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Ovaj diplomski rad sastavljen je od tri poglavlja s naslovima Rješavanje matematičkih problema, Nejednakosti i Aritmetika. Prvo poglavlje započinje opisom pojma matematički problem te nekoliko različitih klasifikacija matematičkog zadatka. Opisan je nastavni sustav problemska nastava te su prezentirane smjernice za izvođenje nastavnog sata ovog tipa. Ukratko je opisana i metoda rješavanja matematičkog problema koju je predstavio George P’olya u svom djelu “Kako ću riješiti matematički zadatak”. Na kraju poglavlja proučen je i nastavni sustav heuristička nastava te su nabrojane i opisane heuristike koje se koriste za rješavanje matematičkih problema. U drugom i trećem poglavlju rješavaju se matematički problemi iz područja nejednakosti i aritmetike. Prije rješavanja navedenih problema, potrebno je prisjetiti se nekoliko najvažnijih i najpoznatijih teorema iz ovog područja. Među njima su Cauchy-Schwarzova nejednakost te nejednakosti između aritmetičke, geometrijske, kvadratne i harmonijske sredine brojeva. Nadalje, nakon prisjećanja osnovnih pojmova vezanih za djeljivost brojeva, teorema o dijeljenju s ostatkom i Euklidovog algoritma, dokazan je osnovni teorem aritmetike te uveden pojam kongruencije i dokazan kineski teorem o ostatcima. Time su usvojene određene tehnike koje omogućuju da se zadaci koji su možda teško rješivi, riješe na znatno jednostavniji način. This master thesis is composed of three chapters with the titles Solving Mathematical Problems, Inequalities and Arithmetic. The first chapter begins with a description of the term mathematical problem and several different classifications of a mathematical task. The problem-based teaching method is described and the guidelines for performing this type of lessons are presented. Also, the method of solving a mathematical problem, presented by George P´olya in his book “How to solve it”, is briefly described. At the end of the chapter, the heuristic teaching method is explained, and the heuristics used to solve mathematical problems are listed and described. In the second and third chapters, mathematical problems in the field of inequality and arithmetic are solved. Before solving the mentioned problems, it is necessary to recall several of the most important and well-known theorems in these fields. Among them are the Cauchy-Schwarz inequality and inequalities between arithmetic, geometric, quadratic and harmonic means of numbers. Furthermore, after summarizing the basic concepts related to the divisibility of numbers, the divison theorem and Euclidean algorithm, the fundamental theorem of arithmetic was proved, the concept of congruences was introduced and the Chinese remainder theorem was proved. Thereby, certain techniques have been mastered, which enable mathematical tasks that may be difficult and demanding to solve, to be solved in a much simpler way.

U radu su, kroz različite primjere, detaljno opisane operacije zbrajanja i oduzimanja prirodnih brojeva na sorobanu te su u kratkim crtama opisani načini množenja i dijeljenja na navedenom pomagalu. Na sorobanu se operira s prirodnim brojevima kojima se bavi posebna grana matematike – aritmetika. Na samome početku prikazan je sadržaj vezan uz aritmetiku. Navedeni su osnovni pojmovi vezani uz broj, različiti brojevni sustavi, način usvajanja brojeva i izvođenje operacija s brojevima. Nakon poglavlja o aritmetici, slijedi poglavlje o sorobanu. Dan je prikaz nastanka i razvoja abakusa. Abakus se smatra prvom računaljkom te je korišten prije razvoja kalkulatora i računala. Postoje različite vrste abakusa, a neke od njih su: kineski abakus, japanski abakus, ruski abakus, Lee Kai-chen, digicus itd. U ovom radu naglasak je na japanskom abakusu sorobanu. Na sorobanu, moguće je izvoditi osnovne računske operacije. Prilikom zbrajanja i oduzimanja na japanskom pomagalu najprije se provjerava mogu li se izravno dodati ili oduzeti kuglice. Ako ne mogu onda se poziva na komplementarne parove do broja 5, a ako ni to nije moguće onda se poziva na komplementarne parove do broja 10. Također, postoje i pojedina odstupanja u računanju za koja nije potrebno učiti nova pravila već u različitim situacijama primijeniti naučena pravila. Množenje na sorobanu zahtijeva predznanja u vidu tablice množenja jednoznamenkastih brojeva i vješto izvođenje zbrajanja na sorobanu. S druge strane, dijeljenje na