Univerza v Mariboru Fakulteta za naravoslovje in ... progami... · Reviji Matematika v šoli in...

Univerza v MariboruUniversity of Maribor

OPIS PREDMETA / SUBJECT SPECIFICATIONPredmet: Delo z učenci s posebnimi potrebamiSubject Title: Working with children with special needs

Študijski program Study programme

Izobraževalna matematika, dvopredmetni študij, 2. stopnja Univerzitetna koda predmeta / University subject code:

Predavanja Lectures

Seminar Seminar

Sem. vajeTutorial

30 Nosilec predmeta / Lecturer:

Jeziki / Languages:

Predavanja / Lecture:Vaje / Tutorial:

Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti: Vsebina: • Sistem družbene pomoči in skrbi

namenjen otrokom s posebnimi potrebami (PP);

• Terminologija povezana z motnjami, primanjkljaji, ovirami;

• Zgodnje odkrivanje in zgodnja obravnava motenj ter vzroki nastanka motenj;

• Opredelitev inkluzivne vzgoje in izobraževanja, filozofska izhodišča inkluzije, socialni model obravnave v inkluziji;

• Razvojne, učne in socialno-emocionalne značilnosti posameznih kategorij otrok z motnjami ter prilagoditve pri delu z njimi;

• Nova koncepcija vzgoje in izobraževanja v Sloveniji, usmerjanje, programi vzgoje in izobraževanja za otroke s PP, dodatna strokovna pomoč, individualizirani programi, vloga šol in zavodov za vzgojo in izobraževanje otrok s PP v integraciji/inkluziji;

• Sodelovanje in timsko delo učiteljev, specialnih pedagogov in drugih strokovnjakov v integraciji/inkluziji;

• Sodelovanje s starši otrok s posepotrebami (ovire v procesu sodelovanja s starši, pomoč staršem, sodelovalno – partnerski model vključevanja staršev);

• Študija primera s poudarkom na kvalitativnem

Univerza v Mariboru University of Maribor

Fakulteta za naravoslovje in matematiko / Faculty of Natural

Sciences and Mathematics

OPIS PREDMETA / SUBJECT SPECIFICATION čenci s posebnimi potrebami

Working with children with special needs

Študijska smer Study field

LetnikYear

1.

Univerzitetna koda predmeta / University subject code:

Sem. vaje Tutorial

Lab. Vaje Lab. Work

Teren. vaje Field work

Samost. deloIndivid. work

30

Majda SCHMIDT

Predavanja / Lecture: slovenski / Slovenian Tutorial: slovenski / Slovenian

itev v delo oz. za opravljanje

Prerequisites:

Contents (Syllabus outline):

či in skrbi namenjen otrokom s posebnimi

Terminologija povezana z motnjami,

Zgodnje odkrivanje in zgodnja obravnava motenj ter vzroki nastanka motenj; Opredelitev inkluzivne vzgoje in izobraževanja,

a inkluzije, socialni model

emocionalne ilnosti posameznih kategorij otrok z

motnjami ter prilagoditve pri delu z njimi; Nova koncepcija vzgoje in izobraževanja v Sloveniji, usmerjanje, programi vzgoje in izobraževanja za otroke s PP, dodatna

, individualizirani programi, vloga šol in zavodov za vzgojo in izobraževanje

čiteljev, specialnih

pedagogov in drugih strokovnjakov v

Sodelovanje s starši otrok s posebnimi potrebami (ovire v procesu sodelovanja s starši,

partnerski model

Študija primera s poudarkom na kvalitativnem

• System of societal support and care for

children with special needs (SN) • Terminology associated with disabilities,

impairments, handicaps, • Early identification and early intervention

of disabilities and causes,• Inclusive education, philosophical

backgrounds of inclusion, social model of treatment in inclusion;

• Developmental, educational and socioemotional characteristics of children with several categories of disabilities and adaptations in work with them;

• New conception of education in Slovenia, direction (assessment), educational programmes for children with SN, additional individualized education programmes,the role of schools and educational institutions of children with SN in integration/inclusion;

• Cooperation and team work of teachers, special educators and other professionals in integration/inclu

• Cooperation with parents of children with SN (barriers in the process of cooperation, support to parents, cooperation-partnership model of



Letnik Year

Semester Semester

1. Zimski/Winter

Samost. delo Individ. work

ECTS

60 4


System of societal support and care for children with special needs (SN) Terminology associated with disabilities, impairments, handicaps, Early identification and early intervention of disabilities and causes, Inclusive education, philosophical backgrounds of inclusion, social model of treatment in inclusion;

educational and socio-emotional characteristics of children with several categories of disabilities and adaptations in work with them; New conception of education in Slovenia, direction (assessment), educational programmes for children with SN, additional professional support, individualized education programmes, the role of schools and educational institutions of children with SN in integration/inclusion; Cooperation and team work of teachers, special educators and other professionals in integration/inclusion; Cooperation with parents of children with SN (barriers in the process of cooperation, support to parents,

partnership model of

pristopu. inclusion the parents); • Case study with emphasis on qualitative

approach.

Temeljni študijski viri / Textbooks: -Schmidt, M. (2001). Socialna integracija otrok s posebnimi potrebami v osnovno šolo. Maribor: Pedagoška fakulteta.

-Schmidt, M., Čagran, B. (2006). Gluhi in naglušni učenci v integraciji/inkluziji. Zbirka Zora, 43. Slavistično društvo, Maribor.

-Lipec-Stopar, M. (1999). Vloga defektologa pri timskem delu z učenci s posebnimi potrebami v osnovni šoli. V: Hytonen, J., Razdevšek-Pučko, C., Smyth, G. (ur.). Izobraževanje učiteljev za prenovljeno šolo. Ljubljana: pedagoška fakulteta, str. 65-72.

-Integracija, inkluzija v vrtcu, osnovni in srednji šoli (2003). Sodobna pedagogika, 54, (120), Posebna izdaja. -Upoštevanje drugačnosti – korak k šoli enakih možnosti (2006). Sodobna pedagogika, 57 (123), Posebna izdaja. Cilji: Objectives:

• Cilj tega predmeta je seznaniti študente s sistemom družbene podpore za otroke s PP, s termini in razumevanjem le-teh, seznaniti s procesom odkrivanja posebnih potreb ter predstaviti možne prilagoditve vzgojno-izobraževalnega procesa, ponuditi znanje o osnovnih specialno-pedagoških načelih in pristopih pri delu z učenci s PP, uvesti v poznavanje inkluzivne vzgoje in izobraževanja ter izpostaviti novosti koncepcije izobraževanja otrok s PP, osvetliti pomen timskega dela in sodelovanja na področju inkluzije, vzpodbuditi znanje o temeljnih pristopih pri delu s starši otrok s PP ter predstaviti uporabo študije primera na področju integracije/inkluzije.

• The objective of this course is: to acquaint

students with system of societal support for children with SN, with terminology and comprehension of it, to acquaint with the process of early intervention and present the possibilities for adaptations of educational process, to offer the knowledge about the basic special education principles and approaches when working with students with SN, to initiate the knowledge about inclusive education and to expose the novels of educational conception of children with special needs, to highlight the importance of the team work and cooperation on the field of inclusion, to encourage the knowledge about the basic approaches when working with parents of children with SN and introduce practical use of case study in the field of integration/inclusion.

Predvideni študijski rezultati:

Intended learning outcomes:

Znanje in razumevanje: Po zaključku tega predmeta bo študent sposoben:

• Izkazati znanje o sistemu družbene podpore za osebe s PP,

• Prepoznavati in upoštevati interindividualne razlike in posebne potrebe učencev,

• Predstaviti inkluzivni sistem vzgoje in izobraževanja ter novosti koncepcije izobraževanja,

• Razumeti vpliv inkluzije za otrokov razvoj in napredek,

• Poznati in razumeti uvajanje prilagoditev v vzgojno-izobraževalni proces,

• Poznati in upoštevati pomen timskega dela

Knowledge and Understanding: On comletion of this course the student will be able to:

• Demonstrate knowledge about the system of societal support for persons with SN,

• Recognise and consider interindividual differences and special needs of students,

• Present inclusive education system and novels of the concept of education,

• Understand the influence of inclusion on childs' development and progress,

• Recognize and understand the meaning of adaptations in educational process,

• Recognise and understand the importance of team work and cooperation in inclusion,

in sodelovanja v inkluziji, • Poznati in upoštevati posebnosti

sodelovanja s starši otrok s PP

• Recognize and consider exceptionalities of cooperation process with parents of children with SN

Prenesljive/ključne spretnosti in drugi atributi: Pri študiju in kasnejši poklicni karieri bo študent sposoben:

• Izbrati prilagoditve vzgojno-izobraževalnega procesa glede na posebne potrebe učencev,

• Upoštevati individualiziran pristop pri delu z učenci s PP,

• Razvijati inkluzivno kulturo v neposredni praksi,

• Identificirati, analizirati probleme s področja vzgoje in izobraževanja skupaj s specialnimi pedagogi in drugimi strokovnjaki,

• Povezati osnovna specialno-pedagoška znanja z znanji iz razvojne psihologije in znanji drugih področij ter jih uporabiti pri delu z učenci in starši,

• Stalnega strokovnega izpopolnjevanja, • Izgrajevati profesionalno etiko.

Transferable/Key Skills and other attributes: In studing process and in later professional career the student will be able to:

• Select adaptations of educational process with regard on special needs of students,

• Consider individualized approach in working with students with SN,

• Develop inclusive culture into direct practice,

• Identify, analyse the problems of the field of education together with special educators and others professionals,

• Link the basic special education knowledge together with the knowledge of developmental psychology and with the knowledge of other professional areas and use them when working with children and parents,

• Permanent professional training, • Complete professional ethics.

Metode poučevanja in učenja:

Learning and teaching methods:

• predavanja z interaktivno udeležbo študentov,

• seminarji, študija primera, sodelovalno učenje in timsko delo

• individualne konsultacije

• lectures with interactive participation of students,

• seminars, the case study, cooperative learning and team work

• individual consultation Načini ocenjevanja: Delež (v %) /

Weight (in %) Assessment:

• Seminarska naloga, • Pisni izpit

-opravil/ni opravil seminarsko nalogo -izpitna ocena 6-10 (pozitivno), 1-5 (negativno)

30 % 70 %

• seminar work, • written exam

-passed/failed seminar work -examin's mark 6 – 10 (positive), 1 – 5 (negative)

Materialni pogoji za izvedbo predmeta : Material conditions for subject realization

• Učilnica z ustrezno AV opremo • Classroom with appropriate AV equipment

Obveznosti študentov: Students’ commitments: (pisni, ustni izpit, naloge, projekti) (written, oral examination, coursework, projects):

• seminarska naloga – predstavitev v skupini, • pisni izpit

• seminar work – presentation in group,

• written exam

UČNI NAČRT PREDMETA / COURSE SYLLABUS

Predmet: Didaktika osnovnošolske matematike

Course title: Didactics of Elementary School Mathematics

Študijski program in stopnja

Study programme and level

Študijska smer

Study field

Letnik

Academic

year

Semester

Semester

Izobraževalna matematika,

dvopredmetni študij, 2. stopnja 1. 1.

Educational mathematics, double

major 2nd

degree 1. 1.

Vrsta predmeta / Course type

Univerzitetna koda predmeta / University course code:

Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

45

45

90 6

Nosilec predmeta / Lecturer: Alenka LIPOVEC

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE

Vaje / Tutorial: SLOVENSKO/SLOVENE

Pogoji za vključitev v delo oz. za opravljanje

študijskih obveznosti:

Prerequisits:

Jih ni. There are none.

Vsebina: Content (Syllabus outline):

- Didaktični elementi izvajanja pouka

matematike (oblike in metode dela, didaktična

načela, pisna priprava, učna gradiva …) in

didaktično načrtovanje vzgojno-

izobraževalnega procesa (dolgoročno in

kratkoročno).

- Taksonomija matematičnih znanj.

- Učni načrt za matematiko v osnovni šoli.

- Izbrane vsebine osnovnošolske matematike

od 6. do 9. razreda. Učne priprave.

- Motivacija pri pouku matematike v osnovni

šoli.

- Reševanje problemskih nalog, strategije in

- Didactic elements of mathematics education

(forms and methods of instruction, didactic

principles, lesson planning model, educational

references and resources …), and planning of

the curriculum (long-range and short-range

planning).

- Taxonomy of mathematical knowledge.

- Mathematics curriculum in elementary school.

- Selected contents of elementary school

mathematics from 6th to 9th class (grades 6 to

9). Unit planning.

- Mathematical motivation in elementary

school.

hevristike.

- Učna gradiva v osnovni šoli (učbeniki,

priročniki, DVD-ji, knjige, e-učna gradiva …).

- Pomen in uporaba tehnologije (IKT) ter e-

učenja pri pouku osnovnošolske matematike.

- Diferenciacija v osnovni šoli.

- Preverjanje in ocenjevanje znanja v osnovni

šoli.

- Šolska zakonodaja, vodenje pedagoške

dokumentacije v osnovni šoli.

- Učenci s posebnimi potrebami in posebej

učenci z učnimi težavami v osnovni šoli.

- Pedagoško delo v razredu v osnovni šoli:

komunikacija, odnosi, vzgoja, razredništvo,

reševanje konfliktov.

- How to solve mathematical problems: solving

strategies and heuristics.

- Educational resources in elementary school

(textbooks, handbooks, books, DVD's, e-

learning materials …).

- Technology (ICT) and e-learning for

enhancing mathematics instruction.

- Differenciation in elementary school.

- Assessment in elementary school.

- School legislation and pedagogical

documentation in elementary school.

- Children with special needs, particularly

children with learning difficulties in elementary

school.

- Pedagogical class management in elementary

school: communication, relations, education,

class teacher work, conflict solving.

Temeljni literatura in viri / Readings:

B. Marentič Požarnik, Psihologija učenja in pouka, DZS, 2003.

J. A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, Sixth

Edition, Allyn & Bacon, 2007.

Učni načrt za osnovno šolo.

Učbeniki, priročniki in druga učna gradiva za osnovno šolo.

Reviji Matematika v šoli in Presek.

Spletni portal E-um: www.e-um.si.

Nekateri dodatni študijski viri / Some additional sources

K. R. Harris, S. Graham, Teaching Mathematics to Middle School Students with Learning

Difficulties, The Guilford Press, 2006.

N. Jaušovec, Naučiti se misliti, Educa, 1993.

A. Orton, Learning Mathematics: Issues, Theory and Classroom Practice, Third Edition,

Continuum, 2004.

P. J. Palmer, Poučevati s srcem: raziskovanje notranjih pokrajin učiteljevega življenja, Educy,

2001.

M. Pašnik [et al.], Razrednik v osnovni in srednji šoli, ZRSŠ, 2002.

A. S. Posamentier [et al.], Problem-Solving Strategies for Efficient and Elegant Solutions: A

Resource for the Mathematics Teacher, Corwin Press, 1998.

R. R. Skemp, The Psychology of Learning Mathematics, Penguin Books, 1986.

F. Strmčnik [et al.], Didaktika, visokošolski učbenik, Visokošolsko središče Novo Mesto, 2003.

D. Vtič Tršinar, Iskalci biserov: priročnik za razredne ure, Društvo Za boljši svet, 2004.

Z. Zalokar-Divjak, Vzgajati z ljubeznijo, Gora, 2000.

B. Žorž, Razvajenost: rak sodobne vzgoje, Mohorjeva družba, 2002.

Pedagoška strokovna in znanstvena periodika.

Cilji in kompetence:

Objectives and competences:

- Uporaba in preverjanje didaktičnih metod in

pedagoških načel v neposredni pedagoški

praksi, poznavanje in uporaba izbrane

- Application and verification of educational

methods and principles in class practice,

application of specific taxonomy on lessons

http://www.e-um.si/

taksonomije matematičnih znanj pri pripravi

vzgojno-izobraževalnega procesa (nastopi pred

kolegi študenti).

- Korektno obvladovanje vsebin in konceptov

osnovnošolske matematike (od 6. do 9.

razreda).

- Poznavanje motivacijskih tehnik in uporaba

strategij poučevanja pri pouku matematike v

osnovni šoli.

- Seznanitev z obstoječimi učnimi gradivi,

učnimi načrti, šolsko zakonodajo za osnovno

šolo ter s sistemskimi značilnostmi

osnovnošolskega izobraževanja.

- Kritično vrednotenje pomena in uporaba IKT

pri pouku matematike. Iskanje in uporaba virov

pri načrtovanju in izvajanju pouka.

- Obvladovanje različnih načinov preverjanja

in ocenjevanja znanja v osnovni šoli.

- Seznanitev z možnostmi oblikovanja

projektnih dni, z medpredmetnimi povezavami,

vodenjem krožka in mentorstvom pri

raziskovalnih nalogah v osnovni šoli.

- Seznanitev s smernicami za delo z učenci z

učnimi težavami v osnovni šoli in z možnostmi

dela z nadarjenimi učenci.

- Privzgajanje pozitivnega odnosa do

vseživljenjskega izpopolnjevanja v

pedagoškem poklicu (zavedanje o nujnosti le

tega in veselje nad njim).

planing (pedagogical appearance before

colleagues).

- Mastering the contents and concepts of

elementary school mathematics (from 6th to 9th

class).

- Knowledge of motivational techniques and

strategies for learning mathematics in

elementary school.

- Acquaintance with educational resources,

curricula, and school legislation in elementary

school.

- Critical evaluation of using ICT in elementary

school to enhance mathematics instruction.

- Mastering assessment methods in elementary

school.

- Discovering school subjects connections,

learning to work in team, leading mathematics

club and research themes in elementary school.

- Acquaintance with instructions for work with

children with learning difficulties in elementary

school. Engaging gifted children.

- Development of positive attitude to the

teaching profession and to the lifelong learning.



Znanje in razumevanje:

- usvojenost specialnih matematičnih,

didaktičnih, pedagoških in psiholoških znanj,

potrebnih za učinkovito poučevanje v osnovni

šoli, ki so predstavljena med Vsebinami in

Cilji.

