) of an arbitrary homographicfunction H are determined and the necessary and sufficient conditions for a solution ofthe equation y m + 1 = H ( y m ) , m ∈ N to be an infinite n-periodic sequence are given. Based on the results from this paper one can easily determine some particular solutionsof the Babbage functional equation

Dekadencja geografii i zapomniane idee Wacława Nałkowskiego

Jacek May (2011)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The article is inspired by Wacław Nałkowski’s writings and is a reflection on the nature of geography and its place in the system of knowledge. Moreover, the author considers the status of geography and its social relevance. The article seeks to justify the thesis that the low prestige of geography in social consciousness is an outcome of the attitudes of scholars and teachers of geography themselves, the media propaganda and also the state policy. The main objective of the paper is...

Quasi-arithmetic means

Jan Górowski, Adam Łomnicki (2015)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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We present a list of geometric problems with solutions that lead to knownor less known means. We also prove, by elementary means, some property for so-calledquasi-arithmetic means. We use the proved result to justify some inequalities betweenthe means.

Symbolika matematyczna związana z pojęciem funkcji

Izabela Jóźwik, Małgorzata Terepeta (2016)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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Mathematical texts are characterised by the occurrence of specificsymbols. They may be letters (as applied, for example, by Euclid), numbersymbols or graphic signs with context-independent meaning. Mathematicalsymbols store knowledge of previous generations that provides frameworkswith which to examine history of mathematical problems. In this paper, wedeal with symbols related to the concept of function and related concepts likelimit, and some other set theoretical or logical concepts....

Metoda Frobeniusa, czyli o rozwiązywaniu pewnej klasy równań różniczkowych

Damian Wiśniewski, Krzysztof Żyjewski (2016)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The article is a comprehensive research paper which provides theoreticalas well as practical aspects of the Ferdinand George Frobenius method.This method is based on seeking infinite series solutions for certainclass of differential equations. It is a generalization of the power series methodand allows us to solve the differential equations at least near somesingular points.

Z badań nad analizą rysunku i odkrywaniem własności funkcji

Joanna Major, Zbigniew Powązka (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The article presents remarks regarding analysis of drawings and the use of the data contained therein in resolving mathematical tasks. A particular type of drawing is considered, which is a graph of a function. The presented findings are the result of several years of research conducted among the Mathematics students at the Pedagogical University of Cracow.

Z badań nad trudnościami studentów w rozumieniu pojęcia miary i całki

Zbigniew Powązka (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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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...

Praktyka kształtowania matematycznej twórczości uczniów

Antoni Pardała (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The subject of this work, teacher’s practice forming mathematical activity and creativity of the gifted pupils, is submerged in the issue of one of the contemporary trends in researching the methodology of teaching maths, which is called: creativity in teaching mathematics – theory, diagnosis and methodology, prospects. In this work I refer to certain synthesis of the knowledge to that point, see A. Pardała (2003; 2004; 2006), and I also articulate one of its aspects – the crucial importance...

Geography in Russia: education and public interest

Valery Solomin, Viaczeslaw Suchorukov (2013)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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In the article, the terms of the public interest in Russia are discussed, among which are the natural resources and the humanitarian component, including the geographic entity, the state plays an important role.

Ski and spa tourism as local development strategy – the case of Krynica Zdrój (Poland)

Sławomir Dorocki, Paweł Brzegowy (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The socio-economic, cultural and technological transformations, characteristic features of the globalization era, result in altering the ways in which tourist and recreational services are rendered, leading to the formation of the so-called mega-tourism trends. At present, tourism has become one of the fastest developing economic branches in many countries around the world. This dynamic development includes, among others, SPA and wellness and outdoor physical activity sectors. Slowly...

A Purely Algebraic Proof of the Fundamental Theorem of Algebra

Piotr Błaszczyk (2016)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely...

Teoretyczne podstawy dzielenia z resztą liczb naturalnych wraz z uwagami o dzieleniu z resztą liczb całkowitych, wymiernych i rzeczywistych

Antoni Chronowski (2019)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this article, I analyze the theoretical foundations of the division with remainder in the arithmetic of natural numbers. As a result of this analysis I justify that the notation a:b=c r s, where a, b, c, s are natural numbers and r denotes, is correct at school mathematics level and does not lead to a contrediction suggested by the author of the article (Semadeni, 1978). As a generalization of the division with remainder of natural numbers, I consider the division with remainder of...

