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The goal of this study was to examine students’ work in online assessments to gain more understanding for designing continuous assessments in blended learning environment for prospective primary school teachers. The study took place during emergency remote teaching due to the COVID-19 pandemic. Course work for prospective primary school teachers in the didactics of mathematics course included continuous, obligatory, non-graded, online assessments. We performed a qualitative, content analysis of their answers. Their work was examined regarding the content knowledge and requirements in the questions, based on the categories of Subject Matter Knowledge and Mathematical Assessment Task Hierarchy taxonomy. The results showed that students’ study approach was strategic, relying heavily on peer support. Their work differed concerning the content and requirements in the questions. Students were more engaged in questions that required creating examples, discussing definitions and properties, and solving contextual problems. Questions related to horizontal content knowledge were most challenging for students. We discussed how the results of our study can affect the design of continuous assessment in a blended learning environment for prospective primary school teachers.

The emergency remote teaching implemented in various ways during COVID-19 global pandemic brought many challenges. The knowledge gained could be beneficial for regular and online prospective teachers’ education. In this presentation, we present analysis of students work in online exercises during emergency remote teaching. Content analysis coupled with the MATH taxonomy provided powerful tool for categorizing students answers and classifying questions in mathematical tasks. Students approached the assignments strategically, relying heavily on peer support. We discuss how the results of our study can affect the design and assessment in the course. For example, online assignments for students, with individual feedback and synchronized with the course content, as means of continuous formative assessment, could complement lectures into a blended learning environment.

Asymptotes in the upper secondary mathematics curriculum are related to graphs and limits of elementary functions. We analyzed fourth-grade mathematics textbooks for general upper secondary education in Croatia concerning the study of asymptotes in Calculus. Asymptotes occurred in several praxeologies confined to two units ; one related to function limit and the other to applying derivatives in function graphing. The study of asymptotes was organized around two didactic moments: the institutionalization of formulas and working on determining asymptotes. We propose exploring the asymptotic behaviour of a function to motivate the construction of formulas with a limit for the coefficients of slant asymptote and generalized asymptotic relations.

Didactic materials can be used to introduce abstract mathematical concepts. Various studies found positive effects of using manipulatives in mathematics teaching and learning. Teachers’ knowledge and attitudes towards manipulatives influence the profound and helpful implementation of manipulatives in the didactic process. There is a variety of ready-made and self-made manipulatives that can be utilized in mathematics education. Following that perspective, we questioned what role manipulatives play in Croatian primary education. For that purpose, we conducted research with teachers on the use of didactic materials in mathematics classroom teaching. Results showed that teachers use all sorts of manipulatives. They acknowledged the benefits of using manipulatives and most of them claimed they used manipulatives often in the didactic process. However, some answers indicate the lack of knowledge about the characteristics of manipulatives and the implementation of manipulatives in the didactic process.

Graphical representation is one of the fundamental and widely spread representations. We performed a comprehensive research of the didactic transposition of asymptote and asymptotic behaviour in the upper secondary education in Croatia, within the framework of the anthropological theory of the didactics. Our study included textbook analysis, questionnaires with university students and interviews with two mathematicians. In this poster, we present a part of our research with an emphasis on university students’ graphing praxeologies. Results showed that students’ graphing praxeologies differ from scholarly praxeologies. Further, students grounded their praxeologies mostly on their high-school graphing knowledge, even after being exposed to advanced mathematics that could foster their autonomous thinking.

Teškoće učenika na svakoj razini matematičkog obrazovanja često se povezuju s neadekvatnim znanjima stečenim na prethodnoj razini obrazovanja. Matematika je kumulativna znanost. To znači da se matematički koncepti i kompetencije razvijaju se u dva smjera: vertikalno, povećavajući njihovu složenost, i horizontalno, okupljajući više pojmova i procedura u jedan novi objekt. Mreža znanja i učeničke kompetencije razvijaju se povezivanjem koncepata i procedura te njihovom primjenom u smislenim kontekstima. S jedne strane, matematički udžbenici trebaju održavati i odražavati (vertikalnu i horizontalnu) povezanost matematičkih sadržaja. S druge strane, rutinski matematički sadržaji viših razina obrazovanja predmet su problemskih i istraživačkih aktivnosti na nižim razinama obrazovanja. Obratno, konkretne aktivnosti s apstraktnim sadržajima na nižim razinama obrazovanja olakšavaju razumijevanje i primjenu apstraktnih matematičkih ideja na višim razinama obrazovanja. U ovom predavanju dat ćemo teorijsku osnovu i konkretne primjere kako ostvariti vertikalnu korelaciju i suradnju učitelja matematike.

Relacija “biti jednak” i znak jednakosti sastavni je dio kurikuluma predmeta matematike od početka obrazovanja. Međutim, istraživanja su pokazala da učenici zadržavaju zablude o jednakosti i znaku jednakosti tijekom njihovog matematičkog obrazovanja. Pregledavanjem udžbenika iz matematike za razrednu nastavu (učenici starosti 7-10 godina) mogu se pronaći loše osmišljeni zadaci, matematičke nedosljednosti, pogrešno označavanje i drugi metodički propusti. U ovom ćemo radu predstaviti zadatak iz udžbenika sa sadržajem koji je u suprotnosti s konceptom jednakosti. Ispitali smo kako su učitelji razredne nastave (N = 51, između jedne i 34 godine radnog iskustva) i studenti, koji su budući učitelji razredne nastave (N = 340), pristupili tom zadatku. Gotovo su svi sudionici u našem istraživanju protumačili znak jednakosti kao "izvrši izračun s lijeva na desno". Uglavnom su komentirali prirodu i strukturu zadatka, a rijetko matematički sadržaj. U ovom ćemo radu predstaviti njihove prosudbe, iskustva i zapažanja vezana uz zadatak. Primjećujući da je većina budućih učitelja pokazala pogrešnu predodžbu o jednakosti, moramo razmotriti rješavanje ovog pitanja kroz reviziju udžbenika, ali i kroz obrazovanje budućih učitelja te stručno i metodičko usavršavanje učitelja u praksi.