Prenesljive/ključne spretnosti in drugi atributi:

- pridobljena znanja in spretnosti, ki so

navedene med Vsebinami in Cilji, so podlaga

za uspešno izvajanje neposredne pedagoške

prakse in za didaktično-matematične predmete

v nadaljevanju študija (posebej za predmet

Didaktika srednješolske matematike).

Pri didaktiki matematike bomo stremeli k

usvojenosti naslednjih zmožnosti učitelja

matematike:

- Poznavanje aktualnega učnega načrta za

Knowledge and Understanding:

- Adoption of special mathematical, didactic,

pedagogical and psychological knowledge for

effective elementary classroom teaching,

presented in rubrics Contents and Objectives.

Transferable/Key Skills and other attributes:

- The obtained knowledge and skills are basis

for effective pedagogical class practice and for

subject Didactics of Secondary School

Mathematics.

At didactics of mathematics we will strive to

develop the following competences of

mathematics teacher:

- Knowing and understanding the current

mathematics syllabus and professional mastery

of contents and concepts of elementary school

mathematics in order to achieve learning

matematiko in profesionalno obvladovanje

matematičnih konceptov v osnovni šoli z

namenom oblikovanja takšnega učnega okolja,

ki učencem omogoča učinkovito izgradnjo

znanja ter njegovo trajnost, prenosljivost in

celovitost.

- Zmožnost oblikovanja učnih ciljev in

načrtovanja pouka matematike ter vrednotenja

znanja na podlagi ene od taksonomij znanj;

zmožnost vzpostavljanja vzpodbudnega učnega

okolja, ki pri učencu omogoča uravnotežen

razvoj konceptualnih, proceduralnih in

problemskih znanj.

- Zmožnost uporabe in kritičnega vrednotenja

obstoječih učnih gradiv in materialov.

- Obvladovanje različnih oblik pouka in metod

dela (vključno s kombiniranim izobraževanjem)

ter izbira takšnega poučevalnega pristopa, ki je

najbližje izbrani skupini učencev in učitelju

samemu.

- Poznavanje in uporaba psiholoških in

didaktičnih spoznanj različnih teorij učenja pri

načrtovanju in izvajanju pouka ter uporaba

raznolikih poučevalnih pristopov, ki se najbolje

prilegajo kognitivni zrelosti ter spoznavnim in

učnim stilom učencev.

- Zmožnost učinkovitega ugotavljanja znanja

učencev, samoevalvacije ter morebitnih

izboljšav ocenjevalnih pristopov.

- Zmožnost empatične medosebne

komunikacije skupaj z zmožnostjo pisnega in

ustnega izražanja v maternem jeziku.

- Zmožnost opismenjevanja učencev za

temeljno matematično pismenost.

- Zmožnost študija in upravljanja z viri v enem

od tujih jezikov.

- Zmožnost učinkovite uporabe informacijsko-

komunikacijske tehnologije pri pouku, sledenja

njenemu razvoju in kritičnega vrednotenja

njenega pomena za vzgojno-izobraževalni

proces.

- Zmožnost evalvacije lastnih poučevalnih

pristopov (metakognicija) ter povezovanja

spoznanj teorij učenja z učno prakso z

namenom vseživljenjskega osebnega razvoja na

poklicnem področju.

- Pozitiven odnos do življenja in smisel za

humor.

conditions which enable learners to acquire

knowledge (durability, transferability,

wholeness);

- Ability to form aims, to plan and to teach

Mathematics and evaluation of the knowledge

according to one of the taxonomies; ability to

provide an encouraging environment for

balanced development of learners’ conceptual,

procedural and problem-solving knowledge.

- Ability to use and evaluate existing math

study materials.

- Mastering different learning forms and

methods (also some newer approaches, eg. e-

learning) and adopting the best fitting approach

for students and teacher himself.

- Expertise in psychological and didactic

aspects of teaching theories in a way that the

teacher can use different teaching strategies

adjusted to different learning styles and age of

the learners.

- Familiarity with and use of different forms of

checking and evaluating knowledge together

with evaluation and necessary improvements.

- Skills of good interpersonal communication

together with skills of written and oral

expression in mother tongue.

- Ability to introduce language awareness-to

help learners to become mathematically literate.

- Ability to study and manage resources in one

of the foreign languages.

- Ability to work with information-

communicational technology, to follow its

development and autonomously evaluate the

meaning of different media and discoveries for

effective learning process.

- Ability to evaluate one’s own teaching and

learning methods (metacognition), connecting

theory of teaching with teaching experience to

ensure personal growth in the professional field.

- Positive attitude towards life and a sense of

humor.



Predavanje,

razgovor in diskusija,

demonstracija,

metoda pisnih in grafičnih del,

uporaba IKT,

reševanje problemov in preiskovanje,

delo z viri.

Oblike dela: individualno delo, skupinsko

delo (kooperativno učenje), timsko delo,

delo v dvojicah, frontalno delo.

Lecture,

conversation and discussion,

demonstration,

method of written and graphic products,

usage of ICT,

problem solving and investigation,

work with resources.

Learning forms: individual work,

teamwork, group learning (cooperative

learning), work in pair, frontal instruction.

Načini ocenjevanja:

Assessment:

Način (pisni izpit, ustno izpraševanje,

naloge, projekt)

Sprotno ocenjevanje:

- pisni test,

- mikronastop pred kolegi študenti,

- portfolij.

Vsaka izmed naštetih obveznosti mora

biti opravljena s pozitivno oceno.

Delež (v %) /

Weight (in %)

45%

10%

45%

Type (examination, oral, coursework,

project):

Ongoing assessment:

- written test,

- one pedagogical appearance in front of

the colleagues,

- portfolio.

Each of the listed obligations must have

positive grade.

Reference nosilca / Lecturer's

references:

1. LIPOVEC, Alenka, ANTOLIN, Darja, VAUPOTIČ, Alenka. Ulomki v vrtcu = Fractions in

kindergarten. Revija za elementarno izobraževanje, apr. 2012, letn. 5, št. 1, str. 67-77, ilustr.

[COBISS.SI-ID 19114248]

2. JERENEC, Simona, REPOLUSK, Samo, LIPOVEC, Alenka. Medpredmetno načrtovanje vsebin

pri pouku matematike v srednjih šolah = Intercurricular planning of learning contents by

instruction of mathematics in secondary schools. Mat. šol., 2011, letn. 17, št. 3/4, str. 71-89, graf.

prikazi. [COBISS.SI-ID 1739900]

3. ANTOLIN, Darja, LIPOVEC, Alenka. Uporaba spletne učilnice pri matematiki v okviru

izobraževanju bodočih učiteljev = The use of virtual classroom at mathematical course during pre-

service elementary teacher education = Korištenje virtualne učionice kod matematike u kontekstu

obrazovanja budućih učitelja razredne nastave. Metodički obzori, 2011, vol. 6, no. 13, str. 55-68.


4. LIPOVEC, Alenka, BERLIČ, Martina. Učenje in poučevanje matematike skozi kretnje =

Teaching and learning mathematics through gestures. Revija za elementarno izobraževanje, dec.

2010, letn. 3, št. 4, str. 25-39, ilustr. [COBISS.SI-ID 18059272]

5. LIPOVEC, Alenka, PANGRČIČ, Polonca. Elementary preservice teachers' change. Acta

didactica napocensia, 2008, vol. 1, no. 2, str. 31-36. [COBISS.SI-ID 16598280]

http://cobiss.izum.si/scripts/cobiss?command=DISPLAY&base=COBIB&RID=19114248






Predmet: Didaktika srednješolske matematike

Course title: Didactics of Secondary School Mathematics



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd

degree 2. 3.



Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

45

45

60 5


Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



- Učni načrti za matematiko v srednjih šolah.

- Izbrane vsebine srednješolske matematike.

Učne priprave.

- Motivacija pri pouku matematike v srednjih

šolah.

- Učna gradiva v srednjih šolah (učbeniki,

priročniki, didaktični materiali, knjige, internet

in e-učna gradiva …).

- Kombinirano e-izobraževanje pri pouku

matematike v srednji šoli.

- Individualizacija pri pouku matematike v

srednji šoli.

- Učenci z učnimi težavami v srednji šoli.

- Mathematics curricula in secondary schools.

- Selected contents of secondary school

mathematics Unit planning.

- Mathematical motivations in secondary

schools.

- Educational resources in secondary schools

(textbooks, handbooks, books, didactic

materials, internet and e-learning materials …).

- Blended e-learning at mathematics instruction

in secondary school.

- Individualisation in mathematics instruction in

secondary schools.

- Children with learning difficulties in

- Medpredmetne povezave in delo v timu v

srednjih šolah, vodenje krožka in mentorstvo

pri raziskovalnih nalogah.

- Preverjanje in ocenjevanje znanja v srednjih

šolah: oblike, sestava preizkusov, vrednotenje.

Splošna in poklicna matura v Sloveniji in

primeri zaključnih izpitov v tujini.

- Šolska zakonodaja, vodenje pedagoške

dokumentacije v srednji šoli, doba

pripravništva.

- Pedagoško delo v razredu v srednji šoli:

komunikacija, odnosi, vzgoja, razredništvo,

reševanje konfliktov.

- Nasilje v šoli.

- Umeščenost in vizija pedagoškega poklica v

družbi.

secondary school.

- School subjects connection and teamwork,

mathematics club, research themes and tutor's

role in secondary school.

- Assessment in secondary schools: forms,

exam composition, and grading. Leaving

examinations (finishing secondary schools) in

Slovenia and other countries.

- School legislation and pedagogical

documentation in secondary schools, teaching

probation.

- Pedagogical class management in secondary

school: communication, relations, education,

class teacher work, conflict solving.

- Violence in school.

- Meaning and the vision of teaching

profession in our society.


A. S. Posamentier [et al.], Teaching Secondary Mathematics: Techniques and Enrichment Units.

7th Edition, Pearson Prentice Hall, 2006.


Učni načrti za srednje šole.

Učbeniki in druga učna gradiva za srednje šole.

Reviji Matematika v šoli in Presek.



W. P. Berlinghof, Math through the Ages: A gentle history for teachers and others, Oxton House

Publishers, 2002.

D. Fomin [et al.], Mathematical Circles (Russian Experience), AMS, 1996.



H. A. Hauptman [et al.], 101 Great Ideas for Introducing Key Concepts in Mathematics: A

Resource for Secondary School Teachers, Corwin Press, 2001.

S. G. Krantz, How to Teach Mathematics, Second Edition, AMS, 1999.

R. B. Nelsen, Proofs without Words, MAA, 1993.

A. Orton, Learning Mathematics: Issues, Theory and Classroom Practice. Third Edition,

Continuum, 2004.

P. J. Palmer, Poučevati s srcem: raziskovanje notranjih pokrajin učiteljevega življenja, Educy,

2001.

M. Pašnik [et al.], Razrednik v osnovni in srednji šoli, ZRSŠ, 2002.



M. A. Sobel, Evan M. Maletsky, Teaching Mathematics: A Sourcebook of Aids, Activities and

Strategies, 3rd Edition, Allyn & Bacon, 1999.

F. Strmčnik [et al.], Didaktika, visokošolski učbenik, Visokošolsko središče Novo Mesto, 2003.

D. Vtič Tršinar, Iskalci biserov: priročnik za razredne ure, Društvo Za boljši svet, 2004.

Z. Usiskin [et al.], Mathematics for high school teachers: an advanced perspective, Pearson

Education (Prentice Hall), 2003.

http://www.e-um.si/

Z. Zalokar-Divjak, Vzgajati z ljubeznijo, Gora, 2000.

Pedagoška strokovna in znanstvena periodika.



- Korektno obvladovanje vsebin in konceptov

srednješolske matematike, podkrepljeno z

izkušnjami visokošolske matematike.

- Uporaba in preverjanje didaktičnih metod in

pedagoških načel v srednješolski učni praksi

(nastopi med letom, pedagoška praksa).

- Poznavanje motivacijskih pristopov in

strategij poučevanja pri pouku matematike v

srednjih šolah.

- Seznanitev z obstoječimi učnimi gradivi,

učnimi načrti, šolsko zakonodajo za srednje

šole ter s sistemskimi značilnostmi

srednješolskega izobraževanja.

- Priprava na učinkovito e-poučevanje

srednješolske matematike.

- Obvladovanje načinov preverjanja in

ocenjevanja znanja v srednjih šolah.

- Seznanitev z možnostmi oblikovanja

projektnih dni, z medpredmetnimi povezavami,

vodenjem krožka in mentorstvom pri

raziskovalnih nalogah v srednjih šolah.

- Seznanitev s smernicami za delo z učenci z

učnimi težavami v srednji šoli.

- Izdelava učnih gradiv za neposredno uporabo

pri pouku (študentov portfolij).

- Privzgajanje pozitivnega odnosa do

vseživljenjskega izpopolnjevanja v

pedagoškem poklicu (zavedanje o nujnosti le

tega in veselje nad njim).

- Mastering the contents and concepts of

secondary school mathematics with some

connections to university mathematics.

- Application and verification of educational

methods and principles in secondary class

practice (class appearances, pedagogical class

practice).

- Knowledge of motivational techniques and

learning strategies in secondary schools.

- Acquaintance with educational resources,

curricula, and school legislation in secondary

schools.

- Preparation on effective e-learning of

secondary school mathematics.

- Mastering the verification and assessment

methods in secondary schools.

- Discovering school subjects connections,

learning to work in team, leading mathematics

club and research themes in secondary school.

- Acquaintance with instructions for work with

children with learning difficulties in secondary

school.

- Preparation of own educational resources

(student's portfolio).

- Development of positive attitude to the

teaching profession and to the lifelong learning.




- usvojenost specialnih matematičnih,


potrebnih za učinkovito poučevanje v srednjih

šolah, ki so predstavljena med Vsebinami in

Cilji.




za uspešno izvajanje neposredne pedagoške

prakse, za prihodnje poučevanje v šoli in za

vseživljenjsko osebnostno izpopolnjevanje za


- Adoption of special mathematical, didactic,


effective secondary classroom teaching,




for effective pedagogical class practice, for

future class teaching and for permanent self-

eduaction.

At didactics of mathematics we will strive to

dobrega učitelja matematike ter vzgojitelja

otrok in mladostnikov.

Pri didaktiki matematike bomo stremeli k

usvojenosti naslednjih zmožnosti učitelja

matematike:



matematičnih konceptov v srednji šoli z

namenom oblikovanja takšnega učnega okolja,

ki učencem omogoča učinkovito izgradnjo

znanja ter njegovo trajnost, prenosljivost in

celovitost.







problemskih znanj.




dela (vključno s kombiniranim e-

izobraževanjem) ter izbira takšnega

poučevalnega pristopa, ki je najbližje izbrani

skupini učencev in učitelju samemu.

- Poznavanje in uporaba psiholoških in

didaktičnih spoznanj različnih teorij učenja pri

načrtovanju in izvajanju pouka ter uporaba

raznolikih poučevalnih pristopov, ki se najbolje

prilegajo kognitivni zrelosti ter spoznavnim in

učnim stilom učencev.

- Zmožnost holističnega pogleda na vzgojno-

izobraževalni proces ter medpredmetnega

povezovanja, načrtovanja in izvajanja pouka.

- Zmožnost učinkovitega ugotavljanja znanja

učencev, samoevalvacije ter morebitnih

izboljšav ocenjevalnih pristopov.





temeljno matematično pismenost.


od tujih jezikov.









of contents and concepts of secondary school

mathematics in order to achieve learning

conditions which enable learners to acquire

knowledge (durability, transferability,

wholeness);








study materials.





- Expertise in psychological and didactic

aspects of teaching theories in a way that the

teacher can use different teaching strategies

adjusted to different learning styles and age of

the learners.

- Ability to employ a holistic view of the

educational process and renew the forms of the

discipline with inter-subject connections.

- Familiarity with and use of different forms of

checking and evaluating knowledge together

with evaluation and necessary improvements.




- Ability to introduce language awareness-to

help learners to become mathematically literate.












- Positive attitude towards life and a sense of

humor.

proces.






- Pozitiven odnos do življenja in razvijanje

smisla za humor.



Predavanje,


demonstracija,


uporaba IKT,

reševanje problemskih nalog in

preiskovanje,

delo z viri.




Lecture,


demonstration,


usage of ICT,







Assessment:


naloge, projekt)

Sprotno ocenjevanje:

- pisni test,

- mikronastop pred kolegi študenti,

- portfolij.



Delež (v %) /

Weight (in %)

45%

10%

45%


project):

Ongoing assessment:

- written test,

- one pedagogical appearance in front of

the colleagues,

- portfolio.

Each of the listed obligations must have

positive grade.


references:













Predmet: Fraktali

Course title: Fractals



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester


dvopredmetni študij, 2. stopnja Modul D2 1. ali 2. 2. ali 4.


major 2nd

degree Module D2 1. or 2. 2. or 4.



Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

30 15 45 3

Nosilec predmeta / Lecturer: Dušan PAGON

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:


Vsebina:

Content (Syllabus outline):

Metričen prostor, različne vrste

podprostorov, prostor fraktalov.

Afine transformacije, skrčitve, sistemi

iterirajočih funkcij.

Teoretično in eksperimentalno določanje

dimenzije fraktala, Hausdorff-Bezikovičeva

dimenzija.

A metric space, different types of subspaces,

the space of fractals.

Affine transformations, contraction

mappings, systems of iterating functions.

The theoretical and experimental

determination of the fractal dimension,

Hausdorff-Besicovitch dimension.


Barnsley, M. F.: Fractals Everywhere. Academic Press, Boston (1988); Second edition (1993)

Barnsley, M. F.: Superfractals. Cambridge University Press, Cambridge (2006)

Devaney. R. L.: Chaos, Fractals and Dynamics - Computer Experiments in Dynamics, Addison-

Wesley (1990)

Edgar, G: Classics on Fractals. Westview Press, Boulder (1992)

Falconer, K. J.: The Geometry of Fractal Sets. Cambridge University Press,

Cambridge (1985)



Študenti se seznanijo s strukturo podprostora

fraktalov v metričnem prostoru in z osnovnimi

načini generiranja fraktalov (družine

iterirajočih preslikav). Spoznajo tudi definicijo

dimenzije fraktala.