sorobanu zahtijeva također znanje tablice množenja, ali i vješto oduzimanje na pomagalu. Na kraju je prikazana primjena abakusa u praksi. Navedene su prednosti korištenja abakusa, ali i njegova uporaba u praksi. Bez obzira na to što se najviše koristi u azijskim zemljama i u Hrvatskoj postoje škole za učenje rada na abakusu. Također, postoje i različite aplikacije i web stranice koje omogućavaju rad na njemu., In the paper, through various examples, the operations of adding and subtracting natural numbers on the soroban are described in detail, and the methods of multiplication and division on the mentioned aid are briefly described. Soroban is operated with natural numbers, which are dealt with by a special branch of mathematics - arithmetic. At the very beginning, content related to arithmetic is shown. The basic terms related to number, different number systems, the method of acquiring numbers and performing operations with numbers are listed. After the chapter on arithmetic, there is a chapter on soroban. The origin and development of the abacus is presented. The abacus is considered the first calculating device and was used before the development of calculators and computers. There are different types of abacus, and some of them are: Chinese abacus, Japanese abacus, Russian abacus, Lee Kai-chen, digicus, etc. In this paper, the emphasis is on the Japanese soroban abacus. On soroban, it is possible to perform basic arithmetic operations. When adding and subtracting on the Japanese aid, it is first checked whether balls can be added or subtracted directly. If they cannot, then they refer to complementary pairs up to the number 5, and if that is not possible, then they refer to complementary pairs up to the number 10. Also, there are some deviations in the calculation for which it is not necessary to learn new rules, but to apply the learned ones in different situations. Multiplication on the soroban requires prior knowledge in the form of a table of multiplication of single-digit numbers and skillful execution of the addition on the soroban. On the other hand, dividing by soroban also requires knowledge of the multiplication table, but also skillful subtraction on the aid. At the end, the application of the abacus in practice is presented. The advantages of using the abacus, as well as its use in practice, are listed. Regardless of the fact that it is mostly used in Asian countries, there are also schools in Croatia for learning to work on the abacus. Also, there are different applications and websites that allow working on it.

U ovom diplomskom radu prvo opisujemo razvoj društvenih igara tijekom povijesti. Prikazani su primjeri prvih igraćih ploča i komponenata, razne zanimljivosti te su uvedeni termini tipova igara. Prikazujemo povijesni razvoj modernih društvenih igara predvođenih euroigrama poput Catana, jedne od igara koja je u drugom dijelu rada odabrana kao temelj za razna otvorena pitanja koja potiču matematička istraživanja. U glavnom dijelu rada izdvajamo odabrane moderne društvene igre primjenjive u nastavi matematike: ”Backgammon”, ”City of Zombies”, ”Bohnanza”, ”Love Letter”, ”Hanabi”, ”Catan”, ”SET”. Navodimo njihova pravila i tijek igranja, a zatim analiziramo kako bi se pojedina igra ili njeni elementi mogli koristiti u radu s učenicima. U analizi navedenih igara susrećemo i opisujemo sljedeće matematičke teme: aritmetiku, osnove vjerojatnosti i kombinatorike, logičko zaključivanje. This thesis begins with the description of the development of board games throughout history. We mention examples of first game boards and components, a variety of interesting facts, and we introduce board game types. We present the historical development of modern board games led by Eurogames like Catan, one of the games that features in the second part of the paper as a basis for various open-ended questions which encourage mathematical research. In the main part of the paper, we present a selection of modern tabletop games applicable in teaching of mathematics: ”Backgammon”, ”City of Zombies”, ”Bohnanza”, ”Love Letter”, ”Hanabi”, ”Catan”, ”SET”. We state their rules and the course of play, and then analyze how a particular game or its elements could be used in working with students. In the analysis of these games we describe the following mathematical contents: arithmetics, basic probability and combinatorics, and logical reasoning.