Preporuke Vijeća Europe o ključnim kompetencijama za cjeloživotno obrazovanje uključuju matematičku kompetenciju koja se temelji na sposobnostima računanja i znanju o brojevima. Teorijski okviri međunarodnih istraživanja PISA i TIMSS te hrvatski nacionalni kurikulu sadrže komponentu rada s brojevima. Brojevi su važan dio matematičkog obrazovanja u nižim razredima osnovne škole i neizostavni su u brojnim aspektima profesionalnog i društvenog života. Pojam osjećaj za broj (eng. number sense) motivira istraživanja s različitim fokusom i pristupom. Psiholozima je zanimljiv urođeni osjećaj za broj. Metodičari su ispitivali kako je osjećaj za broj povezan s uspjehom i sposobnostima u matematici te istraživali i dizajnirali aktivnosti koje razvijaju osjećaj za broj. Dunphy (2007) je pisao kako je osjećaj za broj pojam koji se teško definira. Osjećaj za broj odnosi se na razumijevanja, vještine i stavove o broju koji nadilaze sposobnosti rada s brojevima u svakodnevnim situacijama. Rezultati hrvatskih učenika na međunarodnim istraživanjima i nacionalnim ispitima te odgovori studenata učiteljskih studija na testu koji je osmislila autorica, ukazuju na slabije sposobnosti u radu s brojevima. Iskustvo autora s alternativnim obrazovnim sustavima, Montessori i Waldorf, potaknulo je ispitivanje kako pojedini kurikuli podržavaju razvijanje osjećaja za broj u kontekstu matematičkog obrazovanja u nižim razredima osnovne škole u Hrvatskoj. Pregledom znanstvene literature pronađeno je nekoliko teorijskih okvira koji tumače pojam osjećaja za broj (Andrews i Sayers, 2015 ; Dunphy, 2007 ; Howell i Kemp, 2010 ; Yang, 2005). Uspostavljen je okvir za usporedbu kurikula koji se sastoji od deset prepoznatih komponenti osjećaja za broj. Rezultati pokazuju kako hrvatski kurikulu u usporedbi s alternativnim kurikulima podbacuje u komponentama povezanima s veličinom broja, uzorcima i stavovima o broju. Nadalje, ishodi učenja u hrvatskom kurikulu, a koji pripadaju komponentama povezanima s odnosima, prikazima, računanjem i primjenom broja, nefleksibilni su, ograničeni i rutinski. Razvijanje osjećaja za broj, u odnosu na uvježbavanje procedura, može se ostvariti aktivnostima važnima za komponente osjećaja za broj, a podzastupljenima u hrvatskom kurikulu. Smatramo kako obrazovanje budućih učitelja i učitelja u praksi o alternativnim pristupima matematičkom obrazovanju može doprinijeti učenju i poučavanju koje razvija osjećaj za broj.

An exponential function is one of the elementary functions in mathematical modelling and a common part of the upper secondary mathematics curriculum. A horizontal asymptote is a prominent feature of an exponential function graph. We have performed a comprehensive study within the Anthropological theory of the didactics on asymptotes and asymptotic behaviour in the context of upper secondary mathematics education in Croatia. Therein, we administered three questionnaires with prospective mathematics teachers. One of the tasks from the questionnaires was a priori designed to explore the role of an asymptote in graphing and describing an exponential function. In this paper, we present results of an a posteriori analysis of students' work on the modified contextual textbook task with an exponential-type function. We discuss the affordances of the textbook versus redesigned task and provide suggestions for task selection and design to promote coherency and discourse in students' available mathematical knowledge

Recommendation from the Council of Europe on the key competences for lifelong learning features mathematical competence that builds “on a sound mastery of numeracy” and includes “sound knowledge of numbers”. Numbers are an important part of primary mathematics education and they permeate many aspects of professional and social life. The notion number sense motivated studies with a different focus and approach. Educators examined how number sense correlated with achievement and abilities in mathematics and investigated and designed activities to develop students’ number sense. Number sense concerns “understandings, skills and attitudes about number” that are beyond being numerate in the everyday workaround mathematics. The achievement of the Croatian pupils in the international surveys and national exams and the prospective primary school teachers’ answers in a test developed by the author indicate low-level abilities in the working with numbers. Having experience with the alternative education, Montessori and Waldorf, we questioned how the corresponding curriculums promote the development of the number sense in the context of the primary mathematics education in Croatia. We created a tool for curriculum analysis that inc luded ten common components of the numbers sense from the literature. Our results showed that the Croatian curriculum failed by comparison in the number sense components related to the size of numbers, patterns and attitudes, and the outcomes which pertain to the components related to the relationship, representation, calculation and application, are rigid, limited and routine. We believe that educating prospective and practising teachers about alternative approaches to mathematics education might promote teaching and learning to develop the number sense.

