The role of notation in mathematics

In this paper, the historical process of the Cardan's rule notation is presented. There are some descriptions with respect to the linear and/or structured form of texts compared there, and also the exactness of these descriptions is discussed. It is interesting to look at the way of finding the solution. There is the English transaltion of the first general method for solving cubic equation in Cardan's Ars Magna introduced in this contribution. This comparison could be useful to presentation of a new mathematical problem to students or to writing a mathematical text for students.

Paweł Borkowski

Learning system based on cellular automata

In this paper we present the CELLS2 system which is a learning system based on cellular automata. In the learning process CELLS2 used the Quine-McCluskey method which minimalizes boolean formulas.

Jaroslava Brincková

Interdisciplinary projects of geometry

We will present an illustration from interdisciplinary project which has been done with 12 years-old as well as 20 years-old students. Traditionally projects comprise several subjects and mathematics is considered to be a tool solving problems in them. In our interpretation, mathematics itself has become the core of student projects which then comprise various activities in which students discover mathematical concepts outside and/or in which they learn about possibilities of the use of mathematical concepts outside mathematics.

Yakov Brodskyy

On the structure of probability-statistical education at school

The proposed structure of mathematics education in general - and probability in particular - has been realized in the lyceum at Donetsk University. This structure enables the students to acquire a thorough mathematic preparation, it influences positively the development of their motivated sphere, it takes into account the difference in the general and mathematic preparation of the students.

The principle of successive modeling in the teaching process of the specialist's activity is the basic one underlying this structure.

The foundation of the probability-statistical preparation makes up the respective parts of the basic course in mathematics. They must be understandable for all students and that is why they should be based, for example, on statistical foundation. Their peculiarity lies in the fact that they are aimed at the development of the mode of thinking required from a respective specialist, at the formation of the skill to use probability-statistical methods at modeling real processes and phenomena.

Optional courses play an important part in the formation of a specialist, they give the possibility to take into consideration different requirements and abilities of the students. In different years the students of the lyceum were offered such courses as: Ëlementary probability models", Ëlements of theory of combinations", "Geometrical probabilities", Ëlements of Mathematical statistics".

Research gives the opportunity of modeling actual professional activity. We have at our disposal a bank of similar probability-statistical tasks.

Grzegorz Bryll, Robert Sochacki

A method of comparison of Post-complete propositional logics

In this paper we consider a necessary and sufficient condition for every propositional calculus to be Post-complete. We will show, how this condition may be used to compare consequence operations for these calculi. The completeness notion may be characterized in the following way:

Theorem 1

The next theorem states connection between some propositional calculi and consequences determined by them:

Theorem 2
(R₁ , X₁ ),(R₂ , X₂ ) Î Cpl ŮR₁ ČR₂ Í V( M )ŮX₁ ČX₂ Í E( M )Ţ C₁ = C₂.

where C₁ (Y) = Cn(R₁ , X₁ ČY), C₂ (Y) = Cn(R₂ , X₂ ČY).

Grzegorz Bryll

Pedagogical University

Institute of Mathematics

and Computer Science

Armii Krajowej 13/15

42-200 Częstochowa

Poland

Robert Sochacki

Opole University

Institute of Mathematics

and Computer Science

Oleska 48

45-052 Opole

Poland

E-mail: sochacki@math.uni.opole.pl

Grzegorz Bryll, Robert Sochacki

On a test concerning the set notion

Many students have problems with solving exercises concerning sets which elements are also sets. Many wrong solutions are due to incorrect understanding of the term set. We suggest a test which is aimed at drawing attention to the above problems.

Grzegorz Bryll

Pedagogical University

Institute of Mathematics

and Computer Science

Armii Krajowej 13/15

42-200 Częstochowa

Poland

Robert Sochacki

Opole University

Institute of Mathematics

and Computer Science

Oleska 48

45-052 Opole

Poland

E-mail: sochacki@math.uni.opole.pl

Danuta Drygała

Power set-problems with understanding of the idea

The article concerns choosen symbols introduced in the first year of maths studies and some problems connected with understanding them students. On this scale I have devoted myself to the idea of power set. I have discussed the course of didactic activities made with students, concerning discussion of join and meet property in a power set of natural numbers.

I am describing the offer of a specified procedure of introducing the "power set" idea during the activities connected with this subject, which will exclude any appearing problems.