Students get familiar with the structure of the

subset of fractals in a metric space and with the

main ways of generating fractals (iterated

functions systems). They also learn the

definition of the fractal dimension.




aktivno obvladanje strukture metričnega

prostora in prepoznavanje fraktalnih

podmnožic

teoretično in eksperimentalno določanje

dimenzije fraktalov


sposobnost generiranja fraktalov

izračun dimenzije fraktalne množice


active knowledge of metric space structure

and the ability to recognize its fractal subsets

theoretical and experimental ways for

finding the dimension of a fractal


the abbility to generate fractals

the calculation of fractal dimension



Predavanja

Seminarske vaje

Individualno delo

Lectures

Tutorial

Individual work


Assessment:


naloge, projekt)

seminarska naloga

pisni izpit – praktični del

ustni izpit – teoretični del

Delež (v %) /

Weight (in %)

20%

40%

40%


project):

coursework

written exam – practical part

oral exam – theoretical part


references:

1. PAGON, Dušan, REPOVŠ, Dušan, ZAICEV, Mikhail. On the codimension growth of simple

color Lie superalgebras. J. Lie theory, 2012, vol. 22, no. 2, str. 465-479.

http://www.heldermann.de/JLT/JLT22/JLT222/jlt22017.htm. [COBISS.SI-ID

16070233]

2. PAGON, Dušan. Simplified square equation in the quaternion algebra. International journal of

pure and applied mathematics, 2010, vol. 61, no. 2, str. 231-240. [COBISS.SI-ID 17718024]

3. GUTIK, Oleg, PAGON, Dušan, REPOVŠ, Dušan. On chains in H-closed topological pospaces.

Order (Dordr.), 2010, vol. 27, no. 1, str. 69-81. http://dx.doi.org/10.1007/s11083-010-

9140-x. [COBISS.SI-ID 15502169]

http://www.heldermann.de/JLT/JLT22/JLT222/jlt22017.htm



http://dx.doi.org/10.1007/s11083-010-9140-x

http://dx.doi.org/10.1007/s11083-010-9140-x


4. GUTIK, Oleg, PAGON, Dušan, REPOVŠ, Dušan. The continuity of the inversion and the

structure of maximal subgroups in countably compact topological semigroups. Acta math. Hung.,

2009, vol. 124, no. 3, str. 201-214. http://dx.doi.org/10.1007/s10474-009-8144-8, doi:

10.1007/s10474-009-8144-8. [COBISS.SI-ID 15212121]

5. PAGON, Dušan. The dynamics of selfsimilar sets generated by multibranching trees.

International journal of computational and numerical analysis and applications, 2004, vol. 6, no.

1, str. 65-76. [COBISS.SI-ID 14037081]

http://dx.doi.org/10.1007/s10474-009-8144-8

http://dx.doi.org/10.1007/s10474-009-8144-8




Predmet: Diferencialne enačbe v kontekstu

Course title: Differential equations in the context of use



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd

degree 1. 2.



Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

15

30

75 4

Nosilec predmeta / Lecturer: Blaž ZMAZEK

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



1. Osnovni pojmi: Konstrukcija NDE,

grafično reševanje, enačbe z ločljivima

spremenljivkama, naravna rast.

2. Navadne diferencialne enačbe: Osnovni tipi

NDE, parametrično reševanje, singularni

integrali, uporaba v geometriji in fiziki,

Modeliranje sprememb z diferencialnimi

enačbami.

3. Sistemi linearnih diferencialnih enačb,

1. Basics: Construction of ODE, graphical

solutions, equations with separable variables,

natural growth.

2. Ordinary differential equations: Basic types

of ODE, parametric solving, singular

integrals, applications in geometry and

physics, Modeling changes with differencial

equations.

3. Systems of linear differential equations,

linearna diferencialna enačba reda n.

4. Osnovni primeri in zgledi numeričnega

reševanja diferencialnih enačb.

5. Variacijski račun: Naloge variacijskega

računa, posplošitve.

linear differential equation of n-th order.

4. Basic cases and examples of numerically

solving differential equations.

5. Calculus of variations: Calculus of variations

tasks, generalizations.


E. Zakrajšek, Analiza III, DMFA Slovenije, Ljubljana, 1998.

F. Križanič, Navadne diferencialne enačbe in variacijski račun, DZS, Ljubljana 1974.

W. Kaplan, Advanced Calculusi, Fourth Edition. Addisson-Wesley Publishing Company, Redwood

City, California, 1991.



Spoznati navadne diferencialne enačbe, njihovo

uporabo in variacijski račun. To know ordinary differential equations, their

implementations and calculus of variations.




Poznavanje in razumevanje

diferencialnih enačb in metod za

njihovo reševanje.

Razumevanje in uporaba diferencialnih

enačb in variacijskega računa.


- Pridobljena znanja so podlaga za mnogo

predmetov v nadaljevanju študija.


Knowledge and understanding of

differential equations and methods of

their solution .

Be able to understand and implement

differential equations and calculus of

variations.


- The obtained knowledge is a basis for many

of the later subjects.



Predavanja

Laboratorijske in seminarske vaje

Individualno delo

Praktična demonstracija

Uporaba IKT

Lectures

Lab- and seminar exercises

Individual work

Practical demonstration

Applications of IT


Assessment:


naloge, projekt)

Pisni test – praktični del

Izpit (ustni) – teoretični del



Pozitivna ocena pri pisnem testu je

pogoj za pristop k izpitu.

Delež (v %) /

Weight (in %)

50%

50%


project):

Written test – practical part

Exam (oral) – theoretical part

Each of the mentioned commitments

must be assessed with a passing grade.

Passing grade of the written test is

required for taking the exam.


references:

1. PRNAVER, Katja, ZMAZEK, Blaž. On total chromatic number of direct product graphs. J.

appl. math. comput. (Internet), 2010, issue 1-2, vol. 33, str. 449-457.

http://dx.doi.org/10.1007/s12190-009-0296-8, doi: 10.1007/s12190-009-

0296-8. [COBISS.SI-ID 17523720]

2. ZMAZEK, Blaž, ŽEROVNIK, Janez. The Hosoya-Wiener polynomial of weighted trees. Croat.

chem. acta, 2007, vol. 80, 1, str. 75-80. [COBISS.SI-ID 11338518]

3. ZMAZEK, Blaž, ŽEROVNIK, Janez. Weak reconstruction of strong product graphs. Discrete

math.. [Print ed.], 2007, vol. 307, iss. 3-5, str. 641-649.

http://dx.doi.org/10.1016/j.disc.2006.07.013. [COBISS.SI-ID 14184025]

4. ZMAZEK, Blaž, ŽEROVNIK, Janez. On domination numbers of graph bundles. J. Appl. Math.

Comput., Int. J., 2006, vol. 22, no. 1/2, str. 39-48. [COBISS.SI-ID 10636822]

5. ZMAZEK, Blaž, ŽEROVNIK, Janez. On generalization of the Hosoya-Wiener polynomial.

MATCH Commun. Math. Comput. Chem. (Krag.), 2006, vol. 55, no. 2, str. 359-362. [COBISS.SI-

ID 13990745]

http://dx.doi.org/10.1007/s12190-009-0296-8

http://dx.doi.org/10.1007/s12190-009-0296-8

http://dx.doi.org/10.1007/s12190-009-0296-8



http://dx.doi.org/10.1016/j.disc.2006.07.013





OPIS PREDMETA / SUBJECT SPECIFICATIONPredmet: E-učenje Subject Title: E-learning


Izobraževalna matematika, dvopredmetni študij, 2. stopnja Univerzitetna koda predmeta / University

Predavanja Lectures

Seminar Seminar


Jeziki / Languages:

Predavanja / Lecture:Vaje / Tutorial:

Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti: Vsebina:

• Teoretične osnove - struktura e-učenja

• strokovni in didaktični principi e• tipi in pristopi e-učenja• Pristopi in orodja za pripravo e

gradiv. • Sistemi za vodenje in upravljanje e

učenja (LMS, LCMS, CMS..)• Standardi priprave in izvedbe u

gradiv za e-učenje (SCORM…)• Sodobne IKT in e-učenje.

Temeljni študijski viri / Textbooks:Osnovno / primary:

• Gerlič. I. Sodobna informacijska tehnologija v izobraževanju. DZS, Ljubljana, 2000.• W, Horton, K, Horton, E-• Palloff R. M., Pratt K. Building Online Learning Communities• Revije: Computer education, Monitor, Moj mikro, Presek

• E-študijska gradiva Cilji: • seznaniti se s teoretičnimi izhodiš• Pridobiti osnovne izkušnje z uporabo in

upravljanjem učnega okolja • spoznati osnovne principe e-• seznaniti se z orodji za izdelavo e• Pridobiti osnovne izkušnje uporabe elektronske

table • seznaniti se z drugimi novejšimi tehnologijami s




OPIS PREDMETA / SUBJECT SPECIFICATION

learning


Letnik


Sem. vaje Tutorial

Lab. Vaje Lab. Work



30

Ivan GERLIČ

Predavanja / Lecture: slovenski / Slovenian Vaje / Tutorial: slovenski / Slovenian


Prerequisites:


pojmi in pojmovna

čni principi e-učenja čenja

Pristopi in orodja za pripravo e-učnih

Sistemi za vodenje in upravljanje e-enja (LMS, LCMS, CMS..)

Standardi priprave in izvedbe učnih enje (SCORM…)

čenje.

• Theoretical principles conceptual structure of e

• Professional and didactic principles of elearning

• Authoring tools for e• LMS/CMS systems• Standards for making e• Advanced ICT technologies and e

Temeljni študijski viri / Textbooks:

. I. Sodobna informacijska tehnologija v izobraževanju. DZS, Ljubljana, 2000.learning Tools and Technologies, Wiley 2003

Building Online Learning Communities, Wiley 2007Computer education, Monitor, Moj mikro, Presek

Objectives: nimi izhodišči e-učenja

Pridobiti osnovne izkušnje z uporabo in -gradiv

seznaniti se z orodji za izdelavo e-gradiv uporabe elektronske

seznaniti se z drugimi novejšimi tehnologijami s

• to get acquaint with the learning

• to acquire basic experience of using the LMS/CMS systems

• to get acquaint with basic principles of ematerials

• mastering the authoring tools for the e• to acquire basic experience with electronic



Letnik Year

Semester Semester

1. ali 2. 2. ali 4.

Samost. delo Individ. work ECTS

45 3


Theoretical principles – concepts and conceptual structure of e-learning Professional and didactic principles of e-

Authoring tools for e-learning materials LMS/CMS systems Standards for making e-materials

technologies and e-learning

. I. Sodobna informacijska tehnologija v izobraževanju. DZS, Ljubljana, 2000.

, Wiley 2007

to get acquaint with the theoretical origins of e-

to acquire basic experience of using the

to get acquaint with basic principles of e-

mastering the authoring tools for the e-materials to acquire basic experience with electronic

področja e-učenja whiteboard • to get acquaint with other new technologies from

the e-learning field.



Znanje in razumevanje: • Strokovno-teoretično ozadje s področja e-

učenja • Prednosti in slabosti uporabe e-izobraževanja • Organizacija distribucije in prenosa znanja

Knowledge and Understanding: • Theoretical background of e-materials. • Advantages and disadvantages of using

e-materials. • Organization of knowledge distributions and

knowledge transmission. Prenesljive/ključne spretnosti in drugi atributi: • Uporaba znanj pri izdelavi kakovostnih e-učnih

gradiv • Organiziranje in vodenje projektov za izdelavo e-

učnih gradiv

Transferable/Key Skills and other attributes: • Knowledge for development of quality e-learning

materials. • Organizing and manage projects for produce e-

learning materials. Metode poučevanja in učenja:


• Predavanje, razgovor in diskusija, demonstracija, metoda pisnih in grafičnih del, uporaba IKT, reševanje problemskih nalog in preiskovanje,, oblike dela (individualno delo, skupinsko delo - kooperativno učenje, timsko delo, delo v dvojicah, frontalno delo), delo z viri.

• Lecture, conversation and discussion, demonstration, method of written and graphic products, usage of ICT, problem solving and investigation, learning forms (individual work, teamwork, group learning (cooperative learning, work in pair, frontal instruction), work with sources.

Načini ocenjevanja: Delež (v %) / Weight (in %)

Assessment:

– Portfolio s pisnimi izdelki (seminarska naloga, e-gradivo);

– opravljene vaje – ustni izpit

30% 20% 50%

– Portfolio with student's works (seminar work, e-learning material);

– completed didactics/laboratory work – oral exam

Materialni pogoji za izvedbo predmeta : Material conditions for subject realization Predavalnica, prenosni računalnik, LCD-projektor z interaktivno tablo projekcijsko platno, internet, računalniška učilnica.

Lecture hall, notebook, LCD-projector with e-table, projector screen, internet, computer classroom .

Obveznosti študentov: Students’ commitments: (pisni, ustni izpit, naloge, projekti) (written, oral examination, coursework, projects): ● Samostojni študij izbranih vsebin po predlaganih virih, učnih listih in e-gradivih; • Opravljeno preverjanje znanja navedeno pod

»Načini ocenjevanja«.

● Self-study presented in selected topics, textbooks and E-materials.

• Finished checking of knowledge presented in rubrics “Assessment”


OPIS PREDMETA / SUBJECT SPECIFICATIONPredmet: Kognicija in osebnost v procesu učenjaSubject Title: Cognition and personality in the learning process


Izobraževalna matematika, dvopredmetni študij, 2. stopnja Univerzitetna koda predmeta / University subject

Predavanja Lectures

Seminar Seminar


Jeziki / Languages:

Predavanja / Vaje / Tutorial:

Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti: Vsebina:

• Razvoj možganov in učenje. • Različne strategije in načini učenja; kognitivni in učni

stili; individualne razlike. • Koncept »učenje učenja«. • Spomin: struktura, delovanje in razvoj; implikacije za

učenje in poučevanje. • Mišljenje: reševanje problemov, presojanje in

odločanje, metakognicija. • Inteligentnost: modeli in teorije intelekta;

ustvarjalnost; modrost; koncept nadarjenosti.• Teorije kognitivnega razvoja ter njihove implikacije za

učenje.

• Struktura, dinamika in razvoj osebnosti.• Samopodoba, samoregulacija.• Motivi in emocije; storilnostna

emocije; vpliv emocij na kognitivne procese.• Osebnost in psihične obremenitve (frustracije,

konflikti, stres); soočanje s stresom.• Osebnost in učna uspešnost.

Temeljni študijski viri / Textbooks: Byrnes, J.P. (2000). Cognitive development and learning in instructional contexts. Papalia, D. E., Olds, S. W. & Feldman, R. D. (2003). Otrokov svet. Ljubljana: EducyMusek, J. (2006). Psihološke dimenzije osebnosti. Ljubljana, FFSternberg, R.J. & Zhang, L. (2001). Perspectives on thinking, learning, and cognitive styles. Associates Blakemore, S.J. & Frith, U. (2005). The learning brain: Lessons for education. Blackwell Publishing




OPIS PREDMETA / SUBJECT SPECIFICATION Kognicija in osebnost v procesu učenja

and personality in the learning process


Letnik


Sem. vaje Tutorial

Lab. Vaje Lab. Work



15

Karin BAKRAČEVIČ VUKMAN



Prerequisites:


strategije in načini učenja; kognitivni in učni

Spomin: struktura, delovanje in razvoj; implikacije za

Mišljenje: reševanje problemov, presojanje in

igentnost: modeli in teorije intelekta; ustvarjalnost; modrost; koncept nadarjenosti. Teorije kognitivnega razvoja ter njihove implikacije za

Struktura, dinamika in razvoj osebnosti. Samopodoba, samoregulacija. Motivi in emocije; storilnostna motivacija; učne emocije; vpliv emocij na kognitivne procese. Osebnost in psihične obremenitve (frustracije, konflikti, stres); soočanje s stresom.

• Brain development and learning.• Different strategies and ways of learning; cog

learning styles, individual differences.• “Learning to learn” concept.• Memory: structure and development; instructional

implications. • Thinking: problem solving, judgment and decision

making, metacognition.• Intelligence: models and theories of int

wisdom; concept of giftedness.• Theories of cognitive development and learning.

• Structure, dynamics and development of personality.• Self-concept and self-regulation.• Motivation and emotions; learning motivation;

emotions in learning; incognitive processes.

• Frustration, conflict and stress; coping strategies.• Personality and school performance.


Byrnes, J.P. (2000). Cognitive development and learning in instructional contexts. Allyn & Bacon Papalia, D. E., Olds, S. W. & Feldman, R. D. (2003). Otrokov svet. Ljubljana: Educy Musek, J. (2006). Psihološke dimenzije osebnosti. Ljubljana, FF Sternberg, R.J. & Zhang, L. (2001). Perspectives on thinking, learning, and cognitive styles. Mahwah: Lawrence Erlbaum

Blakemore, S.J. & Frith, U. (2005). The learning brain: Lessons for education. Blackwell Publishing



Letnik Year

Semester Semester

1. ali 2. 2. ali 4.


ECTS

45 3


Brain development and learning. Different strategies and ways of learning; cognitive and learning styles, individual differences. “Learning to learn” concept. Memory: structure and development; instructional

Thinking: problem solving, judgment and decision making, metacognition. Intelligence: models and theories of intellect; creativity; wisdom; concept of giftedness. Theories of cognitive development and learning.

Structure, dynamics and development of personality. regulation.

Motivation and emotions; learning motivation; emotions in learning; influence of emotions on

Frustration, conflict and stress; coping strategies. Personality and school performance.

Mahwah: Lawrence Erlbaum

Blakemore, S.J. & Frith, U. (2005). The learning brain: Lessons for education. Blackwell Publishing

Cilji: Objectives: Študentje in študentke: • Poglobljeno spoznajo kognitivne in osebnostne vidike

človekove narave in njihovo interakcijo v procesu učenja; • spoznajo in razumejo pomen razvojnih in individualnih

razlik pri učenju; • obvladajo pomembne teorije in modele kognicije in

osebnosti ter novejše izsledke na področju strukture, dinamike in razvoja osebnosti ter kognitivnega razvoja.