U ovom su diplomskom radu sustavno, prema gradivu matematike za pojedine razrede osnovnih i srednjih škola, opisane razne primjene matematike na nogomet. U prvom poglavlju, koje se tiče razredne nastave matematike, dajemo niz primjera osnovne aritmetike i geometrije u nogometnom kontekstu. U drugom se poglavlju bavimo primjenom matematike viših razreda osnovne škole na nogomet: račun s razlomcima, osnove vjerojatnosti i Pitagorin teorem samo su neke od tema koje se ovdje primjenjuju na situacije iz nogometnog svijeta. U trećem poglavlju opisujemo kako srednjoškolsko (gimnazijsko) matematičko gradivo povezati s nogometom, te ovdje na nogomet, među ostalim, primjenjujemo funkcije, trigonometriju i statistiku. In this diploma thesis we systematically, according to the mathematics content for individual classes of primary and secondary schools, describe various applications of mathematics to football. In the first chapter, which concerns the teaching of mathematics in the first four years of primary school, we give several examples of basic arithmetic and geometry in football context. In the second chapter, we apply upper primary school mathematics to football: fraction arithmetic, basic probability, and the Pythagorean theorem are just some of the topics that we here apply to situations from the football world. In the third chapter, we describe how to connect high school (gymnasium) mathematical content with football, and here we describe, among other, football applications of functions, trigonometry and statistics.

Budući da aritmetički sadržaji čine najveći dio matematičkih sa držaja u razrednoj nastavi te da uspješnost usvajanja temeljnih aritmetičkih pojmova značajno ovisi o načinu podučavanja, u prvome dijelu rada prikazani su različiti pristupi formiranju temeljnih aritmetičkih pojmova (prirodni broj, brojenje, relacije na skupu prirodnih brojeva, zbrajanje, oduzimanje, množenje i dijeljenje). Središnji dio ovoga rad bavi se modularnom aritmetikom koja se definira uvoĎenjem relacije kongruencije na skupu cijelih brojeva. Modularna aritmetika može se koristiti u kreiranju umjetničkih uzoraka: zvijezde s m krakova, (m, n) – uzorka te uzoraka dobivenih pomoću tablica zbrajanja i množenja modulo m. Za potrebe diplomskoga rada provedena je radionica Aritmetikom do dizajna u kojoj su sudjelovali studenti 4. godine Fakulteta za odgojne i obrazovne znanosti, dislociranoga Učiteljskoga studija u Slavonskome Brodu te učenici sedmoga razreda OŠ „Ivan Goran Kovačić'“ iz Slavonskoga Broda. Uzorci kreirani pomoću tablica zbrajanja i množenja modulo 6 pridonijeli su ostvarenju cilja ovoga rada: pokazati da se spojivši znanje o osnovnim računskim radnjama, kongruencijama i izometrijama ravnine mogu kreirati zaista mala umjetnička djela. Since the arithmetic contents make the largest part of mathematical contents in class teaching and the success in acquiring basic arithmetic terms depends on the teaching methods significally, the first part of the paper shows different approaches to forming basic arithmetic terms (natural number, counting, relations on a set of natural numbers, addition, subtraction, multiplication and division). The central part of the paper deals with modular arithmetic which defines introducing congruence relation on a set of integers. Modular arithmetic can be used to create artistic patterns: an m-pointed star, an (m, n) residue pattern and patterns created by using addition and multiplication tables for least residues modulo m. A workshop Design ideas through arithmetic was organized and conducted for the purpose of the thesis. Participants were fourth year students of the Faculty of Education, Slavonski Brod branch and seventh grade students of primary school Ivan Goran Kovačić from Slavonski Brod. Student works – patterns created by using addition and multiplication tables for least residues modulo 6 have contributed to achieving the goal of the paper: to show that by combining knowledge about basic mathematical operations, congruences and plane isometries little p ieces of art can be created.

