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Medpredmetno povezovanje je pomemben del poučevanja v šoli, vendar se zaradi pomanjkanja časa, znanja in idej izvaja redkeje, kot bi si želeli. Raziskave tudi kažejo, da imajo učitelji večkrat zmotno predstavo o tem, kaj medpredmetno povezovanje sploh je. Pri učencih s pomočjo medpredmetnih povezav namreč razvijamo načine povezovalnega razmišljanja, ki ga ne bodo uporabljali zgolj med šolanjem, temveč tudi v nadaljnjem življenju. Da pri pouku izvedemo smiselne medpredmetne povezave, moramo te prej skrbno načrtovati. Pri načrtovanju se moramo najprej vprašati, kaj želimo s tem načinom podajanja znanja sploh doseči, katere cilje različnih predmetov naj učenci dosežejo, in na podlagi tega načrtovati smiselne dejavnosti, ki bodo učence motivirale za delo. V magistrskem delu so opredeljene medpredmetne povezave, predstavljeni so različni modeli medpredmetnega povezovanja, pa tudi načrtovanje in izvajanje omenjenih povezav ter prednosti in pomanjkljivosti medpredmetnega povezovanja. Ker smo se osredotočili na prvo vzgojno-izobraževalno obdobje, so v teoretičnem delu predstavljene tudi značilnosti otrok v omenjenem obdobju. Prav tako so v zadnjem delu teoretičnega dela predstavljene medpredmetne povezave matematike z ostalimi učnimi predmeti prvega vzgojno-izobraževalnega obdobja. Empirični del je zasnovan na kvantitativni raziskavi, katere glavni namen je raziskati pogostost izvajanja medpredmetnih povezav z matematiko, načine načrtovanja in izvajanja medpredmetnih povezav z matematiko ter kompetentnost in motiviranost učiteljev za izvajanje medpredmetnih povezav z matematiko. Navsezadnje nas je zanimalo tudi, katere predmete in katere matematične vsebine učitelji povezujejo s posameznimi predmeti prvega vzgojno-izobraževalnega obdobja. V raziskavo je bilo vključenih 127 učiteljev in učiteljic prvega vzgojno-izobraževalnega obdobja osnovne šole. Rezultati anketnega vprašalnika so pokazali, da velika večina učiteljev pri svojem poučevanju pogosto izvaja medpredmetne povezave z matematiko. Ugotovili smo, da se med učitelji z različno delovno dobo ne pojavljajo razlike v pogostosti izvajanja medpredmetnih povezav. Učitelji medpredmetne povezave z matematiko najpogosteje načrtujejo v tedenskih in dnevnih pripravah ter so pri tem najbolj pozorni na same dejavnosti, šele nato svojo pozornost usmerijo na učne cilje, vsebine, pripomočke ipd. Njihovo načrtovanje pa največkrat temelji na učnih načrtih. Rezultati anketnega vprašalnika so pokazali, da učitelji medpredmetne povezave najpogosteje izvajajo v procesu ponavljanja snovi, hkrati pa se izogibajo uporabe medpredmetnih povezav v procesu preverjanja in ocenjevanja znanja. Najpogostejši predmet, s katerim učitelji izvajajo medpredmetne povezave z matematiko v prvem vzgojno-izobraževalnem obdobju, je spoznavanje okolja, medtem ko najredkeje matematiko povezujejo s tujim jezikom, to je z angleščino, kar je lahko posledica tega, da tuji jezik na razredni stopnji najpogosteje poučuje drug učitelj. Med razlogi, zakaj izvajajo medpredmetne povezave z matematiko, jih je največ izpostavilo, da želijo s tem načinom poučevanja spodbujati učenje za vsakodnevno življenje in na podlagi tega tudi sami izhajajo iz življenjskih situacij, ki vsebujejo medpredmetne povezave. Pričakujemo, da bodo pridobljene ugotovitve dale učiteljem uvid v trenutno stanje izvajanja medpredmetnih povezav ter spodbudo za njihovo nadaljnjo uporabo in izobraževanje na tem področju. Učitelji lahko na podlagi podanih dejavnosti anketirancev za izvajanje medpredmetnih povezav z matematiko tudi sami dobijo ideje za uvajanje teh povezav v svoj pouk in posledično dvignejo kakovost svojega poučevanja ter navsezadnje poglobijo in osmislijo znanje svojih učencev. Cross-curricular link are important part of teaching in schools, but it is less often done than we would like due to lack of time, knowledge and ideas. Research also show that teachers often have misconceptions about what cross-curricular links are. With the help of cross-curricular links we want to develop pupils' integrative thinking that they will use not only during their schooling, but also in later life. In order to make meaningful cross-curricular links in the classroom, they need to be carefully planned in advance. When planning, we first need to ask ourselves what we want to achieve by using cross-curricular links in our classroom, what are the goals of the different subjects that pupils need to achieve, and then plan meaningful activities that will motivate the pupils to do their work. The master thesis defines cross-curricular links, presents different models of cross-curricular integration, discusses the planning and implementation of these links, and outlines the advantages and disadvantages of cross-curricular integration. As the focus is on the first educational period, the characteristics of children in this period are also presented in the theoretical part of the thesis. In the last part of the theoretical part of the thesis there are also presented cross-curricular links between mathematics and other subjects in the first educational period. The empirical part is based on a quantitative survey, the main purpose of which is to investigate the frequency of implementation of cross-curricular links in mathematics, the ways in which cross-curricular links in mathematics are planned and implemented, and the competence and motivation of teachers to implement cross-curricular links in mathematics. Finally, we were also interested in which subjects and which mathematics content teachers make links with each subject in the first educational period. The survey involved 127 teachers of the first educational period of primary school. The results of the questionnaire showed that the vast majority of teachers often make cross-curricular links with mathematics in their teaching. We found that there are no differences in the frequency of cross-curricular links between teachers with different length of service. Teachers most often plan cross-curricular links in mathematics in weekly and daily preparation, paying most attention to the activities themselves, before focusing their attention on learning goals, content, tools, etc. Their planning is most often based on the curriculum. The results of the questionnaire showed that teachers most often use cross-curricular links when practising the material and avoid using them in the process of assessment and evaluation. The most frequent subject with which teachers make cross-curricular links to mathematics in the first educational period is science, and the least frequent subject with which they make cross-curricular links to mathematics is the foreign language English, which may be due to the fact that the foreign language English is most often taught by a different teacher at the primary level. Among the reasons why they implement cross-curricular links with mathematics, most teachers indicated that they want to promote learning for everyday life through this way of teaching and, as a result, they themselves arise from life situations that involve cross-curricular links. The findings are expected to give teachers insight into the current state of the implementation of cross-curricular links and to stimulate further use and training in this area. Teachers themselves can draw on the respondents' activities for cross-curricular links in mathematics to get ideas for introducing these links into their own classrooms and, as a consequence, to raise the quality of their teaching and, ultimately, to deepen and deepen their pupils' knowledge of mathematics.

V pričujočem gradivu za študente bomo predstavili dele nekaterih časopisnih člankov, ki se nanašajo na profesorja dr. Albina Belarja (1864–1939), velikega znanstvenika, ki je deloval na naših tleh. Predvsem se je proslavil kot seizmolog. Na ljubljanski realki je po potresu, ki je leta 1895 prizadel Ljubljano, ustanovil seizmološko opazovalnico in mednarodni znanstveni seizmološki časopis.