Students: • get acquainted with cognitive and personality

characteristics of individuals in the process of learning, and their interaction;

• become able to understand developmental and individual differences in learning;

• become able to understand theories and models of cognition and personality and get familiar with new findings in the field of personality and cognitive development.



Znanje in razumevanje: Poznavanje in poglobljeno razumevanje kognitivnih in osebnostnih, motivacijskih in čustvenih značilnosti posameznika, ki vplivajo na način in uspešnost učenja – tako z razvojnega vidika, kot s stališča individualnih razlik.

Knowledge and Understanding: familiarity with and understanding of cognitive, personality, motivational and emotional characteristics of individuals, which influence ways and success of learning – from the developmental, as well as “individual differences” point of view.

Prenesljive/ključne spretnosti in drugi atributi: Sposobnost kritične presoje in uporabe znanstvenih in strokovnih spoznanj o kogniciji in osebnosti v procesu učenja na področju drugih ved ter v praksi.

Transferable/Key Skills and other attributes: ability to critically judge and apply scientific and professional findings about cognitive and personality characteristics in learning process in other fields and in the praxis.



• Predavanja • Seminarske vaje • Individualno delo

• Lectures • Excersises • Individual work


Assessment:

• seminarska naloga • pisni izpit

30 70

• coursework

• written examination


• Predavalnica- multimedijsko opremljena

• Lecture hall with multimedia equipment


• seminarska naloga • pisni izpit

• coursework

• written examination


OPIS PREDMETA / SUBJECT SPECIFICATIONPredmet: Magistrsko delo in magistrski izpitSubject Title: Master work


Izobraževalna matematika dvopredmetni študij, 2. stopnja


Predavanja Lectures

Seminar Seminar

Nosilec predmeta / Lecturer:

Jeziki / Languages:

Predavanja / Vaje / Tutorial:

Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti: Vsebina: 1. Ovitek 2. Notranja naslovna stran 3. Izjava kandidata o avtorstvu 4. Zahvala 5. Izvleček magistrskega dela v slovenskem in angleškem jeziku in ključne besede 6. Pregled vsebine - kazalo 7.Uvod 8Jedro magistrskega dela 9. Zaključek 10. Literatura in viri 11. Priloge (po potrebi) Temeljni študijski viri / Textbooks:Osnovno / primary: • Katz, M. J., 2007: From research to manuscript. A guide to scientific writting. Springer, 152 str.

Cilji: Magistrsko delo je pisni dokument, s katerim študent dokaže sposobnost uporabe teoretiznanj in v praksi pridobljenih izkušenj za rešitev problema, ki si ga je izbral za magistrsko delo. Priprava na magistrski izpit pomeni študentovo zbrano ponovitev ključnih matem


Fakulteta za naravoslovje in matematiko / Faculty of Natural Sciences and

MathematicsOPIS PREDMETA / SUBJECT SPECIFICATION

Magistrsko delo in magistrski izpit Master work and master exam

Študijska smer

Study field Letnik

dvopredmetni študij, 2. stopnja


Sem. vaje Tutorial

Lab. vaje Lab. work



Habilitirani visokošolski učitelji matematike

Predavanja / Lecture: slovenski / Slovenian Vaje / Tutorial:


Prerequisites:


v slovenskem in ne besede

1. Cover 2. Inside title page 3. Statement of the candidate 4. Acknowledgement 5. Abstract of the master work in Slovene and English and key words 6. Review of the subject – 7. Introduction 8.The core of the master thesis9. Summary 10. Literature and sources11. Supplements (if needed)


Katz, M. J., 2007: From research to manuscript. A guide to scientific writting. Springer, 152 str.

Objectives:

je pisni dokument, s katerim študent dokaže sposobnost uporabe teoretičnih znanj in v praksi pridobljenih izkušenj za rešitev

za magistrsko delo..

Priprava na magistrski izpit pomeni študentovo nih matematičnih vsebin, ki

The master work is a written document by means of which the student proves ability to use the theoretical knowledge and in his experiences achieved in practical work resolving a problem chof his master work. Preparing for the masters

Fakulteta za naravoslovje in matematiko / Faculty of Natural Sciences and

Mathematics

Letnik Year

Semester Semester

2 4


ECTS

240 8


3. Statement of the candidate

work in Slovene and

index

8.The core of the master thesis

sources . Supplements (if needed)

Katz, M. J., 2007: From research to manuscript. A guide to scientific writting. Springer, 152 str.

is a written document by means of which the student proves ability to use the theoretical knowledge and in his experiences achieved in practical work resolving a problem chosen the theme

exam provides a student

so za učitelja matematike še posebej pomembna. a repetition of key mathematical content particularly important for mathematics teachers.



Znanje in razumevanje: Znanje širšega strokovnega področja, v katerega spada tema magistrskega dela in ožje znanje razumevanje strokovnih pojmov za celovit prikaz magistrskega dela. Poudarek je na teoretičnih in praktičnih znanjih ter priredbah metodologij zajemanja, obdelovanja in prikazovanja podatkov.

Knowledge and Understanding: Knowledge of the broader professional field to which belongs the master work, and special knowledge of the glossary as well as the creation of new conceptions. The emphasis is on the theoretical and practical skills in using and adjustment of methods of collecting, processing and presenting data.

Prenesljive/ključne spretnosti in drugi atributi: Strokovno zapisovanje in izražanje vsebine, obvladanje reševanja problemov, suverena predstavitev ključnih spoznanj in spretnost argumentiranja.

Transferable/Key Skills and other attributes: Documenting and expressing the subject, mastering the solving of the scientific problems, independent presentation of the key conclusions and ability in arguing.



Mentor • Na konzultacijah preverja vsebinski in strukturni

vidik magistrskega dela; • Usmerja kandidata pri predstavitvi dela in na

verjetna okvirna vprašanja pri zagovoru.

The supervisor: • Checks and verifies the thematic and structural

aspect of master work in consultations; • Directs the student in preparation of an

appropriate presentation of his work and to the most probable qustions with respect to the commision of defense.


Assessment:

• Magistrski izpit • Magistrsko delo; • Zagovor.

20% 60% 20%

• Mastrer exam • Master work; • Presentation.

Materialni pogoji za izvedbo predmeta : Material conditions for subject realization Seminarska soba z osnovnimi audiovizualnimi pripomočki.

Study room with the multimedia equipment


• Magistrski izpit • Izdelava magistrskega dela; • Zagovor pred komisijo.

• Master exam • Production of the master work; • Its defense in the presence of the boaer of

examiners.


Predmet: Matematične krivulje

Course title: Mathematical Curves



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd




Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

15 15 15

45 3

Nosilec predmeta / Lecturer: Iztok BANIČ

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:

Opravljen izpit iz Osnov analize in Analize Exam in Basic Analysis, Analysis


Krivulje v ravnini. Sistematizcija krivulj.

Parametrizacija, tangenta, ločna dolžina.

Primeri ravninskih krivulj: stožnice, krivulje

tretje stopnje, krivulje četrte stopnje, cikloidne

krivulje, transcendentne krivulje.

Singularna točka. Ogrinjača.

Planar curves. Systematization of curves.

Parametrization, tangent, arc length.

Examples of planar curves: curves of degree 2,

curves of degree 3, curves of degree 4, cyclic

curves, transcendental curves.

Singular point. Hull.

Šestnajsti Hilbertov problem.

Hilbert's sixteenth problem.


M. Razpet: Ravninske krivulje. Ljubljana: Knjižnica sigma, DMFA, 1998.

I. Vidav: Eliptične krivulje in eliptične funkcije. Ljubljana: DMFA, 1991.

M. Dobovišek: Rešene naloge iz analize II. Ljubljana: DMFA, 1996.

B. Hvala: Zbirka izpitnih nalog iz analize. Ljubljana: DMFA, 1996.

D. Benkovič: Analiza II (dodatna gradiva na spletu)

http://matematika-racunalnistvo.fnm.uni-mb.si/dodatna_gradiva/analiza_II.html



Poglobiti znanje glavnih dejstvev o krivuljah.

Poglobiti znanje o ravninskih krivuljah.

Poglobiti znanje o konstrukcijah krivulj in

njihovem zgodovinskem razvoju.

Deepening the knowledge of basic facts about

curves.

Deepening the knowledge of planar curves.

Deepening the knowledge of constructions of

curves and their historical development.




Študent poglobi znanje o osnovah diferencialne

geometrije krivulj v ravnini.

Študent poglobi znanje o ravninskih krivuljah,

njihovih lastnostih in konstrukcijah.


Prenos znanja v zvezi s krivuljami na druga

področja (geografija, astronomija, fizika)


Deepening the knowledge of the basic facts

about differential geometry of curves in plane.

Deepening the knowledge of the concepts of

planar curves, their properties and constructions.


Knowledge transfer of the concepts,

connected with curves into other fields

(geography, astronomy, physics).



Predavanja

Seminarji

Seminarske vaje

Individualno delo

Lectures

Seminars

Tutorial

Individual work


Assessment:


naloge, projekt)



Seminarska naloga

Delež (v %) /

Weight (in %)

40%

40%

20%


project):



Seminar










references:

1. BANIČ, Iztok, ČREPNJAK, Matevž, MERHAR, Matej, MILUTINOVIĆ, Uroš, SOVIČ, Tina.

Ważewski's universal dendrite as an inverse limit with one set-valued bonding function. Preprint

series, 2012, vol. 50, št. 1169, str. 1-33. http://www.imfm.si/preprinti/PDF/01169.pdf.


2. BANIČ, Iztok, ČREPNJAK, Matevž, MERHAR, Matej, MILUTINOVIĆ, Uroš. Paths through

inverse limits. Topol. appl.. [Print ed.], 2011, vol. 158, iss. 9, str. 1099-1112.

http://dx.doi.org/10.1016/j.topol.2011.03.001. [COBISS.SI-ID 18474504]

3. BANIČ, Iztok, ŽEROVNIK, Janez. Wide diameter of Cartesian graph bundles. Discrete math..

[Print ed.], str. 1697-1701. http://dx.doi.org/10.1016/j.disc.2009.11.024, doi:

10.1016/j.disc.2009.11.024. [COBISS.SI-ID 17543176]

tipologija 1.08 -> 1.01

4. BANIČ, Iztok, ČREPNJAK, Matevž, MERHAR, Matej, MILUTINOVIĆ, Uroš. Limits of



5. BANIČ, Iztok, ERVEŠ, Rija, ŽEROVNIK, Janez. Edge, vertex and mixed fault diameters. Adv.

appl. math., 2009, vol. 43, iss. 3, str. 231-238.

http://dx.doi.org/10.1016/j.aam.2009.01.005, doi:

10.1016/j.aam.2009.01.005. [COBISS.SI-ID 13396502]

http://www.imfm.si/preprinti/PDF/01169.pdf


http://dx.doi.org/10.1016/j.topol.2011.03.001







http://dx.doi.org/10.1016/j.aam.2009.01.005




Predmet: Novejši pristopi k poučevanju matematike

Course title: Recent Methods of Teaching Mathematics



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd




Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

15

30 45 3

Nosilec predmeta / Lecturer: Blaž ZMAZEK

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



- Teorije učenja in pouk matematike

(behaviorizem, socialni konstruktivizem …).

- Novejše metode poučevanja matematike z

uporabo učnih tehnologij (IKT): npr.

sodelovalno učenje, e-učenje, matematična

preiskovanja in reševanje problemov, uporaba

računalniških matematičnih programov.

- Internet, elektronska učna gradiva in

predstavitvene tehnologije pri pouku

- Theories of learning and mathematics

education (behaviourism, social-constructivism

…).

- Recent methods of learning mathematics with

learning technologies (ICT): e. g. cooperative

learning, e-learning, mathematical investigations

and problem solving, mathematical computer

programs.

- Internet, e-learning materials and

matematike. -

- Matematični računalniški programi pri pouku

matematike.

- Izdelava e-učnih gradiv.

- Matematika v kontekstu (npr. matematično

modeliranje) pri pouku matematike z uporabo

računalnika (osnove)

- Delo z učenci z učnimi težavami in z

nadarjenimi učenci ob pomoči učnih tehnologij

(IKT).

- Znanstveno-raziskovalno delo pri didaktiki

matematike.

representational technologies at mathematics

instruction. Didactics of e-learning.

- Mathematical computer programs at

mathematics instruction.

- Creation of e-learning materials.

- Mathematics in context at mathematics

instruction (e.g. mathematical modelling) with

computer (basics).

- Scaffolding children with learning difficulties

and gifted children with learning technologies

(ICT).

- Scientific research in mathematics education.


A. Orton, Learning Mathematics: Issues, Theory and Classroom Practice, Third Edition,

Continuum, 2004.



J. A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, Sixth


Spletni portal E-um: www.e-um.si in drugi internetni portali za učenje matematike.

Matematični učni računalniški programi (za dinamično geometrijo, obdelavo podatkov, simbolno

računanje, risanje grafov funkcij …).


J. A. Ameis, Mathematics on the Internet: a resourse for K-12 teachers, Third edition. Pearson

Prentice Hall, 2006.

R. C. Clark, R. E. Mayer, e-Learning and the Science of Instruction, Second Edition, Pfeiffer,

2008.



S. G. Krantz, How to Teach Mathematics, Second Edition, AMS, 1999.




M. A. Sobel, E. M. Maletsky, Teaching Mathematics: A Sourcebook of Aids, Activities and




Revije: Journal for Research in Mathematics Education, Educational Studies in Mathematics,

Logika & razvedrilna matematika, Matematičko-fizički list, Matematika i škola, Matematika u

škole, Mathematics Teaching, Micro Math, Obzornik za matematiko in fiziko, Poučak, Teaching

Children Mathematics, Mathematics Teacher, On-Math, Matematika v šoli, Presek …



- Seznanitev s teorijami učenja s poudarkom

na socialnem konstruktivizmu in uporabi

spoznanj v neposredni učni praksi.

- Preizkušanje novejših metod poučevanja

- Acquaintance with learning theories (social-

constructivism) and their applications in

classroom practice.

- Testing different methods of learning

http://www.e-um.si/

matematike z uporabo različnih učnih

tehnologij (IKT).

- Uporaba interneta in izdelava elektronskih

učnih gradiv pri pouku matematike.

Laboratorijske vaje v računalniški učilnici z

uporabo različnih predstavitvenih tehnologij.

- Poznavanje in uporaba matematičnih učnih

programov pri pouku matematike: za

dinamično geometrijo, simbolno računanje,

risanje grafov funkcij, obdelavo podatkov itd.

Laboratorijske vaje v računalniški učilnici.

- Obravnava različnih možnosti dela z učenci z

učnimi težavami in z nadarjenimi učenci ob

podpori učnih tehnologij (IKT).

- Seznanitev z znanstvenimi članki pri

didaktiki matematike: razumevanje vsebine in

predstavitev članka.

mathematics with ICT.

- Using internet and creating e-learning

materials at mathematics instruction. Lab. work

in computer classroom with different

representational technologies.

- Teaching with mathematical programs at

mathematics instruction: for dynamic geometry,

symbolic computations, ploting functions,

statistics, etc. Lab. work in computer classroom.

- Engaging children with learning difficulties

and gifted children with help of ICT (scaffolding

possibilities).

- Understanding and presenting the content and

the meaning of one scientific article (about

mathematics education).




- usvojenost zahtevnejših matematičnih,


potrebnih za učinkovito poučevanje, ki so

predstavljena med Vsebinami in Cilji.




za uspešno izvajanje pedagoške prakse.

Pri tem predmetu bomo stremeli k usvojenosti

naslednjih zmožnosti učitelja matematike:

- Profesionalno obvladovanje matematičnih

konceptov z namenom oblikovanja takšnega

učnega okolja, ki učencem omogoča učinkovito

izgradnjo znanja ter njegovo trajnost,

prenosljivost in celovitost.







problemskih znanj.

- Razvijanje algoritmičnega mišljenja.


obstoječih elektronskih učnih gradiv in

tehnologij.


dela (vključno s kombiniranim izobraževanjem)


- Adoption of advanced mathematical, didactic,


effective classroom teaching, presented in

rubrics Contents and Objectives.



for effective pedagogical class practice.

We will strive to develop the following

competences of mathematics teacher:

- Professional mastery of contents and concepts

of school mathematics in order to achieve

learning conditions which enable learners to

acquire knowledge (durability, transferability,

wholeness);







- Development of algorithmic thinking.

- Ability to use and evaluate existing e-learning

materials and technologies.


methods (also some newer approaches, e.g. e-



- Ability to help learners to become

ter izbira takšnega poučevalnega pristopa, ki je

najbližje izbrani skupini učencev in učitelju

samemu.


temeljno matematično in digitalno pismenost.


od tujih jezikov.





proces.






mathematically and digitally literate.



- Ability to work with learning technologies

(ICT), to follow theirs development and

autonomously evaluate the meaning of different

media and discoveries for effective learning

process.







Predavanje,


demonstracija,


uporaba IKT,

reševanje problemskih nalog in

preiskovanje,

delo z viri.




Lecture,


demonstration,


usage of ICT,







Assessment:


naloge, projekt)

Teoretični del:

- predstavitev članka.

Praktični del:

- portfolij laboratorijskih vaj.

Delež (v %) /

Weight (in %)

opravil/passed,

100 %.


project):

Theoretical part:

- representation of the article.

