Preliminary assessment: homework (10%), written tests (20%)
Final assessment: oral exam (70%)
During the semester, students must obtain at least 4/10 from the preliminary assessment in order to be eligible for the oral exam. Failure to meet this requirement automatically results in an FX grade.
- n-dimensional affine space A^n and Euclidean space E^n;
- coordinate systems;
- affine maps;
- orientation of affine space;
- subspaces / linear varieties in E^n: parametric description and implicit equations, relative positions, distances and angles of some subspaces;
- invariants of affine maps (fixed points, eigenvectors);
- isometries, reflections as generators of the group of isometries of the Euclidean plane
Master the analytical methods of studying the geometric properties of subspaces of n-dimensional affine (or Euclidean) space and its maps
Preliminary assessment: two written tests (30%)
Final assessment: oral exam (70%)
During the semester, students must obtain at least 4/10 from the preliminary assessment in order to be eligible for the oral exam. Failure to meet this requirement automatically results in an FX grade.
Divisibility of integers, greatest common divisor, Euclidean algorithm, least common multiple. Prime numbers, decomposition into the product of prime numbers. Congruences, Euler's theorem and its applications, Lagrange's theorem. Number systems and divisibility criteria. Selected arithmetic functions. Rational and irrational numbers. G-adic development of real numbers. Criteria of rationality of real numbers.
Students master the basics of divisibility theory in the field of integers and its applications and will be able to actively use this knowledge to solve various problems. Furthermore, they will control the expression of real numbers using g-adic developments and selected criteria for the rationality (irrationality) of real numbers.
Kapitoly z diskrétní matematiky: Jiří Matoušek, Jaroslav Nešetřil. Praha: Karolinum, 2009 Kombinatorika a teória grafov: Martin Knor. Bratislava: Vydavateľstvo UK, 2000
Lecture notes.
Continuous evaluation: homework
Final examination: written exam
Grades: A 90%, B 80%, C 70%, D 60%, E 50%
slovak, english
Basic combinatorial tools, permutations, combinations, binomila coeficients and Pascal triangle, binomial an multinomial theorem, combinatorial identities, principle of inclusion and exlusion, Dirichlet principle.
50/50
Gaining comprehensive overview of basic combinatorial problems and skills to solve them.
Preliminary assessment: homework (20%), written tests (40%)
Final assessment: oral exam (40%)
Grading: A 90%, B 80%, C 70%, D 60%, E 50%
- history of axiomatics of geometry, Euclidean constructions
- axioms of incidence, incidence geometry models
- axioms of order, ordered plane models
- axioms of congruence, theorems about the congruence of triangles, arithmetics of line segments and angles, Hilbert plane
- controversy of the axiom of parallelism
- axioms of continuity and circle continuity principles
- some of Apollonius' problems
The student gets familiar with the axiomatic construction of planimetry. He learns partly Euclid's, but especially Hilbert's axiomatic system. They will practice thorough mathematical argumentation and get knowledge of several models of different groups of axioms.
Preliminary assessment: homework
Final assessment: written exam
Grading: A 90%, B 80%, C 70%, D 60%, E 50%
- introduction to theory of polyhedral, Euler's theorem, Platonic solids
- incidence (sections of solids) and metric (distances and angles) problems in stereometry
- principles of parallel projection, orthogonal projection (Monge projection), oblique projection
- ellipse as an affine image of a circle
- central projection, linear perspective, basics of projective space
- non-linear projection: stereographic projection, other cartographic representations
The main goal of the course is to strengthen the spatial imagination. The students learn the basics of polyhedral theory, the possibilities of displaying three-dimensional objects in a plane and also maps of two-dimensional spaces, which require embedding into three dimensions. They get practice in solving stereometric problems.
Dítě, škola a matematika: Konstruktivistické přístupy k vyučování / Milan Hejný, František Kuřina. Praha : Portál, 2001
Moderní vyučování / George Petty Praha : Portál, 1993
Dvacet pět kapitol z didaktiky matematiky / Milan Hejný, Jarmila Novotná, Naďa Stehlíková (Eds.) Praha, 2004
Komunikácia v inkluzívnej škole / Marta Horňáková. Bratislava: Univerzita Komenského v Bratislave. 2017
Elementary and Middle School Mathematics: Teaching Developmentally. / John A. Van de Walle, Karen Karp, Jennifer M. Bay-Williams. Pearson.