V gradivu je zbranih nekaj nalog, ki so podobne tistim, ki so bile dane v reševanje tekmovalcem na raznih olimpijadah, pa tudi druge. Takoj za vsako nalogo je tudi rešitev. Morda boste našli tudi svojo rešitev, kar bo samo pohvalno. Nekatere naloge so težje, nekatere se zlahka rešijo z nekim trikom, vse pa je odvisno od znanja in izkušenosti reševalca. Za razumevanje nalog in njihovo reševanje se predpostavlja solidno znanje elementarne matematike, redkokdaj višje matematike. Kjer se le da, je dovoljeno uporabljati tudi geometrijo.

Povod za nastanek dela Strokovno didaktične usmeritve za poučevanje matematike v prilagojenem izobraževalnem programu z nižjim izobrazbenim standardom (v nadaljevanju PP NIS) je v letu 2022 potrjen učni načrt za matematiko za prilagojeni izobraževalni program z nižjim izobrazbenim standardom. K oblikovanju usmeritev nas je spodbudilo tudi dejstvo, da za učence in učitelje v tem programu primanjkuje didaktičnih gradiv. V podporo razvoju inkluzivne šole smo se odločili oblikovati strokovno didaktične usmeritve za poučevanje matematike v PP NIS. Učiteljem, ki poučujejo matematiko v tem programu, kot tudi ostalim učiteljem, ki se v procesu poučevanja srečujejo z učenci z lažjo motnjo v duševnem razvoju in tistimi, pri katerih je prisotno sopojavljanje več motenj in so vključeni v druge izobraževalne programe, želimo ponuditi specialno didaktične rešitve pri poučevanju matematičnih pojmov, strategij in spretnosti. Strokovno didaktične usmeritve smo organizirali v dve razčlenjeni poglavji: specialno pedagoška zasnova pouka matematike v PP NIS in strokovno didaktična obravnava matematičnih vsebin po sklopih iz učnega načrta. Usmeritve poučevanja smo zasnovali na novejših specialno didaktičnih in didaktično matematičnih spoznanjih, ki v čim večji meri upoštevajo posebne potrebe učencev in opolnomočijo učitelje za kakovostno vnašanje sprememb v proces poučevanja. Kakovostno poučevanje matematike omogoča učitelju izvajanje individualizacije in diferenciacije, ki temeljita na posebnih potrebah učencev in identifikaciji predhodnih znanjih, ki so opredeljena v učnem načrtu, saj so posamezni pojmi in postopki sistematično nanizani od prvega razreda naprej. Enotno rabo matematičnih izrazov podpirajo opredelitve izrazov pred matematičnimi vsebinskimi sklopi, kot so opredeljeni v učnem načrtu. Strokovno didaktične usmeritve učitelje spodbujajo k uvajanju inovativnih rešitev in raznolikih prilagoditev v skladu s posebnimi potrebami učencev.

Didaktični pripomočki omogočajo konkretizacijo zahtevnih in abstraktnih vsebin pri pouku in so zato na razredni stopnji poučevanja vsakdanji spremljevalec učitelja. Pri matematiki so nepogrešljivi, saj učencem omogočajo boljšo predstavo in lažje razumevanje matematičnih pojmov. V magistrskem delu so zajete vsebine s področja didaktičnih pripomočkov. Predstavljena je vloga didaktičnih pripomočkov in razlike v terminih, ki se pojavljajo v literaturi. Navedene in opisane so teorije, ki zagovarjajo uporabo didaktičnih pripomočkov pri pouku nasploh (Piagetova teorija o spoznavnem razvoju, teorija izkustvenega učenja) in nekatere, ki se nanašajo na rabo didaktičnih pripomočkov predvsem pri poučevanju matematike (Hejnyjeva teorija, Dienesova in Goldingova načela idr.). Opisani so tudi dejavniki izbire didaktičnih sredstev in njihov vpliv na stopnjo motivacije učencev za pouk in učenje. Predstavljeni so didaktični pripomočki, ki jih za pouk matematike priporoča Zavod Republike Slovenije za šolstvo. Podrobneje so opisani nekateri didaktični pripomočki za razvoj številskih predstav v 2. in 3. razredu (številski trak in stotični kvadrat). Predstavljene so tudi dinamične različice teh didaktičnih pripomočkov, ki so izdelane iz materialov ponovne rabe in drugih tehničnih materialov. V empiričnem delu magistrskega dela so prikazane ugotovitve raziskave izvedene na vzorcu 128 učencev. Namen raziskave je bil ugotoviti, kakšne so številske predstave drugošolcev in tretješolcev in ali se te razlikujejo glede na uporabo različnih didaktičnih pripomočkov (priporočenih in dinamičnih). Cilj je bil tudi ugotoviti, ali se pojavljajo morebitne razlike v stopnji motivacije za delo s priporočenimi in z dinamičnimi didaktičnimi pripomočki. V kontrolnih skupinah so bili učenci deležni rabe priporočenih, v eksperimentalnih skupinah, dinamičnih didaktičnih pripomočkov. Rezultati so pokazali, da so drugošolci in tretješolci eksperimentalne skupine glede številskih predstav napredovali v višji meri kot učenci kontrolnih skupin. Prav tako so rezultati pokazali, da je stopnja motivacije za delo z dinamičnimi didaktičnimi pripomočki višja kot za delo s priporočenimi oblikami le teh. Ugotovitve raziskovalnega dela so pomemben podatek za učitelje razrednega pouka, kajti usmerjajo nas v vključevanje čim bolj konkretnih pripomočkov pri poučevanju. Didactic tools enable concretization of abstract subject matters. That is the reason why they are an indispensable companion of classroom teachers in the first years of primary school. In math they enable the comprehension of abstract concepts. The thesis deals with content-related didactic tools. Their role in math, the review of different terms, the classification and the key factors of selection for a teaching lesson are featured. Some theories and researches which advocate the use of didactic tools in teaching are listed, such as Piaget’s theory of cognitive development, the importance of experiential learning, Hejney’s theory of teaching math, Dienes and Golding’s principles, etc. The thesis illustrates some didactic tools recommended to develop the comprehension of number representations in the second and third grade of primary school. They are listed and published in the catalogue of the National Educational Institute of the Republic of Slovenia and a term typical didactic tools has been applied for them. Furthermore, dynamic versions of these same didactic tools made of reused and technical materials are presented and illustrated with pictures. In the thesis the term dynamic didactic tools refers to these. The main difference between the two groups of didactic tools is the level of concreteness. The empirical part of the thesis shows the findings of a research based on a sample of 128 pupils. The purpose of the study was to compare the understanding of number representations in second and third graders as regards the use of either typical or dynamic didactic tools. In addition to this, the thesis aims at determining potential differences in the level of students’ motivation for the use of each type of didactic tool. The control groups consisted of students working with typical didactic tools, whereas the experimental groups were offered dynamic ones. The results showed that the second and third graders who worked with dynamic didactic tools in the period of three weeks showed a better understanding of number representations and a higher level of motivation as opposed to their control group peers. Key findings of the role of dynamic didactic tools in the empirical part of the thesis provide an important indication for classroom teachers, namely to integrate didactic tools which are as concrete as possible in their teaching.