Practical part:

- portfolio of laboratory work.


references:

1. PRNAVER, Katja, ZMAZEK, Blaž. On total chromatic number of direct product graphs. J.

appl. math. comput. (Internet), 2010, issue 1-2, vol. 33, str. 449-457.

http://dx.doi.org/10.1007/s12190-009-0296-8, doi: 10.1007/s12190-009-

0296-8. [COBISS.SI-ID 17523720]

2. ZMAZEK, Blaž, ŽEROVNIK, Janez. The Hosoya-Wiener polynomial of weighted trees. Croat.

http://dx.doi.org/10.1007/s12190-009-0296-8

http://dx.doi.org/10.1007/s12190-009-0296-8

http://dx.doi.org/10.1007/s12190-009-0296-8


chem. acta, 2007, vol. 80, 1, str. 75-80. [COBISS.SI-ID 11338518]

3. ZMAZEK, Blaž, ŽEROVNIK, Janez. Weak reconstruction of strong product graphs. Discrete

math.. [Print ed.], 2007, vol. 307, iss. 3-5, str. 641-649.

http://dx.doi.org/10.1016/j.disc.2006.07.013. [COBISS.SI-ID 14184025]

4. ZMAZEK, Blaž, ŽEROVNIK, Janez. On domination numbers of graph bundles. J. Appl. Math.

Comput., Int. J., 2006, vol. 22, no. 1/2, str. 39-48. [COBISS.SI-ID 10636822]

5. ZMAZEK, Blaž, ŽEROVNIK, Janez. On generalization of the Hosoya-Wiener polynomial.

MATCH Commun. Math. Comput. Chem. (Krag.), 2006, vol. 55, no. 2, str. 359-362. [COBISS.SI-

ID 13990745]







Predmet: Optimizacijske metode

Course title: Optimization methods



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd




Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

30

15 45 3

Nosilec predmeta / Lecturer: Drago BOKAL

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



Optimizacijska naloga. Matematični

program. Vrste matematičnih programov.

Mešani celoštevilski matematični program.

Hevristike in metahevristike. Lokalna

optimizacija. Iskanje brez vračanja.

Simulirano ohlajanje. Genetski algoritmi.

Linearni program. Dualni linearni program.

Osnovni izrek dualnosti. Simpleksna

metoda.

Optimization problem. Mathematical

program. Types of mathematical programs.

Mixed integer mathematical program.

Heuristics and metaheuristics. Local

optimization. Tabu search. Simulated

annealing. Genetic algorithms.

Linear program. Dual linear program.

Fundamental theorem of duality. Simplex

method.

Primeri uporabe. Applications of the above methods.


J.Žerovnik: Osnove teorije grafov in diskretne optimizacije, (druga izdaja), Fakulteta za strojništvo,

Maribor 2005. B. Korte, J. Vygen: Combinatorial Optimization, Theory and Algorithms, Springer,

Berlin 2000.

D. Cvetkovič, V. Kovačević-Vujčić: Kombinatorna optimizacija, DOPIS Beograd 1996.

E. Zakrajšek: Matematično modeliranje, DMFA, Ljubljana 2004.



Pridobiti znanje in razumevanje osnovnih

optimizacijskih metod.

Razviti sposobnost reševanja realnih

problemov z uporabo osnovnih

optimizacijskih metod.

Obtain the knowledge and understanding of

the basic optimization methods.

Develop the ability to apply basic

optimization methods to real life problems.




Osnovnih hevrističnih metod (lokalno

vzpenjanje, tabu iskanje, simulirano

ohlajanje, genetski algoritmi).

Linearnega programiranja in simpleksne

metode.


- Formuliranje problema kot optimizacijske

naloge, izbor ustrezne metode za njeno

reševanje ter reševanje z ustreznimi orodji.


Basic heuristic methods (local hillclimbing,

tabu search, simulated annealing, genetic

algorithms).

Linear programming and simplex method.


Formulating a real life problem as an

abstract optimization problem, selecting

a suitable method to obtain a solution,

applying a suitable solver with the

chosen method.



Predavanja

Laboratorijske vaje v računalniški učilnici.

Izdelava seminarske naloge.

Lectures

Laboratory excersises in computer

classroom,

Seminar project thesis


Assessment:


naloge, projekt)

Seminarska naloga





Delež (v %) /

Weight (in %)

50%

50%


project):

Seminar project work










references:

1. BOKAL, Drago, BREŠAR, Boštjan, JEREBIC, Janja. A generalization of Hungarian method

and Hall's theorem with applications in wireless sensor networks. Discrete appl. math.. [Print ed.],

2012, vol. 160, iss. 4-5, str. 460-470. http://dx.doi.org/10.1016/j.dam.2011.11.007. [COBISS.SI-ID

16191577]

2. KOS, Andrej, PRISTOV, Damijan, SEDLAR, Urban, STERLE, Janez, VOLK, Mojca,

VIDONJA, Tomaž, BAJEC, Marko, BOKAL, Drago, BEŠTER, Janez. Open and scalable IoT

platform and its applications for real time access line monitoring and alarm correlation. Lect. notes

comput. sci., str. 27-38, ilustr. [COBISS.SI-ID 9370964]


3. BOKAL, Drago, DEVOS, Matt, KLAVŽAR, Sandi, MIMOTO, Aki, MOOERS, Arne Ø.

Computing quadratic entropy in evolutionary trees. Comput. math. appl. (1987). [Print ed.], 2011,

vol. 62, no. 10, str. 3821-3828. http://dx.doi.org/10.1016/j.camwa.2011.09.030. [COBISS.SI-ID

16059481]

4. ŽUNKO, Matjaž, BOKAL, Drago, JAGRIČ, Timotej. Testiranje modelov VaR v izjemnih

okoliščinah. IB rev. (Ljubl., Tisk. izd.). [Tiskana izd.], 2011, letn. 45, št. 3, str. 57-67, tabele, graf.


5. BOKAL, Drago, CZABARKA, Éva, SZÉKELY, László, VRT'O, Imrich. General lower bounds

for the minor crossing number of graphs. Discrete comput. geom., 2010, vol. 44, no. 2, str. 463-

483. http://dx.doi.org/10.1007/s00454-010-9245-4. [COBISS.SI-ID 15636057]

http://dx.doi.org/10.1016/j.dam.2011.11.007



http://dx.doi.org/10.1016/j.camwa.2011.09.030



http://dx.doi.org/10.1007/s00454-010-9245-4



Predmet: Osnove teorije grafov

Course title: Basic graph theory



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd

degree 2. 3.



Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

15 15 15

45 3

Nosilec predmeta / Lecturer: Boštjan BREŠAR

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



Osnovni pojmi in primeri: graf, stopnja,

izomorfizem grafov, podgrafi, povezanost,

poti in cikli, dvodelni grafi, drevesa, tetivni

grafi.

Prirejanja: prirejanja in pokritja, prirejanja

v dvodelnih grafih, prirejanja v splošnih

grafih, Hallov poročni izrek.

Ravninski grafi: risbe grafov, zemljevidi,

dualni graf, Eulerjeva formula.

Basic concepts and examples: graph, degree,

graph isomorphism, subgraphs, paths and

cycles, trees, bipartite graphs, chordal

graphs.

Matchings: matchings and covers, matchings

in bipartite graphs, matchings in general

graphs, Hall’s marrriage theorem,

Planar graphs: graph drawings, maps, graph

dual, Euler’s formula,

Barvanja grafov: barvanja vozlišč,

Brooksov izrek, barvanja povezav, barvanja

zemljevidov, izrek 4 barv, sodobni koncepti

barvanj.

Eulerjevi in Hamiltonovi grafi: problem

Konnigsbergških mostov in Eulerjev izrek,

Fleuryjev postopek, Hamiltonovi cikli in

poti, potrebni in zadostni pogoji za

hamiltonskost, usmerjeni grafi in turnirji,

problem trgovskega potnika, problem

kitajskega poštarja.

Del snovi bo prilagojen interesom in pobudam

študentov ali sproti se porajajočim trendom v

teoriji grafov in razvedrilni diskretni

matematiki.

Colourings of graphs: vertex colourings,

Brooks’ theorem, edge colourings, map

colourings, 4 colour theorem, modern

colouring concepts.

Eulerian and hamiltonian graphs: bridges of

Konnigsberg problem and Euler’s theorem,

Fleury’s procedure, Hamilton cycles and

paths, necessary and sufficient conditions for

hamiltonicity, digraphs and tournaments,

traveling salesman problem, Chinese

postman problem.

A part of the contents will be adjusted to

interests and initiative of students or to newly

appearing trends in graph theory and

recreational discrete mathematics.


D.B. West: Introduction to Graph Theory, Prentice Hall, New Jersey, 2001.

R. J. Wilson, J. J. Watkins: Uvod v teorijo grafov, DMFA, Ljubljana, 1997.

R. J. Wilson: Introduction to graph theory, Longman, New York, 1987.

J.A. Bondy and U.S.R. Murty: Graph Theory, Springer, London, 2008.



Cilj predmeta je seznaniti študente z

najpomembnejšimi koncepti teorije grafov in

njene uporabe. V okviru seminarja se študent

samostojno nauči izbrano snov in pripravi

seminarsko predstavitev.

The objective of this course is to acquaint

students with the most important concepts in

graph theory and its application. For the seminar

a student self-reliantly learns a chosen topic and

prepares a presentation.




Po zaključku tega predmeta bo študent

sposoben izkazati razumevanje osnov teorije

grafov, reševati probleme, ki se v teoriji grafov

pojavljajo ter pridobljeno znanje uporabljati.


Spretnosti komuniciranja: ustno izražanje in

javni nastop pri seminarju, ustno in pisno

izražanje na izpitih

Reševanje problemov: reševanje

kombinatoričnih in ekstremalnih problemov

v teoriji grafov.


On completion of this course the student will be

able to demonstrate understanding of graph

theory basics, solve problems that appear in

graph theory and apply the obtained knowledge.


Communication skills: public performance at

seminar presentation, manner of expression

at exams.

Problem solving: solving combinatorial and

extremal problems in graph theory.



Predavanja Lectures

Seminar

Individualno delo

Seminar

Individual work


Assessment:


naloge, projekt)

Seminar

Pisni izpit (naloge)

Izpit (teorija)



Pozitivna ocena pri seminarju in pisnem

izpitu sta pogoja za pristop k izpitu iz

teorije

Delež (v %) /

Weight (in %)

30%

30%

40%


project):

Seminar

Written Exam (exercises)

Exam (theory)



Passing grade of the seminar and of the

written are required for taking the exam.


references:

1. BOKAL, Drago, BREŠAR, Boštjan, JEREBIC, Janja. A generalization of Hungarian method

and Hall's theorem with applications in wireless sensor networks. Discrete appl. math.. [Print ed.],

2012, vol. 160, iss. 4-5, str. 460-470. http://dx.doi.org/10.1016/j.dam.2011.11.007.


2. BREŠAR, Boštjan, CHALOPIN, Jérémie, CHEPOI, Victor, GOLOGRANC, Tanja, OSAJDA,

Damian. Bucolic complexes. Preprint series, 2012, vol. 50, št. 1171, str. 1-24.

http://www.imfm.si/preprinti/PDF/01171.pdf. [COBISS.SI-ID 16207961]

3. BALAKRISHNAN, Kannan, BREŠAR, Boštjan, CHANGAT, Manoj, KLAVŽAR, Sandi,

PETERIN, Iztok, SUBHAMATHI, Ajitha R. Almost self-centered median and chordal graphs.

Taiwan. j. math., 2012, vol. 16, no. 5, str. 1911-1922.

http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2393/1403.


4. BREŠAR, Boštjan, KARDOŠ, František, KATRENIČ, Ján, SEMANIŠIN, Gabriel. Minimum k-

path vertex cover. Discrete appl. math.. [Print ed.], 2011, vol. 159, iss. 12, str. 1189-1195.

http://dx.doi.org/10.1016/j.dam.2011.04.008. [COBISS.SI-ID 15929689]

5. BREŠAR, Boštjan, KRANER ŠUMENJAK, Tadeja, TEPEH, Aleksandra. The geodetic number

of the lexicographic product of graphs. Discrete math.. [Print ed.], 2011, vol. 311, iss. 16, str.

1693-1698. http://dx.doi.org/10.1016/j.disc.2011.04.004. [COBISS.SI-ID

15929945]





http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2393/1403







Predmet: Verižni ulomki

Course title: Continued Fractions



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd




Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

30

15

45 3

Nosilec predmeta / Lecturer: Daniel EREMITA

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



Končni verižni ulomki

Neskončni verižni ulomki

Periodični verižni ulomki

Diofantska aproksimacija

Pellova enačba

Faktorizacija z uporabo verižnih

ulomkov

Fermatov izrek o vsotah dveh kvadratov

Finite continued fractions

Infinite continued fractions

Periodic continued fractions

Diophantine approximation

Pell’s equation

Factoring using continued fractions

Fermat’s theorem on sums of squares


Burton, D. M.: Elementary Number Theory, 6th ed., McGraw-Hill, New York, 2007

Grasselli, J.: Osnove teorije števil, 2. predelana izdaja, DZS, Ljubljana, 1975

Grasselli, J.: Diofantske enačbe, DMFA, Ljubljana 1984

Grasselli, J.: Diofantski približki, DMFA, Ljubljana 1992

Rockett, A. M., Szüsz, P.: Continued Fractions, World Scientific Publishing Co. Pte. Ltd.,

Singapore, 1992

Rosen, K. H.: Elementary Number Theory and its applications, 5th ed., Pearson/Addison

Wesley, Boston, 2005



Razumevanje osnovnih konceptov in rezultatov

klasične teorije navadnih verižnih ulomkov.

Understanding basic concepts and results of

classical theory of simple continued fractions.




konceptov in rezultatov klasične teorije

navadnih verižnih ulomkov

nekaterih aplikacij verižnih ulomkov


- pridobljena znanja se dopolnjujejo z znanji

iz drugih področij teorije števil in z znanji s

področja algebre, kombinatorike, analize,

računalništva, …


concepts and results of classical theory

of simple continued fractions

some applications of continued fractions.


- the obtained knowledge supplements with

the knowledge of other fields of number

theory and also with the knowledge of

algebra, combinatorics, analysis, computer

science, …



Predavanja

Seminarske vaje

Individualno delo

Lectures

Tutorial

Individual work


Assessment:


naloge, projekt)







Delež (v %) /

Weight (in %)

50%

50%


project):








references:

1. EREMITA, Daniel, ILIŠEVIĆ, Dijana. On (anti-)multiplicative generalized derivations. Glas.

mat., 2012, vol. 47, no. 1, str. 105-118. http://dx.doi.org/10.3336/gm.47.1.08.


2. BENKOVIČ, Dominik, EREMITA, Daniel. Multiplicative Lie n-derivations of triangular rings.

Linear algebra appl.. [Print ed.], 2012, vol. 436, iss 11, str. 4223-4240.

http://dx.doi.org/10.1016/j.laa.2012.01.022. [COBISS.SI-ID 16278361]

3. BENKOVIČ, Dominik, EREMITA, Daniel, VUKMAN, Joso. A characterization of the centroid

of a prime ring. Stud. sci. math. Hung. (Print), 2008, vol. 45, no. 3, str. 379-394.

http://dx.doi.org/10.1556/SScMath.2008.1069, doi:

10.1556/SScMath.2008.1069. [COBISS.SI-ID 16236040]

4. EREMITA, Daniel, ILIŠEVIĆ, Dijana. On additivity of centralisers. Bull. Aust. Math. Soc.,

2006, 74, str. 177-184. [COBISS.SI-ID 14915336]

5. VUKMAN, Joso, KOSI-ULBL, Irena, EREMITA, Daniel. On certain equations in rings. Bull.

Aust. Math. Soc., 2005, vol. 71, str. 53-60. [COBISS.SI-ID 13721096]

http://dx.doi.org/10.3336/gm.47.1.08


http://dx.doi.org/10.1016/j.laa.2012.01.022


http://dx.doi.org/10.1556/SScMath.2008.1069

http://dx.doi.org/10.1556/SScMath.2008.1069





OPIS PREDMETA / SUBJECT SPECIFICATIONPredmet: Izbrana poglavja šolske discipline Subject Title:

Selected topics in school discipline


Izobraževalna matematika

dvopredmetni študij, 2. stopnja Univerzitetna koda predmeta / University subject code:

Predavanja Lectures

Seminar Seminar

30 15 Nosilec predmeta / Lecturer:

Jeziki / Languages:

Predavanja / Lecture:Vaje /

Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti:

Pogojev ni.

Vsebina:

• Opredelitev osnovnih pojmovin pomen v vzgojno-izobraževalnih institucijah

• Disciplina v šoli v kontekstu oziroma otrokovih pravic

• Formalne in neformalne ovzgojno-izobraževalnih institucijah

• Izhodišča in načela disciplinskega pristopa• Modeli vodenja razreda • Šola in problem nasilja


• Ayers, H, Gray, F (2002): Vodenje razreda, Priro• Jones V. F., Jones S. J. (2005): Comprehensive classroom management: Creating communities of

support and solving problems, Allyn and Bacon, Boston.• Devjak, T., ur. (2007): Pravila in vzgojno delovanje šole, Evropski socialni sklad, Ljubljana.• Pšunder, M. (2004): Disciplina v sodobni šoli, ZRSŠ, Ljubljana.• Pušnik, M. (1999): Vrstniško nasilje v šolah• Pšunder, M. (2006): Na

Sodobna pedagogika, št. 1.




OPIS PREDMETA / SUBJECT SPECIFICATION Izbrana poglavja šolske discipline (izbirni predmet)

Selected topics in school discipline (Elective subject)


Letnik

dvopredmetni študij, 2. stopnja


Sem. vaje Tutorial

Lab. Vaje Lab. Work



Mateja PŠUNDER



Prerequisites:

None.


Opredelitev osnovnih pojmov, cilji discipline izobraževalnih

Disciplina v šoli v kontekstu človekovih

Formalne in neformalne oblike discipline v izobraževalnih institucijah

ela disciplinskega pristopa

• Definition of basic principles, goals of discipline and its importance in educational institutions

• Discipline in the contemporary school in the context of children’s rights and human rights

• Formal and informal methods of education discipline in educational institution

• Basic and principles of discipline approach• Classroom management models• School and problem of violence

Temeljni študijski viri / Textbooks: Ayers, H, Gray, F (2002): Vodenje razreda, Priročnik za učitelje, Educy, Ljubljana.Jones V. F., Jones S. J. (2005): Comprehensive classroom management: Creating communities of support and solving problems, Allyn and Bacon, Boston.

