## Apstrakcija i primena matematičke analize u nastavi na tehničkim fakuletima

Abstraction and application of mathematical analysis in teaching on technical faculties

##### Doktorand

Albijanić, Miloljub##### Mentor

Mateljević, Miodrag##### Članovi komisije

Milovanović, GradimirArsenović, Miloš

##### Metapodaci

Prikaz svih podataka o disertaciji##### Sažetak

Osnovni koncept rada jeste povezanost apstraktne teorije i primenjene matematičke analize
u univerzalnom matematičkom sistemu. Matematika je doživela veliku praktičnu primenu, na
primer, primenu matematičke statistike, numeričke analize, primene u elektrotehnici, doprinos
razvoju računarstva i dr. Istovremeno, u naučnom pogledu, uzdigla se do neslućenih apstrakcija
(topološki prostori, vektorski prostori i drugo). Zbog ovih činjenica neophodno je unapredivati
nastavu matematičke analize na tehničkim fakultetima ali i same metode nastave.

The main concept of this thesis is the connection of abstract theory and applied mathematical
analysis in the universal mathematical system. The Mathematics is being applied practically in
great number of cases, for example, in the application of mathematical statistics, numerical analysis,
application in electrical engineering, contribution to computer developing etc. At the same time, in
scientific perspective, it reached the level of incredible abstractions (topological spaces, vector spaces
et al.). Due to these facts, it has become necessary to advance the teaching of mathematical analysis in
technological universities, but also the very methods of teaching as well.
This thesis contains both a theoretical and an empirical research. The theoretical research sheds
light onto the notions of abstraction and application and gives examples from next teaching subjects:
Lagrange’s Theorem, Convexity and Consequences. Taylor’s Formula. Hardy’s Approach to Calculating
Surfaces of ...Flat Shapes. Fourier’s Series and their Application. Banach Fixed-Point Theorem and
its’ Application. The empirical research consists of two parts: a questionnaire and a test. This research
shows how the relation between theory and application is seen by students, how they perceive the teaching
of mathematics and which learning instruments they use. The research also shows how students
solve simple problems and which type of problems they solve more sucessfully.
The sample consists of 429 students of electrical engineering, construction and mechanical engineering
for the Questionnaire and 450 students of the same universities for the Test. Students that
participated in the research are studying at the University of Belgrade, University of Novi Sad and
University of Niˇs.
The results of the research confirmed that students from technological universities have a favourable
attitude towards mathematics and that they see its significance in its application, i.e. in its use
value. Students have clearly defined attitude on the idea that a good lecture by a professor is one which
can be understood, which is well articulated, and which motivates students to take part in it. They point
out the significance of examples that have elements of application. Visual presentations also enhance
the success in solving problems. The research shows that students did not acquire the skill to apply
their knowledge of mathematical analysis in solving tasks and problems.
Theoretical clarification of notions of abstraction and application, followed by a displayfive topics
of mathematical analysis confirm that abstract theory and applied mathematical analysis are interconnected
and conjoint in the universal mathematical system.
On the basis of results recommendations are defined concerning innovational approaches to
teaching, such as planning the lectures and meliorating the contents, asking questions, and also an
intelligent prospect, lecture improvement and learning instruments application.
This way, it is confirmed that a methodically well organized lecture helps a better understanding
of the relation between abstraction and application of mathematical analysis.