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Assessment is among the inevitable components of a curriculum and directs students’ learning. E- assessment, as prepared and administered with the use of ICT, provides opportunities to make the process easier in some aspects, but also brings certain challenges. This paper presents an e- assessment framework from a student perspective. Our study lasted for one academic year and incorporated a total of 631 students’ responses to questionnaires conducted three times in two different mathematics courses. The courses are compulsory for undergraduate informatics students. The research included a three-step factor analysis: two exploratory factor analyses and a confirmatory factor analysis. It yielded a model of a student perspective on e-assessment consisting of four factors: Transparency and fairness of assessment, Formative and summative assessment and feedback, Meaningful use of technology in assessment and Difficulty of learning outcomes. The students’ opinion about the use of e-assessment was generally positive. The students appreciated the student-centred approach based on sound pedagogical alignment of all teaching and learning elements with assessment. They found it important that technology should not distract students during assessment tasks and considered cheating in a controlled e-assessment environment to be no more frequent than in face- to-face assessment.

Zbog pandemije COVID-a obrazovne institucije diljem svijeta (pa tako i u Republici Hrvatskoj) privremeno su zatvorile svoje zgrade i učionice. To je rezultiralo brzim prelaskom na učenje i poučavanje na daljinu te online vrednovanje. Veliki izazov je u hibridnom ili potpuno online modelu osigurati i održati kvalitetnu nastavu matematike, posebno prilikom uvođenja novih koncepata koji su studentima zahtjevni. Posebno je zahtjevna priprema, organizacija i provođenje online vrednovanja koje mora udovoljavati stručnim i metodičkim zahtjevima, ali i biti vjerodostojno. Kritički ispitujemo ulogu učenja na daljinu i online vrednovanja u stjecanju znanja, vještina i stavova. Zalažemo se za konstrukciju cjelovitog programa vrednovanja s pomno pripremljenim skupom metoda vrednovanja, ali i njegovu evaluaciju prema sveobuhvatnom okviru. U tu svrhu odabrali smo takozvani Model korisnosti vrednovanja koji su uveli Van der Vleuten i Schurwith (2005). Spomenuti model sadrži 5 kriterija: pouzdanost, valjanost i prihvatljivost vrednovanja, utjecaj vrednovanja na učenje, te troškovi modela vrednovanja. Fokus našeg rada je na pouzdanosti koju vrlo općenito shvaćamo kao razinu točnosti odluka za prolaz i neuspjeh. Analizu pouzdanosti provodimo na razini ispita, ali i na razini cijelog programa vrednovanja za pojedini predmet. Preciznije, konstruiramo kompozitni indeks za program vrednovanja na razini cijelog predmeta na dva načina. Taj kompozitni indeks predstavlja moguću mjeru pouzdanosti cijelog programa vrednovanja. Za testiranje modela koristili smo analitike učenja dostupne u sustavu za upravljanje e-učenjem (Moodle) za posljednjih nekoliko akademskih godina. Podatke o rezultatima vrednovanja analizirali smo uz pomoć statistika testa u Moodleu te odredili koeficijent interne konzistentnosti (tzv. Crombach alpha) na razini pojedinog dijela u planu vrednovanja (npr. za kolokvij) koje smo zatim težinski usrednjili. Statistike testa u Moodleu, omogućavaju upotrebu i drugih parametra kvalitete ispitnih zadataka na razini testa, ali i na razini pojedinog pitanja.

Technology can support teaching and learning processes in many ways. In particular, e-assessment can provide a meaningful upgrade to standard approaches. We will present a model of a student perspective on e-assessment in mathematics in higher education. The focus of our presentation will be on e-assessment supported by a Learning Management System (LMS). Within our mathematics courses paper-based assessments were replaced with computer-based ones. E-assessment questions were individualized, taken from a systematized online question bank, in which questions are organized according to a taxonomy of learning outcomes. E-assessments were conducted in a controlled online environment. We collected 631 students' responses to questionnaires on three occasions, in two different mathematics courses during one academic year. The research included two exploratory factor analyses and a confirmatory factor analysis. It yielded a model of a student perspective on e-assessment, consisting of four factors: Transparency and fairness of assessment, Formative and summative assessment and feedback, Meaningful use of technology in assessment and The difficulty of learning outcomes.