Pravila in vzgojno delovanje šole, Evropski socialni sklad, Ljubljana.Pšunder, M. (2004): Disciplina v sodobni šoli, ZRSŠ, Ljubljana. Pušnik, M. (1999): Vrstniško nasilje v šolah, ZRSŠ, Ljubljana. Pšunder, M. (2006): Načela disciplinskega pristopa: izhodišča in stališSodobna pedagogika, št. 1.

naravoslovje in matematiko / Faculty of Natural


Letnik Year

Semester Semester

2 Poletni

2 Summer


ECTS

45 3

Contents (Syllabus outline): Definition of basic principles, goals of discipline and its importance in educational institutions Discipline in the contemporary school in the context of children’s rights and human rights

methods of education discipline in educational institution Basic and principles of discipline approach Classroom management models School and problem of violence

itelje, Educy, Ljubljana. Jones V. F., Jones S. J. (2005): Comprehensive classroom management: Creating communities of

Pravila in vzgojno delovanje šole, Evropski socialni sklad, Ljubljana.

a in stališča nekaterih avtorjev,

Cilji: Objectives:

Cilj tega predmeta je seznaniti študente s sodobnimi pogledi na disciplino in vodenje razreda in jih usposobiti za samostojno odkrivanje in reševanje vzgojno-disciplinskih vprašanj in problemov sodobne pedagoške prakse.

The objective of this course is to encourage the acquisition of modern views on discipline and classroom management and to enable students to explore autonomous solutions for educational-disciplinary questions and problems in contemporary pedagogical practice.



Znanje in razumevanje: Po zaključku tega predmeta bo študent sposoben:

• izkazati znanje in razumevanje sodobnih pogledov na disciplino in vodenje razreda,

• izkazati znanje in razumevanje formalnih in neformalnih oblik discipline v vzgojno-izobraževalnih institucijah,

• ovrednotiti in uporabiti dejavnike učinkovitega disciplinskega pristopa.

Knowledge and Understanding: On completion of this course the student will be able to do the following:

• demonstrate knowledge and understanding of modern views on discipline and classroom management,

• demonstrate knowledge and understanding of formal and informal methods of educating/disciplining in educational institutions,

• evaluate and use methods of efficient discipline approach.

Prenesljive/ključne spretnosti in drugi atributi: • Spretnost komuniciranja, • uporabo informacijske tehnologije, • kombinirana uporaba različnih znanj za

reševanje praktičnih problemov, • delo v skupini.

Transferable/Key Skills and other attributes: • Communication skills, • usage of IT, • combined use of different skills for solution of

practical problems • team work.



• Predavanje, • seminar, • metoda razgovora, • skupinska diskusija, • metoda reševanja problemov, • kooperativno in individualno učenje.

• Lectures, • seminar, • conversation, • group discussion, • problem-based approach, • cooperative and individual learning.


Assessment:

Končna ocena je sestavljena iz: • pisnega izpita, • izdelave in zagovora seminarske

naloge, • aktivnega sodelovanja v seminarju.

50% 40% 10%

The final mark consists of the: • written exam, • seminar paper and its presentation, • active collaboration in the seminar.


• predavalnica z multimedijskimi pripomočki

• lecture room with multimedia facilitie


• Aktivna udeležba v seminarju, izdelava in predstavitev seminarske naloge ter opravljen pisni izpit.

• Študent lahko pristopi k izpitu, ko opravi obveznosti seminarskega referata.

• Active participation in the seminar, seminar paper and it presentation, written exam.

• Student can attend an exam when he successfully finishes his seminar work.


Predmet: Izbrana poglavja iz algebre

Course title: Selected topics in algebra



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd




Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

30 30 60 4

Nosilec predmeta / Lecturer: Dušan PAGON

Jeziki /

Languages:

Predavanja/Lectures: SLOVENSKO/SLOVENE




Prerequisits:

Ne. None.


Grupe in podgrupe. Simetrične grupe.

Konjugiranost elementov in podgrup.

Homomorfizmi in izomorfizmi grup.

Podgrupe edinke in faktorske grupe.

Delovanje grupe na množico.

Sylowske podgrupe, izreki Sylowa.

Kolobar, ideal, obseg.

Karakteristika kolobarja. Končna polja.

Groups and subgroups. Symmetric groups.

Conjugated elements and subgroups.

Group homomorphisms and isomorphisms.

Normal subgroups and factor groups.

Action of a group on a set.

Sylow subgroups, Sylow theorems

Ring, ideal, division ring.

The characteristics of a ring. Finite fields.


W. Y. Gilbert, W. K. Nicholson, Modern Algebra with Applications, Wiley, Chichester 2004

S. Lang, Undergraduate Algebra, Springer, 2005

A. I. Kostrikin, Introduction to Algebra, Springer-Verlag, New York 1982

I. Vidav, Algebra, DMFA, Ljubljana 1980

N. Božović, Ž. Mihajlović. Uvod u teoriju grupa. Naučna knjiga, Beograd 1983



Študentje spoznajo osnove teorije grup in polj,

skupaj s spremljajočimi pojmi kot so

podstruktura,homomorfizem, kvocientna

struktura.

The students get familiar with the fundamentals

of the theory of groups and fields, including

such related topics as substructure,

homomorphism and factor structure.




Razumevanje temeljnih pojmov algebrske

strukture, njene podstrukture in izomorfnih

struktur.

Poznavanje osnovnih značilnosti in tipičnih

primerov grup, kolobarjev in polj.


- Algebrske strukture z eno in dvema

notranjima binarnima operacijama so

osnova za razumevanje sodobne

matematike.


Understanding the basic notions about an

algebraic structure, its substructure and

isomorphic structures.

To recognize the typical properties and main

examples of groups, rings and fields.


- Algebraic structures with one and two inner

binary operations are of principal

importance for understanding the modern

mathematics.

Metode poučevanja in učenja: Learning and teaching methods:

Predavanja

Seminarske vaje

Individualno delo

Lectures

Tutorial

Individual work

Načini ocenjevanja: Assessment:

Način (pisni izpit, ustno izpraševanje, naloge,

projekt)

Pisni izpit – praktični del

Ustni izpit – teoretični del

Pisni izpit – praktični del se lahko nadomesti

z dvema delnima testoma (sprotni

obveznosti).

Delež (v %) /

Weight (in %)

50%

50%

Type (examination, oral,

coursework, project):

Written exam – practical part

Oral exam – theoretical part

Written test – practical part can be

replaced by two partial tests (mid-

term testing).

Reference nosilca / Lecturer's references:

1. PAGON, Dušan, REPOVŠ, Dušan, ZAICEV, Mikhail. On the codimension growth of simple

color Lie superalgebras. J. Lie theory, 2012, vol. 22, no. 2, str. 465-479.

http://www.heldermann.de/JLT/JLT22/JLT222/jlt22017.htm. [COBISS.SI-ID

16070233]

2. PAGON, Dušan. Simplified square equation in the quaternion algebra. International journal of

pure and applied mathematics, 2010, vol. 61, no. 2, str. 231-240. [COBISS.SI-ID 17718024]

3. GUTIK, Oleg, PAGON, Dušan, REPOVŠ, Dušan. On chains in H-closed topological pospaces.

http://www.heldermann.de/JLT/JLT22/JLT222/jlt22017.htm



Order (Dordr.), 2010, vol. 27, no. 1, str. 69-81. http://dx.doi.org/10.1007/s11083-010-9140-x. [COBISS.SI-ID 15502169]

4. GUTIK, Oleg, PAGON, Dušan, REPOVŠ, Dušan. The continuity of the inversion and the

structure of maximal subgroups in countably compact topological semigroups. Acta math. Hung.,

2009, vol. 124, no. 3, str. 201-214. http://dx.doi.org/10.1007/s10474-009-8144-8, doi:

10.1007/s10474-009-8144-8. [COBISS.SI-ID 15212121]

5. PAGON, Dušan. The dynamics of selfsimilar sets generated by multibranching trees.

International journal of computational and numerical analysis and applications, 2004, vol. 6, no.

1, str. 65-76. [COBISS.SI-ID 14037081]

http://dx.doi.org/10.1007/s11083-010-9140-x

http://dx.doi.org/10.1007/s11083-010-9140-x


http://dx.doi.org/10.1007/s10474-009-8144-8

http://dx.doi.org/10.1007/s10474-009-8144-8




Predmet: Izobraževalni programski jeziki

Course title: Educational programming languages

Študijski program in stopnja Study programme and level


Letnik Academic

year

Semester Semester

Izobraževalna matematika, dvopredmetni študij, 2. stopnja

Modul D2 1. ali 2. 2. ali 4.

Educational mathematics, double major 2nd degree

Module D2 1. or 2. 2. or 4.



Predavanja Lectures

Seminar Seminar

Sem. vaje Tutorial

Lab. vaje Laboratory

work


Samost. delo Individ.

work ECTS

30

15 45 3

Nosilec predmeta / Lecturer: Aleksander VESEL

Jeziki / Languages:

Predavanja / Lectures:

SLOVENSKO/SLOVENE



Prerequisits:



Osnovnih elementi in koncepti programskih jezikov. Zgodnje učenje programskih jezikov. Programski jeziki glede na starostna obdobja, stopnjo razvoja in predznanje. Koncepti postopnega nadgrajevanja izobraževalnih programskih jezikov. Različni primeri postopnega nadgrajevanja:

Basic programming languages’ elements and concepts. Early learning of programming languages. Programming languages by age, stage of developments and background knowledge. Koncepts with sequences of programming languages where a student takes a course from easy to

Java, SmallTalk, Lisp. Izobraževalni programski jeziki in programski vzorci. Primeri izobraževalnih programskih jezikov.

understand to complex environment. Various examples: Java, SmallTalk, Lisp. Educational programming languages and programming paradigms. Examples of educational programming languages.


Michael Kolling, Introduction to Programming with Greenfoot: Object-Oriented Programming in Java with Games and Simulations, Prentice Hall, 2009. Jerry Lee Ford, Jr. ,Scratch Programming for Teens, Course Technology PTR, 2008. Jerry Lee Ford, Jr. , Program Programming for the Absolute Beginner, Course Technology PTR, 2008. Warren Sande, Carter Sande, Hello World! Computer Programming for Kids and Other Beginners, Manning Publications, 2009.



spoznati koncepte izobraževalnih programskih jezikov

spoznati primere izobraževalnih programskih jezikov

to know concepts from educational programming languages

to know examples of educational programming languages




Poznavanje elementov programskih jezikov.

Razmevanje pomena zgodnjega učenja programskih jezikov

Poznavanje konceptov postopnega nadgrajevanja

Prenesljive/ključne spretnosti in drugi atributi: - Prenos znanja na druga področja

izobraževanja (naravoslovje, tehnika, matematika,...)


Knowing programming languages’ elements.

Understanding the importance of early learning of programming languages.

Knowing concepts of learning paths for educational programming languages.

Transferable/Key Skills and other attributes: - Transfer of knowledge to other areas

education (science, technology, mathematics, , ...)



Predavanja

Računalniške in teoretične vaje

Lectures

Computer and theoretical exercises


Assessment:

Delež (v %) /

Weight (in %)

Izpit:

Pisni izpit – problem

Pisni izpit – teorija



Opravljen pisni izpit - problemi je pogoj

za pristop k pisnemu izpitu - teorija.

50%

50%

Exams:

Written exam - problems

Written exam - theory



Passing grade of written exam -

problems is required for taking the

written exam – theory.


1. KORŽE, Danilo, VESEL, Aleksander. A note on the independence number of strong products of odd cycles. Ars comb., 2012, vol. 106, str. 473-481. [COBISS.SI-ID 16138006]

2. TARANENKO, Andrej, VESEL, Aleksander. 1-factors and characterization of reducible faces of plane elementary bipartite graphs. Discuss. Math., Graph Theory, 2012, vol. 32, no. 2, str. 289-297, doi: 10.7151/dmgt.1607. [COBISS.SI-ID 19104264]

3. SALEM, Khaled, KLAVŽAR, Sandi, VESEL, Aleksander, ŽIGERT, Petra. The Clar formulas of a benzenoid system and the resonance graph. Discrete appl. math.. [Print ed.], 2009, vol. 157, iss. 11, str. 2565-2569. http://dx.doi.org/10.1016/j.dam.2009.02.016. [COBISS.SI-ID 15142489]

4. VESEL, Aleksander. 4-tilings of benzenoid graphs. MATCH Commun. Math. Comput. Chem. (Krag.), 2009, vol. 62, no. 1, str. 221-234. [COBISS.SI-ID 16886536]

5. TARANENKO, Andrej, VESEL, Aleksander. Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs. Discrete appl. math.. [Print ed.], 2008, vol. 156, iss. 10, str. 1711-1724. http://dx.doi.org/10.1016/j.dam.2007.08.029, doi: 10.1016/j.dam.2007.08.029. [COBISS.SI-ID 16140552]


http://dx.doi.org/10.7151/dmgt.1607









Predmet: Reflektivno poučevanje

Course title: Reflective Teaching



Letnik Academic

year

Semester Semester


1 ali 2 2 ali 4


1 or 2 2 or 4

Vrsta predmeta / Course type Izbrni / Elective


Predavanja Lectures

Seminar Seminar

Vaje Tutorial

Lab. vaje Lab. work

Druge oblike študija


work ECTS

30 15 45 3

Nosilec predmeta / Lecturer: Dr. Milena Ivanuš Grmek

Jeziki / Languages:

Predavanja / Lectures: slovenski /Slovene

Vaje / Tutorial: slovenski /Slovene


Prerequisits:

/ /

Vsebina:

Content (Syllabus outline):

•Pomen refleksije na področju vzgoje in izobraževanja. •Refleksija pri poučevanju naravoslovja •Značilnosti reflektivnega poučevanja. •Pristopi reflektivnega poučevanja. •Vzpodbujanje in ovire za refleksijo v poučevanju. •Listovnik profesionalnega razvoja.

•The meaning of reflection in education and schooling. •Reflection in teaching of science. •Characteristics of reflective teaching. •Approaches of reflective teaching. •Stimulation and obstacles for reflection in teaching. •Professional portfolio.


•Bolton, G. (2005). Reflective Practice. London: Sage Publications. •Campbell, D.M. et. Al. (2006). How to develop professional portfolio. A manual for teachers. Boston, MA: Pearson Education. •Klenowski, V. 2004. Developing portfolios for learning and assessment. London, New York: Routledge Falmer, Taylor & Francis. •Pollard, A. (2002). Reflective teaching. London, New York: Continuum. •Aktualni članki iz domače in tuje periodike



Študent/ka: •spozna namen, teoretična izhodišča in operativne pristope za reflektivno poučevanje; •spozna pomen refleksije pri poučevanju naravoslovja; •spozna značilnosti reflektivnega poučevanja; •spozna različne pristope k reflektivnemu poučevanju; •se usposobi za načrtovanje in izvajanje reflektivnega poučevanja.

A student: •gets familiar with the meaning, theoretical bases and operative approaches for reflective teaching; •gets familiar with the meaning of reflection in teaching of science; •gets familiar with characteristics of reflective teaching; •gets familiar with different approaches of reflective teaching; •becomes qualified to plan and carry out reflective teaching.



Znanje in razumevanje. Študent/ka: • zna opredeliti temeljni namen, izhodišča in značilnosti refleksije v poučevanju; • zna opredeliti različne modele refleksije v poučevanju naravoslovja; • zna povezati kompleksnost vsebine in spoznanja lastne discipline in pedagoške prakse z elementi drugih področij; • zna prevesti in implicitrati različna teoretična spoznanja v prakso in obratno. Prenesljive/ključne spretnosti in drugi atributi: Študent-ka: • zna uporabiti različne pristope in tehnike reflektivnega poučevanja; • pozna tehnike vzpodbujanja refleksije v poučevanju naravoslovja; • zna sestaviti, izdelati listovnik; • sposoben je kritičnega razmisleka o svojem delu; • sposoben je komunicirati s pripadniki drugih profesij, kolegi...

Knowledge and Understanding. A student: • knows how to define a basic purpose, bases and characteristics of reflection in teaching; • knows how to define different models of reflection in science teaching; • is able to connect the complexity of content and recognition of his own branch and pedagogical practice with the elements from other fields; • knows how to transfer and imply various theoretical recognitions into practice and vice versa. Transferable/Key Skills and other attributes: Student: • knows how to use different approaches and techniques of reflective teaching; • is familiar with the techniques of enforcing reflection in teaching of science; • knows how to compose, prepare a professional portfolio; • is capable of critical reflection about his work;

• is capable to communicate with people of other professions, colleagues etc.



• visokošolsko predavanje; • metoda razgovora; • študije primerov in kritičnih dogodkov; • metoda reševanja problemov; • vzajemno opazovanje; • mikropouk; • kooperativno učenje.

• higher education lecture; • the method of discourse; • case studies and critical event studies; • problem solving; • mutual observing; • microteaching; • cooperative learning.


Delež (v %) / Weight (in %)

Assessment:

•ustni izpit; •aktivno sodelovanje pri predavanjih in seminarskem delu •seminarska naloga

50 % 20 % 30 %

•oral examination; •active participation at lesson and seminar work, the results of which are •a seminar paper and a professional portfolio


1. JAVORNIK KREČIČ, Marija, VRŠNIK PERŠE, Tina, IVANUŠ-GRMEK, Milena. Pedagoški delavci v strokovnem in poklicnem izobraževanju kot aktivni oblikovalci in usmerjevalci lastnega poklicnega razvoja = Educational professionals in VET as active designers and guides of their own professional development. Revija za elementarno izobraževanje, ISSN 1855-4431. [Tiskana izd.], jul. 2015, letn. 8, št. 3, str. 77-93, tabeli. [COBISS.SI-ID 21486600] 2. RIZMAN HERGA, Nataša, IVANUŠ-GRMEK, Milena, DINEVSKI, Dejan. Virtual laboratory as an element of visualization when teaching chemical contents in science class. Turkish online journal of educational technology, 2014, vol. 13, iss. 4, str. 157-165, ilustr. [COBISS.SI-ID 20894728]. 3. ŽAKELJ, Amalija, IVANUŠ-GRMEK, Milena. Ability grouping and pupils' results on the national assessment of knowledge. Hrvatski časopis za odgoj i obrazovanje, ISSN 1848-5189. [Tiskana izd.], 2013, vol. 15, no. 2, str. 439-463, tabele. [COBISS.SI-ID 2045308]. 4. LEŠNIK, Sabina, BRUMEN, Mihaela, IVANUŠ-GRMEK, Milena. Attitudes of parents toward learning foreign languages : a Slovene case study. The new educational review, ISSN 1732-6729, 2013, vol. 34, no. 4, str. 52-62. http://www.educationalrev.us.edu.pl/vol/tner_4_2013.pdf. [COBISS.SI-ID 20324616]. 5. LEŠNIK, Sabina, IVANUŠ-GRMEK, Milena, BRUMEN, Mihaela. Učenje tujih jezikov z vidika staršev iz različnih slovenskih regij = Views of parents from different regions across Slovenia on learning

foreign languages. Revija za elementarno izobraževanje, ISSN 1855-4431. [Tiskana izd.], sep. 2013, letn. 6, št. 2/3, str. 31-46, tabele. [COBISS.SI-ID 20102408].