Petr Eisenmann

Students' ideas about infinite

This contribution deals with the description of the students' ideas about the terms finite and infinite. It is based on the results of the research we carried out in the years 1998-2000 at thirty elementary schools and secondary grammar schools in the North Bohemian Region.

Vasyl Fedorchuk

On new differential equations of the first order in the space M(1,4)×R(u) with nontrivial symmetries

The subgroup structure of the generalized Poincaré group P(1,4) has been studied. The differential equations of the first order in the space M(1,4)×R(u) which are invariant under splitting subgroups of the group P(1,4) have been constructed. Among the equations obtained, there are ones which can be used in relativistic and nonrelativistic physics.

E-mail: fedorchuk@wsp.krakow.pl

Roman Frič

Duality: random variables versus observables

We discuss some aspects of the duality, between random variables and observables - one of the basic questions of probability ([1], [2], [3]). Besides a historical background ([4]), we mention the role of elementary events ([5]) and the algebraic (i.e. pointless) approach to probability ([6]). We present some categorical constructions leading to possible generalizations of the field of events, probability measures, and random variables ([3], [1]).

References

[1] Dvurecenskij, A. and S. Pulmannova: New Trends in Quantum Structures. Kluwer Academic. Publ., Dordrecht, 2000.
[2] Fric, R.: Sequential structures in probability: categorical reflections. In: Mathematik-Arbeitspapiere 48 (Editor: H.-E. Porst), Universitat Bremen, Bremen, 1997, 157-169.
[3] Fric, R.: Convergence and duality. Appl. Categorical Structures. (To appear.)
[4] Halmos, P. R.: The foundations of probability. Amer. Math. Montly 51, 1944, 497-510.
[5] Los, J.: Fields of events and their definition in the axiomatic treatment of probability. Studia Logica 9, 1960, 95-115 (Polish), 116-124 (English summary), 125-132 (Russian summary).
[6] Mundici, D. and B. Riecan: Probability on MV-algebras. In: Handbook of Measure Theory (Editor: E. Pap), North-Holland, Amsterdam. (To appear.)

Ján Gunčaga

Remarks on continuous fractions

A lot of interesting examples from the history of mathematics are seldom if at all used in school mathematics. Continuous fractions were used in the ancient Greece to calculate values of the irrational numbers. In the paper we show possible applications to the limit problems, problems with sequences and series.

Pavol Hanzel

The finite subsets natural numbers

The article is about relations between triangle and square numbers. The stress is laid upon numerical equality. The finite set of these numbers and its thin subsets are defined by software. Finding natural numbers with specific sing has more didactical meaning. It is enlisted into the course of preparing next teachers working at elementary school.

Karel Hrach

Two problems solved with MS EXCEL

The first problem, arising from the theory of graphs, is to find the maximum flow in a network. This is equivalent to the problem of finding a minimum cut, according to Ford-Fulkerson theorem. Basic terms (network, capacity, flow, cut, matrix presentation) are briefly explained. The second, the probabilistic problem solved by Bernoulli, is to calculate the probability that a certain sum is obtained, when throwing a dice several times. This contribution briefly presents a recurrent solution for generalized "dice" with any number of possible values and concerns presenting the way to solve these two problems (the maximum flow and the thrown dice) simply by application of the commonly used SW MS-Excel.

The methodics of working out practical exercises in teaching in primary and secondary school

The review of bibliography. Contemporary requirements in mathematical teaching. The classification of exercises occured in maths school - books. The analysis of chosen primary and secondary school - books considering the use of school mathematics in surrounded reality. The trial of working out practical exercises skills formulation in examples.

Henryk Kąkol

Application of Technology in the process of teaching of mathematics

In my lecture I will show a few examples which should explain problems connected with application of Technology in the process of learning mathematics.

Pavel Klenovčan

Pythagorean triples

Students should learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

We describe all integer solutions the equation x²+y² = z² by utilizing elementary knowledge from algebra, geometry and number theory.

Jan Kopka

Helping students to become active learners

Elementary number theory provides a variety of topics suitable for guided investigation in mathematics (the scheme of investigation is diagramed). We demonstrate this with the help of non-traditional and fascinating topic - the Fibonacci sequence. We present here two general strategies of investigation: the strategy of äccepting the given" and the strategy of "not accepting the given". (In our case the first strategy leads to the investigation of the Fibonacci sequence and the second the investigation of different kinds of sequences which are closely related to Fibonacci sequence, such as pseudo-Fibonacci sequences and the Tribonacci sequences.) This article also demonstrates the method of guided investigations through several mini-investigations.

