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.
Tatyana Kirillova (2012)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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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
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. ...
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.
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.
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.
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...
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.
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...
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...
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...
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...
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, Maciej Klakla, Adam Łomnicki (2012)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In the paper, on the basis of elementary geometrical content, the authors present examples of students' reasoning, which permits, in the first place, to formulate a hypothesis, and further to find a proof. The essential fact here is that the same hypothesis can lead students to follow different possible ways of reasoning.
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...
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.
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.
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.
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...
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...
Antoni Chronowski, Joanna Major, Zbigniew Powązka (2009)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In this article we present some didactic ideas of introducing theabsolute value of a real number by means of functional equations and aninequality. The basic properties of the absolute value of a real number arewritten as functional equations and an inequality. The function "absolutevalue" is a solution of a suitable functional equation or an inequality.
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...
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.
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 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...
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...
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 (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.
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...
Maciej Klakla, Renata Liszka, Vladimir Mityushev (2009)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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In the paper, we describe how to introduce the trigonometric functions using their functional characteristics and the Eisenstein series.
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.
Sławomir Kurek, Tomasz Rachwał, Mirosław Wójtowicz (2014)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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The subject of this paper is the issue of urban sprawling in the Kraków Metropolitan Area (Poland). Particular attention is paid on the changing role of industry as one of the key elements of development and metropolization of cities in post-socialist, emerging economies. The role of the industry in terms of activation of labor resources has been diminishing, which is connected to the automation of manufacturing processes and relocations of labor-intensive activities to regions with...
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.
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...
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.
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...
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...
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.
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...
Piotr Błaszczyk (2019)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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Recenzja
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...
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....
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...
Piotr L. Wilczyński (2011)
Annales Universitatis Paedagogicae Cracoviensis Studia Geographica
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The expression „geographical course of history”, currently seen as old-fashioned or obsolete, seems to be typical in geographical works of Nałkowski. It combines two fields that were very close to him just from the very beginning of his scientific career: geography, to which he has devoted his whole life, and history, which was for him a source of information necessary to explain the diversity of landscapes. The origins of the concept can be found in works of German geographer, educator...
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...
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...
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.