U sklopu ovogodišnjeg Festivala znanosti održana je radionica s učenicima dvaju odjela četvrtog razreda osnovne škole u Osijeku pod nazivom „Pozdrav iz prošlosti – tajne geometrije“. Cilj radionice je bio da učenici dožive osjećaj matematičkog otkrića oslanjajući se na vlastite matematičke kompetencije. U ovom radu opisat će se vođeno učenje otkrivanjem kao strategija poučavanja, aktivnosti koje su provedene s učenicima te vrednovati provedba i ishodi radionice iz perspektive učitelja. Vođeno učenje otkrivanjem je strategija poučavanja primjerena u radu s učenicima mlađe školske dobi. Odgovornost učitelja je pripremanje materijala i uputa za rad te po potrebi usmjeravanje učenika u njegovom samostalnom radu prema vlastitom otkriću. Tijekom radionice učenici su istraživali pravilnosti u geometriji koje nisu dio redovitog nastavnog programa poput Pitagorinog poučka, Talesovog teorema o obodnom kutu nad promjerom kružnice, nejednakosti trokuta i zbroja unutrašnjih kutova trokuta. Učenici su trebali mjeriti duljinu, crtati okomite i paralelne pravce te osnovne geometrijske likove, poznavati odnos pravaca u ravnini, svojstva trokuta, pravokutnika i kvadrata te kutove i drugo. Sve planirane aktivnosti su uspješno provedene i učenici su svladavali postavljene zadatke. Smatramo kako je učenicima bilo korisno poznate koncepte i procedure primijeniti u istraživanju i otkrivanju matematike.

Asimptota i asimptotsko ponašanje objekti su matematičkog znanja prisutni u različitim područjima matematike. Asimptotsko ponašanje istaknuto je svojstvo nekih funkcija i krivulja te važan alat za opisivanje i rješavanje raznovrsnih matematičkih i izvanmatematičkih problema. Različitost pristupa kojima se ostvaruje te širok spektar sadržaja i područja nastave matematike s kojima se može povezati, odnosno u kojima je relevantan objekt znanja, asimptotu čine zanimljivim objektom istraživanja. U ovom doktorskom radu istražena je i opisana didaktička transpozicija tog objekta znanja u gimnazijskom obrazovanju u Republici Hrvatskoj. Teorijski okvir unutar kojega je provedeno ovo istraživanje jest antropološka teorija didaktike (ATD), koju je razvio francuski matematičar Yves Chevallard početkom 1980-tih godina za potrebe istraživanja matematičkog obrazovanja. Pregled relevantne svjetske znanstvene literature ukazuje na to da su malobrojna istraživanja fokusirana na opisivanje asimptote kao objekta znanja. Osobito nedostaje onih u kojima se sagledavaju različiti aspekti tog matematičkog pojma te njihov međusobni utjecaj, odnosno različiti konteksti u kojima je taj pojam relevantan. Osim toga, ovo je prvo istraživanje u Republici Hrvatskoj kojim se tema iz gimnazijskog matematičkog obrazovanja teorijski utemeljeno adresira i povezuje s inicijalnim sveučilišnim obrazovanjem nastavnika matematike. Istraživanje je provedeno u skladu s odabranim teorijskim okvirom i obuhvaća tri faze: analizu udžbenika, upitnike sa studentima nastavničkog smjera matematike i intervju sa znanstvenicima. Izgrađen je referentni epistemološki model za objekt znanja asimptote koji je usklađen sa zahtjevima znanja za poučavanje te podržan akademskim znanjem i saznanjima epistemoloških istraživanja. Introduction: Asymptote and asymptotic behaviour are bodies of knowledge that have an important role in many traditional and modern mathematical disciplines and are supported by a rich and well-established abstract theory. Asymptotic behaviour is a significant property of some functions and curves and a tool for describing and solving various problems in theoretical and applied mathematics. Some basic aspects of these concepts must be utilized already in elementary mathematics as powerful tools in graphing and analyzing behaviour of elementary functions at infinity and near singularities and in graphing simple plane curves such as hyperbola. Therefore, this body of knowledge is a common part of upper secondary mathematics curricula worldwide. The notion of an asymptote appears as a theoretical concept, a part of a procedure or as a self-sufficient task in different contexts and different educational cycles. Hence, connection with a wide range of teaching contents and mathematical bodies of knowledge makes it an interesting object of research. This doctoral thesis investigates and elaborates the didactic transposition of asymptote as a body of knowledge in general secondary education in Croatia. The research is conducted within the theoretical framework of the anthropological theory of the didactic (ATD), developed by a French mathematician Y. Chevallard especially for research in mathematics education. Scientific literature review shows that studies focused on describing the body of knowledge asymptote are scarce within ATD and other theoretical frameworks. Corpus of scientific research especially lacks those that consider various aspects of this body of knowledge, their mutual influences and connections or various contexts in which this body of knowledge is relevant. Further, research conducted within this thesis pioneers in mathematics education research in Croatia since it addresses and connects, in a theoretically founded manner, initial university education of mathematics teachers and general secondary education. The main idea of ATD is to determine the relation \(R_I(p, O)\) between a body of knowledge \(O\) and a person that occupies position \(p\) in an institution \(I\). For this purpose mathematical knowledge and activities are described in terms of a praxeology \( [T, \tau, \theta, \Theta]\), where its practical component is represented with task \(T\) and technique \(\tau\) and discursive or theoretical component with technology \(\theta\) and theory \(\Theta\). ATD enables a researcher to develop a working model for a body of knowledge, that is reference epistemological model (REM). REM is a coherent structure consisted of complete and mutually connected praxeologies. It should be compliant, questioned and evaluated with respect to curricular and cultural demands, as well as academic and epistemological knowledge. Methodology : The comprehensive research conducted for this thesis is founded on a REM for the body of knowledge asymptote, \(A\). Asymptote is a body of knowledge consisted of a set of praxeologies for which it is a component of practical or theoretical block. Hence, the developed REM included praxeologies of different range and complexity that exist or can be constructed within curriculum and textbook themes connected to the notion of an asymptote. The relations \(R_B(p,A)\), \(R_S(p,A)\) and \(R_M(p,A)\) are analyzed and compared, where institutions considered are: \(B\) for the two Croatian mathematical gymnasium textbooks, \(S\) for the cohort of 40 the final, fifth year mathematics education students at the largest mathematical department in Croatia and \(M\) for the institution of academic mathematicians. The methodology of the implemented research provided information from all steps of the didactical transposition of the body of knowledge asymptote. It included (1) a praxeological analysis of the textbooks as representatives of the knowledge to be taught, (2) three questionnaires with open-ended questions for the prospective mathematics teachers to provide an insight in related knowledge available to students as a potential taught knowledge and (3) a semi-structured interview with two academics who represent scholarly knowledge. Results: Based on the results gained from three phases of the research, the proposed REM for the body of knowledge asymptote is verified and improved in order to develop an asymptote as a body of knowledge and to make suggestions for the teaching practice especially within the ongoing curricular reform in Croatia. In general, textbook organization is incoherent and contains mainly practical blocks, praxeologies are not connected, and available discourses are either not relevant or not connected to practical activities, except for algebraic manipulation with formulae and equations. Knowledge available to prospective math teachers is loaded with components of the secondary knowledge to be taught, techniques students chose are not the most efficient for the task in question and they do not utilize discursive components. For knowledge to be taught and knowledge available to students asymptote and asymptotic behaviour are available but not fully utilized as praxis or logos in relevant praxeologies. Even though praxeological organization of textbooks and students‟ praxeological equipment do not fully correspond to the proposed REM, we find that the implementation of such REM is possible with a proper support of the noosphere, i.e. the institutions responsible for the mathematics education. Considering this, it is our suggestion for the teaching practice especially within the ongoing curricular reform in Croatia that: all relevant properties of an object (e.g. asymptote, function, curve) should be emphasized and implemented in practical, applied and formal context; common properties of the object should be interrelated and diversely represented; simple, isolated and incomplete praxeologies should be organized into more coherent and complex praxeologies; and asymptotic behaviour should be more emphasized, adequately graphically represented and described by formal and informal mathematical discourse.

