prof. RNDr. Marie Demlová, CSc.

The recent transfer to distance education caused by the pandemic forced higher education to reshape courses and adopt new educational tools. Many teachers became aware of and acquired valuable first-hand experience with alternative techniques of teaching. In this contribution we look at advantages and disadvantages of some popular approaches, assessing them not only from the point of view of distance education (which could still be useful), but also regarding their potential for improving on-site education. This is based on the authors' personal observations and student surveys at our institution (a kind of case study in our department and university), influenced by discussions with colleagues in formal and informal networks across Europe and supplemented with published results when possible.

In mathematics education, especially for engineering students, two main problems occur, the first one being unfamiliarity with the language of mathematics, the second one being students’ motivation. Very often students, especially engineering ones, perceive mathematics as an obstacle to their engineering degree, which results in surface learning. The contribution deals with one example of a part of a discrete mathematics course that can serve well to increase motivation of students to work with mathematical notions and develop some mathematical competencies. Experience from teaching discrete mathematics course for 1st and 2nd year students is given.

In 2017, we proposed a European project: Rules_Math, in order to find new rules to assess mathematical competencies. This proposal was a consequence of what we were facing separately in our daily classes. We teach mathematics in several engineering schools, and we want to change the way of teaching and learning for engineering students. Some university teachers from different departments usually teach mathematics as we have learned mathematics. They only give master classes but students usually want to write numbers and formulas to learn and practice mathematical reasoning and to distinguish mathematical symbols. To communicate in, with, and about Mathematics, they need mathematical thinking, and to use aids and tools for mathematical activity. We have included in this papers our proposal to make this possible, and to assess mathematical competencies.

Arguably the largest obstacle freshmen face in their mathematics courses is their unfamiliarity with the language of mathematics. Addressing this problem right at the start seems like a sensible strategy, as comprehension of mathematical communication helps students in all mathematics courses they will take. In this paper we discuss general strategies that can be used when addressing the competency in understanding and speaking the language of mathematics. In particular we focus on benefits of teaching students to prove statements and difficulties related to such endeavour. We introduce discrete mathematics as a particularly suitable course for such activity. Then we look closer at practical experiences we had when teaching comprehension of mathematical language and proofs in discrete mathematics courses.

The team that has developed this project is part of a consortium of several European institutions that have joined with a common goal: to help the current engineering teaching learning system, so that to make possible a competencies-based assessment. Moreover advocacy and networking will be part of the consortium activities during the whole project. RULES_MATH acronym corresponds to 'New rules for assessing mathematical competencies', and that is exactly what we propose: to work with the mathematical competencies, specifically try to incorporate them into the assessment process. We focus on science and engineering degrees, since in all these studies several Mathematics courses are part of their curriculum, mainly in the first couple of years. In this paper we present one of our first activities that we want to achieve under the project: the selection of several tools and rules that will make possible an assessment and evaluation according to our high education common European space. These ideas are part of the objectives of promoting and institutionalizing good teaching practices and the enhancement of teaching materials. Other goals are to improve the dissemination of academic activities to the business world, administrations and society in general.

In recent the acquisition of mathematical competencies and engineers training in mathematics is a major concern of university teachers. It is well known that mathematics is not a favorite subject of engineering students and students in general sciences. We present and analyze in this paper the specific situation that we have encountered in two different European universities: The University of Salamanca (Spain) and the Czech Technical University in Prague. As in many other universities Discrete Mathematics courses are parts of the engineering bachelor degree curriculum. We will detail both situations, compare the assessment methodology and propose new goals for the future.

Even though congruences on a finite automaton play an important role when investigating the structure of a given automaton, not much attention has been given to finding a fast algorithm for constructing them. This contribution brings a fast algorithm for construction the smallest congruence containing a given relation, which can be used for finding elementary congruences, i.e. congruences merging two distinct states.

A written test with closed books (i.e. without additional sources) is a widely used form of a final assessment of a mathematics course. When the written test is not accompanied by an oral examination, it should guarantee that successful passing of the test requires that students not only understand basic mathematical concepts but also gain some of mathematical competencies. The paper brings experience acquired at the Faculty of Electrical Engineering Czech Technical University in Prague in basic mathematics courses-Calculus, Mathematical Logic, and Discrete mathematics.

In this discussion paper we firstly summarise the c urrent offering of the SEFI Mathematics Working Group with regard to orientation for those who are interested in the mathematical education of engineers. Based on this summary we id entify directions for further work. Finally, we present some ideas of how progress might be made in these directions.

During the last two decades access to higher educat ion increased considerably. One of the effects caused by this rapid expansion is that the students entering a HE institution do not have similar knowledge, background and/or willingness to study. A lot of research has been carried out concerning teaching students with not enough kn owledge and/or skills from their secondary education. Less attention has been paid to the ques tion how to educate bright students. The paper brings some experience gained at the Czech Te chnical University in Prague concerning teaching bright students.