Predmet: Praktično usposabljanje za poučevanje matematike I

Course title: Pedagogical practice for teaching mathematics I



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd

degree 1. 2.



Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

3 4 113 4


Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



- Učni načrt za osnovno šolo.

- Nastopi v šoli.

- Cilji pedagoške prakse v osnovni šoli.

- Pedagoška praksa: priprava, nastopi,

hospitacije, analize, pedagoška dokumentacija,

temeljni šolski pravilniki, pedagoško delo v

razredu.

- Dnevnik pedagoške prakse.

- Analiza nastopov in pedagoške prakse.

- Mathematics curriculum for elementary

school.

- Pedagogical class appearances in school.

- Goals of pedagogical practice in elementary

school.

- Pedagogical practice: preparation,

instructions, observations, analysis, pedagogical

documentation, school legislation, pedagogical

class managament.

- Diary of pedagogical practice.

- Evaluation of class appearances and

pedagogical class practice.


Učni načrt za osnovno šolo.

Učbeniki in druga učna gradiva za osnovno šolo.


Dodatni študijski viri / Additional Sources


J. A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally,Sixth


Drugi viri s primeri aktivnosti in učnih enot pri pouku matematike.



- Načrtovanje vzgojno-izobraževalnega

procesa – priprava na nastope v razredu.

- Uporaba in preverjanje teoretičnih spoznanj v

neposredni pedagoški praksi.

- Pridobivanje pedagoških izkušenj in

razvijanje kompetenc učitelja matematike.

- Analiza in vrednotenje nastopov in

pedagoške prakse.

- Planing of educational process – preparing for

class appearances.

- Application and verification of theoretical

knowledge in class practice.

- Getting experienced on classroom teaching

and developing the competencies for

mathematics teacher.






- usvojenost matematičnih, didaktičnih,

pedagoških in psiholoških znanj, potrebnih za

učinkovito poučevanje v osnovni šoli, ki so





za nadaljnje uspešno delo v razredu.

Pri praktičnem usposabljanju bomo stremeli k

usvojenosti naslednjih zmožnosti (kompetenc)

učitelja matematike:



matematičnih konceptov z namenom

oblikovanja takšnega učnega okolja, ki

učencem omogoča učinkovito izgradnjo znanja

ter njegovo trajnost, prenosljivost in celovitost.







problemskih znanj.


- Adoption of mathematical, didactic,







At pedagogical practice we will strive to





of contents and concepts of school mathematics

in order to achieve learning conditions which

enable learners to acquire knowledge (durability,

transferability, wholeness);








study materials.

http://www.e-um.si/












od tujih jezikov.





proces.


























Razgovor in diskusija,

demonstracija,


uporaba IKT,

drugo.




Conversation and discussion,

Demonstration,

Method of written and graphic products,

Usage of ICT,

Other.





Assessment:


naloge, projekt)

Praktični del:

- nastopi v razredu med letom,

- pedagoška praksa,

- hospitacije študentov.

Delež (v %) /

Weight (in %)

2 × 15 %,

70 %,

opravil/passed


project):

Practical part:

- pedagogical class appearances in the

school,

- pedagogical practice,

- observations (students).


references:






Predmet: Praktično usposabljanje za poučevanje matematike II

Course title: Pedagogical practice for teaching mathematics II



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd

degree 2. 4.



Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

3 4 113 4


Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



- Učni načrti za srednje šole.

- Nastopi v šoli.

- Cilji pedagoške prakse v srednji šoli.

- Pedagoška praksa: priprava, nastopi,

hospitacije, analize, pedagoška dokumentacija,

temeljni šolski pravilniki, pedagoško delo v

razredu.

- Dnevnik pedagoške prakse.

- Analiza nastopov in pedagoške prakse.

- Zakon o pripravništvu.

- Mathematics curriculum for secondary school.

- Pedagogical class appearances in school.

- Goals of pedagogical practice in secondary

school.

- Pedagogical practice: preparation,

instructions, observations, analysis, pedagogical

documentation, school legislation, pedagogical

class managament.

- Diary of pedagogical practice.



- Law of probation for teachers.


Učni načrti za srednje šole.

Učbeniki in druga učna gradiva za srednje šole.

Spletni portal E-um: www.e-um.si. Šolska zakonodaja.

Dodatni študijski viri / Additional Sources






- Načrtovanje vzgojno-izobraževalnega

procesa – priprava na nastope v razredu.

- Uporaba in preverjanje teoretičnih spoznanj v

neposredni pedagoški praksi.

- Pridobivanje pedagoških izkušenj in

razvijanje kompetenc učitelja matematike.

- Analiza in vrednotenje nastopov in

pedagoške prakse.

- Planing of educational process – preparing for

class appearances.

- Application and verification of theoretical

knowledge in class practice.

- Getting experienced on classroom teaching

and developing the competencies for

mathematics teacher.






- usvojenost matematičnih, didaktičnih,

pedagoških in psiholoških znanj, potrebnih za

učinkovito poučevanje v osnovni šoli, ki so





za nadaljnje uspešno delo v razredu.

Pri praktičnem usposabljanju bomo stremeli k

usvojenosti naslednjih zmožnosti (kompetenc)

učitelja matematike:



matematičnih konceptov z namenom

oblikovanja takšnega učnega okolja, ki

učencem omogoča učinkovito izgradnjo znanja

ter njegovo trajnost, prenosljivost in celovitost.







problemskih znanj.


- Adoption of mathematical, didactic,







At pedagogical practice we will strive to





of contents and concepts of school mathematics

in order to achieve learning conditions which

enable learners to acquire knowledge (durability,

transferability, wholeness);








study materials.

http://www.e-um.si/












od tujih jezikov.





proces.


























Razgovor in diskusija,

demonstracija,


uporaba IKT,

drugo.




Conversation and discussion,

Demonstration,

Method of written and graphic products,

Usage of ICT,

Other.





Assessment:


naloge, projekt)

Praktični del:

- nastopi v razredu med letom,

- pedagoška praksa,

- hospitacije študentov.

Delež (v %) /

Weight (in %)

2 × 15 %,

70 %,

opravil/passed.


project):

Practical part:

- pedagogical class appearances in the

school,

- pedagogical practice,

- observations (students).


references:






Predmet: Pedagoška komunikacija

Course title: Communication in education



Študijska

smer

Study field

Letnik

Academic year

Semester

Semester

Izobraževalna tehnika – 2. Stopnja dvopredmetna

Izobraževalna tehnika – 2. Stopnja enopredmetna

Izobraževalna fizika - 2. Stopnja dvopredmetna

Izobraževalno računalništvo - 2. Stopnja dvopredmetna

Izobraževalna biologija - 2. Stopnja dvopredmetna

Izobraževalna matematika - 2. Stopnja dvopredmetna

Izobraževalna matematika – 2. Stopnja enopredmetna

/

1 Zimski/

Winter Educational Design – 2. Cycle two stream

Educational Design – 2. Cycle single major

Educational Physics - 2. Cycle two stream

Educational Computer Science – 2. Cycle two stream

Educational Biology - 2. Cycle two stream

Educational Mathematics – 2. Cycle two stream

Educational Mathematics – 2. Cycle single major

/

Vrsta predmeta / Course type Didaktični izbirni/didactic elective


Predavanja

Lectures

Seminar

Seminar

Vaje

Tutorial

Lab. vaje

Laboratory

work

Terenske vaje

Field work

Samost. delo

Individ. work ECTS

15 15 15 45 3

Nosilec predmeta / Lecturer: Mateja Ploj Virtič, Boris Aberšek

Jeziki /

Languages:

Predavanja / Lectures: slovenski / slovene

Vaje / Tutorial: slovenski / slovene



Prerequisits:

Ni posebnih pogojev. No special prerequisites.

2


Predavanja:

o Komunikacija v izobraževanju

Udeleženci komunikacije

Elementi komunikacijskega

procesa

Osnovne življenjske naravnanosti

in komunikacijski položaji

Dinamika skupinske komunikacije

o Upravljanje konfliktov v

šoli

Ego stanja

Načini komunikacije skozi različna

ego stanja

Strategija upravljanja konfliktov s

pomočjo ego stanj

Upravljanje konfliktov v šoli

o Retorika

Zgodovina retorike

Razporeditev vsebin za

povečanje prepričljivosti

Sredstva prepričevanja:

etos, patos in logos

Uporaba registrov za

povečanje prepričljivosti

govora

Seminar:

Nastopi in njihove analize;

seminarski referati s področja retorike.

Lectures:

o Communication in education

Participants of communication

The elements of the communication

process

Basic life orientation and

communication positions

The dynamics of group

communication

o Conflict management at

school

Ego states

Methods of communication using

different ego states

The strategy of conflict

management using ego states

conflict management in school

o Rhetoric

History of rhetoric

Distribution of content to increase

the persuasiveness

The means of persuasion: ethos,

pathos and logos

Using the registry to increase the

persuasiveness of speech

Tutorials and seminar:

Performance and

guided observation and

analyses;

Seminar works from

area of rhetoric.


Aberšek, B.(2005), Tehnologija sporazumevanja, Fakulteta za strojništvo, Maribor.

Ule M.(2005): Psihologija komuniciranja. Ljubljana.

Kunst Gnamuš O. (1992): Sporazumevanje in spoznavanje jezika. Ljubljana.

Kavčič B. (2004): Osnove poslovnega komuniciranja. Ljubljana.

Blažič M. (2002): Razsežnosti komunikacije, Novo mesto, 2002.

Thompson, P. Persuading Aristotle, Kogan Page Limited, 1999

Fabiani, Petra Atteya: Vzgojni ukrepi in restitucija. Šolski razgledi. Letnik LXV, št. 15, Ljubljana, 2014

Harris, Thomas A.: Jaz sem v redu – ti si v redu. Karantanija, 2007

Humar, Ines: Vrstniška mediacija kot način reševanja konfliktov v osnovni šoli. Diplomsko delo,

mentor: Robi Kroflič, Ljubljana, 2011 (dostopno na: http://www.pedagogika-

http://www.pedagogika-andragogika.com/files/diplome/2011/2011-Humar-Ines.pdf

3

andragogika.com/files/diplome/2011/2011-Humar-Ines.pdf)

Iršič, Marko: Umetnost obvladovanja konfliktov. Zavod RAKMO, Ljubljana, 2004

Zidar Gale, Tatjana: Retorika, veščina prepričevanja. Planet, Ljubljana, 2007



Kompetenca sporazumevanja: uporaba

strategij, metod in oblike komuniciranja v

skupini - udeležence usposobiti za uspešno,

samozavestno, spretno in pravilno

komuniciranje s skupino in posameznikom

kritično mišljenje;

uporaba metodologije načrtovanja makro

in mikroizvedbe predstavitve.

Communication competences: application of

strategies, methods and concepts of

communication in team - to develop skills

for successful communication with the

group and single person;

critical thinking;

using the methodology for planning macro

and micro presentation.




Poznavanje splošnih napotkov in pravil za

izbiro ustreznih oblik komunikacije v

skupini;

razumevanje sovisnosti različnih načinov

komuniciranja in motiviranja;

reševanje praktičnih problemov povezanih

s komunikacijo v skupini in med

posamezniki;

z uporabo strategij komuniciranja

upravljanje konfliktov v šoli;

uporaba sredstev prepričevanja pri

govorih.


Kombinirana uporaba znanj načrtovanja,

izvajanja in vrednotenja določene

situacije;

poglabljanje znanja metod za kreativno

delo in kritičen razvoj novih pristopov;

kritično mišljenje.

Knowledge and understanding:

Knowledge of general instructions and rules

for planning and selecting apropriate way of

communication in team;

understanding of relationships between

different way of communication and

motivation;

efficient solutions of practical problems

connected with communication in team/

single person;

conflict management at school, using

communication strategies;

using the persuasion in speeches.


Combined use different skills for planning,

executing and (self)evaluation of concrete

situation;

knowledge's of methods for creative work

and critical generation new ideas;

critical thinking.



frontalna predavanja,

delo v majhnih skupinah,

primeri dobrih praks,

vodeno opazovanje;

samostojno načrtovanje nastopa.

frontal lectures,

work in a small groups,

examples of good practice,

guided observation,

autonomous planning of presentation.


Delež (v %) /

Weight (in %)

Assessment:

http://www.pedagogika-andragogika.com/files/diplome/2011/2011-Humar-Ines.pdf

4

pisni izpit;

domače naloge;

seminarska naloga;

prisotnost na predavanjih in

seminarjih.

40 %

20 %

20 %

20 %

written exam;

home work;

seminar work;

presence at lectures and seminar works.


PLOJ VIRTIČ, Mateja, REPNIK, Robert. Improving quality of the educational process by raising

teachers' communication skills. V: LAMANAUSKAS, Vincentas (ur.). Philosophy of mind and

cognitive modelling in education - 2012, (Problems of education in the 21st century, ISSN 1822-

7864, vol. 46). Siauliai: Scientific Methodological Center Scientia Educologica, 2012, str. 109-115.


PŠUNDER, Mateja, PLOJ VIRTIČ, Mateja. The problem of cyberbullying among youth : what can we

do? = Problem cyberbullinga među mladima : što možemo učiniti? : lecture at The international

scientific conference 13th Mate Demarin Days, Education for development, Juraj Dobrila University

of Pula, Department of educational sciences, Pula, Croatia, 12 April 2013. 2013. [COBISS.SI-ID

20208392] Kordigel Aberšek, Metka. Didaktika mladinske književnosti. 1. izd. Ljubljana: Zavod

Republike Slovenije za šolstvo, 2008. 436 str., ilustr. ISBN 978-961-234-649-2.

JAVORNIK KREČIČ, Marija, KOVŠE, Suzana, PLOJ VIRTIČ, Mateja. The role and meaning of school

counseling when dealing with peer violence = Uloga i značenje školskog savjetovanja u slučaju

vršnjačkog nasilja. Hrvatski časopis za odgoj i obrazovanje, ISSN 1848-5189. [Tiskana izd.], 2013,

vol. 15, no. 2, str. 521-541. http://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=155265.


PŠUNDER, Mateja, PLOJ VIRTIČ, Mateja. Future teachers' critical view on integration of information

and communication technology into teaching and learning. V: 2nd International Scientific

Conference on Philosophy of Mind and Cognitive Modelling in Education, May 26-28, 2014,

Maribor, Slovenia. ABERŠEK, Boris (ur.). Conference abstract proceedings. Maribor: Faculty of

Natural Sciences and Mathematics, [2014], str. 53-54. [COBISS.SI-ID 20616200]

PLOJ VIRTIČ, Mateja. Pedagoška komunikacija : skripta. Maribor: Fakulteta za naravoslovje in

matematiko, 2015. 87 str., ilustr. [COBISS.SI-ID 21909768]



http://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=155265





Predmet: Kreativno reševanje matematičnih nalog

Course title: Creative mathematical problems solving



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd

degree 1. 2.



Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

15 30 75 4

Nosilec predmeta / Lecturer: Uroš MILUTINOVIĆ

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



Matematični problemi in problemski pristop pri

pouku matematike. Kreativno reševanje

matematičnih nalog; uporaba hevristik,

strategije reševanja problemov, matematičnih

eksperimentov in indukcije.

Primeri nalog in problemov z različnih

matematičnih področij: neenakosti, teorije

števil, geometrije idr.

Matematična tekmovanja in matematični

krožki. Raziskovalne naloge.

Mathematical problems and investigative

approach in teaching mathematics. Creative

problem solving; the use of heuristics, problem-

solving strategies, mathematical experiments

and induction.

Examples from different mathematical areas:

inequalities, number theory, geometry etc.

Mathematical competitions and mathematical

circles. Research projects.


A. S. Posamentier [et al.], Problem-Solving Strategies for Efficient and Elegant Solutions, Grades

6-12: A Resource for the Mathematics Teacher (Second Edition), Corwin Press, 2008.



G. Polya, Kako rešujemo matematične probleme, DMFA založništvo, Ljubljana, 1989.

A. Engel, Problem-solving strategies, Springer, 1998.

L. C. Larson, Problem-Solving Through Problems, Springer, 1990.

H. A. Hauptman [et al.], 101+ Great Ideas for Introducing Key Concepts in Mathematics: A

Resource for Secondary School Teachers (Second Edition), Corwin Press, 2006.

M. A. Sobel, Evan M. Maletsky, Teaching Mathematics: A Sourcebook of Aids, Activities and




Naloge z matematičnih tekmovanj.



Opredeliti matematični problem in

problemski pristop pri pouku

matematike.

Spoznati metode kreativnega reševanja

matematičnih nalog, predvsem uporabo

hevristik, analogije, matematičnih

eksperimentov in indukcije.

Obravnavati primere izbranih

problemskih nalog z različnih

matematičnih področij, ki jih lahko

vključimo v pouk matematike ali v

druge interesne dejavnosti.

Odkriti možnosti dela z matematično

nadarjenimi učenci in študenti.

Prikaz možnosti nadgradnje in

obogatitve pedagoškega dela učitelja

matematike z vodenjem matematičnega

krožka, pripravami na matematična

tekmovanja in z mentorstvom učencem

pri načrtovanju in izvedbi matematičnih

raziskovalnih nalog v osnovni in srednji

šoli.