Komputer jako narzędzie badania sytuacji zadaniowej w długoterminowej pracy domowej ucznia

Renata Wojtuś (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the article I submit a fragment of my research carried out in 2004-2007, which was connected with the role of mathematical computer programs in the independent extracurricular work of the pupil. These examinations were conducted in a group of 119 pupils from the second grade of secondary schools. The article concerns the specific form of homework in mathematics – a long-term homework. The purpose of this homework was to answer the questions: a) In what mathematical situations do pupils...

Wyniki badań nad stanem nauczania statystyki opisowej w Polsce

Zofia Aleksandra Tomaszewska (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This article presents the results of the study of teaching descriptive statistics in Poland nowadays. After a short review of the history of teaching this branch of school mathematics in the second part of the 20th century, I made an attempt at diagnosing the stage of teacher’s preparation for teaching descriptive statistics. The diagnosis was based on the analysis of: - the survey conducted among working teachers; - teaching standards for mathematics at universities; - curriculum and...

Development and present situation of slovenian didactics of geography

Tatjana Resnik-Planinc (2013)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The paper presents the development of Slovenian didactics of geography in the frame of the development of geographical education from the Middle Ages until nowadays. Important milestones are discussed through the prism of the work of important men who influenced the development of geographical education from its very beginning.

Gry miejskie jako innowacyjne produkty turystyczne

Witold Warcholik, Konrad Leja (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Urban games and other forms of field games, such as Questing, are innovative ideas for presentation, disclosure and interpretation of natural and cultural heritage. The results of the survey presented in the study show that soon urban games can become an alternative to a traditional guided sightseeing of an urban area. There may be several reasons for such a choice: an interesting way of exploring a city, aspects of play and physical activity, better memorization of information, as well...

O roli prac dyplomowych z dydaktyki matematyki w kształceniu przyszłych nauczycieli matematyki

Joanna Major, Bożena Olik-Pawlik, Tadeusz Ratusiński, Lidia Zaręba (2017)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the training process of students – future teachers of mathematics, an important role is played by, among others, participation in the diploma seminar, during which the student (being at the final stage of studies)is tasked with preparing a bachelor’s or master’s thesis. This dissertation may be purely mathematical in nature or refer to problems in didactics of mathematics. The article aims to develop and illustrate some thoughts from previous works of the authors (Zaręba, 2009; Major,...

The problem of human development and liberal education

Viaczeslav Sukhorukov, Bożena Wójtowicz (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The modern world is characterized by rapid changes in all spheres of social life. These changes are amplified by extremely rapid dissemination of information, knowledge, values, tastes, and ideas from one society to another. Some values are invisible to the eye, other can be noticed quickly and easily. The result is a new cultural phenomena, which are called “civilization“ and “ the modern way of life.” To change the values and ideological climate in today’s world it is necessary to...

Z badań nad sylwetką absolwenta matematycznych studiów w Akademii Pedagogicznej w Krakowie w latach 2001-2007

Danuta Ciesielska, Joanna Major, Zbigniew Powązka (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The main goal of our studies was eliciting the opinion of graduates and students of the last year of Master studies on the course for teachers of mathematics at the Pedagogical University of Cracow. The research method was a survey questionnaire.We were interested in opinions on the level of studies at the college, and professional skills acquired in the study process. Moreover, we asked the questions concerning the professional career.In the article we present the analysis of the replies...

Czym jest zmysł geograficzny?

Witold Wilczyński (2011)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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„Geographical sense” is one of the original ideas of Wacław Nałkowski. He defined it as oneof the characteristics of human perception and thinking, which manifests itself in such a way, that the whole variety of human experience is transformed in the act of perception to shape the overall, general and comprehensive situations or figures. It is a process based on a spontaneous, unconscious selection of incoming sensory data and its evaluation, interpretation and synthesizing, to create...

