Displaying similar documents to “O nieskończonych ciągach liczb parami względnie pierwszych raz jeszcze”

O nieskończonych ciągach liczb naturalnych, parami względnie pierwszych

Jan Górowski, Adam Łomnicki (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the first part of the paper the authors, using general formulas, determine and describe a class of infinite series of natural numbers pairs of which are relatively prime. The second part of the paper contains - as a proposition - a set of problems concerning prime numbers and pairs of relatively prime numbers suggested for use during the process of work with Mathematics students, as well as some didactic comments concerning these problems.

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.

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...

Around the Wilson's theorem

Jan Górowski, Adam Łomnicki (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper some known conditions and new congruences characterising prime numbers are given. Some of them are obtained by the generalised Wilson theorem given by Gauss. The elementary proof of this theorem is also presented.

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...

O pewnej implikacji

Jan Górowski, Adam Łomnicki (2016)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The purpose of this paper is to show how the process of provinga theorem in different ways or proving generalized versions of the theorem,after learning one its proofs, influences the development of the skills of provingtheorems and analysing proofs by the students of mathematics. Toillustrate this process we use an elementary theorem about numbers and itsgeneralizations, giving fourteen proofs. Proving theorems we use methodsand facts which are available to high school students. ...

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...

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.

Waloryzacja środowiska geograficznego Nadleśnictwa Rymanów pod kątem atrakcyjności turystycznej

Piotr Skóra, Wanda Wilczyńska-Michalik, Maciej Szpiech (2010)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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A sudden increase in various tourism forms in Poland in the second half of the twentieth century brought development of natural environment evaluation methods for tourist purposes. Natural and cultural environment values and slight anthropogenic transformations of the Forestry Rymanów created tourists’ interest in this region. The paper presents the scope and ways of evaluating individual components of the natural environment in connection with historical and political changes. Evaluating...

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...

O cechach podzielności liczb i odkrywaniu twierdzeń

Jan Górowski, Adam Łomnicki (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The paper contains some theorems (with proofs) from numbers divisibility theory, and suggestions how to use them in the process of teacher training, which provide good opportunities for developing students’ creative attitude in the process of discovering new facts in elementary mathematics.

The role of visualisation in the process of generalisation

Lidia Zaręba (2015)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In my paper, I discuss the results of individual observations of pupils aged 13-14. These children undertook an attempt to solve problems which were designed in order to bring about an inductive type of generalizing. The main aim of the study was to classify typical methods of proceeding, which represent the pupils’ ways of reasoning and to determine what influence a particular method of proceeding may have on the final outcome of their work. These results indicate the general pupils’...

High School Identities

Katarzyna Słomczyńska (2015)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In 1969, Polish mathematician and logician, Alfred Tarski asked ifall the identities true in the set of natural numbers involving the constant 1,addition, multiplication, and exponentiation can be derived from the elevenaxioms that are taught at the high school level (High School Identities). In1981 Alex Wilkie negatively solved this problem by constructing an identitythat cannot be proved using these axioms. In this paper we survey resultsconnected with Tarski’s problem.

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...

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. ...

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...

Teoretyczne i dydaktyczne aspekty nauczania o największym wspólnym dzielniku i najmniejszej wspólnej wielokrotności w zbiorze liczb naturalnych

Antoni Chronowski (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This article contains a collection of didactic ideas concerning teaching the elementary arithmetic. Although they are firmly based on abstract mathematics, they can be realized at different levels of teaching school mathematics. The source of these didactic propositions is the fact that using the set of natural numbers and suitable relations it is possible to construct the models of structures which are called lattices. In this paper we consider the two models of lattices: the lattice...

Walory turystyczne wybranych regionów Spitsbergenu

Piotr Dolnicki, Łukasz Gawor (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The paper presents different attractions of Spitsbergen: geological, geomorphological and glaciological objects and processes, as well as industrial sites especially connected with coalmining, which constitute the group of geotourist attractions in Hornsund, Isfjorden and North East Land. It also describes logistic and infrastructure conditions in Svalbard. The evaluation of possibilities to develop geotourism in Spitsbergen is included as well.