Tijekom pandemije je veliki izazov osigurati i održati kvalitetnu nastavu matematike i drugih predmeta koji su studentima konceptualno zahtjevni, u hibridnom ili potpuno online modelu. Posebno je zahtjevna priprema, organizacija i provođenje online vrednovanja koja moraju udovoljavati stručnim i metodičkim zahtjevima, ali i biti vjerodostojna. Principi kojima se vodimo u dizajnu nastave i online vrednovanja su jasna veza vrednovanja s planiranim ishodima učenja predmeta, upotreba kompleksnih i/ili problemskih zadataka strukturiranih pomoću prikladne taksonomije u velike baze zadataka i pitanja (tako da svaki student dobije “svoje zadatke”). Uz nabrojeno, razumno su odabrane i druge mjere za prevenciju i sprečavanje varanja na (online) ispitima.

Assessment is one of the central components of a curriculum and guides students' learning. E- assessment involves the use of ICT to present and deliver assessment tasks, receive and record students' answers, as well as to provide feedback to students. E-assessment has been used considerably more in many higher education institutions since the onset of the COVID-19 pandemic and therefore our aim is to explore the specifities of e-assessment in mathematics. We will present the benefits and downsides of e- assessment in mathematics in higher education. Our study focused on students' perspectives on e- assessment supported by a Learning Management System (LMS) in mathematics courses. It incorporated 631 students' responses to questionnaires conducted three times and it lasted for one academic year. We used descriptive statistics for close-ended questions and qualitative analysis for open-ended questions. The students' perspectives on e- assessment were generally positive. The students appreciated the student-centered approach based on sound pedagogical alignment of all teaching and learning elements with assessment. However, they found it important that technology should not distract them during assessment tasks. Finally, they did not find that there was more cheating and plagiarism in e- assessment in comparison to on-campus face-to-face assessments.

The assessment should be constructed and analysed holistically. The role of assessment should be critically examined in acquisition of knowledge, skills and attitudes. We postulate two aspects: 1. The complete assessment programme with carefully prepared assessment set should be considered. 2. Assessment programme should be evaluated according to an overarching framework. According to (Van der Vleuten & Schuwirth, 2005) the utility framework for assessment depends on 5 factors: Validity (the most frequent quality criterion), Reliability (the accuracy of pass and fail decisions), Educational impact, Acceptability and Cost of assessment. Our aim is to use learning analytics to support evaluation of assessment set according to the utility framework.

Due to the COVID{;19 pandemic the educational institutions around the world have been temporarily closed by their governments which resulted in rapid transition to remote learning and online assessment. We critically examine it's role in acquisition of knowledge, skills and attitudes. We advocate construction of the complete assessment programme with carefully prepared assessment set, but also its evaluation according to an overarching framework. For that purpose we have chosen the so-called utility framework introduced by Van der Vleuten & Schurwith, 2005. We will present the utility framework which consists of five elements (validity, reliability, educational impact, acceptability and costs). The focus will be on the reliability which we broadly understand as the accuracy of pass and fail decisions. More precisely, we will introduce and analyse several approaches for construction of the composite index for the assessment programme on the course level as a possible measure of the reliability. To do that, we use data available in the learning management system for past two academic years.

We use the theory of point and branching processes to study asymptotic distribution of the maximal claim size for marked Hawkes processes, in which marks determine the size and other characteristics of the individual claims and influence arrival rate of the future claims. We present sufficient conditions under which the maximal claim size tends in distribution to a max-stable random variable.