Matematika ima kot eden izmed najpomembnejših učnih predmetov temeljno vlogo pri razvoju ključnih kompetenc, ki jih učenec potrebuje za razvoj celovite osebnosti. Ne gre zgolj za obvladovanje rutinskih postopkov, temveč tudi za sposobnost reševanja problemov, iskanje ustreznih strategij, razvijanje ustvarjalnosti itd. (Žakelj idr., 2011). Otrok prve izkušnje z matematiko pridobi že v predšolskem obdobju (Kurikulum za vrtce, 1999), še bolj načrtno pa se kompetence začnejo razvijati v 1. razredu osnovne šole, v katerem naj bi pouk izhajal iz učenčevih konkretnih izkušenj in znanj, pri čemer si za boljšo motivacijo in razumevanje učitelji lahko pomagajo z realnimi primeri, konkretnimi ponazorili in raznimi didaktičnimi pripomočki (Žakelj idr., 2011). Zadnji dve desetletji se kot dober didaktični pripomoček za podporo učencem pri razumevanju matematičnih konceptov vse bolj uveljavljajo slikanice (Hong, 1996 Haury, 2001 National Council of Teachers of Mathematics [NCTM], 2000). V magistrskem delu smo pisali o matematični slikanici z naslovom 70 češenj, ki je bila ustvarjena v okviru projekta na Pedagoški fakulteti Univerze v Ljubljani, ni pa še bila preizkušena v praksi. Zanimalo nas je, ali matematična slikanica v praksi dosega cilje, za katere je bila ustvarjena. Teoretični del je sestavljen iz petih poglavij. V prvem poglavju smo predstavili poučevanje predmeta matematika in se osredinili predvsem na predstavitev didaktičnih pripomočkov. V drugem delu smo predstavili slikanice na splošno, v tretjem delu pa smo se usmerili bolj specifično, in sicer na matematične slikanice. Opisali smo njihove značilnosti ter opredelili njihovo uporabnost pri pouku. Združili in povzeli smo glavna spoznanja različnih raziskav s področja uporabe matematičnih slikanic za namen učenja matematike. V četrtem poglavju smo predstavili matematično slikanico z naslovom 70 češenj, ki je uporabljena v empiričnem delu in je osrednji predmet tega dela. Opisali smo, kako je nastala, njeno vsebino in teoretično ozadje. V zadnjem sklopu smo predstavili koncept števila nič, ki je tema matematične slikanice. Opisali smo zgodovino njegovega razvoja, kako učenci razumejo število nič in navedli njihove napačne predstave. Tudi tukaj smo povzeli nekatera spoznanja najvidnejših raziskav s tega področja. V empiričnem delu smo raziskali, kako matematična slikanica 70 češenj kot didaktični pripomoček pomaga pri razumevanju koncepta števila nič pri prvošolcih, kako se le-ti odzovejo na pouk matematike ob uporabi slikanice ter kako slikanico sprejemajo učitelji prvega triletja. Uporabili smo kvantitativni in kvalitativni raziskovalni pristop. Rezultati raziskave so pokazali, da se z uporabo matematične slikanice 70 češenj izboljša učenčevo razumevanje različnih vidikov števila nič. Učenci se pri učni uri z vključeno matematično slikanico počutijo prijetno in nimajo občutka, da bi se učili (zgolj) matematiko, ampak tudi druge učne predmete. Učenci prepoznavajo tudi njen vzgojni vidik. V raziskavi smo tudi ugotovili, da učitelji prvega triletja matematično slikanico 70 češenj prepoznavajo kot uporaben didaktični pripomoček pri poučevanju matematike in tudi drugih učnih predmetov. Prepoznavajo motivacijsko in vzgojno funkcijo tega didaktičnega pripomočka, težave pa so se pokazale pri prepoznavanju izobraževalne funkcije. Učitelji slikanici sicer pripisujejo pomembno vlogo, a se med njimi pojavljajo tudi napake pri razumevanju matematične vsebine. Raziskava je pomembna za večjo prepoznavnost tega, v Sloveniji še relativno novega, a zelo uporabnega didaktičnega pripomočka. Pričakujemo, da bo vplivala na uporabo matematičnih slikanic v vlogi didaktičnega pripomočka pri pouku matematike. Menimo, da bi bilo potrebno tako učitelje razrednega pouka kot tudi študente te smeri seznaniti z matematičnimi slikanicami in jim ponuditi smernice za uporabo. As one of the most important subjects, mathematics has a fundamental role in the development of key competencies that a student needs to develop a holistic personality. It is not just about mastering routine procedures, but also about the ability to solve problems, finding appropriate strategies, developing creativity, etc. (Žakelj et al., 2011). The child acquires the first experience with mathematics as early as the pre-school period (Kurikulum za vrtce, 1999) even more systematically, competencies begin to develop in the 1st grade of primary school, where lessons should be based on students' concrete experience and knowledge. Teachers can improve the students’ motivation and understanding by means of real examples, concrete illustrations, and various didactic aids (Žakelj et al., 2011). In the last two decades, picture books have become increasingly popular as a good didactic tool to enhance the students’ understanding of mathematical concepts (Hong, 1996 Haury, 2001 National Council of Teachers of Mathematics [NCTM], 2000). In our master's thesis, we write about a mathematical picture book, entitled ‘70 češenj,’ which was written as part of a project at the Faculty of Education, University of Ljubljana, but has not yet been tested in practice. We are interested in whether the mathematical picture book achieves its practical goals. The theoretical part consists of five chapters. In the first part, we present the teaching of the subject of mathematics and are focused mainly on the presentation of didactic aids. In the second part, we offer a presentation picture books in general, while in the third part, we aim our focus more specifically on the mathematical picture books. We describe their characteristics and define their usefulness in teaching. We have combined and summarized the main findings of various research studies in the field of the use of mathematical picture books for the purpose of learning mathematics. In the fourth part, we present a mathematical picture book, entitled ‘70 češenj,’ which is used in the empirical part and is the central subject of this paper. We describe the manner in which it was created, including both its content and the theoretical background. In the last part, we present the concept of the number zero, which is the theme of the mathematical picture book. We explain the history of its development, the way students understand the number zero, and list their misconceptions. We have summarized some of the findings of the most prominent research studies in this field. In the empirical part we investigate how the mathematical picture book ‘70 češenj’ as a didactic tool helps to understand the concept of the number zero in first graders, how they respond to mathematics lessons using the picture book, and how the picture book is received by the teachers of the first three years. We use a quantitative and qualitative research approach. Research results show that using the mathematical picture book ‘70 cherries’ improves the student's understanding of various aspects of the number zero. Students feel comfortable during lessons when mathematical picture book is included and do not have the feeling of learning (merely) mathematics, but other subjects also. Students likewise recognize its educational aspect. Our research also finds that teachers in the first three years recognize the mathematical picture book ‘70 češenj’ as a useful didactic tool in teaching mathematics and other subjects. They recognize the motivational and educational function of this didactic tool, and difficulties have been shown in recognizing the educational function. Teachers attach great importance to it, but there are also errors in understanding the mathematical content among them. Our research is important for rendering more recognizable of what is a relatively new but very useful didactic tool in Slovenia. It is expected to influence the use of mathematical picture books as a didactic tool in mathematics lessons. We believe that it would be necessary to acquaint both classroom teachers and students of this field with mathematical picture books and offer them guidelines for their use.