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.

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...

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...

Idee Wacława Nałkowskiego a współczesna dydaktyka

Elżbieta Szkurłat (2011)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The answer to the question whether teaching of geography nowadays respects the universal ideas of the great geographer, is not clear. A critical look at teaching achievements of Wacław Nałkowski, as well as at the contemporary state of geographical teaching, seems to be difficult due to the individual views and personal experiences. This paper is an attempt to assess the extent to which the important educational concepts of Nałkowski are respected in contemporary educational practice....

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.

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.

Aktywności matematyczne studentów inspirowane grami Penneya

Maciej Major, Barbara Nawolska (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This paper proposes a way to adapt theories of countable probabilistic spaces in order to use them to educate mathematical students trained to be teachers. It presents many examples of how to develop students’ mathematical activities using stochastic games called Penney games. Introducing chance games delivers the opportunity of creating and examining probabilistic spaces. The fairness of games is a problem that inspires and motivates mathematicians to formulate problems and tasks of...

Kontrasty w użytkowaniu przestrzeni jako wynik wielowiekowej degradacji środowiska przyrodniczego – na przykładzie obszaru miasta Mysłowice (Wyżyna Śląska)

Tomasz Parusel, Dominik Karkosz (2010)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The town Mysłowice is located in the southern mesoregions of the Silesian Upland: Katowice Upland (341.13) and Jaworzno Hills (341.14). Mysłowice terrain has been intensively used by man for centuries and still is nowadays. This process left permanent signs on the natural environment; the mosaic forms of land use are the effect of this activity, among others. In the town area contrasts are frequent – direct proximity of various land use forms is observed. Initial characterization of...

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...

Применение методов математической статистики в научных исследованиях

Владимир В. Афанасьев (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the paper, the author presents the basic concepts and theorems concerning methods of mathematical statistics and its application in the domain of empirical research. Several examples presented in the paper can be useful as patterns of using statistical tool in didactic research.

O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej

Agata Matczyk, Vladimir Mityushev, Zbigniew Powązka (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The lattice sum S_2 for the square array conditionally converges. Having used physical arguments, Rayleigh chose an order of summation in such a way that S_2 = π. The Eisenstein summation method applied to S2 yields the same result. This paper is devoted to a rigorous proof of S_2 = π for the Eisenstein summation method. The study can be used in class for students as an interesting example which illustrates different types of convergence.

Atrakcyjność turystyczna regionów na przykładzie Francji

Sławomir Dorocki, Małgorzata Zdon-Korzeniowska, Anna Delekta (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Technological and economic transformation of the past few decades resulted in a significant re-evaluation of the factors affecting the location of economic activity. General trend in these changes is the declining importance of the so-called “hard factors”, “cost factors” (such as transport costs, labour costs and the amount of taxes), as opposed to the so-called “soft factors”; special emphasis among these factors is placed on human capital. An important aspect of the human factor in...

Studia geograficzne w Uniwersytecie Jagiellońskim w XIX wieku

Mariola Tracz (2011)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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In time of Wacław Nałkowski’s studies in Kraków (1871–1876), the chair of geography at the Jagiellonian University did not exist. Despite this, in addition to mathematics, astronomy and physics, he attended lectures in geography, astronomy and cartography. Geographical learning opportunities within the Faculty of Philosophy were enabled there as early as1765, thanks to the reform of the University made by Hugo Kołłątaj. In late 1811 and 1812among 12 departments in the Faculty of Philosophy...

Wielomiany Fibonacciego stopnia k

Jan Górowski, Adam Łomnicki (2014)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper, we give formulas determining the Fibonacci polynomials of order k using the so-called generalized Newton symbols, i.e., the coefficients in the expansion of (1+z +z^2 +. . .+z^{k−1})n with respect to the powers of z.