Štefan Kováčik;

Computers and teaching mathematics

Ireneusz Krech

Waiting for a series of colours

This paper presents methods of calculating the probability of events related to Markov's homogeneous chains theory. The main means of argumentation being the stochastic graph.

František Kuřina

Wallas-Polya-Hadamard and creativity

Comparison of the theories of G. Wallas and G. Polya with the school practice.

Wallas:

1. Preparation (hard work on a problem).
2. Incubation (concious praparation is changed into unconscious mechanism of searching for the solution).
3. Illumination (an idea emerges into conscious).
4. Verification of the right idea.

Polya:

1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Look back.

The observation of the process of problem solving on some examples.

Paweł Kwiatkowski

Why do we teach trigonometry?

This paper is about certain idea for teaching methods and educational research. It describes my conclusions based on research carried out in classes with mathematics/informatics profile, in Polish high schools.

The concept is "Why do we teach trigonometry".

The author of this paper shows trigonometry as a tool in solving geometry problems.

Ludmiła Macedońska

Introductory suggestion to the criteria of analysis of student's books in maths (levels 4-6 of the primary school)

My presentation concerns problems of the criteria of an integrated analysis of student's books in mathematics. This work presents an introductory suggestion to the criteria of evaluating students books in math, levels 4-6 of the primary school. This suggestion is based on considering the student's book as a set of announcement language (R. Jacobson 1975, 1989: J. Skrzypczak 1990: G. Treli 1995). Taking into consideration that those language announcements can be an object of analysis and evaluations held from the point of main functions of announcements. According to this concept, every announcement, not depending on presenter's intentions, is taken by the sender and receiver on such levels as emotion, appeal, poetry, metalingual aspect, reference and fact. This touch enables considering the book from many different points - from the author's side, pupil's and teacher's ones, so from and math's one.

The criteria, presented in this work, are conclusions from investigations, the aim of which was establishing psycho-didactic conditions of schoolbooks in mathematics for the 4-6 levels of the primary schools.

Maciej Major

Solutions to a certain problem ilustrating the idea of fusionism in the probability theory

The solutions presented above may serve as an illustration of ''the principle of internal integration'', known as the idea of fusionism.

Three balls are selected simultaneously from an urn containing b white balls and c black ones. If all balls are of the same colour, then one of the players wins, in different situation the other player is the winner. For which values of b and c is this game fair?

Questions caused by the probability problems may serve as a considerable source of such opportunities.

Gennady A. Medvedev

No arbitrage conditions for markets with inflation and for segmented markets

Under consideration of the continuous time mathematical models of the price dynamics in the financial market it is assumed that the processes of the interest rates in the market follow the stochastic differential equations. The no arbitrage condition in that market with inflation is obtained for a portfolio with any number of assets. Also it is derived from explicit form the no arbitrage condition in the segmented market (without inflation), where it is considered, that in the market simultaneously there are some segments, in which the bonds with hardly distinguishing maturities enter, and each segment has its own risk free interest rate. It is considered that in this situation the investor has a possibility to purchase the bonds of any segment.

Jan Melichar

Mathematical logic and mother tongue in elementary school

Logical thinking is closely connected to the term of proclamation and identification if it is valid or not. Sentence is language expression of thought. It means that the logical thinking is closely connected to mother tongue. Proclamation can be introduced even at the elementary school as an proclamation sentence which clearly expresses something what can be valid or not. If we use one of many options in the proclamation the opposite proclamation must contain all other options. In the article I study proclamations about numbers of people and things and their opposite proclamations. There are also examples of interconnection between logic and mother tongue in the education at elementary school.

Barbara Nawolska

The probability distribution of random variables as a problem in the theory of combinations

This paper presents the process of discovering, by means of induction, certain facts (concerning the theory of combinations) required in order to construct the probability distribution of a series of certain random variables in classic probabilistic spaces.

Bohumil Novák

On the use of mathematical problems in teacher training

Mathematical problems and thier solution tarditionally play an important part in mathematical education. The problems are used for various didactic purposes in the professional teacher training and in the educational reality of primary and secondary schools. They form and develop necessary teaching skills.