Šolski leti 2019/2020 ter 2020/2021 sta bili zaznamovani z epidemijo COVID-19. Zaradi ukrepov za zajezitev epidemije je pouk v šolskem letu 2019/2020 od marca do maja ter v šolskem letu 2020/21 od oktobra do konca januarja potekal na daljavo. Tako so bili učitelji primorani uporabiti popolnoma drugačen pristop, kot so ga bili vajeni. Poučevanje se je iz učilnic prestavilo na splet, klopi pa so zamenjali računalniki. V magistrskem delu smo se osredotočili na motivacijo nadarjenih učencev med poučevanjem na daljavo. Teoretični del magistrskega dela je sestavljen iz treh sklopov. V prvem sklopu smo predstavili značilnosti pouka matematike. V drugem delu smo predstavili nadarjene učence pri pouku matematike. Tako kot učenci s primanjkljaji na posameznih področjih potrebujejo posebno pozornost učitelja, dopolnilni pouk in prilagojene naloge, da lahko napredujejo pri snovi, pa nadarjeni učenci potrebujejo dodatne spodbude, problemske naloge, dodatni pouk, da v celoti razvijejo svoje potencialne. V zadnjem, tretjem delu, smo se osredinili na poučevanje na daljavo. Dotaknili smo se uporabe IKT v izobraževanju, na kratko opisali zgodovino izobraževanja na daljavo, posebno pozornost pa smo namenili tudi poučevanju matematike na daljavo. V empiričnem delu smo raziskovali, na kakšen način je potekal pouk matematike v 4. in 5. razredih osnovnih šol pri poučevanju na daljavo ter na kakšen način so učitelji prilagajali pouk matematike nadarjenim učencem. Raziskava je bila sestavljena iz dveh delov. V prvem delu smo uporabili kvantitativni pristop. Z anketami smo preverili mnenje učiteljic in učiteljev glede učinkovitosti poučevanja na daljavo. Zanimalo nas je, ali je pouk na daljavo vplival na način dela učiteljev z nadarjenimi učenci pri pouku matematike. Prav tako smo želeli preveriti, ali obstajajo razlike pri učiteljih 4. in 5. razreda glede mnenja o njihovi usposobljenosti za prilagajanje pouka na daljavo nadarjenim učencem marca 2020 in danes ter ali obstajajo razlike pri učiteljih 4. in 5. razreda glede mnenja o njihovi usposobljenosti za prilagajanje pouka na daljavo nadarjenim v razredu in na daljavo. V drugem delu smo z intervjuji skušali dobiti primere dobre prakse za poučevanje nadarjenih učencev na daljavo pri pouku matematike. Rezultati so pokazali, da je pouk na daljavo vplival na delo učiteljev z nadarjenimi učenci pri pouku matematike. Delež učiteljev, ki je prilagajal pouk matematike nadarjenim učencem na daljavo je bil namreč manjši kot delež učiteljev, ki prilagajajo pouk matematike nadarjenim učencem v razredu. Pokazale pa so se tudi razlike med samim načinom prilagajanja. Učitelji namreč pri prilagajanju pouka matematike v razredu uporabljajo predvsem dodatni pouk in vrstniško pomoč, pri delu na daljavo pa se teh dveh načinov prilagajanja v večini niso posluževali. Pri analizi vprašanj iz intervjuja smo dobili tudi primere dobre prakse, na kakšne načine prilagajati pouk matematike nadarjenim učencem na daljavo. Nekatere od teh smo tudi predstavili. The 2019/2020 and 2020/2021 school years were marked by the COVID-19 epidemic. As a result of the epidemic, lessons in the school year 2019/2020 from March to May and in the school year 2020/21 from October to the end of January were transferred online. Thus, teachers were forced to take a completely different approach than they were used to. Teaching moved from classrooms to the web and desks were replaced by computers. In this master’s thesis we focused on the motivation of talented pupils while teaching remotely. The theoretical part of the master's work consists of three sets. In the first set we presented the characteristics of mathematics lessons. In the second part we introduced talented students in maths class. Just as students with deficits in specific areas need special attention from the teacher, complementary instruction and tailored tasks to progress in substance, talented students need additional incentives, problem tasks, and additional instruction to fully develop their potential. In the last, third part, we focused on distance teaching. We touched on the use of ICT in education, briefly described the history of distance education, and we paid special attention to distance mathematics teaching. In the empirical work we explored how the math lessons took place in the 4th and 5th classes during distance teaching and how teachers adapted math lessons for gifted pupils. The study consisted of two parts. In the first part we used a quantitative approach. The surveys examined the opinion of teachers and teachers on the effectiveness of distance teaching. We wanted to know whether remote lessons influenced the way teachers work with talented students in maths lessons. However, we wanted to check whether there are differences in teachers in grade 4 and 5 in terms of their competence to adapt to remote training for talented pupils in March 2020 and today, and whether there are differences in teachers in grade 4 and 5 in terms of their ability to adapt distance lessons for talented pupils in classrooms and at a distance. In the second part, we tried to obtain examples of good practice for teaching talented pupils at a distance through interviews. The results showed that distance learning had an impact on the work of teachers with gifted pupils in maths lessons. The proportion of teachers adapting maths lessons for gifted pupils at a distance was lower than the proportion of teachers adapting to maths classes for gifted pupils in the classroom. In addition, there were differences between the ways of adaptation itself. Teachers mainly use supplementary lessons and peer support when adapting maths lessons in the classroom, but mostly they did not use these two methods of adaptation during distance learning. In analysing the questions from the interview, we received examples of good practice in how to adapt math lessons for gifted pupils at a distance. We have presented some of these.

V pričujočem gradivu bomo spoznali Filona iz Bizanca, ki je bil za današnje pojme vojaški inženir, lahko pa rečemo, da je bil tudi fizik in pisatelj. Pri svojem delu je naletel na nekatere matematične probleme in jih dokaj uspešno rešil. Po njem se v matematiki imenuje Filonova premica, ki jo bomo obravnavali v gradivu. Filonova premica je v zvezi s starodavnim problemom podvojitve kocke, pa tudi z nekaterimi ekstremalnimi problemi.

Števnike vseh vrst in števila srečujemo tako v domači kot tudi v svetovni književnosti. V gradivu bomo skušali predstaviti tiste, ki jih je Homer uporabil v Iliadi. Števila v tej obsežni pesnitvi niso nič posebnega, kajti večina je namreč takih, ki jih še dandanes uporabljamo v vsakdanjih pogovorih in katerih velikost je običajnim ljudem zlahka predstavljiva.

