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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.

Mathematical textbooks, as part of a potentially implemented curriculum have been increasingly analysed, mostly from the point of their structural features, performance expectations and mathematical content. In the previous work, we have performed such kind of analysis of Croatian mathematical textbooks for the 4th grade of secondary schools. During this process, the question of language used in the books came out. In this paper, we use the framework proposed by Lisa O’Keefe and John O’Donoghue (2014), based on the Halliday’s functional grammar analysis (Halliday (1985), Halliday and Martin (1993), Morgan (1998), to perform a preliminary analysis of two Croatian mathematical textbooks.

This paper presents an analysis of Croatian State Matura exam tasks relating to the mathematical domain of functions. The analysis is conducted on the 2015 and the 2016 State Matura, both higher-level mathematics exams. Requirements of the tasks are explored and classified based on the instrument designed for this purpose. The instrument is conceptually based on the theory of “basic ideas” (originally: Grundvorstellungen) and Mathematical Proficiency theory. The objectives of the research were to explore the possibility of using such an instrument for task analysis, to determine the characteristics of the tasks in the Croatian State Matura, and to use student achievement to value their efficiency in the mathematical domain of functions in terms of specific classification provided by the instrument.

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 paper, authors describe their own teaching experience using Maxima and GNU Octave software packages as computational tools in math courses at studies for Electrical and Civil Engineering of Zagreb University of Applied Sciences and discuss possibilities of achieving the same learning outcomes using software package Giac/Xcas. Finally, a way of modernizing math courses by introduction and usage of suitable computer software is proposed, emphasizing the importance of using free software in education.

Goal of our research is to present a method for selection of a group of students. The selected group of students is supposed to receive additional teaching attention in order to improve their performance in the course. Guiding line used, for the group selection, is: value of an action is proportional to the benefit that produces to a customer. In our case customers are students. The selection method is based on multinomial logistic regression, Poisson-type discrete variables modelling the number of points achieved in a term exam, and transition probability matrix. We use data of a study progress during monitoring program in an undergraduate mathematical course. Demographic data and other attributes about previous performance were not included in the analyses. In the first part of the paper we present methodology, while in the second part we introduce data used for demonstration of the proposed method. At the end of the paper, an individual approach for the final selection is proposed. Criteria for selection is clear: increased probabilities of obtaining desired final grade for a student. Weighing of criteria is subjective depending on the goals of the decision makers.

Technology is slowly entering the curriculum of mathematics in Croatian primary and secondary schools, and a little faster at the faculty level. This entry often depends on the teacher. Mathematical education at universities has witnessed daily progress, not only because of the introduction of e-learning systems (Moodle, MudRi) but also because of the availability of different technologies and computer programs, which are used for improving teaching, such as Mathematica, Matlab, GeoGebra, and the like. Experiences of teachers show that overcoming certain teaching units can be successfully and easily improved by implementing modern technology in their teaching. The result of this is additional student motivation for adopting mathematical knowledge by connecting various disciplines. Observing the selected problem through mathematical as well as other sciences’ perspectives, such as physics, chemistry or biology, has an impact on building a more well-rounded individual who will be able to handle today’s challenges. In this way, understanding mathematics as a science, which is a purpose to itself, can be overcome. This in turn contributes to its successful popularization and creates a positive atmosphere about this science as an important tool, which can help in resolving many interdisciplinary problems. The aim of this paper is to present the way in which mathematics can be improved in higher education with the help of selected examples.

This work deals with the impact of using digital textbooks consisting of GeoGebra interactive applets on students' conceptual and procedural knowledge in mathematics. The authors propose the model of computer-guided discovery learning by using dynamic geometry software which includes three basic elements: learning objects, tasks for students and learning outcomes. The model is primary based on the theory of constructivism, Pólya’s heuristic strategy and Schoenfeld’s problem-solving model. The experimental research plan with experimental and control groups was used and there were 703 students in twelve elementary schools in Croatia participating in the research. The experimental group of students was taught by the model of computer-guided discovery learning and the control group of students was taught by teachers using traditional teaching methods. The covariance analysis determined the impact of the model of computer-guided discovery learning on students' conceptual and procedural knowledge and the results have shown statistically significant differences between the groups in favor of the experimental group. The qualitative and quantitative analysis of students' and teachers' questionnaires and forum discussions was used to assess their attitudes towards the proposed learning model. The importance of teachers’ experience in teaching using this model has also been noted. This learning model indicates the potential of computers and dynamic geometry software GeoGebra for scaffolding support in learning mathematics.