Wokół liczb i szeregów harmonicznych

Damian Wiśniewski (2015)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The harmonic series is one of the most celebrated infinite series ofmathematics. From a pedagogical point of view, the harmonic series providesa wealth of opportunities. Applications such as Gabriel’s wedding cake andEuler’s proof of the divergence of prime numbers can lead to some verynice discussions. The main idea of this article is to survey some of unusual,insightful and inspiring divergence proofs. First of all, this article is addressedat first-year calculus students. ...

Jan Flis – współtwórcą współczesnej polskiej dydaktyki geografii

Mariola Tracz (2013)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The study shows the role of professor Jan Flis in the development of geography education in the post-war era. His academic input into the discipline is expressed in 134 publications, out of which 40 concern the topic of the didactics of geography. Professor’s interest in the field of geography education focused on: the selection and structure of geographical material, forming of theories in teaching and learning geography, function of geography in education and cognitive development,...

Roma ethnic minority risk of social exclusion in Poland

Agnieszka Świętek (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Roma people are one of the most colourful minorities in Poland and arouse interest and controversy among contemporary Polish society, nearly monolithic in terms of ethnic origin and nationality. At present, the socio-economic situation of Roma people in Poland and in other European countries is difficult and marked with social exclusion. In the article the author identifies and describes the following symptoms of social exclusion experienced by Roma people in Poland: lack of tolerance...

Geografia na maturze – zróżnicowanie merytoryczne i ocena zadań… z arkuszy egzaminacyjnych w latach 2005–2011

Jan Wójcik (2013)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The paper presents a substantive and methodological assessment of the tasks and the trends in changes in the scope of knowledge and skills tested during the final examinations (matura) at Polish high schools in the years 2005–2011. The examination sheets contained a various number of tasks (27–38). They included open-ended and closed-ended tasks. With a total of 14 sheets, closed-ended tasks predominated in 9 sheets, open-ended tasks prevailed in 4 sheets, while the number of open-ended...

Náměty na globální rozvojové vzdělávání dětí mladšího školního věku

Petra Karvánková, Dagmar Popjaková, Vlasta Kovaříková (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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This paper initially deals with the origins of the idea of global education. It points to the multitude of internationally existing concepts and business terms of global education which makes it difficult to implement the global dimension into real-life education. This “international” multitude is, as a result, obvious in the Czech terminology of global development education (GRV in the Czech text) where it originates as particular authors followed one or the other concept of global...

Koncepcja geografii malowniczej

Katarzyna Mentel (2011)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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„Picturesque geography” is an original expression of Wacław Nałkowski, which probably has no counterpart in the whole history of geographical thought. It clearly shows Nałkowski’s belief that the materials utilized in the process of education should have a beautiful shape. This is also a very convincing expression of his romantic sensibility and the testimony of his great literary talent. Romantic aspects in works of Nałkowski are thought to be very important, since geography is a field...

“Nature of Saint-Petersburg” electronic educational edition as a mean of shaping ecological awareness in schoolchildren

T.V. Vileyto, E.M. Rebko (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The article discusses in detail the contents of electronic educational edition of “Nature of St. Petersburg”. It shows the scientific and methodological benefits, as well as the principles underlying the publication. Much attention is paid to the analytic section of the manual, especially to enable the development of ecological awareness of schoolchildren.

Przykłady zagadnień z różnych działów matematyki niezbędnych do studiowania teorii miary

Antoni Chronowski, Zbigniew Powązka (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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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...

Актуальные проблемы современной методики обучения математике

Роза А. Утеева (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This article analyzes the development of the concept of methodical system of teaching mathematics. Various approaches (A. Pyshkalo, G. Sarantsev, T. Ivanova, N. Stefanova, etc.) to the concept of methodical system of teaching mathematics are analyzed.

Etapy rozwoju osadnictwa zachodniej części Płaskowyżu Nałęczowskiego i ich zapis w osadach dna doliny Bystrej

Jan Reder, Józef Superson, Tadeusz Król (2010)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The article analyzed the impact of old (prehistoric, medieval) and modern settlement on the nature of the Holocene alluvial sediments at the bottom of the Bystra valley. It was shown that the transition from peat to mineral deposits was the result of Neolithic farming. Nomadic culture of the late Neolithic and early Bronze Age have helped to stop the denudation slope sand increasing erosion in the bottom of the river valley. Restarting the denudation in the basin and the beginning of...