Theory of didactical situations in mathematics /Guy Brousseau. Springer, 1997
Textbook explanations: Modes of reasoning in 7th grade Israeli mathematics textbooks. / Silverman B, and Even R., CERME 9. Charles University in Prague. Faculty of Education. 2015, pp.205-212
Od obsahu vzdělávání k žákově znalosti: Kritická místa na cestě do školy a ze školy / T. Janik, Arnica 8, 2018, 1–8. Západočeská univerzita v Plzni, Plzeň.
Refining teacher design capacity: Mathematics teachers’ interactions with digital curriculum resources / Birgit Pepin, Ghislaine Gueudet, Luc Trouche. In. ZDM Mathematics Education, 2017, 49, 799–812 https://doi.org/10.1007/s11858-017-0870-8
Own electronic materials published via the subject's website (eg course in LMS Moodle)
Continuous assessment: group homework, reading of scientific articles, discussion of articles and the specified topic, group work on a lesson plan described for each group in the assignment.
The student must obtain at least 50% of the continuous assessment and each assignment must be evaluated with a non-zero number of points.
Final evaluation: oral exam, individual preparation of a lesson plan and its presentation
Indicative assessment scale: A 94%, B 86%, C 79%, D 70%, E 60%, Fx <60%
slovak, englihs
The language of mathematics, its historical development and didactic significance. A parallel between phylogeny and ontogeny of mathematical thinking. Conceptual and cognitive process in mathematics. Principles, means and forms of teaching in mathematics. Objectives of the teaching process in mathematics. Learning theories. Lesson preparation and its starting points. Resources and their use in preparation, during teaching in evaluation. Hypothetical learning trajectories.
60/40
The student will gain an overview of didactic theories, will be able to critically evaluate them and use the appropriate methods and forms. Through activities at the seminar, they will acquire modern teaching techniques that are a prevention of formal knowledge and misconceptions in mathematics. The study of scientific and research articles and the subsequent discussion or authentic experience will give graduates a better insight into the issues of inclusive school, creating preparations for the lesson (or sequence) and asking "good questions".
Dítě, škola a matematika: Konstruktivistické přístupy k vyučování / Milan Hejný, František Kuřina. Praha : Portál, 2001
Moderní vyučování / Geoffrey Petty Praha : Portál, 1993
Dvacet pět kapitol z didaktiky matematiky / Milan Hejný, Jarmila Novotná, Naďa Stehnlíková (Eds.) Praha, 2004
Thinking mathematically / John Mason, Leone Burton, Kaye Stacey. Pearson, 2010
Explanation and Proof in Mathematics. Philosophical and Educational Perspectives / G. Hanna, H. N. Jahnke, H. Pulte (Eds.), Springer 2010
Matematika a svet okolo nás : Zbierka úloh / Zbyněk Kubáček ... [et al.]. Bratislava : Pavol Cibulka, 2008
Učebnice matematiky pre gymnáziá / Zbyněk Kubáček
Matematika : 1 : zbierka úloh pre stredné školy / Iveta Kohanová ... [et al.]. Bratislava : Orbis Pictus Istropolitana, 2011
Own electronic materials published via the subject's website (eg course in LMS Moodle)
Continuous assessment: individual preparation of a lesson plan and its placement in the thematic plan, microteaching, preparation of written tests (2 equal groups), peer assessment, didactic analysis of teaching texts, reading and discussion of scientific articles.
The student must obtain at least 50% of the continuous assessment and each assignment must be evaluated with a non-zero number of points.