Using examples, we demonstrate the acquisition of the mathematics teacher's competence to reflect their own activity during the analysis of various aspects of some types of mathematical problems and their solutions.

Bożena Pawlik

On the understanding of geometric transformations by mathematics freshmen

Two questions will be discussed in my presentation.

The first one concerns difficulties the first year university students meet while learning definition of the geometrical transformation.

The second question - the incorrect way of searching for the image of a line segment in different transformations by mentioned above groups of students.

Zdeněk Pejsar

E-learning in education of arithmetics for future primary schools teachers - combined form of study

Possibilities of e-learning implementation in distant future teachers study at primary schools with orientation on arithmetic course in second year of study. Study artwork preparation for individual course modules, text conversion into electronic form of presentation. Motivations components. Creation tests with the backward effect, final test creation. On-line and off-line education, his possibilities, exploitation and combination.

Štěpán Pelikán

"Probability and statistics" - a modular system of instruction at the University of J.E. Purkynĕ in Ústí nad Labem

Probability and Statistics courses are offered at three colleges of the University of J.E. Purkynĕ in Ústi nad Labem and at the Institute of Engineering Technology. These courses are a part of required curricula for several majors at different educational levels (baccalaureate, master's and engineering). Such wide range of coverage requires extensive personnel, technical and financial support. The Department of Mathematics utilized the Fund for Development of College Education in Czech Republic to develop a system of modular instruction for the courses of Probability and Statistics. This systems integrates requirements of individual majors with interests of all involved parties. At the same time, the system provides sufficient flexibility for all stakeholders to meet their specific needs and requirements.

Jaroslav Perný

The space imagination on the cube

The contribution presents some results of the space imagination research on the cube carried out on 9-to-12-year old students.

The experiment is based on a task situation "Tilting a dice" in which a student "tilts" the die over its edges according to a plan in his/her mind only. A group of tasks of a different type, length and difficulty is composed. This research is the continuation before mentioned exercises "Walking on the cube" and Ïmagining a cube from its given net".

There is investigated successfulness, different between boys and girls, between older and younger students and so on. Some common phenomena occurred in students solving strategies. These are for example, difficulties in understanding the direction ahead-to the back, the use of the movement and the regularity in tilting.

Adam Płocki

Probability in street gambling

The concept of probability is shown based on the example of a street gambling game as a mathematical means of estimating the expected gains and losses in a gambling game. This gambling game illustrates the decision process in conditions of risk.

Jurij Povstenko

Non-riemannian geometry and the theory of lattice imperfections

Creation and development of continuum theory of imperfections of a crystal structure in solids is closely associated with ideas and methods of non-Euclidean and non-Riemannian geometry. In the continuum theory of lattice defects the Cartan torsion tensor is identified with the dislocation density and the Riemann-Christoffel curvature tensor is identified with the disclination density. To give a differential-geometrical interpretation of the imperfection kinematics it is necessary to take the time into consideration. We discuss a three-dimensional space of affine connection with time as a parameter and introduce the material time derivative using the "time-connection" tensor.

Jana Přihonská

The sets of "generated problems" with a view to use the graph encoding close to solving of given problems

The Graph Theory is not a component of standard syllabus of the elementary school, nevertheless, some ideas of this theory are used in solving different problems, for example to visualization or systematization some situations described in word or done by geometry. We are going to discuss the possibility of taking advantage of graph encoding during the process of solving the problems of the labyrinth's type in this article. We take steps gradually to the other types of problems. There are compiled the sets of "generated problems" with making the best of previously obtained our own experiences during the basic and "pilot" research focusing on to useing the graphs in primary school. Solving these problems contribute to developing of the solving strategies of the pupils. The improving the quality of the ability to operate with encoded symbol systems and connecting through the different areas of math in primary school is our aim.

Magdalena Prokopová

Pupil's conception of some elementary objects of geometry

My contribution deals with pupil's conception of some elementary objects of geometry, like a point, a straight line and a plane. Pupils (of primary and secondary schools) have often many problems with understanding notions, where the problems spring from a variance between a phylogeny and ontogeny of these images. The variances often look like an ignorance or a mistake, but they claim a natural evolution of notions of pupils.

Tadeusz Ratusiński

Use of computer in dissolving of mathematical problems

In raport I will present results of preliminary didactic investigations consecrete meeting of part of computer in process of dissolving mathematical assignments.

