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
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...
Jan Górowski, Adam Łomnicki (2013)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In the paper a set of strategy games is presented. It is shown how the “manipulative” developing of the winning strategy of a known and simple game can lead to conceptual reasoning based on reduction; then it is suggested how to formalise and generalise such a game, similar games and the procedure of finding the winning strategy.
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.
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...
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.
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...
Tatyana Kirillova (2012)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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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. ...
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.
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.
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.
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.
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.
Jan Górowski, Adam Łomnicki (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The paper presents some non-standard (in the sense: not yet presented in the literature), necessary and sufficient conditions for a triangle to be isosceles.
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...
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.
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...
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.
Anna K. Żeromska (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The purpose of this paper is to draw attention to the still current (as compared to the reform of higher education) issue: how should we train mathematics teachers in order to attain the desired outcomes? The article presents an extensive range of thought-provoking questions about what knowledge of well-educated mathematics teachers should be. References to the already existing didactic literature are included. Reflections on this question, containing some arguments as to the direction...
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...
Antoni Chronowski (2009)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The subject-matter of this article can be divided into three main parts. The first one includes the theoretical problems concerning the fields of algebraic fractions and rational functions. In the second part we deal with didactic questions of algebraic fractions and rational functions in some chosen school textbooks. In the third part deep ideas and formal models of algebraic fractions and rational functions are discussed.
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...
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.
Martina Škodová, Alfonz Gajdoš (2010)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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Forest edges generally represent a very important element of ecological stability. The goal of this paper is to classify the types of anthropogenic forest edges in the area of Starohorské Mountains and to analyze anthropogenic impacts on the selected attributes of forest edges. We evaluated the attributes of a biocomplex, type of forest edge, its structure, consistence, species composition and other factors. The differences in environment variables and anthropic impact were influenced...
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.
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...
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...
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...
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...
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’...
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”.
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.
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.
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....
S. Paszkowski (1954)
Applicationes Mathematicae
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Tadeusz Rams (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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Asking questions is one of the most common activities of mathematics teachers at various levels of education. Moreover, the way the teachers formulate questions has profound impact on the results of teaching, especially on their pupils' reasoning process. Erotetic logic is frequently presented in a very formal manner. In this paper, we restrict our attention to these aspects of erotetic logic that can turn it into a useful framework to describe various didactic situations. In particular,...
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.
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...
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...
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.
Petra Karvánková (2013)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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This paper deals with the issue of the didactics of the Czech lands geography process and the position of geography as a scientific branch in its system. It describes new trends and styles in teaching geography in the Czech Republic. At the same time, it mentions some current teaching difficulties at Czech primary and secondary schools. The first stage of teaching developing theory, the period from 1948 to 1965, is tagged “applied period” (Hájek, 2003). The theory of teaching geography...
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.
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.
Teresa Sadoń-Osowiecka (2011)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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Although the very term “constructivism” appeared rather recently, its concept was knownin science long before now. According to modern cognitive psychology, the reality is created individually in person’s mind. An individual transforms and interprets the flow of information which he receives in his own way, depending on cultural, social, biological or situational context. Nałkowski’s statements concerning perception and thinking from the beginning of the twentieth century, connected...
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...
Wiktor Osuch (2012)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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Although Poland has gone through a period of many various trials regarding the introduction of different changes in the educational system (involving everything from types of schools, through levels of teaching, changes in curricula for each school subject, syllabuses and textbooks, to the modification of teaching aims and content for each subject), very little, in fact, has been done to promote the necessary remodelling of work that would ensure the shaping of a truly modern teacher....
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....