The theory of point processes constitutes an important part of modern stochastic process theory and is widely recognized as a useful and elegant tool for modelling. Point processes are well understood models and have applications in a wide range of applied probability areas, especially in risk theory which is important for understanding non-life insurance mathematics. It deals with the modelling of claims and gives answers on premium amount. Elegant mathematical analysis of the classical Cramér - Lundberg risk model has an important place in non-life insurance theory. The theory yields precise or approximate computations of the ruin probabilities, appropriate reserves, distribution of the total claim amount and other properties of an idealized insurance portfolio. In recent years, some special models have been proposed to account for the possibility of clustering of some events, for instance the Hawkes processes. We study asymptotic distribution of the total claim amount in the setting where Cramér - Lundberg risk model is augmented with a marked Poisson cluster structure. Marked Hawkes processes are then a special case and have an important role as the key example in our analysis. To make this more precise, we model arrival of claims in an insurance portfolio by a marked point process, say \[N = \sum_{k=1}^{\infty}\delta_{\tau_k,A^k}\] where \(\tau_k\)'s are non-negative random variables representing arrival times with some degree of clustering and \(A^k\)'s represent corresponding marks in a rather general metric space \(\mathbb{M}\). For each marked event, the claim size can be calculated using a measurable mapping of marks to non-negative real numbers, \(f: \mathbb{M}\) \rightarrow \mathbb{R}_{\geq 0}\) say. So that the total claim amount in the time interval \([0, t]\) can be calculated as \[S(t) = \sum_{\tau_k \leq t}f(A^k).\] We determine the effect of the clustering on the quantity \(S(t)\), as \(t\to \infty \), even in the case when the distribution of the individual claims does not satisfy assumptions of the classical central limit theorem. Besides new results regarding the case when second moments do not exist, we use different approach based on the limit theory for two dimensional random walks which stems from the classical Anscombe's theorem and not on martingale central limit theorem which was commonly used. We present the central limit theorem for the total claim amount \(S(t)\) in our setting under appropriate second moment conditions and prove a functional limit theorem concerning the sums of regularly varying non-negative random variables when subordinated to an independent renewal process. Based on this, we prove the limit theorem for the total claim amount \(S(t)\) in cases when individual claims have infinite variance. Moreover, we apply these results to three special models. In particular, we give a detailed analysis of the marked Hawkes processes which are extensively studied in recent years. In the last chapter we move our attention to the maximal claim size and present our results regarding limiting behaviour of maximum when claims belong to the maximum domain of attraction of one of the three extreme value distributions (Fréchet, Weibull and Gumbel). We also apply those results to three special models which we studied in previous chapter. Besides that, we try to clarify the notion of stochastic intensity which can be described in several different ways. The understanding of the stochastic intensity is important because of it's usage in the implicit definition of Hawkes processes. Teorija točkovnih procesa utemeljuje važan dio moderne teorije stohastičkih procesa i široko je prepoznata kao koristan i elegantan alat za modeliranje. Točkovni procesi su model koji se dobro razumije i koriste u mnogim područjima primijenjene vjerojatnosti, posebno u teoriji rizika (matematika neživotnih osiguranja). Teorija rizika se bavi modeliranjem zahtjeva za isplatom u svrhu određivanja visine premije. Elegantna matematička analiza Cramér-Lundbergovog modela rizika ima važnu ulogu u teoriji neživotnih osiguranja. Spomenuta teorija nam daje precizne ili aproksimativne izračune vjerojatnosti propasti, odgovarajućih rezervi, distribuciju sume zahtjeva za isplatama i druga svojstva idealiziranog portfelja osiguravatelja. Posljednjih godina predloženi su neki specijalni modeli koji uključuju mogućnost klasteriranja događaja, na primjer Hawkesovi procesi. Proučavamo asimptotske distribucije ukupnog iznosa zahtjeva za isplatom u označenim Poissonovim procesima s klasterima u kojima oznake određuju visinu, ali i druge karakteristike pojedinih zahtjeva te potencijalno utječu na stopu dolazaka budućih zahtjeva. Označeni Hawkesovi procesi u tom slučaju postaju specijani slučaj općenitog modela označenih Poissonovih procesa s klasterima. Malo preciznije, dolaske zahtjeva za isplatom u promatranom portfelju modeliramo označenim točkovnim procesom, npr. oblika \[N = \sum_{k=1}^{\infty}\delta_{\tau_k,A^k}\] pri čemu su \(\tau_k\) nenegativne slučajne varijable koje predstavljaju vremena dolazaka s nekim stupnjem klasteriranja, a \(A^k\) pripadne oznake u nekom metričkom prostoru \(\mathbb{M}\). Visina zahtjeva za isplatom u svakom označenom događaju može se izračunati upotrebom izmjerivog preslikavanja \(f(A^k)\) iz prostora oznaka u nenegativne realne brojeve. Tada se suma zahtjeva za isplatom u intervalu \([0, t]\) može izraziti kao \[S(t) = \sum_{\tau_k \leq t}f(A^k).\] Promatramo učinak klasteriranja na \(S(t),\) kada \(t\to \infty \) čak i u slučaju kada distribucija individualnih zahtjeva ne zadovoljava pretpostavke klasičnog centralnog graničnog teorema. Osim novih rezultata u slučaju kada drugi moment nije konačan, u izračunima koristimo drugačiji pristup koji se temelji na graničnim teoremima za zaustavljene dvodimenzionalne slučajne šetnje (koji proizlaze iz Anscombeovog teorema), a ne na martingalnom centralnom graničnom teoremu koji je često korišten. Prezentirat ćemo dovoljne uvjete uz koje ukupan iznos zahtjeva zadovoljava centralni granični teorem ili alternativno teži po distribuciji stabilnoj slučajnoj varijabli s beskonačnom varijancom. Diskutirat ćemo nekoliko Poissonnovih modela s klasterima, pri čemu će označeni Hawkesovi procesi biti naš ključni primjer. U posljednjem poglavlju fokus prebacujemo na maksimalan iznos zahtjeva za isplatom u intervalu \([0,t].\) Prezentirat ćemo rezultate vezane uz granično ponašanje maksimuma u slučaju kada pojedinačni zahtjevi za isplatama pripadaju Fréchetovoj, Weibullovoj ili Gumbelovoj maksimalnoj domeni privlačnosti. Ponovo primjenjujemo dobivene rezultate na tri specijalna modela (s posebnim naglaskom na Hawkesove procese). Osim graničnog ponašanja suma i maksimuma, pokušali smo razjasniti pojam stohastičkog intenziteta, posebno jer se u literaturi može pronaći nekoliko različitih definicija spomenutog stohastičkog intenziteta. Razumijevanje stohastičkog intenziteta nam je važno jer se koristi prilikom definiranja Hawkesovih procesa.