Asymptote and asymptotic behaviour are bodies of knowledge present in elementary as well as in theoretical and applied mathematics. Some aspects of these concepts make a common part of upper secondary mathematics curricula worldwide. The notion of an asymptote appears as a theoretical concept, a part of a procedure or as a self-sufficient task in different contexts and different educational cycles of upper secondary mathematics education. In this paper, we present results of a textbook analysis on asymptote and asymptotic behaviour in two most common series of gymnasium mathematics textbooks in Croatia. It is a part of a larger comprehensive study regarding this body of knowledge in general upper secondary education in Croatia. The research is conducted within the theoretical framework of the Anthropological theory of the didactics (ATD), developed by a French mathematician Y. Chevallard. This framework provides praxeology[T, τ, θ, Θ] as a tool for describing mathematical knowledge and activities, with its practical component – a task T and a technique τ, and discursive or theoretical component – a technology θ and a theory Θ. The results show that the asymptote is a complex body of knowledge consisted of several praxeologies for which it is a part of practical or theoretical component. Hence, we present the related praxeological organization of considered textbooks along with its direct implications to mathematics teaching and learning.

Prepoznavanje i podrška darovitim učenicima je odgovornost obrazovnog sustava kako stoji u Nacionalnom okvirnom kurikulumu. Margita Pavleković je 2003. godine utemeljila Malu matematičku školu kao izvanškolsku aktivnost za desetogodišnje učenike s posebnim interesom za matematiku pri Fakultetu za odgojne i obrazovne znanosti u Osijeku. Studenti završne godine učiteljskog studija rade s učenicima u sklopu nastave izbornog kolegija Matematika i nadareni učenici. Pavleković, Zekić-Sušac i Đurđević su izradili ekspertni sustav Mat-dar za prepoznavanje matematičke darovitosti kod desetogodišnjih učenika. Ekspertni sustav svrstava učenika u jednu od četiri kategorije: (1) potencijalno daroviti učenik, (2) učenik iznadprosječnih matematičkih sposobnosti, (3) učenik prosječnih matematičkih sposobnosti ili (4) učenik s nedovoljno razvijenim sposobnostima za matematiku. Prikazat ćemo učenička rješenja jednog geometrijskog zadatka u kontekstu procjene matematičke darovitosti učenika. Rezultati su pokazali odstupanja u procjeni studenata i ekspertnog sustava te kako su učenici svrstani u više kategorije uglavnom bolje riješili zadatak nego učenici svrstani u niže kategorije. Potrebno je detaljnije istražiti razloge odstupanja i načine kako poboljšati procjene studenata te kako rješavanje geometrijskih zadataka može ukazati na matematičku darovitost učenika.

In this poster we will present assessment of mathematical giftedness of ten -year-old pupils with special interest in mathematics and their solutions of a geometric task. The research was carried out within an extracurricular activity at the Faculty of Education in Osijek with students in their final year of teacher studies and pupils in their fourth grade of elementary school, named Little School of Mathematics. This kind of cooperation between schools and the Faculty was initiated in 2003 by Margita Pavleković and it is carried out as a part of the elective course Mathematics and gifted pupils. Little School of Mathematics supports developing pupils’ mathematical competencies and provides students with an insight into recognizing, supporting and educating mathematically gifted pupils. Pavleković and associates developed an expert system named MathGift for detecting ten-year-old pupils potentially gifted in mathematics [Pavleković, M., Zekić-Sušac, M., Đurđević, I. (2010). Recognizing mathematically gifted children by using expert systems', teachers', and psychologists' estimations. Društvena istraživanja, 19 (3), 487- 510.]. By using this expert system, pupils are classified into one of the four categories of giftedness – from potentially gifted pupils to pupils with insufficiently developed mathematical skills. The results showed that the students ranked pupils lower in the categories than the expert system MathGift. Furthermore, a review of pupils’ solution to the given geometric task showed a connection between the established category of pupils’ giftedness and the accuracy of the solution.