Formativno spremljanje je pomemben del poučevanja v šoli, vendar je zaradi časovne stiske večkrat opuščeno ali nestrukturirano. Namen formativnega spremljanja je podajanje povratnih informacij učencem, s čimer lahko učencem pomagamo pri doseganju zastavljenih ciljev. V magistrskem delu je predstavljeno preverjanje in ocenjevanje znanja pri matematiki s poudarkom na formativnem spremljanju kot načinu za izboljšanje učne motivacije in uspeha učencev kot tudi možnosti za boljše prilagajanje nadaljnjega učnega procesa učencem. Pri tem smo se sklicevali na že opravljene raziskave s tega področja in ostalo strokovno literaturo. Empirični del je osnovan na kvantitativni raziskavi, katere glavni namen je bil raziskati načine, vzroke in pogostost izvajanja formativnega spremljanja pri pouku matematike učiteljev in učiteljic 2. in 5. razreda javne osnovne šole. Zanimalo nas je, ali učitelji in učiteljice v 5. razredu pogosteje kot v 2. razredu izvajajo formativno spremljanje pri matematiki ter ali za to uporabljajo drugačne načine izvedbe, torej druge tehnike. Poleg tega smo raziskovali, kako pogosto in na kakšne načine učitelji in učiteljice izvajajo formativno spremljanje pri pouku matematike, s katerim namenom izvajajo formativno spremljanje pri matematiki, ali in kako prilagajajo formativno spremljanje različnim dejavnikom (individualnim značilnostim učencev, specifiki učne vsebine, pouku na daljavo) ter kako kompetentne se počutijo učitelji in učiteljice za izvajanje formativnega spremljanja pri pouku matematike. Potrebne podatke pa smo pridobili s spletno anketo Enka, torej z anketnim vprašalnikom. Ugotovili smo, da se med učitelji in učiteljicami 2. in 5. razreda pojavljajo statistično pomembne razlike le v pogostosti izvajanja formativnega spremljanja, ne pa tudi v načinih izvedbe. Učitelji in učiteljice 5. razredov izvajajo formativno spremljanje pogosteje, kot učitelji in učiteljice 2. razredov. Učitelji in učiteljice najpogosteje uporabljajo tehniki KŽN in Semafor, formativno spremljanje pa izvajajo z namenom ugotavljanja usvojenega znanja učencev. Rezultati ankete so pokazali, da učitelji in učiteljice načine izvedbe formativnega spremljanja prilagajajo predvsem individualnim značilnostim učencem, medtem ko specifiki učne vsebine in pouku na daljavo v večini prilagajajo le z različno pogostostjo izvedbe formativnega spremljanja. Učitelji in učiteljice se ne počutijo zelo kompetentne za izvajanje formativnega spremljanja pri pouku matematike in se v večini ne udeležujejo dodatnih izobraževanj na to temo. Temu sledi, da pri izvedbi samega formativnega spremljanja ne uporabljajo formalno pridobljenega znanja, ampak ga izvajajo po svoje. Pričakujemo, da bodo dobljene ugotovitve učiteljem in drugim pedagoškim delavcem osmislile (strukturirano) uporabo formativnega spremljanja, s pomočjo katerega lahko prilagajajo pedagoški proces individualnim značilnostim učencev, specifiki učne vsebine in drugim okoliščinam. Formative assessment is an important part of school teaching, but it is often abandoned or unstructured due to time constraints. The purpose of formative assessment is to provide feedback, which can help students achieve their goals. The master's thesis presents the pre-assessment and assessment of knowledge in mathematics with an emphasis on formative assessment as a way to improve students' learning motivation and success, as well as present an opportunity to more efficiently adapt the further learning process to students. The thesis refers to the already conducted research in this field as well as to other professional literature. The empirical part is based on quantitative research, the main purpose of which was to investigate the ways, causes and frequency of formative assessment in the process of teaching mathematics to 2nd and 5th grade students. The main interest is in whether teachers in the 5th grade perform formative assessment in mathematics more often than teachers in the 2nd grade, and whether they use different methods and techniques of implementation. In addition to that, we researched how often, in what ways, and with what purpose teachers carry out formative assessment in mathematics lessons, if and how they adapt formative assessment to different classroom factors (individual characteristics of students, specifics of learning content, distance learning) and the level of competence to perform formative assessment in mathematics lessons teachers feel they have. Teachers of 2nd and 5th grades were included in the research, and the necessary data were obtained with the Enka online survey, i.e. with the survey questionnaire. The results show statistically significant differences in the frequency of formative assessment between 2nd and 5th grade teachers- 5th grade teachers perform formative assessment more often than 2nd grade teachers, but not in the methods of implementation. Teachers most often use KŽN and Semafor techniques, and they carry out formative assessment in order to determine the acquired knowledge of students. The results of the survey showed that teachers adapt the methods of formative monitoring to the individual characteristics of students, while the specifics of learning content and distance learning are mostly adjusted only with different frequency of formative assessment. Teachers acknowledge that they may not always feel competent enough to carry out formative assessment in mathematics lessons, however in most cases they do not attend any additional training on this topic. The conclusion is that the teachers do not use formally acquired knowledge when performing formative assessment, but rather implement it in their own way. It is hoped that the obtained findings will inspire teachers and other pedagogical workers to structurally start using formative assessment, through which they can adapt the pedagogical process to individual characteristics of students, the specifics of learning content and to other circumstances.

Given a bipartite cubic graph with a certain degree of symmetry two covering constructions that provide infinitely many tetravalent graphs admitting half-arc-transitive groups of automorphisms are introduced. Symmetry properties of constructed graphs are investigated. In the second part of the paper the two constructions are applied to the Heawood graph, the well-known incidence graph of the Fano plane.

S predloženim matematičnim modelom je predstavljeno gozdnogospodarsko območje, ki je v ekonomskem in organizacijskem smislu zaokrožen sistem z vsemi specifičnostmi proizvodnega procesa v gozdu. Za upravljanje in razvoj tako definiranega sistema od začetnega stanja k ciljnemu so v modelu iskane in oblikovane tiste strategije, ki imajo za posledico maksimalen in trajen donos. Ker pa je proces za usmerjanje razvoja sistema izrazito dolgoročen in večstopenjski, je zgradba modela zasnovana na uporabi kombinacije diskretnega dinamičnega in linearnega programiranja. Je preizkušen na majhnem homogenem elementu gozdne površine in nato na poenostavljenem primeru celotnega gozdnogospodarskega območja. The work establishes a mathematical model for forest management. A forest management area, which - from the economic and with all the particularity of wood production, is presented through this model. In order to develop and manage a model which has been defined in this way - from the beginning to the end, the work deals with strategies of forest management which result in maximal sustained yield. Since the process that direct the development of the system is rather long-term oriented and model has been created by the combination of discrete dynamic and linear programming. The model presented in this work was first tested on a small, homogeneous element of forest area and then on a simplified example area.