V diplomskem delu z naslovom Razvijanje matematičnih sposobnosti v predšolskem obdobju s pomočjo tipanke so predstavljene dejavnosti z matematičnimi tipankami, ki vključujejo matematične pojme s treh matematičnih področij: geometrije, algebre in aritmetike. Za vsako matematično področje smo izdelali veččutno tipanko, s pomočjo katere smo izvedli dejavnosti v vrtcu. Pri tem nas je zanimalo, v kolikšni meri bodo otroci preko dejavnosti s tipankami pridobili nova matematična spoznanja. Dejavnosti v vrtcu smo izvedli v dveh skupinah otrok, starih od 5 do 6 let, in jih natančneje opisali v praktičnem delu, kjer smo na podlagi pridobljenega vzorca evalvirali učinkovitost tipank pri razvoju matematičnih sposobnosti pri predšolskem otroku. Predstavili smo rezultate, in sicer koliko so otroci že imeli razvite matematične sposobnosti pred aktivnostmi s tipankami in po izvedenih aktivnostih s tipankami. Rezultate smo pridobili na podlagi vnaprej pripravljenega ocenjevalnega lista ter tako ugotovili, v kolikšni meri je tipanka pri otrocih razvijala matematične sposobnosti. In the diploma work entitled Development of mathematical skills in the pre-school period with the help of the tactile book, the activities with the mathematical tactile books are presented. They include mathematical concepts from three mathematical areas: geometry, algebra and arithmetics. For each mathematical area a multi-sensory tactile book has been made, and with its help activities in the nursery were carried out. In this context we were interested in the extent to which the children through activities with the tactile books gain new mathematical insights. Activities in the nursery were carried out in two groups of children, aged from five to six, and more specifically described in the practical part, where on the basis of the obtained sample the efficiency of tactile books in the development of mathematical skills at pre-school children was evaluated. We have introduced the results of how much the children have already developed the mathematical skills before the tactile book activities and after the activities have been carried out. The results have been obtained on the basis of predefined evaluation sheet. This way we have determined the extent to which the tactile book has helped to develop the children's mathematical skills.

Učenci s primanjkljaji na področju učenja aritmetike se spoprijemajo s pomanjkljivo in ovirano avtomatizacijo matematičnih dejstev ter postopkov, ki je pogojena s težavami semantičnega spomina, proceduralnimi ovirami in vidno-prostorskimi primanjkljaji. Zadnje ima ključen vpliv na obvladovanje pisnega računanja in usvajanje pisnih algoritmov pri aritmetiki. Namen magistrskega dela je ugotoviti, kako pri učencu s primanjkljaji pri aritmetiki izboljšati uspešnost pri izvajanju računskih operacij pisnega množenja in deljenja. V magistrskem delu je predstavljen 30-urni trening, ki je bil izveden z učencem šestega razreda. Diagnostična ocena je bila pridobljena z neformalnimi razgovori z učencem, njegovimi starši, učiteljicami, izvajalko učne pomoči ter specialno in rehabilitacijsko pedagoginjo, ki je učenca obravnavala, z analizo učenčeve dokumentacije, vprašalnikom za ugotavljanje prevladujočega učnega stila učenca ter matematičnimi preizkusi in testi za oceno obvladovanja aritmetičnih znanj in spretnosti. Znotraj treninga je prikazan pomen povezovanja aritmetičnega proceduralnega, konceptualnega in deklarativnega znanja ter smiselnost poučevanja, kjer se prepletajo načela dobre poučevalne prakse in specialnopedagoški načini pomoči (poučevanje strategij reševanja nalog z odkrivanjem povezav med dejstvi in zakoni oziroma pravili, uporaba mnemotehnik, učenje kompenzacijskih strategij reševanja aritmetičnih nalog, raba ponazoril, pomoč učencu v obliki tutorstva itd.). Primerjava začetnih in končnih rezultatov testiranja kaže, da je učenec s pomočjo treninga bolje razumel in usvojil deklarativna in konceptualna aritmetična znanja, ki so pomembna za izvajanje postopka pisnega množenja, ter napredoval na področju zapomnitve in izvedbe korakov v postopku pisnega deljenja. Pridobljeni podatki bodo v pomoč učiteljem, izvajalcem učne ali dodatne strokovne pomoči ter specialnim in rehabilitacijskim pedagogom pri oblikovanju ustreznih pristopov in načinov pomoči za