Darovita djeca veliko su blago svakog društva, ključni potencijal za kreativnost, inovacije i napredak. Važno je na vrijeme prepoznati darovito dijete, a zatim sustavno raditi na razvoju te darovitosti kako ona ne bi ostala neostvarena. Kroz ovo predavanje prikazat će se nekoliko načina kako informacijska tehnologija može pomoći u prepoznavanju matematičke darovitosti kod djece osnovnoškolskog uzrasta i potaknuti interes za matematiku i njezinu kreativnu primjenu. Pokazat će se ekspertni sustav MATDAR napravljen prema modelu prof.dr.sc. Margite Pavleković, te nekoliko primjera multimedijalnih aplikacija i simulacijskih igara za poticanje interesa za matematiku, koje se koriste u “Maloj matematičkoj školi” na Učiteljskom fakultetu u Osijeku. Cilj je potaknuti nastavnike osnovnih škola na veću uporabu ovakvih alata kako bi darovita djeca bila prepoznata i ostvarila svoje potencijale.

Asymptote and asymptotic behavior are bodies of knowledge that are present in various areas of mathematics. The concept of an asymptote can be found in different grades of secondary mathematics education and in various aspects. The research conducted at the University of Zagreb within the framework of the Anthropological theory of the didactics(ATD) investigates the didactic transposition of this body of knowledge. In this posterwe present someresults of this research in connection to a hyperbola.

Numeričke sposobnosti su temelj razvijanju matematičke kompetencije učenika. Promatraju se numeričke sposobnosti učenika četvrtih razreda osnovne škole sukladno zahtjevima matematičkog kurikuluma. Potrebno je posvetiti više pažnje realizaciji numeričkih sposobnosti.

Primary school teachers' and students' of teacher studies self-reported creativity in diverse domains and relationships of domain-specific creativity to self-ratings of the creativity in teaching mathematics, language, physical education, visual arts, music, and science, were explored in this study. Teachers in primary schools (N = 105) and students of teacher studies (N = 111) were the study participants (sample age ranged 21–64 years ; M = 31.2, SD = 11.6 ; 95.4% women). Sixty-two lesson plans for teaching mathematics in grades 1–4 written by students were analyzed regarding their contents (i.e. chosen activities that map onto different creativity domains). The results show that when the students of teacher studies rated themselves as generally creative, they also rated themselves as creative to a different degree, for example, the area that had the lowest and not significant correlation with general creativity ratings was mathematics. This internally experienced subjective structure of creativity in students, as well as the meaning of being creative (excluding mathematics), may have implications for teaching mathematics in the primary school, such as the choice of activities during mathematics lessons. Age- and expertise-related differences between teachers and students of teacher studies were found as well, bringing up numerous questions on (creative) teaching of mathematics and its relationship to teaching excellence and student outcomes.

Nacionalni okvirni kurikulum za prvo obrazovno razdoblje matematičkog obrazovanja predviđa razvijanje kompetencija prikazivanja i komunikacije. Od učenika se očekuje predvidjeti rezultat rastavljanja i sastavljanja ravninskih oblika te prepoznavanje i prikazivanje ravninskih oblika u različitim položajima i odnosima. U tekstu je opisan nerutinski zadatak za učenike mlađe školske dobi i rezultati provedene aktivnosti s učenicima.

Faculties of teacher education which bring together prominent scientists in the field of education should respresent a fruitful soil for innovative solutions in modern teaching. However, the achievements of scientists working at faculties of teacher education are often not directly applicable in teaching practice. Such situation is to a larger degree caused by the lack of unmediated and long-term cooperation of scientists with primary school administration, the classroom environment, and especially pupils and their parents. Little School of Mathematics is the original project involving the afore-mentioned type of cooperation initiated by the university professors of mathematics employed at the Faculty of Teacher Education in Osijek (Pavleković et al.). Accordingly, the Little School of Mathematics was established in the academic year 2003/2004 on the premises of the Faculty of Teacher Education in Osijek, and it constitutes a meeting and collaboration point for university professors / instructors and the students of the Faculty on the one hand, and ten-year-olds, their teachers and their parents on the other. Mentored by the instructors of the Faculty of Teacher Education, the students create, prepare and host workshops with fourth grade pupils of primary schools with special interest in mathematics. Every week during ninety minute sessions throughout two semesters in one academic year pupils expand their knowledge in mathematics and the related fields, and develop abilities to implement the acquired knowledge in everyday situations. On the other hand, through direct contact with pupils the students acquire specific competences required for their future profession. In previous years the Little School of Mathematics has significantly contributed to bringing forth a number of scientific results, one of which is the expert system Mat-Dar designed to detect mathematically gifted children at the age of ten. During the past two years the Little School of Mathematics has also welcomed children with disabilities who exhibit a special interest in mathematics. Moreover, it has also introduced different types of e-learning. By applying the free distance learning management system Moodle, we have provided access to various digital learning materials for students, pupils and teachers, regardless of time or place. Our students complement their workshops with educational simulations and computer games, and explore their efficiency in the classroom setting. The project Little School of Mathematics encourages scientists in the field of teaching mathematics at the Faculty of Teacher Education to pursue research, support a higher quality development of student competences, provide high quality extracurricular support in developing a mathematical gift for all interested ten-year-olds, as well as efficient professional support concerning lifelong learning of teachers, and always with a special emphasis on implementation of innovative and modern-day teaching methods. The Faculty of Teacher Education in Osijek web site contains the created expert system matdar http://matdar.ufos.hr/ for detecting mathematically gifted children at the age of ten, which is currently being tested.