Na univerzitetno smer fakultetnega programa Elektrotehnika UN na Fakulteti za elektrotehniko Univerze v Ljubljani (UL FE) se vsako leto vpišejo dijaki z različnimi matematičnimi podlagami. V največjem številu se vpisujejo dijaki s splošnih ali strokovnih gimnazij oz. s srednjih strokovnih šol. Vpisanih je torej veliko dijakov z različnimi matematičnimi predznanji. Pomanjkljivo matematično predznanje na UL FE povzroča slabo razumevanje snovi pri najpomembnejših osnovnih predmetih, ki so ključnega pomena pri razumevanju nadaljnjih kompleksnejših tematik. Učenje nezadostnega znanja je učiteljem časovno zamudno, študentom pa primanjkljaj znanja povzroča težave pri izobraževanju. Nezadostno matematično znanje se razvija že od osnovne šole, zato je bilo v tej magistrski nalogi ključnega pomena začeti analizirati težavo že pri osnovnih šolah. Narejena je bila primerjava in analiza spreminjanja osnovnošolskih učnih načrtov od leta 1983 dalje, srednješolskih učnih načrtov od leta 1991 dalje in primerjava maturitetnih nalog od leta 2004 dalje. V srednjih šolah je analiza učnih načrtov potekala za gimnazije in srednje strokovne šole (SSI). Izvedena je bila tudi primerjava nalog pri predmetu matematike na splošni in poklicni maturi. V nadaljevanju magistrskega dela je prav tako pregled trenutnih ukrepov na UL FE, s katerimi si pomagajo pri poenotenju matematičnega predznanja. Analizirane tematike so bile razdeljene v tri sklope – Geometrija in merjenja, Aritmetika in algebra in Druge vsebine. Ta način razdelitve se pojavi že v nekaterih osnovnošolskih učnih načrtih, zato so na ta način razdeljene tudi srednje šolske tematike in maturitetne naloge. Eden izmed ukrepov na UL FE je t. i. Matematični uvod v študij elektrotehnike. V njem so zbrane najpomembnejše tematike, potrebne za uspešen prvi letnik na prvi stopnji. Tematike za Matematični uvod so izbrali profesorji, ki na UL FE poučujejo vse predmete v prvem letniku. Vse tematike, ki so obravnavane v tem predmetu, spadajo pod sklop Aritmetika in algebra, zato je tem tematikam pri analizi posvečeno več časa. Pri primerjavi sklopa Aritmetika in algebra pri osnovnošolskih učnih načrtih je bilo ugotovljeno, da se je z vsakim učnim načrtom količina snovi povečevala oz. se je zmanjšalo število ur, namenjenih matematiki. V gimnazijah se je kljub enakemu številu ur količina snovi z novejšim učnim načrtom povečala. V primerjavi s starejšim učnim načrtom SSI, se je z novejšim učnim načrtom iz leta 2007 število ur zmanjšalo. Poleg tega so velik del obveznih vsebin iz leta 1991 pri učnem načrtu leta 2007 premaknili med izbirne vsebine. Tako se je kot v gimnazijah in osnovnih šolah tudi v SSI število ur zmanjšalo. SSI se v primerjavi z gimnazijami in osnovnimi šolami razlikujejo v tem, da se število snovi z novejšim učnim načrtom ni povečalo, ampak zmanjšalo. Z zmanjšanjem snovi oz. premikom snovi med izbirne vsebine, je velik del ključnih tematik, potrebnih na UL FE, odpadel. To se pokaže tudi pri primerjavi nalog na poklicni in splošni maturi, saj izbirnih tematik s SSI na poklicni maturi niso obravnavali. UL FE je leta 2017 v prvem tednu začetka fakultetnega izobraževanja uvedla test matematičnega predznanja, na katerem učitelji dobijo vpogled v matematično predznanje vpisanih študentov. Iz rezultatov tega testa se vidi, kolikim študentom osnovne naloge povzročajo težave. Prav tako se vidi, da študentje s SSI v povprečju dosegajo slabše rezultate v primerjavi z gimnazijci. Manjšanje obsega ur ali povečava obsega snovi neugodno vplivata na znanje učencev. Z manjšim obsegom ur ostane manj časa za poglabljanje znanja oz. utrjevanje naučenega, s povečavo obsega snovi pa pride do prenasičenja podatkov, s katerimi učenci razpolagajo. Rezultat tega je pomanjkljivejše znanje, nezadostno za ustrezen začetek študija na UL FE. Problem bi se zmanjšal s povečanjem števila ur v osnovnih šolah, saj bi s tem učencem zagotovili več časa za ponavljanje in utrjevanje novih znanj. Hkrati se količina snovi ne bi smela povečati oz. še večji poudarek bi moral biti na sklopu Aritmetika in algebra. Tudi s tem bi se učencem olajšal prehod na srednjo šolo in kasneje na fakulteto. Each year, students with different knowledge levels of mathematics enroll in the Academic study programme of Electrical Engineering at the Faculty of Electrical Engineering, University of Ljubljana (UL FE). The majority of students come from general or vocational secondary schools which means there are a lot of different students with different levels of knowledge. Insufficient mathematical knowledge at the UL FE leads to poor understanding of the most important basic subjects that are crucial in understanding further, more complex topics. Reducing these disparities is time consuming for the professors, while students struggle with continuing their education. These differences in knowledge begin in primary school, and that is why it was crucial to analyse the problem of primary schools in this postgraduate thesis. This thesis carries out a comparison, and an analysis of curriculums in primary schools from 1983 on, curriculums in secondary schools from 1991 on, as well as a comparison of exercises in matura from 2004 on. The analysis of curriculums in secondary schools includes general and vocational secondary schools (SSI). Furthermore, an analysis of exercises in splošna and poklicna matura was carried out. An overview of current measures taken by UL FE is included, to show how they are managing a standardization of prior knowledge of mathematics. Analysed topics are divided into three groups – Geometry and measuring, Arithmetics and algebra, and Other topics. This type of categorizing appears in some primary schools’ curriculums that is why secondary school topics and exercises from matura are categorized the same way. One of the measures at UL FE is a course Matematični uvod v študij elektrotehnike. It includes the most important topics a student needs to successfully finish first year of undergraduate study. Topics for this course were chosen by professors who teach all first year classes at UL FE. All topics, covered by this course, belong in Arithmetics and algebra, and that is why we focus more on this group in the analysis. When comparing Arithmetics and algebra in primary school curriculums it was apparent that each curriculum increased the amount of topics to be covered, and at the same time decreased the number of hours of mathematics. General secondary schools generally kept the same number of hours, devoted to mathematics, but increased the amount of topics to be covered. In comparison with older curriculums in vocational secondary schools, the new curriculum from 2007 decreases the number of hours for mathematics. In addition, a large part of obligatory topics in 1991 were made elective with curriculum of 2007. This consequently decreased the number of hours for mathematics in both general and vocational secondary schools. Curriculum in vocational secondary schools, in comparison with general secondary schools, did not increase number of covered topics, but rather decreased it. It changed obligatory topics to elective topics and thusly a large part of topics, needed at UL FE, was not covered. This is apparent in comparison of exercises in splošna and poklicna matura, since elective topics are never among exercises chosen for the matura exam. In 2017, UL FE started a test of prior mathematical knowledge that is carried out in the first week of the first year. This way, the professors get insight into prior mathematical knowledge of new students, and see if basic mathematical tasks cause any difficulties. It is also clear the students from vocational schools in general achieve worse results than students from general schools. Decreasing the amount of hours, or increasing the number of topics to be covered has significant adverse effect on the knowledge of students. Less hours of classes mean less hours of deepening the knowledge, and the increase in topics cause an overabundance of data the students have at their disposal. Consequently, this leads to inadequate knowledge, not sufficient for a good beginning at UL FE. This problem could be diminished by increasing the number of hours for mathematics in primary schools, because it would provide the time for necessary revising and intensifying the knowledge. At the same time, the quantity of topics covered should not increase. Rather, a bigger emphasis should be put on the group Arithmetics and algebra. This would make the transition to secondary schools, and later to universities much easier.

The article deals with mathematical literacy in relation to mathematical knowledge and mathematical problems, and presents the Slovenian project NA-MA POTI, which aims to develop mathematical literacy at the national level, from kindergarten to secondary education. All of the topics treated represent starting points for our research, in which we were interested in how sixth-grade primary school students solve non-contextual and contextual problems involving the same mathematical content (in the contextual problems this content still needs to be recognised, whereas in the non-contextual problems it is obvious). The main guideline in the research was to discover the relationship between mathematical knowledge, which is the starting point for solving problems from mathematical literacy (contextual problems), and mathematical literacy. The empirical study was based on the descriptive, causal and non-experimental methods of pedagogical research. We used both quantitative and qualitative research based on the grounded theory method to process the data gathered from how the participants solved the problems. The results were quantitatively analysed in order to compare the success at solving problems from different perspectives. Analysis of the students’ success in solving the contextual and non-contextual tasks, as well as the strategies used, showed that the relationship between mathematical knowledge and mathematical literacy is complex: in most cases, students solve non-contextual tasks more successfully; in solvingcontextual tasks, students can use completely different strategies from those used in solving non-contextual tasks; and students who recognise the mathematical content in contextual tasks and apply mathematical knowledge and procedures are more successfulin solving such tasks. Our research opens up new issues that need to be considered when developing mathematical literacy competencies: which contexts to choose, how to empower students to identify mathematical content in contextual problems, and how to systematically ensure –including through projects such as NA-MA POTI –that changes to the mathematics curriculum are introduced thoughtfully, with regard to which appropriate teacher training is crucial.