This documents adapts the competence concept to the mathematical education of enginners and explains and illustrates it by giving examples. It also provides information for specifying the extent to which a competency should be acquired.

The next edition of the European Society for Engineering Education (SEFI) Mathematics Working Group’s curriculum document will be based on the notion of mathematical competence. This approach aims particularly to capture higher-level learning goals that go beyond the traditional content-related and often small-scale description of learning outcomes. In this contribution, ‘classical’ and more recent learning arrangements are investigated regarding their potential for competence acquisition. The learning arrangements include lectures, assignments, tutorials, laboratories, projects and technology enhanced learning arrangements. We relate these to the eight competencies identified in the current SEFI curriculum discussion document and explore the contribution each of the different learning arrangements may provide for competency acquisition.

A concrete category is almost universal if its class of non-constant morphisms contains an isomorphic copy of every category of algebras as a full subcategory. This paper characterizes almost universal varieties of commutative semigroups. As a consequence we obtain that for every infinite cardinal κ there exists a commutative semigroup of cardinality κ such that it has exactly two endomorphisms, the identity endomorphism and a single constant endomorphism.

There is no doubt that working with mathematical models and solving problems using mathematical procesures forms an essential part of engineering work.

B.M. Schein has proved that in the semigroup variety NIB = Mod(x(2) = x, xuvy = xvuy) of normal bands one can find at most four pairwise non-isomorphic semigroups with isomorphic monoids of endomorphisms. We prove here the same result for the extension semigroup variety (NB) over tilde = Mod(x(2)y = xy,xuvy = xvuy) of NB, properly separated from N only by the variety of inflations of the N-semigroups.

A variety V is var- relatively universal if it contains a subvariety W such that the class of all homomorphisms that do not factorize through any algebra in W is algebraically universal. And V has an algebraically universal alpha - expansion alpha V if adding a nullary operations to all algebras in V gives rise to a class alpha V of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety V has an algebraically universal alpha- expansion aV. This note contains a more general result that proves this conjecture.

An introductory paper of a special edition of Journal of Teaching Mathematics and its Application where six papers are published selected from the Conference Mathematical Education of Engineers held in Loughborough University during April 2008.

The book deals with propositional logic and predicate logic. The resolution method is presented for both propositional and predicate logic.

The paper deals with investigations and results that were obtained when solving problems addressed by SEFI Mathematics Working group, especially those posed by the Assessment project.

Universality, Q-universality and determinacy are solved for varieties of semigroups that are nearly J-trivial. A semigroup variety V is nearly J-trivial if for every semigroup S in V any J-class is a singleton. It is proved that for a nearly J-trivial variety the following is equivalent: V is 3-determined, V is not var-relatively alg-universal, the lattice of all subquasivarieties of V is finite, V is a subvariety of one of two special finitely generated varieties

In the study of universality in semigroups varieties it was shown that three finite semigroups paly an important role. The paper brings the proof that quasivarieties generated by two of them are neither relatively algo-universal nor Q-universal, but there exist two semigroups generating the same semigroup variety but such that the quasivariety generated by them is Q-universal.

The paper shows a possibility how to use the current SEFI-MWG Core Curriculum into a hierarchical form wich will enable a reader to produce a syllabus of a new mathematical course. It also manifests the inner structure of SEFI-MWG CC

The paper studies different types of weak universality in semigroup varieties. It is proved that three semigroups of finite order play a significant role in the investigations

Kategorialni metody teorie struktur a informatiky

A variety V is subdirectly dominated if every algebra A in V is embeddable into a subdirectly irreducible algebra belonging to V. We characterize the subdirectly dominated semigroup varieties of nil semigroups, of commutative semigroups, and of completely regular semigroups

The aim of this note is to improve the results of T. Watanabe and A. Nakamura. We present algorithms which for a given automaton A decide whether the transition semigroup of A contains left or right identity, or whether the transition semigroup of A is a left or a right group, or permutation group in linear time (i. e. it requires O( vertical Q vertical multiplied by (times) vertical X vertical ) time where Q is the set of states of A, X is the set of inputs of A). Further we give algorithms which for a given automaton A decide whether A is quasi-state independent, or state independent and requires O( vertical Q vertical **2 multiplied by (times) vertical X vertical ) time.

Fast algorithms are presented which find the minimal nontrivial congruences, subalgebras, and ideals of a finite algebra, given by tables of its operations. Our method involves finding objects which are minimal among all nontrivial objects, i.e., objects distinct from the least one. The running times of the presented algorithms are O(nr+1), in case of congreunces, and O(nr), in case of subalgebras and ideals, where n is the number of elements of the algebra and r is the maximal arity of its operations. For the minimal nontrivial congruences in groups and rings, better algorithms are presented working in O(n2) time. The algorithm for minimal nontrivial congruences is used for a test whether a given algebra is simple, or subdirectly irreducible. This algorithm is both a generalization and an improvement of that for sequential automata, published by the present authors (1981). © 1985.