U izlaganju se prezentiraju dobri primjeri iz nastave geometrije s posebnom pozornošću na prilike u kojima se nastavom geometrije potiču bolji ishodi učenja matematike kod učenika osnovne škole.

Online radionica s učiteljicama razredne nastave o primjerima primjene računalnih igara u razrednoj nastavi, provjerenih u Maloj matematičkoj školi (dugogodišnja suradnja nastavnika matematike Učiteljskog fakulteta u Osijeku i desetogodišnjaka s 15 osječkih osnovnih škola koji pokazuju poseban interes za matematiku).

The previous years Croatia has lost a number of distinguished mathematicians who have contributed greatly to teaching mathematics. These contributions can be roughly divided into two crucially different, yet inseparable wholes.

Rad se bazira na istraživanju uporabe informacijsko komunikacijskih tehnologija kao sredstva posredovanja u primjeni različitih suradničkih metoda u poučavanju. U okviru kolegija „Računalo u odmoru i razonodi“, koji se izvodi na Učiteljskom fakultetu u Osijeku, u svrhu olakšavanja usvajanja sadržaja kolegija, kreirali su se zadatci korištenjem suvremene informacijske tehnologije pri čemu su studenti trebali pronaći i analizirati edukativnu igru, učiti istraživanjem koristeći simulacije te kreirati vlastite edukativne igre. Cilj ovog rada bio je otkriti učinak danih metoda poučavanja na: uspješnost studenata, motivaciju te zadovoljstvo studenata prilikom izvršavanja zadataka.

Od školske 2003./04. godine u prostorijama Učiteljskog fakulteta u Osijeku usustavljena je Mala matematička škola. Pod mentorstvom nastavnika Učiteljskog fakulteta studenti osmišljavaju, organiziraju realiziraju radionice s učenicima, čime stječu posebne kompetencije i vještine u kreiranju kurikula za učenike četvrtih razreda osnovne škole s posebnim interesom za matematiku. Tijekom rada studentima učiteljskih studija i učenicima - polaznicima Male matematičke škole, omogućen je besplatan pristup Internetu u računalnom praktikumu Učiteljskog fakulteta, osiguran im je kompletan pribor za rad, dostupni su im didaktički kompleti, matematički časopisi i literatura primjerena dobi učenika. Učenici sudjelovanjem u Maloj matematičkoj školi proširuju svoje znanje iz matematike i prirodoslovlja, razvijaju kompetencije rješavanja problema iz svakodnevnog života matematičkim modeliranjem, promatraju objekte i situacije, te formiraju i provjeravaju točnost pravila i tvrdnji koji nadilaze zahtjeve predviñene planom i programom nastave matematike. Trudom voditelja i studenata, učenici doživljavaju matematiku kroz perspektivu njezine važnosti u svakodnevnom životu, te kroz njezina lijepa (oku ugodna) i zanimljiva (neobična) svojstva. Prateći trend sveprisutnosti informacijske tehnologije i popularizacije njenog implementiranja u nastavu, prethodne školske godine dio radionica Male matematičke škole izvodili smo u obliku elektroničkog kolegija primjenom sustava za elektroničko obrazovanje Moodle. Uključivanje modela elektroničkog učenja u rad Male matematičke škole provedeno je s ciljevima unaprjeñenja učeničkih postignuća kako na području matematike, tako i posrednim djelovanjem na području primjene informacijskih tehnologija, dalje upoznavanja učenika osnovnih škola s drugačijim nastavnim pristupima i metodama u svrhu poboljšanja procesa učenja, te u konačnici s ciljem ostvarivanja kvalitetnije, učestalije i pouzdanije komunikacije na relaciji učenik – student – voditelj. Prilikom osmišljavanja radionica predviñenih za izvoñenje u računalnom praktikumu studenti su pripremali Hot Potatoes kvizove, različite matematičke zagonetke i pitalice, brojne edukativne igre, Moodle testove i kvizove, primjerice modelirani „Tko želi biti milijunaš?“. Jedna od specifičnih aktivnosti učenja organiziranih u sklopu Moodle elektroničkog kolegija Mala matematička škola bile su edukativne simulacije poznate pod nazivom Gizmo. Uzimajući u obzir da djeca nove generacije popularno zvanog „Computer Age“-a, provode velik dio slobodnog vremena na računalu, nameće se potreba za osmišljavanjem kvalitetnih edukacijskih okruženja. Prednost edukacijskih simulacija je upravo u motiviranju učenika za primjenu informacijske tehnologije u svrhu učenja. Ako k tomu još dodamo besplatan pristup Internetu, koji je svakom učeniku osiguran od strane CARNet-a, te činjenicu da simulacije mogu u odreñenoj mjeri zamijeniti skupe didaktičke materijale, dostupnost, ekonomičnost i motivacija su važne pozitivne strane primjene edukativnih simulacija u nastavi. No, vodeći računa o uzrastu učenika o kojima je ovdje riječ, to ni u kojem slučaju ne isključuje sticanje znanja i vještina uz pomoć konkretnih didaktičkih modela.

The Lifelong Learning Program "Enactive Learning in Mathematics" was implemented for the first time at the Faculty of Mathematics, University of Rijeka, in the winter semester 2022/2023. After the completion of the program, interviews were conducted with mathematics teachers who had participated in the program. In addition to evaluating the program itself, we asked teachers about their motivation to participate in lifelong learning programs and the support they received. This paper provides reflections on the interviews conducted and the program implemented.