učence predmetne stopnje, ki nimajo usvojenih temeljnih matematičnih znanj in spretnosti. Students with the deficiencies in the field of learning arithmetic face deficient and impeded automation of mathematical facts and procedures which is conditioned by the troubles of the semantic memory, procedural impediments, and visual-spatial deficiencies. The latter has an essential influence on mastering computation in writing and assimilating written algorithms within arithmetic. The purpose of the master’s thesis is to ascertain how to improve success in performing arithmetic operations of written multiplication and division in a student with deficiencies in arithmetic. In the master’s thesis, I present 30-hour training which was performed with a student of the sixth grade. The diagnostic evaluation was acquired by means of informal discussion with the student, with his parents, teachers, the performer of the learning assistance, special rehabilitation teacher who treated the student, the analysis of the student’s record, questionnaire for ascertaining the prevailing learning style of the student, and mathematical tests and tests for the evaluation of mastering arithmetic knowledge and skills. Within the training, the significance of connecting arithmetic procedural, conceptual, and declarative knowledge, and the sanity of teaching where the principles of good teaching practice and special teaching approaches of assistance (learning strategies of solving tasks by means of discovering connections between the facts and laws or rules, the use of mnemotechniques, learning of compensation strategies of solving arithmetic tasks, the use of depictors, assistance to the student in form of tuition, etc.) intertwine. The comparison of the initial and concluding results of testing proves that, by means of training, the student better understood and assimilated declarative and conceptual arithmetic knowledge which is important for the performance of the procedure of written multiplication, and improved in the field of retention and performance of the steps in the procedure of written division. The acquired data will be helpful to teachers, performers of learning or additional professional help, and special rehabilitation teachers in the case of forming the appropriate approaches and methods of the assistance for the students of the subject level, which did not assimilate the basic mathematical knowledge and skills.

V diplomskem delu z naslovom Uporaba ploščic za vzorčke na področju aritmetike in delov celot so predstavljene nekatere aktivnosti z uporabo teh ploščic iz seta za vzorčke. Sledili smo aktivnostim, ki so navedene v članku Hungry for Early spatial and Algebraic Reasoning. Otrokom smo najprej predstavili uporabo in pomen ploščic za vzorčke. Za večjo motivacijo in čim bolj zanimivo in razumevajočo razlago za otroke smo uporabili tudi lutko gosenico Berto, ki je otroke še bolj motivirala k sodelovanju. Otroci so se skozi aktivnosti srečali s področjem aritmetike in delov celote. This diploma work, entitled "Using pattern blocks in the field of arithmetic and parts of wholes", presents some activities by using these blocks. We have followed the activities listed in the article “Hungry for Early spatial and Algebraic Reasoning”. The children were first acquainted with how to use the pattern blocks and with their importance. To increase children’s motivation and to ensure the most interesting and sympathetic explanation for the kids, we used a doll-caterpillar named "Berta", which motivated children to participate even more. Through their activities children met with the scope of arithmetic and parts of wholes.

V diplomskem delu so prikazani in dokazani osnovni kriteriji deljivosti. V tretjem poglavju je prikazano, kako je tema deljivosti predstavljena v slovenskih osnovnošolskih učbenikih. V zadnjem poglavju pa so predstavljeni primeri nalog iz osnovnošolskih tekmovanj, ki se rešujejo s pomočjo obravnavanih kriterijev. In this graduation thesis, basic divisibility criteria are shown and proven. The third chapter shows how the topic of divisibility is presented in Slovenian primary mathematics textbooks. The final chapter presents examples of exercises from primary school mathematics competitions, which are solved with the use of divisibility criteria.