Namen te raziskave je preučiti vlogo didaktičnih sredstev pri učenju in poučevanju množenja in deljenja v drugem razredu osnovne šole na Kosovu. Na Kosovu veliko učiteljev še vedno uporablja tradicionalno metodo poučevanja matematike, kar pomeni, brez uporabe didaktičnih sredstev, ampak le knjige, zvezke, mize in krede. Če bi učitelji med poučevanjem matematike uporabljali različna manipulativna orodja, bi to otrokom pomagalo do boljšega in lažjega razumevanja. Uporaba didaktičnih sredstev pozitivno vpliva tudi na motivacijo učencev med poučevanjem in učenjem matematike. Zaradi konkretne narave ponazoril so le-ta še posebej primerna v vrtcih in nižjih razredih osnovne šole. Namen naše raziskave je raziskati vpliv učnega pristopa na razumevanje množenja in delitve učencev: primerjali smo dva različna pristopa: tradicionalnega, ki se uporablja v večini kosovskih osnovnih šol in pristop poučevanja in učenja z uporabo didaktičnih sredstev. Raziskava je bila izvedena v izbrani osnovni šoli, vključena sta bila dva oddelka drugega razreda. Prvi oddelke je predstavljal kontrolno skupino, drugi pa eksperimentalno skupino. Pri delu z učenci so bila v eksperimentalni skupini uporabljana didaktična sredstva, pri kontrolni skupini pa le šolski učbeniki. Podatki so bili pridobljeni iz testov znanja za množenje in deljenje, ki so bili ivedeni pred in po eksperimentu. Poleg tega so bili podatki pridobljeni tudi iz vprašalnika za kosovske učitelje drugega razreda osnovnih šol. Rezultati so pokazali, da uporaba didaktičnih sredstev pozitivno vpliva na dosežke učencev pri reševanju nalog množenja in deljenja. Na osnovi analize vprašalnikov, razdeljenih učiteljem, ugotovljamo, da lokacija prebivališče ne vpliva na to, v kolikšni meri učitelji uporabljajo didaktična sredstva, po drugi strani pa starost učiteljev vpliva na njihovo uporabo med poučevanjem množenja in deljenja v drugem razredu - mlajši učitelji ponavadi uporabljajo več didaktičnih sredstev kot starejši učitelji. The purpose of this research is to examine the role of manipulative tools in learning and teaching multiplication and division in second grades in Kosovo. In Kosovo there are a lot of teachers that still use the traditional method for teaching mathematics to children. Traditional method consists of learning a concept by observing the teacher “do” problems at the board and copying down the steps during note taking. Use of manipulative tools presents a more effective approach towards developing students' understanding of mathematical concepts. The usage of manipulative tools also has positive affects on students’ motivation while teaching and learning mathematics. Concrete nature of manipulatives makes them particularly appropriate for kindergartners and young elementary school children. The purpose of our research is to investigate the influence of the teaching approach on pupils’ understanding of multiplication and division: two different methods were compared, a traditional method which is used in most Kosovo schools and method of teaching and learning through use of manipulative tools. This research was conducted in a selected primary school and two second-grade classes were included. The first one was the experimental group and the other one was the control group. While working with pupils, manipulative tools were used with the experimental group whereas only school textbooks were used with the control group. The data has been derived from the assessment of the knowledge test for multiplication and division that was conducted before and after the experiment. Additionally, data were also extracted from a questionnaire for second grade teachers The results show that the use of manipulative tools positively impacts the students' results in solving multiplication and division tasks. From the analysis of the questionnaires distributed to the teachers, it has been concluded that the place of residence does not affect the extent to which the teachers use manipulative means, but on the other hand the age of teachers influences the use of manipulative tools during the teaching of multiplication and division in the second grade – younger teachers tend to use more manipulatives the elder teachers’.

Eden izmed najpomembnejših in najvplivnejših dejavnikov iz okolja v obdobju poznega otroštva je gibanje, ki učinkuje na celostni razvoj učencev, posledično vpliva na šolsko uspešnost in motivacijo. Gibanje je tudi ena izmed otrokovih osnovnih potreb, zato v prvih letih osnovne šole učenci ne zmorejo daljšega sedenja, s sedenjem upada tudi koncentracija za učenje. Z gibanjem otroci spoznavajo svet, zato bi morali biti učitelji še posebej pozorni na redno in premišljeno vključevanje gibanja v poučevanje. V teoretičnem delu je na začetku predstavljen otrokov razvoj in faze kognitivnega razvoja, ki predstavljajo izhodišče za načrtovanje aktivnosti za učence. Opisani so vplivi gibalnih/športnih aktivnosti na posamezna področja razvoja. Povzete so teorije, ki spodbujajo vključevanje gibalnih aktivnosti v pouk, predstavljeni so načini, kako lahko učitelj gibanje vključi v pouk. Podana so načela poučevanja matematike in razlogi za vključevanje gibalnih aktivnosti v pouk matematike. V empiričnem delu smo predstavili izvedbo 4 ur pouka matematike v treh oddelkih 2. razreda osnovne šole, kar predstavlja skupaj 61 učencev. V učne ure smo vključevali gibalne aktivnosti, saj smo raziskovali, kakšen učinek ima vključevanje gibalnih aktivnosti pri urah matematike na znanje vseh učencev drugega razreda, med drugim nas je zanimal tudi učinek vključevanja gibalnih aktivnosti na znanje različno uspešnih drugošolcev. Uporabili smo kvantitativni pristop pedagoškega raziskovanja. Napredek smo ugotavljali pri aritmetiki, natančneje pri matematičnih vsebinah: naravnih številih do 100 in številu 0, urejanju števil po velikosti, številskih odnosih, seštevanju in odštevanju v množici naravnih števil do 100 brez prehoda ter pri iskanju neznanega člena v obsegu števil do 20. Rezultati raziskave so pokazali, da vključevanje gibalnih aktivnosti pri utrjevanju matematičnih vsebin pozitivno vpliva na napredek v znanju skoraj vseh učencev, saj je na končnem preverjanju znanja večina učencev (85,3 %) napredovala v znanju pri vsaj eni učni vsebini glede na predtest. Pri ugotavljanju učinka vključevanja gibalnih aktivnosti na znanje različno uspešnih učencev, ki smo jih razdelili v skupine glede na uspešnost na predtestu, smo ugotovili, da ima utrjevanje izbranih matematičnih vsebin z gibalnimi aktivnostmi najmanjši učinek na učence z najboljšim matematičnim predznanjem, medtem ko ima največji učinek na učence z najslabšim matematičnim predznanjem. Pri urah utrjevanja smo ugotovili, da gibalne aktivnosti ne vplivajo le na znanje učencev in razumevanje učnih vsebin, ampak tudi na počutje učencev in njihovo motivacijo za učenje. One of the most significant and influential environmental factors in late childhood is movement, which affects the overall development of learners, and consequently affects their performance in school and motivation. Movement is also one of the basic needs of children therefore, learners are unable to sit for longer periods of time during the first years of primary school, and sitting also reduces their concentration for learning. Through movement, children learn about the world, and for that reason, teachers should pay special attention to the regular and thoughtful inclusion of movement in teaching. In the theoretical part, child development and stages of cognitive development are presented, which is the starting point for planning activities for learners. The influence of physical/sports activities on individual areas of development are also described. Furthermore, theories that promote the inclusion of physical activities in lessons are summarised, as well as ways in which the teacher can include movement in their lessons. Finally, the principles of teaching maths and the reasons for the inclusion of physical activities in maths lessons are presented. In the empirical part, we presented the implementation of 4 hours of maths lessons, in which we included physical activities, in three departments of the 2nd year of primary school, which represents a total of 61 pupils, as we investigated the effect of using physical activities in maths lessons on the knowledge of all second year pupils. Among other things, we were also interested in the effect of the inclusion of physical activities on the knowledge of differently performing second year pupils. We used a quantitative approach to the pedagogical research. We measured their progress in arithmetic, more precisely in mathematical content: natural numbers up to 100 and number 0, ordering numbers by size, numerical relations, addition and subtraction in a set of natural numbers up to 100 without transition, and in searching for an unknown term in the range of numbers up to 20. The results of the research showed that the inclusion of physical activities when revising mathematical content had a positive effect on the progress of knowledge of almost all learners, as the majority of learners (85.3%) performed better in at least one learning content on the final exam when compared to the pre-exam. In determining the effect of the inclusion of physical activities on the knowledge of differently performing pupils, who were divided into groups according to their performance on the pre-exam, we found that revising selected mathematical content with physical activities has the least effect on students with the best mathematical knowledge, while it has the most pronounced effect on students with the worst prior knowledge of maths. We found that during lessons, intended for revision, physical activities not only affect the learners’ knowledge and understanding of the learning content, but also learners' well-being and their motivation to learn.