Research presented in the papers collected for this Monograph deals with various aspects of teaching mathematics on different educational levels and across several European countries. The Monograph consists of three chapters. In the first chapter of the Monograph, the authors wrote about students’ knowledge and representations of knowledge in teaching mathematics. The first paper examines types of errors in students dealing with different aspects of fractions. The author cautions for teaching directed towards comprehension to reduce misconceptions and recognize mathematical concepts in a real-life context. The results of recent studies focussing on typical misconceptions when dealing with different aspects of fractions are present in the paper. In the second paper, the author compared geometry tasks in university entrance examinations across a period and found the content of the tasks decreasing in complexity. Throughout his survey, the author considered the commonly used five components of spatial intelligence as well as the mathematical content of the problems. The authors of the third paper in this chapter raise an interesting issue of managing dichotomies in mathematics education. They discuss and advise mindfully combining different teaching approaches to produce the best results depending on the educational context. Using the example of introducing students to the mathematical structure of division, they demonstrate different ways of knowledge transfer, the role and meaning of repetition, and the logic of including concrete representations. The fourth paper reports on teachers’ use of didactic manipulatives in primary mathematics education. In the final paper in this chapter, the authors conducted an experiment showing that visual representation facilitated problem-solving. The second chapter contains papers focused on teaching practice. The author of the first paper presents a theoretically supported method for introducing arithmetic operations in primary education. The concurrent method of teaching multiplication and division in elementary school is discussed. The second paper investigates interactions in children’s mathematical lessons. The authors found that the different classroom setting encourages mathematical communication and connecting pieces of mathematical knowledge. In the third paper, the authors describe the construction process and use of a novel game for mathematics education. It advantages students’ knowledge by describing and connecting different ideas encountered across the mathematics curriculum. The authors of the final paper in this chapter propose football-related tasks and activities across all grades of primary education. They argue for the popularization of mathematics by connecting to interesting and familiar realistic topics. The content of the third and the last chapter of the Monograph relates to technology in teaching mathematics. The first paper reports on prospective teachers’ familiarity and attitudes about participating in the eTwinning programme. The following papers address the consequences of online education during Covid-19 lockdown. The authors of the first paper examined the differences in grades obtained after onsite and online learning and the author of the second paper examined students’ attitudes and satisfaction concerning an online course. The final paper presents a result of mathematical modelling of population density data used for computer mobility simulations.

U ovom radu prikazuju se rezultati dobiveni u prvoj fazi kolaboracijskog akcijskog istraživanja o razvoju algebarskog i funkcijskog mišljenja u nastavi matematike. Posebno se razmatraju različiti načini zaključivanja i vrste generalizacija koje učenici izvode kroz proces rješavanja problema s rastućim uzorcima. U svrhu identifikacije problema i uvida u znanja i vještine učenika, pripremljen je instrument s četiri zadatka s rastućim geometrijskim uzorcima, različite složenosti (prema vrsti funkcijskog pravila i prema načinu prikaza figura od kojih su sastavljeni). Korišteni zadaci s rastućim uzorcima predstavljaju izolirane geometrijske probleme jer nisu eksplicitno vezani niti uz jednu nastavnu cjelinu te je potrebno uložiti veći kognitivni napor za određivanje općeg pravila i povezati različite matematičke ideje i koncepte. Deskriptivnom i kvalitativnom analizom obrađen je proces rješavanja izoliranog problema, na uzorku od dvjesto učenika hrvatskih gradskih srednjih škola u dobi od 15 do 17 godina. Dobiveni rezultati pokazuju da se učenici većim dijelom služe slikovnim zaključivanjem, a tek u manjoj mjeri brojčanim. Kroz učenički tekstualni opis vidljivi su različiti oblici generalizacija pri čemu u manjoj mjeri provode simboličku generalizaciju kao i postavljanje općeg pravila. Na temelju dobivenih rezultata planira se druga faza akcijskog istraživanja, u kojoj će se, u suradnji s nastavnicima matematike provesti rasprave i radionice o ulozi i načinu korištenja (rastućih) geometrijskih uzoraka s ciljem uvođenja promjena u proces učenja i poučavanja matematike. [The paper presents the results obtained in the first phase of collaborative action research on the development of algebraic and functional thinking in teaching and learning mathematics. In particular, different kinds of generalizations and ways of students' reasoning through the problem-solving process with growing patterns are considered. In order to identify problems and get insight into students' knowledge and skills, an instrument was prepared, consisting of four tasks with growing geometric patterns and of different complexity (according to the type of functional rule and the manner of representation of the figures of which they are composed). The used tasks with growing patterns are stand-alone geometric problems because they are not explicitly related to any teaching unit and require a greater cognitive effort to determine the general rule and to connect different mathematical ideas and concepts. A descriptive and qualitative analysis of the process of solving a stand-alone problem on a sample of two hundred students of Croatian urban high schools aged 15 to 17 was performed. The obtained results show that students mostly use figural reasoning, and only to a lesser extent numerical reasoning. Through the students’ textual description, various forms of generalizations are observed, to a lesser extent carrying out symbolic generalization as well as setting a general rule. Based on the obtained results, the second phase of action research is planned. Discussions and workshops on the role and use of (growing) geometric patterns will be conducted in collaboration with mathematics teachers, aimed at introducing changes in learning and teaching mathematics.]

