Jan Górowski, Adam Łomnicki (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In the paper the authors, using general formulas, determine and describe a class of infinite sequences of natural numbers, pairs of which are relatively prime. This result was obtained using the theory of Lucas sequences and recurrent equations of the second degree.
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...
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.
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.
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...
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. ...
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...
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...
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.
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.
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. ...
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’...
Antoni Pardała (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The subject matter of this paper relates to one of the modern research trends in teaching mathematics: the problems of quality of mathematical education of pupils and students in the twenty-first century in different countries and cultural traditions. This analysis throws light on the literature, research results, and observations from the Polish perspective of practice, and outlines the intentions of the range. The work also discusses some theoretical bases for the concept of improving...
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...
Zbigniew Powązka (2013)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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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...
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.
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”.
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.
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....
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...
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...
Bogdan J. Nowecki (2006)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The article constitutes a written version of the Author’s presentation during the special session of the Senate Science Committee held in December 2003. The session was devoted to teacher training. It contains important general ideas concerning teacher training, criticism of the current system and regulations as well as proposals for improving the situation.
Maciej Klakla (2006)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The article contains a didactic project of classes with mathematics students – future teachers. On the example of problems related to Langley’s angles, the author presents differences between two styles of teaching mathematics. One style is based on existing mathematical knowledge, and the second requires active participation of the students in creating this knowledge during the problem solving process.
Joanna Major, Zbigniew Powązka (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In this paper, we understand the term ''creative attitude'' to denote an action of the learner which consists in undertaking creative activity. This activity can take different forms, such as: discovery, invention, a creative act, etc. We understand ''discovery'' to denote a fact which already exists but was unknown to the discoverer. While teaching mathematics, we deal with such an understanding of ''discovery'' e.g. when we use the problem method. Thence, subjectively new theorems...
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...
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...
Joanna Major, Bożena Olik-Pawlik, Tadeusz Ratusiński, Lidia Zaręba (2016)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The paper presents the comments relating to the role of tasks in theprocess of teaching and learning of mathematics in the context of educationof students studying to become teachers. In the considerations part, referencewas made to the fragments of undergraduate and postgraduate works, madeat seminars of didactics of mathematics.
Paolo Bussotti (2015)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In this research, three problems faced by Leonardo Pisano, called Fibonacci,are dealt with. The main purpose is to show that history of mathematics can offerinteresting material for mathematics education. The approach to the use of history ofmathematics in mathematics education cannot be merely historical, but adapted to theneeds of the explanations in a classroom. In the course of this paper, the meaning ofsuch assertion will be clarified. A further purpose is a trying to explore the...
Ewa Lubaś (2006)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In the article the errors in the solutions of geometry problems made by the candidates for the first year of the mathematical studies at the Pedagogical University of Cracow during the entrance examinations in 2003 and 2004 are analysed. Every geometry problem is provided with the examples of its solutions, and the detailed conclusions are drawn from the analysis. In the final part of the article a general didactic analysis of the errors is carried out as well as some aspects of teaching...
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
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.
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...
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....
Ewa Borak (2010)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The MIZAR system is a computer system for representing mathematical proofs in such a way that the computer checks their correctness. The texts written in the MIZAR language are called Mizar articles and are organized into the Mizar Mathematical Library (MML). Since the very beginnig of the development of the MIZAR system, experiments using MIZAR as a tool for teaching mathematics have been conducted. Numerous courses were organized which were based on different versions of the system:...
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...
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...
Tatyana Kirillova (2012)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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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.
Natalia Batsun (2012)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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Maria Dąbrowa (2014)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The aims and functions of mathematics education differ depending on the level of education. Moreover, the role of mathematics itself differs depending on the domain of its application. In this paper, we focus on mathematics education in the course of economics studies. Unfortunately, while mathematical methods prove useful in building economic models and the importance is still growing, the mathematics education plays increasingly smaller role in the course of economics studies at Polish...
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.
Jerzy Pogonowski (2016)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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We introduce the concept of the context of transmission. It coversthe ways in which mathematical knowledge and mathematical abilities aretransmitted in education and popularization of mathematics. We stress therole of intuitive explanations in these processes. Several examples of suchexplanations are presented, related to: linguistic explanations, perception,empirical models, and internal explanations inside mathematics itself.
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...
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...
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...
Danuta Ciesielska, Zbigniew Powązka (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In the last years in Poland new standards for the matriculation examination have been established, especially in mathematics. Subsequently, for many years at the Pedagogical University of Krakow research has been conducted on difficulties that students experience in understanding the basic concepts of mathematical analysis. For the results of this research see: Powązka, Z. (2006, 2009); these are the reports of research conducted in the academic years: 2003/2004 and 2006/2007. Our main...
Antoni Chronowski, Zbigniew Powązka (2016)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In this article we propose ways of developing the active and creativeattitude of students towards solving mathematical problems. The taskswere sourced from various mathematical competitions for secondary schoolstudents, such as the so-called Mathematical Olympiad, and require theuse of Vieta’s formulas for third-degree polynomials. These problems inspirestudents to conduct their own elementary research work and foster theircreative attitude towards mathematics. This article is dedicated...
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.
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.