Živimo v času, ko je vse bolj zaželeno in priporočljivo znanje tujih jezikov. Na področju Evrope raste ozaveščenost, da je učenje tujih jezikov učinkovitejše in trajnejše, kadar je povezano s smiselnim kontekstom. Tudi sodobne teorije usvajanja tujega jezika vse bolj poudarjajo njegovo vlogo. Način učenja tujega jezika moramo prilagoditi starosti učencev in njihovim zmožnostim. Ker je v zgodnjem obdobju otrokom najljubša in najbolj znana dejavnost igra, jo pri poučevanju pogosto uporabljamo. Da si učenci lažje zapomnijo snov in kasneje prikličejo znanje iz spomina, igri dodamo veliko vizualnega in slušnega materiala ter gibanja in petja. Aktivnosti pri zgodnjem poučevanju tujega jezika morajo biti kratke in pestre, a morajo učence intelektualno izzvati za preoblikovanje idej in informacij, da znajo rešiti probleme, pridobiti razumevanje in odkriti nove pomene. Za učinkovito učenje jezika in vsebine pa jim moramo poleg kognitivnega izziva zagotoviti še zadostno jezikovno podporo. V magistrskem delu so bili oblikovani štirje pristopi poučevanja matematičnih vsebin v angleškem jeziku po načelih vsebinsko in jezikovno integriranega učenja (v nadaljevanju VJIU), ki temelji na medpredmetnem povezovanju in vključevanju sodobnih pristopov k poučevanju. Oblikovani učni pristopi, uporabni za različne učne enote, temeljijo na matriki VJIU, sestavljeni iz štirih kvadrantov, ki se razlikujejo glede na visoke ali nizke kognitivne in visoke ali nizke jezikovne zahteve. Ugotavljali smo, katero zaporedje kvadrantov omenjene matrike je najbolj optimalno za poučevanje izbranih vsebin oz. kako učinkovito je posamezno zaporedje matrike z vidika doseganja učnih ciljev pri matematiki in angleščini v 2. in 3. razredu. Uspešnost učnih pristopov je bila določena z vidika doseganja učnih ciljev na preizkusih znanja, zasnovanih glede na Bloomove taksonomske ravni, ki so jih učenci reševali na koncu vsakega vsebinskega sklopa. Oblikovani pristopi so bili v empiričnem delu aplicirani v prakso tako, da smo izbrane matematične vsebine v angleškem jeziku poučevali po različnih zaporedjih kvadrantov matrike VJIU. Vsak vsebinski sklop je sledil določenemu zaporedju kvadrantov. V akcijsko raziskavo je bilo vključenih 14 drugošolcev in 14 tretješolcev. Izvedenih je bilo 12 učnih ur v angleščini za matematične vsebine: naravna števila do 20, liki, deli celote in poštevanka. Vsaki vsebini so bile namenjene tri učne ure, na koncu vsakega vsebinskega sklopa pa so učenci rešili kratek preizkus znanja. Prva dva vsebinska sklopa smo izvedli v drugem razredu, druga dva pa v tretjem. Vsi pristopi poučevanja, ki so sledili različnim zaporedjem matrike VJIU, so se izkazali za uspešne in dosegli namen v smislu, da so učenci dosegli tako cilje pri matematiki kot pri angleščini. We live in a world where it is very important and also recommendable that we can speak in a foreign language. The awareness that the knowledge of a foreign language is more effective and long-lasting when it is learned in a meaningful context is rising in Europe. This kind of foreign language learning is also emphasized by contemporary theories of language learning. The way of learning a foreign language has to be adapted to pupils’ age and capabilities. Using play in early childhood education is crucial because playing is children’s primary and preferred activity. In order to accomplish better information (subject matter) remembrance and retrieval, we add a lot of visual and listening material, movement and singing to children’s play. Activities in early foreign language teaching have to be short and diverse, they have to challenge the children intellectually so they are able to redesign ideas and information, can solve problems, gain understanding and discover new meanings. Beside cognitive challenge we have to provide the children with efficient language support in order to develop effective language and content learning. The thesis focuses on four approaches to teaching mathematics content in English by using the CLIL (Content Language Integrated Learning) method that is founded on cross-curricular integration and adding modern approaches to learning. Formed teaching approaches which can be used in different learning units are based on CLIL Matrix composed of four quadrants differing in high or low cognitive demands and high or low linguistic demands. The main goal of this thesis was to establish which sequence of the CLIL Matrix quadrants is ideal for teaching the chosen contents and how effective the individual sequence is in achieving learning objectives in Mathematics and English in the 2nd and 3rd Grade of Primary School. Effectiveness of these approaches was determined through achieving of learning objectives in tests that pupils took at the end of each content unit. The tests contained different levels of Bloom’s taxonomy. In order to apply the formed approaches in practice, we have taught selected mathematical content units in English language considering different sequences of CLIL Matrix’s quadrants. Each content unit followed a different sequence of quadrants. There were fourteen second-grade and fourteen third-grade pupils included in the action research. We have carried out twelve English lessons with mathematical contents, such as natural numbers to 20, shapes, fractions and multiplication table. Three lessons were used for each content unit and at the end of each one the pupils took a short test. The first two content units were taught in the 2nd grade and the second two were taught in the 3rd grade. All teaching methods that followed different CLIL Matrix sequences were proved as successful and reached its aim in achieving learning objectives in Mathematics and English.