Bożena Rożek

Perception of rectangular and skew structures in a row-column arrangement of figures

The aim of the reported research is identification of difficulties and natural developmental limitations in understanding of a Row-Column Arrangement of Figures (RCAF) by 10-13 years old children.

RCAF includes the arrangements made by regular distribution of any elements in rows and columns. These elements don't have to be the same and adjoining. This idea refers also to diagonal arrangements, where the columns and the rows don't intersect at the right angle.

Analyzing school curricula shows that the multilateral understanding of RCAF is a prerequisite knowledge for many problems; e.g. the product of two positive integers, the commutative law of multiplication, the idea of area, coordinates, matrices, various tables and diagrams...

The aim of the research was to find the answer to the following question: Which elements of Rectangular and Skew Structures are seen by children in RCAF? In their tasks children, had to construct and draw examples of RCAF. My research results show which elements of the given structure are important for children and which of them are dominant in the children's awareness.

The grade of disclosure of the given structure by children in their drawings was various from omitting all the important features to regarding some characteristic ones for given structure.

The research also proves that at the beginning of school, children do not possess well formed structures, necessary for the understanding of some mathematical ideas or operations, while the curriculum assumes their presence. The introduction of concepts based on unformed structures may lead to dramatical failures of some students.

Long-term projects in school

I would like to answer these questions during my presentation:

1. When to start setting long-term projects to students?
2. How to compose projects - problems?
3. How to help student who works on a project?
4. How to mark such projects?

Petr Rys, Tomáš Zdráhal

Problems in computing the limit of some sequences

Computing the limit of a sequence by means of a computer can often bring a surprise. To understand it one should see what happens for various values along the way.

Petr Rys, Tomás Zdráhal

The theorem about implicit function in the calculus infinitesimalis

This paper is a common work of both authors P. Rys and T. Zdráhal: The theorem about implicit functions in the calculus infinitesimalis.

The aim of this paper is formulate Theorem about implicit function in the Calculus Infinitesimalis with its proof. The Calculus Infinitesimalis is a nonstandard analysis. It was published in P. Vopenka: Calculus Infinitesimalis pars prima, Práh, Praha, 1996.

Jiři Soukup

Topological conceptions with support multimedia presentation on PC

Tools for multimedia support of topological conceptions in geometry education for combined form of future primary schools teachers study. This multimedia presentation motivating case for concepts formation such as: "pixel environ", "formation boundary", "overlap of formation" etc. Specific aspect of the methodology of creation, application of multimedia presentations within lifetime education.

Anna Stopenová

Space imagination as a premise in geometry learning

In the contribution, some experience acquired in the university education of primary school prospective teachers of mathematics will be presented. Space imagination is the ability needed by everyone. Prospective teachers should know activities which develop space imagination of pupils and should use them in their teaching.

Jana Šimsová

Wavelet Galerkin method for solving the Fredholm's integral equation of the second kind

Wavelets are discussed today in many fields of mathematics. For this reason, I want to show some application to solve the Fredholm's integral equation of the second kind. The basis of discretisation of such equation will be Galerkin method. In this method, I shall suppose the subspaces as arising from multiresolution analysis. Their base can be either scaling function or wavelets.

Marie Tichá

On the role of translations between modes of representation - case of fractions

We believe that utilisation of various modes of representation is important for the development and deepends on understanding. Many didacticians stress that the level of understanding is related to the continous enrichment of a set of various modes of representation and emphasise the development of student's capability of translation between modes of representation. But it is also possible to utilise the work with representations for recognition (diagnosis) of the level of understanding and also for searching for misunderstandings and obstacles in understanding and further as a tool for re-education.

This contribution focuses on the study of student's formulations of word problems to the given calculation (Pose the word problem which was solved by the calculation ²/₃+¹/₄.) or to the given visual representation. Our present investigation shows that this way we can get a lot of didactically interesting information.

In the contribution some examples of problems formulated by students will be shown. Teacher's reaction to these problems will be mentioned. I will concentrate especially on those problems that contain the most frequent misunderstandings and I will outline suggestions how to re-educate these misunderstandings.

Michael Tikhonenko, Andrey Kolyada, Viktor Revinski

Gradient disbalance method for fingerprint image direction field optimization

Direction field building has an exclusive importance for common process of the classification analysis in up-to-date automatic fingerprint identification systems. All characteristics of this analysis strongly depend on direction field reliability. That's why effective technologies development for the optimization of direct field is the actual problem. In this work we pay attention to the smoothing procedures of relaxation type, which use the gradient disbalance minimization principle.