O fałszywych przekonaniach obserwowanych na zajęciach z analizy matematycznej

Zbigniew Powązka (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper we present the results of our research carried out among the students of the first, second and third years of mathematical studies in the years 2003-2006. The research was focused on the difficulties these students faced in understanding the fundamental notions of the mathematical analysis such as: a limit, continuity, differentiability and integrability. These special problems, described in this paper, are called ”false beliefs”.

Main issues concerning drinking water supply in the Angara River watershed

Elena A. Rush (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Main issues concerning drinking water supply in the Angara River watershed. Water supply for drinking and communal-general purposes plays an important role in every economy all over the world as water is not only widely used for people’s everyday purposes, but is also essential for people’s vital functions. The principle concerning the priorities of supplying drinking and communal water, stipulated in relevant norms and regulations, presumes that in the first place people should be provided...

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.

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.

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...

Jak bardzo przemienna może być grupa nieprzemienna?

Katarzyna Słomczyńska (2017)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper we survey, also in historical perspective, the results connected with the notion of the commutativity degree of a finite group, i.e., the probability that two randomly selected elements of the group commute.

O potrzebie dokształcania nauczycieli akademickich

Bogdan J. Nowecki (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper author underlines the necessity of continuing education of young academic teachers in the domain of methods of students’ activity during their studies.

Dokształcanie i doskonalenie nauczycieli na studiach podyplomowych

Bogdan J. Nowecki (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The project presents an outline of a general concept of teacher training and self-improvement in the postgraduate courses. The general proposals are illustrated with an example of postgraduate Studies in Mathematics.

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...

Nowe cechy równobocznych trójkątów

Michał Biesiada, Jan Górowski, Adam Łomnicki (2014)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper the authors formulate and prove several conditions fortriangle to be equilateral. These conditions are associated with Gergonne,Nagel and Torricelli points and were obtained by composing and solvingthe so-called ‘enforcement tasks’. Both, the method and the results, can beused to trigger some mathematical student activities at different levels ofeducation or even some teacher activities.

Rola portów morskich w rozwoju cruisingu na Karaibach

Renata Rettinger, Anna Urbańska (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Sea tourism brings huge profits to the Caribbean. The cruise market is set mainly for tourists coming from North America. Low travel costs, high income and the long-lasting tradition of sailing excursions are the main reasons. More and more European tourists can also be seen in the Caribbean thanks to numerous air connections. The cruise routes follow the most attractive ports. The ongoing competition causes the increase of quality of passenger’s service and goods. The investment of...

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

O zjawisku zniekształcania obrazu matematyki

Bożena Olik-Pawlik (2015)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper, we identify the phenomenon of false conviction. Weanalyze the definition of false conviction and provide the typology of falseconvictions, including their logical, geometric, cognitive and methodologicaltypes. Every type of false conviction is exemplified by empirical data.

Twierdzenie Bézouta o przecięciu krzywych algebraicznych w pracach Eulera

Danuta Ciesielska (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the paper an early history of the Bézout theorem on algebraiccurves and effective methods in elimination theory is presented. The hypothesis,stated in 1665 by Newton, on the ”intersection number” of algebraiccurves is given. Effective methods on eliminations of one variable in the systemof algebraic variables come from Euler’s papers: Demonstration sur lenombre des points, ou deux lignes des ordres quelconques peuvent se couper(Euler, 1750), Nouvelle methode d’eliminer les quantites...

Profesor Jan Flis – mistrzem dla uczestników studiów doktoranckich z dydaktyki geografii

Michalina Lubelska (2013)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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In the first part of this paper the author presents her own path as a teacher. During her professional career she has met the Professor numerous times. The first time was when she was a geography student at the Jagiellonian University. The subsequent occasions occurred when she was a geography teacher in primary school and also as a PhD student of the didactics of geography. In main part of the paper the author presented the curriculum of the PhD studies in the field of didactics of...