To specify the mathematical problem and

investigative approach in teaching

mathematics

To know methods of creative problem

solving, such as the use of heuristics,

analogy, mathematical experiments and

induction.

To consider examples of selected

mathematical problems, which may be

integrated in the mathematics

curricculum or in other students'

activities.

To identify opportunities to work with

mathematically gifted students.

To demonstrate enrichment

opportunities for mathematics teacher in

mathematics classes: managing math

circles, preparation for mathematical

competitions and mentoring students in

planning and carrying out mathematical

research projects in primary and

secondary school.




Sposobnost originalnega reševanja

matematičnih nalog.

Sposobnost formuliranja domnev v zvezi z

matematičnimi rezultati.

Sposobnost za uporabo hevrističnih metod,

analogije, indukcije, matematičnih

eksperimentov.


Ability to solve mathematical problems

using original approaches.

Ability to state hypotheses regarding

mathematical results.

Ability to use heuristic methods, analogy,

induction, mathematical experiments.

Ability to develop problem-solving

Sposobnost razvijanja poblemskih znanj

(strategij, hevristik, ...) pri učencih in

učinkovitega vodenja učencev pri reševanju

matematičnih problemov.

Sposobnost učinkovitega načrtovanja dela z

matematično radovednimi in nadarjenimi

učenci v obliki vodenja matematičnega

krožka, priprav na tekmovanja in

mentorstva pri izdelavi matematičnih

raziskovalnih nalog.


Pridobljena znanja in sposobnosti so osnova za

kvalitetnejši pouk matematike in raziskovalno

delo tako na področju matematike kot tudi

izobraževanja matematike, s tem pa tudi za

vseživljenjsko učenje.

knowledge (strategies, heuristics, ...) of

students and to guide students in problem

solving effectively.

Ability to plan and to work with

mathematically inquisitive and talented

students in the form of math circles,

preparation for competitions, and mentoring

in the development of mathematical research

projects.


Acquired knowledge and skills are the basis for

higher quality mathematics instruction and for

research work in the fields of mathematics and

mathematics education, and thereby also for

lifelong learning.



Predavanja

Teoretične vaje

Individualno delo

Domače naloge

Lectures

Theoretical exercises

Individual work

Homeworks


Assessment:


naloge, projekt)

Domače naloge

Pisni izpit – problemi

Delež (v %) /

Weight (in %)

20%

80%


project):

Homeworks

Written exam - problems

Reference nosilca:

Lecturer's references:



series, 2012, vol. 50, št. 1169, str. 1-33. http://www.imfm.si/preprinti/PDF/01169.pdf. [COBISS.SI-

ID 16194137]







4. KLAVŽAR, Sandi, MILUTINOVIĆ, Uroš, PETR, Ciril. Stern polynomials. Adv. appl. math.,

2007, vol. 39, iss. 1, str. 86-95. http://dx.doi.org/10.1016/j.aam.2006.01.003. [COBISS.SI-ID

14276441]

5. IVANŠIĆ, Ivan, MILUTINOVIĆ, Uroš. Closed embeddings into Lipscomb's universal space.

Glas. mat., 2007, vol. 42, no. 1, str. 95-108. [COBISS.SI-ID 14338393]











Predmet: Metrični prostori

Course title: Metric Spaces



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd




Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

30

15

45 3

Nosilec predmeta / Lecturer: Iztok BANIČ

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:

Opravljen izpit iz Osnov analize in Analize. Exam in Basic Analysis, Analysis.


Metrični prostori. Primeri metrik. Primeri

ravninskih metrik. Ekvivalentne metrike.

Normirani prostori. Prostori s skalarnim

produktom.

Odprte in zaprte krogle. Odprte in zaprte

množice.

Notranjost, rob, zaprtje in zunanjost množice.

Podprostori metričnih prostorov. Produkti

metričnih prostorov.

Metric spaces. Examples of metrics. Examples

of metrics in the plane. Equivalent metrices.

Normed spaces. Spaces with scalar product.

Open and closed balls. Open and closed sets.

Interior, boundary, closure and exterior of a set.

Subspaces of metric spaces. Product spaces.

Sequences in metric spaces. Convergence and

uniform convergence. Complete metric spaces.

Zaporedja v metričnih prostorih. Konvergenca

in enakomerna konvergenca. Polnost.

Zveznost in enakomerna zveznost.

Kompaktnost in povezanost.

Continuous and uniformly continuous functions.

Compact and connected spaces.


J. Vrabec: Metrični prostori. Ljubljana: DMFA, 1993.

A. Suhodolc: Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija.

Matematični rokopisi 23, Ljubljana: DMFA, 1998.

D. Benkovič: Analiza II (dodatna gradiva na spletu)

http://matematika-racunalnistvo.fnm.uni-mb.si/dodatna_gradiva/analiza_II.html

V. Bryant: Metric Spaces: Iteration and Application. Cambridge: Cambridge University Press,

1985.



Posplošiti rezultate v zvezi z odprtimi, zaprtimi

intervali, s konvergenco realnih zaporedij in z

zveznostjo realnih funkcij na metrične prostore.

To generalize the results about closed intervals,

convergent sequences in real line, and the

continuity of real functions to metric spaces.




Študent obvlada osnovne koncepte v metričnih

prostorih. Zaveda se pomena odprtih, zaprtih

množic, kompaktnosti, polnosti in povezanosti

metričnih prostorov.


Prenos znanja obravnavanih metod na druga

področja, predvsem skozi uporabo metrike in

zveznih funkcij.


To understand basic concepts of metric spaces .

To be aware of the importance of open sets,

closed sets, compactness, completeness and

connectedness of metric spaces


Knowledge transfer of treated methods into

other fields, basically through the use of metric

and continuous functions.



Predavanja

Seminarske vaje

Individualno delo

Lectures

Tutorial

Individual work

Načini ocenjevanja: Assessment:


naloge, projekt)

Izpit:

Pisni izpit – problemi

Ustni izpit – teorija



Opravljen pisni izpit – problemi je pogoj

za pristop k ustnemu izpitu – teorija.

Pisni izpit – problemi se lahko nadomesti z enim delnim testom (sprotne obveznosti).

50% 50%


project):

Exam:

Written exam – problems

Oral exam – theory

Each of the mentioned assessments


Passing grade of written exam –

problems is required to take the oral

exam – theory.

Written exam – problems can be repalced with one mid-term test.


references:



series, 2012, vol. 50, št. 1169, str. 1-33. http://www.imfm.si/preprinti/PDF/01169.pdf.





3. BANIČ, Iztok, ŽEROVNIK, Janez. Wide diameter of Cartesian graph bundles. Discrete math..

[Print ed.], str. 1697-1701. http://dx.doi.org/10.1016/j.disc.2009.11.024, doi:

10.1016/j.disc.2009.11.024. [COBISS.SI-ID 17543176]





5. BANIČ, Iztok, ERVEŠ, Rija, ŽEROVNIK, Janez. Edge, vertex and mixed fault diameters. Adv.

appl. math., 2009, vol. 43, iss. 3, str. 231-238.

http://dx.doi.org/10.1016/j.aam.2009.01.005, doi:

10.1016/j.aam.2009.01.005. [COBISS.SI-ID 13396502]














Predmet: Programska oprema za matematike

Course title: Software for mathematicians



Letnik Academic

year

Semester Semester


Modul D2 1. ali 2. 2. ali 4.


Module D2 1. or 2. 2. or 4.



Predavanja Lectures

Seminar Seminar

Sem. vaje Tutorial

Lab. vaje Laboratory

work



work ECTS

15

30 45 3

Nosilec predmeta / Lecturer: Andrej TARANENKO

Jeziki / Languages:

Predavanja / Lectures:

SLOVENSKO/SLOVENE



Prerequisits:



Oblikovanje matematičnih besedil: uporaba in osnove programa LaTeX

Programi za numerično računanje: uporaba in osnove programa za numerično računanje, npr. SciLab, MatLab, Octave, Sage

Programi za simbolno računanje: uporaba in osnove programa za simbolno računanje, npr. Mathematica, Maxima, Sage

Editing mathematical texts: basics and usage of LaTeX

Software for numerical computing: basics and usage of a numerical computing software like SciLab, Matlab, Octave, Sage

Software for algebraic computing: basics and usage of a algebraic computing software like Mathematica, Maxima, Sage

Programi za statistično obdelavo podatkov: uporaba in osnove programa za statistično obdelavo podatkov, npr. SPSS, R

Software for statistics: basics and usage of a software for statistics like SPSS, R


Odvisno od izbrane programske opreme. Npr.:

Oetiker Tobias in drugi, Ne najkrajši uvod v LaTeX.

Griffiths D. F., Higham D. J., Learning latex, Philadelphia SIAM, 1997.

Abell M. L., Braselton J. P., Mathematica by example, San Diego, Academic press, 1997

Gašperšič M., Matlab do nezavesti, Trzin, 2009.

Morgan G. A. in drugi, SPSS for introductory statistics: use and interpretation, London : Lawrence Erlbaum, 2004



Spoznati osnove oblikovanja matematičnih besedil s paketom LaTeX

Spoznati osnove dela s programom za numerično računanje.

Spoznati osnove dela s programom za simbolno računanje.

Spoznati osnove dela s programom za statistično obdelavo podatkov.

To know basics of mathematical text editing using the LaTeX package.

To know basics of a software for numerical computing.

To know basics of a software for algebraic computing.

To know basics of a software for statistical data manipulation.




Zna uporabljati paket LaTeX pri oblikovanju matematičnih besedil.

Zna uporabljati program za numerično računanje.

Zna uporabljati program za simbolno računanje.

Zna uporabljati program za statistično obdelavo podatkov.


Sposoben poiskati ustrezno programsko opremo za reševanje problemov.

Sposoben določiti vrsto programske opreme za pomoč pri reševanju danega problema.


Knows how to use LaTeX when editing mathematical texts.

Knows how to use numerical computing software.

Knows how to use algebraic computing software.

Knows how to use statistical data manipulation software.


Is capable to find appropriate software for help with solving problems.

Is capable to determine thy type of software needed for solving a certain problem.



Predavanja

Laboratorijske vaje

Samostojno delo

Lectures

Laboratory exercises

Individual work


Assessment:

Sprotno preverjanje:

Domače naloge

Projekt



Delež (v %) /

Weight (in %)

50%

50%

Mid-term testing:

Homework

Project




1. TARANENKO, Andrej, VESEL, Aleksander. 1-factors and characterization of reducible faces of plane elementary bipartite graphs. Discuss. Math., Graph Theory, 2012, vol. 32, no. 2, str. 289-297, doi: 10.7151/dmgt.1607. [COBISS.SI-ID 19104264]

2. TARANENKO, Andrej, ŽIGERT, Petra. Resonant sets of benzenoid graphs and hypercubes of their resonance graphs. MATCH Commun. Math. Comput. Chem. (Krag.), 2012, vol. 68, no. 1, str. 65-77. http://www.pmf.kg.ac.rs/match/content68n1.htm. [COBISS.SI-ID 16051990]

3. KLAVŽAR, Sandi, SALEM, Khaled, TARANENKO, Andrej. Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems. Comput. math. appl. (1987). [Print ed.], 2010, vol. 59, no. 1, str. 506-513. http://dx.doi.org/10.1016/j.camwa.2009.06.011. [COBISS.SI-ID 15383641]

4. TARANENKO, Andrej, VESEL, Aleksander. Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs. Discrete appl. math.. [Print ed.], 2008, vol. 156, iss. 10, str. 1711-1724. http://dx.doi.org/10.1016/j.dam.2007.08.029, doi: 10.1016/j.dam.2007.08.029. [COBISS.SI-ID 16140552]

5. TARANENKO, Andrej, VESEL, Aleksander. On elementary benzenoid graphs: new characterization and structure of their resonance graphs. MATCH Commun. Math. Comput. Chem. (Krag.), 2008, #Vol. #60, #no. #1, str. 193-216, ilustr. [COBISS.SI-ID 1939989]

http://dx.doi.org/10.7151/dmgt.1607


http://www.pmf.kg.ac.rs/match/content68n1.htm


http://dx.doi.org/10.1016/j.camwa.2009.06.011







Predmet: Zgodovina matematike

Course title: History of Mathematics



Študijska smer

Study field

Letnik

Academic

year

Semester

Semester




major 2nd




Predavanja

Lectures

Seminar

Seminar

Sem. vaje

Tutorial

Lab. vaje

Laboratory

work

Teren. vaje

Field work

Samost. delo

Individ.

work

ECTS

75

135 7

Nosilec predmeta / Lecturer: Daniel EREMITA

Jeziki /

Languages:

Predavanja /

Lectures:

SLOVENSKO/SLOVENE




Prerequisits:



Metodologija zgodovine matematike,

zgodovinski viri.

Glavni centri in obdobja razvoja matematike:

mezopotamska matematika, egipčanska

matematika, starogrška in helenistična

matematika, kitajska matematika, indijska

matematika, japonska matematika, matematika

indijanskih civilizacij, arabska matematika,

matematika renesanse, matematika XV., XVI.,

XVII., XVIII., XIX. in XX. stoletja.

Methodology of the history of mathematics,

historical sources.

The main centers and periods of mathematical

development: Mesopotamian mathematics,

Egyptian mathematics, Ancient Greek and

Hellenistic mathematics, Chinese mathematics,

Hindu mathematics, Japanese mathematics,

mathematics of indigenous cultures of the

Americas, Arabic mathematics, Renaissance

mathematics, mathematics of XV., XVI., XVII.,

Razvoj glavnih področij matematike:

geometrije, aritmetike, algebre, teorije števil,

analize, matematične logike, teorije množic,

topologije, teorije grafov, teorije verjetnosti,

statistike, računalništva, metodike matematike,

zgodovine matematike idr. Razvoj osnovnih

matematičnih pojmov.

Pomembni matematiki in njihov prispevek k

razvoju matematike. Slovenski matematiki.

Zgodovina matematike kot del splošne

zgodovine. Filozofski, sociološki, psihološki,

lingvistični in podobni aspekti matematike.

Matematika in druge znanosti.

XVIII., XIX. and XX. centuries.

The development of the major areas of

mathematics: geometry, arithmetic, algebra,

number theory, analysis, mathematical logic, set

theory, topology, graph theory, probability

theory, statistics, computer science,

methodology of mathematics, history of

mathematics, etc. The development of the

fundamental mathematical notions.

Important mathematicians and their contribution

to mathematics. Slovenian mathematicians.

A history of mathematics as a part of a general

history. Philosophical, sociological,

psychological, linguistic and similar aspects of

mathematics. Mathematics and other sciences.


A History of Mathematics. New York: J. Wiley & Sons, 1989.

A History of Mathematics, An Introduction. Reading (Mass.) [etc.] : Addison-

Wesley, 1998

A History of Mathematicad Notation. New York: Dover Publications, Inc., 1993.

Geometry and Algebra in Ancient Civilizations. Berlin: Springer Verlag,

1983.

Kratka zgodovina matematike. Ljubljana: Državna založba Slovenije, 1978.



Spoznati zgodovinski razvoj matematike,

razvoj njenih osnovnih področij in razvoj

osnovnih matematičnih pojmov. Seznaniti se s

pomembnimi matematiki in njihovimi

prispevki k razvoju matematike.

To obtain knowledge of the historical

development of mathematics, the development

of its major areas, and the development of the

fundamental mathematical notions. To get

acquainted with the important mathematicians

and their contribution to mathematics.




zgodovinski razvoj matematike, razvoj

njenih osnovnih področij in razvoj osnovnih

matematičnih pojmov

pomembni matematiki in njihovi prispevki

k razvoju matematike



historical development of mathematics,

the development of its major areas, and

the development of the fundamental

mathematical notions

important mathematicians and their

contribution to mathematics

prenos znanja zgodovine matematike na vse

matematične predmete in na nekatera druga

področja (fizika, astronomija, mehanika,

računalništvo, filozofija, zgodovina, …).


knowledge transfer of history of

mathematics to all mathematical courses and

also to other areas (physics, astronomy,

mechanics, computer science, philosophy,

history, …).



Predavanja

Individualno delo

Lectures

Individual work


Assessment:

Seminarska naloga

Ustni izpit



Opravljena seminarska naloga je pogoj

za pristop k izpitu.

Delež (v %) /

Weight (in %)

20%

80%

Seminar assignment

Oral exam



Passing grade of the seminar assignment

is required to take the exam.


references:

1. EREMITA, Daniel. Functional identities of degree 2 in triangular rings revisited. Linear and Multilinear Algebra, ISSN 0308-1087, 2015, vol. 63, iss. 3, str. 534-553. http://dx.doi.org/10.1080/03081087.2013.877012. [COBISS.SI-ID 17044057] 2. EREMITA, Daniel, GOGIĆ, Ilja, ILIŠEVIĆ, Dijana. Generalized skew derivations implemented by elementary operators. Algebras and representation theory, ISSN 1386-923X, 2014, vol. 17, iss. 3, str. 983-996. http://dx.doi.org/10.1007/s10468-013-9429-8. [COBISS.SI-ID 17043545] 3. EREMITA, Daniel. Functional identities of degree 2 in triangular rings. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2013, vol. 438, iss 1, str. 584-597. http://dx.doi.org/10.1016/j.laa.2012.07.028. [COBISS.SI-ID 16528217] 4. EREMITA, Daniel, ILIŠEVIĆ, Dijana. On (anti-)multiplicative generalized derivations. Glasnik matematički. Serija 3, ISSN 0017-095X, 2012, vol. 47, no. 1, str. 105-118. http://dx.doi.org/10.3336/gm.47.1.08. [COBISS.SI-ID 16341849] 5. BENKOVIČ, Dominik, EREMITA, Daniel. Multiplicative Lie n-derivations of triangular rings. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2012, vol. 436, iss 11, str. 4223-4240. http://dx.doi.org/10.1016/j.laa.2012.01.022. [COBISS.SI-ID 16278361]

http://dx.doi.org/10.1080/03081087.2013.877012


http://dx.doi.org/10.1007/s10468-013-9429-8




http://dx.doi.org/10.3336/gm.47.1.08




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