Oleg Tikhonenko

Generalized Erlang problem for queueing systems with random volume demands and restricted summarized volume

We consider a non-classical M/G/n/0 type queueing system with random volume demands. Each demand requires m servers for its simultaneous service with probability q_m, ĺ_{m = 1}ⁿg_m = 1. Service time of the demand depends on the demand volume and m. The total volume of demands presenting in the system is restricted by memory volume V. For such system the stationary distribution of demands number and loss probability are determined.

Pavel Tlustý

The optimal strategy in one stochastic game

The article deals with the stochastic game "Portion on three fractions". The optimal solution in this game is investigated.

Jerzy Tocki

The systematical construction of aims in mathematical education

In education there are two elementary systems: the students and society, and one auxiliary, intermediary system: the teacher.

The construction of aims in mathematical instruction requires a pre-vious description of the most important educational proprieties and needs of both society and the student who is being gradually included in it.

It follows from that:

1. the basic subject of school education is the individual student and the essence of the educational process should be based on learn-ing; where as the teacher and the student's nearest environment (i.e. school) should be the organizer of learning,
2. school education should be a solid foundation for further educa-tion (including self-education), which equips the learner with relatively long-lasting and useful knowledge as well as with the skill of the widening and applying the knowledge.

Student's intellectual development is possible only in the process of ac-tive and directed work containing receipting, gaining, transformating and applying the information.

So it should be ëducation trough using mathematics, not only mathe-matics for itself" - the students should build their own cognitive schemas.

All statements above lead to three interrelated classifications:

• - the types of aims considering their contents at the level of generality,
• - the types of aims cosidering educational character,
• - the types of aims considering the level of realization (students achievements).

A set of all educational aims mentioned in classification I, II, III creates the didactic dimension of educational aims.

Włodzimierz Trochanowski

Efficiency in mathematical education in early classes in primary school - integrated education

The mathematical education in primary school is the foundation and the beginning of further learning maths. One did research work on efficiency in mathematical education in early classes in primary school, using suitable tests. The results confirm low mathematical knowledge of pupils. It refers especially to solving problems which demand logical thinking from children, as well as to practical knowledge. Good results were accomplished in problems which concerned learned algorithms.

We can draw the conlusion that one ought to integrate subjects both around Polish lessons and maths too. Then the efficiency in mathematical education will increase.

Stefan Turnau

Developing mathematical thinking

There is no way back to the old paradigm of geometry as an academic discipline. The principal objective of teaching geometry, which has been students' systematic knowledge of selected chapters, with an understanding of the deductive method, with a certain skill in solving certain kinds of classical problems in the "correct" way - has to be abandoned. What can it be replaced with?

The principal objective of teaching geometry in the school should be development of students' mathematical activity. By mathematical activity I mean:

- perceiving and generalising perceived (or indicated) regularities,

- perceiving, using and veryfying analogies,

- verifying general statements using adequate examples,

- verifying own procedures (e.g. solutions of problems) and their results,

- defining and axiomatising,

- drawing conclusions from accepted statements,

- using algorithms,

- schematising,

- mathematising i.e. expressing the given situation in mathematical terms,

- modelling,

- interpreting results.

Geometry provides plenty of excellent open problem situations that can be freely investigated by students on various school levels. The investigation process will inevitably include many of the mentioned kinds of activity. There is abundance of problem situations were questions of different kind and level can be asked, investigation in various ways - from physical experiment to abstract reasoning - performed, errors committed and corrected, answers found and verified, and nonbanal facts produced. It is very important that in geometry many questions can be investigated on various levels, which makes it educationally valuable for all students, no matter what their knowledge and reasoning skill is. If the problem situations are well chosen and the teacher consciously guides work on them then students will also acquire important knowledge, of both practical and theoretical applications.

Anna Katarzyna Żeromska

Selected aims of teaching mathematics (mathematical education) and the problem-solving process

The talk will present a study which was meant as an attempt to diagnose the realisation of those aims of teaching mathematics which assume modelling active attitudes and behaviours of students during the process of solving mathematical problems. Therefore, the analysis of the problem-solving process became here a specific tool applied to characterising and describing 'a 14-15-year-old student's attitude towards mathematical problems'.