Didaktički su materijali u nastavi matematike modeli koji se koriste da zorno prikažu određene matematičke sadržaje. Potrebno ih je koristiti zbog kognitivno-razvojnog razdoblja u kojem se učenici razredne nastave nalaze. Razvoj apstraktnog mišljenja kod učenika počinje od manipulacije konkretnim objektima i učenjem putem osjetila. U ovom radu govorimo o značaju korištenja didaktičkih materijala u razrednoj nastavi matematike. Također, opisujemo različite gotove (kupljene) didaktičke materijale (poput abakusa, Cuisenaireovih štapića, Dienes blokova...), virtualne didaktičke materijale i dajemo primjere samostalno izrađenih didaktičkih materijala. Veliki značaj u korištenju didaktičkih materijala daje i Montessori pedagogija. Upitnik o korištenju didaktičkih materijala u razrednoj nastavi matematike poslan je učiteljima primarnog obrazovanja. Uzorak je činilo 80 učiteljica i učitelja iz različitih gradova Republike Hrvatske. Istraživanje je pokazalo da učitelji razredne nastave koriste gotove (kupljene), virtualne i samostalno izrađene didaktičke materijale u jednakoj mjeri. Veliki dio ispitanika smatra da škole trebaju biti bolje opremljene didaktičkim materijalima. Didactic materials in mathematics teaching are models that are used to visualized certain mathematical contents. They are used because of the cognitive-developmental stage of primary school students. The development of abstract thinking in students begins with the manipulation of concrete objects and learning through the senses. In this thesis, the importance of using didactic materials in mathematics classroom teaching is discussed. It also describes various prefabricated (purchased) didactic materials (such as the abacus, Cuisenaire rods, Dienes blocks...), virtual didactic materials, and gives examples of self-made didactic materials. Montessori pedagogy puts a substantial focus on the use of didactic materials. A questionnaire on the use of didactic materials in the teaching of mathematics was distributed to primary school teachers. The sample consists of 80 teachers from different cities of the Republic of Croatia. This research shows that primary school teachers use prefabricated (purchased), virtual, and self-made didactic materials to the same extent. Many participants believe that schools should be better equipped with didactic materials.

Mjerenje je proces uspoređivanja neke veličine s istovrsnom veličinom koja je dogovorena jedinicom mjere. U razrednoj nastavi u Hrvatskoj mjeri se duljina, vrijeme, obujam tekućine, masa, temperatura, opseg i površina. Kako bi djeca usvojila mjerenje, potrebno ih je voditi kroz različite faze. Učenike je potrebno poticati na procjenjivanje rezultata u mjerenju. Dobra procjena ne znači nasumično pogađanje bez smisla, već se odnosi na smisleno i znanjem opravdano procjenjivanje. Procjenjivanje je čovjeku prirodno potrebno za snalaženje u životnim situacijama te identificiranje nelogičnosti rezultata postupka mjerenja. Procjenjivanje se istraživalo na različite načine odnosno s različitim fokusom. Postoje određene strategije koje ljudi koriste kako bi procjenjivali: primjena zamišljenih mjernih instrumenta, ponavljanje jedinica, referentne točke, prethodno znanje, rekompozicija. Pregledom istraživanja o procjenjivanju i analizom nekih udžbenika koji se koriste u razrednoj nastavi matematike, može se zaključiti kako su kompetencije procjenjivanja slabe i zadaci procjenjivanja vrlo rijetko zastupljeni u udžbenicima. Measurement is the process of comparing some size to an equivalent size to the agreed size upon measurement unit. In primary education in Croatia, students measure length, time, volume of fluids, the mass, temperature, perimeter and area. In order for children to adopt the measurement, they need to be guided through different stages. The pupils should be encouraged to estimate the results of measurement. Good estimation does not mean random guessing without meaning, but it refers to knowledgeable and justified assessment. The estimation is necessary for a person to naturally cope with life situations and to identify the illogicality of the results of the measurement procedure. The estimation was explored in different ways and with a different focus. There are certain strategies that people use to estimate measure: the application of imaginary measuring instruments, repetition of units, reference points, prior knowledge, disintegration. By reviewing research on estimation and analysing some mathematics textbooks used in mathematics teaching, it can be concluded that estimation competencies are weak and tasks are very rarely represented in textbooks.

Množenje napamet ili metodama pisanog računa, te primjenjujući međusobne veze računskih operacija, učenička su postignuća koja se odnose na množenje, a koja Nacionalni okvirni kurikulum propisuje za prvi odgojno-obrazovni ciklus. U nastavi matematike u Republici Hrvatskoj poučava se procedura množenja koja obuhvaća nekoliko djelomičnih radnji: računanje djelomičnih umnožaka, potpisivanje s pomicanjem i pismeno zbrajanje međurezultata. U radu su opisani različiti pristupi množenju dostupni u literaturi kao i rezultati znanstvenih istraživanja na ovu temu. Za potrebe diplomskog rada provedeno je istraživanje s ciljem ispitivanja znanja, stavova i mišljenja studenata 3. i 4. godine Učiteljskog studija o različitim pristupima proceduri množenja te njihovoj primjeni u razrednoj nastavi matematike. Rezultati su ukazali da je specijalizirano matematičko znanje, znanje o kurikulumu i znanje o poučavanju darovitih učenika određenom matematičkom sadržaju studenata slabo razvijeno. Među različitim procedurama studenti preferiraju Mrežno množenje. Mental multiplications, multiplying by written calculations or by applying the relations between arithmetic operations, are pupil achievements related to multiplication from the national curriculum for the first educational cycle. In mathematics teaching, in Croatia, the taught multiplication procedure involves several partial actions: computing partial products, signing them while moving accordingly and adding them with written calculation. The paper describes different approaches to multiplication available in sources as well as the results of scientific research on this topic. For the purpose of this thesis, a study was conducted to examine the knowledge, attitudes and opinions of students in their 3rd and 4th year of the Teacher's study on different approaches to multiplication and their application in classroom mathematics. The results indicated that students’ specialized mathematical knowledge, knowledge of the curriculum and the knowledge on teaching gifted students a certain mathematical content are poorly developed. Among some multiplication approaches, students preferred lattice multiplication.