We use the theory of point processes to study asymptotic distribution of the maximal claim size for marked Poisson cluster models, in which marks determine the size and other characteristics of the individual claims and potentially influence arrival rate of the future claims. We present sufficient conditions under which the maximal claim size tends in distribution to a max-stable random variable. We apply these results on three Poisson cluster models with the marked Hawkes processes as our key example.

In this talk we want to present some results regarding asymptotic distribution of the total claim amount for marked Poisson cluster models, in which marks determine the size and other characteristics of the individual claims and potentially influence arrival rate of the future claims. A particular emphasis will be put on the application of these results on some Poisson cluster models with the marked Hawkes processes as our key example.

"A problem is only a problem (as mathematicians use the word) if you don’t know how to go about solving it. A problem that has no “surprises” in store, and can be solved comfortably by routine or familiar procedures (no matter how difficult!) it is an exercise "(A. Schoenfeld). Problem solving has been studied very thoroughly from Polya’s time till today. It plays an important role in mathematics and should have a prominent role in the mathematics education. On the other hand, there is little or no research on implementing problem solving into e-learning. We are going to give a brief overview of problem solving from Polya’s time till today in two directions. In the first are characteristics of good problem solving process – what is good problem and teaching process? In the second are skills of good problem solver – what are the adequate patience level, confidence, willingness and attitude towards problem solving? Then we will present our first steps in dealing with problem solving through e-learning. Our idea was to expose students to greater number of problem tasks (that they didn’t expect in the beginning of the semester) with an opportunity to get the feedback and discussion with teacher through e-learning. We wanted to research three things: how many students would use this way of problem solving, whether this extra engagement would yield better results for similar tasks in mid-term tests, and students attitude towards problem solving. We came to conclusion that students cannot become successful problem solvers overnight. Helping students to become successful problem solvers should be a long-term goal, so effort should be made to reach this goal in every mathematical topic and in every lesson.

Dokaz je važan katalizator u razvoju apstraktnog i logičkog razmišljanja. To su poželjne osobine svih studenata, a ne samo matematičara. Vođeni ovakvim razmišljanjem, počeli smo se baviti ovom temom, s naglaskom na prenosti koje imaju studenti na Fakuletu organizacije i informatike Sveučilišta u Zagrebu. Proveli smo istraživanje u dva navrata te smo analizirali dobivene podatke u svrhu uočavanja studentskih prepreka i problema sa dokazima.

Two tests, a questionnaire and a few questions in each mid-term test were given to second and third year undergraduate students at the Faculty of organization and informatics (FOI) of the University of Zagreb. The data were analyzed to identify and clarify the importance of logic in education of non-mathematicians and their understanding of counterexample as a method of disproving mathematical statements.

Two tests and a questionnaire were given to second and third year undergraduate students at the Faculty of organization and informatics (FOI) of the University of Zagreb. The data were analyzed to identify and clarify some of the pedagogical roles of proof in education of non-mathematicians.

The purpose of this study is to research and validate items of Bandura’s social cognitive theory of self-efficacy among ICT students of the first year at Faculty of organization and informatics at the University of Zagreb. Our research was conducted in 2009/2010 year in summer semester in mathematics on the sample of 202 students. After confirmatory factor analysis we have got five factors which can influence students’ self-efficacy: mastery experience with social persuasion, psychological states, vicarious experience, grade and family support. We have found that there are differences in self-efficacy for those students who passed the mathematics exam and those who failed. There are three common factors which influence self-efficacy of both successful and less successful students – mastery experience with social persuasion, psychological states, and vicarious experience. Two remaining factors are grade factor influences successful students’ self-efficacy, and family support factor influences self-efficacy of less successful students. There are many reasons why is self-efficacy important for school and university students and one of the most important is that academic self-efficacy is the good predictor of academic outcomes.