Exam: written with oral consultation
Indicative assessment scale: A 94%, B 86%, C 79%, D 70%, E 60%, Fx <60%
Specifics of the last year of lower secondary school, transition to the 1st year of upper secondary school. Appropriate activities of teaching mathematics for individual topics of the first and second year of upper secondary school: educational goals, activities, the assumption of the development of student understanding in the context of educational activities and the creation of their hypothetical scenario. Selection of adequate methods, forms and cognitively appropriate activities. Integration of digital technologies into the teaching of mathematics also in order to support argumentation, reasoning, and building a mathematical culture (correctness of concepts, procedures, arguments, etc.).
60/40
The graduate of the course is acquainted with the goals of mathematics education at upper secondary schools, with adequate teaching methods, forms and means, with ways of introducing selected mathematical concepts, is ready for his work at school and outside school, can transfer to pedagogical practice knowledge and skills acquired in individual professional disciplines. Specially, the graduate of the course will gain an overview of didactic procedures for teaching mathematics suitable for the transition from lower to upper secondary school and in the first years of upper secondary school.
seminar work / project
slovak, english
Flipped Classroom, Cooperative teaching, Brain-compatible teaching, method "Break", Realistic teaching (RME), Guided active learning, Jigsaw method, problem solving, rpg, simulation.
100/0
Students will be able to distinguish between different methods of education. They will have an overview of the principles and principles of each method and will be able to apply these principles in practice.
Dějiny matematiky / Dirk J. Struik ; přeložili Jaroslav Folta, Luboš Nový. Praha : Orbis, 1963
Dějiny matematiky ve starověku / Arnošt Kolman. Praha : Academia, 1968
Dějiny matematiky ve středověku / Adolf P. Juškevič. Praha : Academia, 1977
Dejiny matematiky / Ján Čižmár. Bratislava : Perfekt, 2020
The history of mathematics / Roger L. Cooke. Hoboke, NJ : John Wiley, 2003
The history of mathematics / David M. Burton, New York : McGraw-Hill, 2011
Ongoing evaluation: presentation of the prepared lesson (20 points), preparation of the written exam for other participants (10 points), active participation in the evaluation of presentations and written exams of other participants (30 points).
Grading: A (56-60 points), B (51-55 points), C (46-50 points), D (41-45 points), E (36-40 points), Fx (0-35 points).
Weight of the ongoing / final assessment: 100/0
Slovak, English
Mathematics of ancient Egypt and Babylonia. Pre-Euclidean mathematics in ancient Greece. Euclid´s Elements.
100/0
Student will gain an overview of the various periods of development of mathematics in antiquity, including examples of problems solved in individual basic works.
Matematika v proměnách věků III / Editori Jindřich Bečvář, Eduard Fuchs. Praha : Výzkumné centrum pro dějiny vědy, 2004
Dějiny matematiky / Dirk J. Struik ; přeložili Jaroslav Folta, Luboš Nový. Praha : Orbis, 1963
Dějiny matematiky ve starověku / Arnošt Kolman. Praha : Academia, 1968
Dějiny matematiky ve středověku / Adolf P. Juškevič. Praha : Academia, 1977
Dejiny matematiky / Ján Čižmár. Bratislava : Perfekt, 2020
The history of mathematics / Roger L. Cooke. Hoboke, NJ : John Wiley, 2003
The history of mathematics / David M. Burton, New York : McGraw-Hill, 2011
Ongoing evaluation: presentation of the prepared lesson (20 points), preparation of the written exam for other participants (10 points), active participation in the evaluation of presentations and written exams of other participants (30 points).
Grading: A (56-60 points), B (51-55 points), C (46-50 points), D (41-45 points), E (36-40 points), Fx (0-35 points).
Weight of the ongoing / final assessment: 100/0
Slovak, English
Students will choose from the following topics: Ptolemy. Apollonius. Chinese and Arabic mathematics. Fibonacci. Alcuin's problems. Cardano´s Ars Magna. Pascal's Arithmetic Triangle. Huygens´s De Ratiociniis in Ludo Aleae. Bernoulli's Ars Conjectandi. Cavalieri's Geometry of indivisibles. Euler's Introductio and Letters to a German Princess. Venn´s Symbolic Logic.
100/0
The student will gain an overview of the various periods of mathematics development, including examples of problems solved in individual basic works.