The papers in the Monograph addressed different topics related to teaching and learning processes of mathematics which are of great interest to both students and prospective teachers. Some papers opened new questions for research, some showed examples of good practice and others provided more information for the earlier findings. All papers portray the complex role of mathematics education: mathematics education is a science but also a profession. Research in mathematics education is needed to investigate various phenomena, but the research has also the responsibility to inform practitioners of its findings. Mathematics teachers are a critical component in this process. Their collaboration with the researchers is necessary to achieve the shift from the traditional to contemporary teaching mathematics. The Monograph consists of six chapters. In the first chapter of the Monograph the author studied the relation between the surface approach and the strategic approach on the learning outcome. In the second chapter the authors examined the topic of geometric character. They discussed the topic of the importance of the spatial reasoning, as well as the effect of computer technology on geometry education. An emphasis was put on the fact that the strong links between geometry and technology are important because geometry’s need for proficiency in many technical fields exceeds the traditional Euclidean space geometry. In the third chapter the authors argued that teachers’ beliefs about teaching and learning mathematics were significant in the utilization of a particular resource. They examined teachers’ classroom practice and beliefs about mathematics, mathematics education, teaching mathematics and using textbooks as curriculum resources. The importance of the application of ICT in teaching mathematics was discussed in the fourth chapter. The impact of using proposed computer guided discovery learning model on students’ conceptual and procedural knowledge in mathematics was investigated. In the fifth chapter the author examined whether there was a connection between the attitudes towards mathematics and the performance on mathematics exams. In the last chapter of the Monograph the authors suggested some useful approaches to teaching mathematics.

Cilj ovoga rada jest predstaviti različite strategije zbrajanja i oduzimanja, kao i ukazati na važnost korištenja različitih pri rješavanju zadataka. Odabir strategija je adaptivan, on ovisi o djetetovom poznavanju određenih strategija i o obilježjima zadataka koje dijete rješava. Cilj rada je, također, predstaviti kognitivne strukture, odnosno mentalne slike, kao važan dio računanja. Način na koji djeca zamišljaju i predstavljaju manje brojeve, pridonosi lakšem prikazu većih brojeva i računskih radnji. Materijali Rechenschiffchen i Zwanzigerfeld pridonose izgradnji tih kognitivnih struktura. Istraživanje provedeno za potrebe ovoga rada temelji se na problematici izgradnje kognitivnih modela, a izvedeno je na materijalu Das Zwanzigerfeld mit Wendeplättchen. Istraživanje je provedeno u OŠ Mitnica Vukovar, a sudjelovalo je 37 učenika. Rezultati istraživanja pokazali su da su djeca radne skupine sposobna lakše prikazati i prepoznati određenu strukturu broja i/ili računske radnje. Manje strukture djeca puno lakše prepoznaju na pogled, a materijal je osobito pogodan za rad s djecom sa specifičnim poteškoćama u matematici i vizualnim tipovima učenika. Djeca češće koriste prikaz broja i računske radnje u redu. The purpose of this paper is to present different strategies of addition and subtraction as well as to point out the importance of using different strategies when solving tasks. The selection of strategies is adaptive, it depends on child's knowledge of about specific strategies, as well as on characteristics of the task which is being solved. Moreover, the purpose of the paper is to present cognitive structures, that is, mental images, as an important part of calculating. The way in which children imagine and present smaller numbers contributes to easier presentation of larger numbers and mathematical operations. The materials Rechenschiffchen and Zwanzigerfeld contribute to the construction of those cognitive structures. The research conducted for the purpose of this paper is based on the issue of construction of cognitive models, and it is carried out on the material Das Zwanzigerfeld mit Wendeplättchen. The research was conducted in primary school Mitnica Vukovar, with 37 pupils engaged. The results of the research have shown that the children of the working group are capable to easily II present and recognize the specific number and/or operation structure. Children easily recognize the smaller structures on sight, and the material is particularly suitable for working with children with specific difficulties in mathematics as well as for visual types of pupils. Children use the number presentation and operations in a line more often.

In this paper, the author examines current problems in the teaching of mathematics and mathematics education and shows how theoretical and practical findings underestimate the role of the social dimension in the teaching of mathematics. Mathematical education is perceived in this research as a kind of a social construct in contrast to the traditional definition of (teaching) mathematics as a purely scientific discipline. The author focuses on students’ emotional reactions, classroom environment and teacher competencies as indicators of quality in the teaching of mathematics. In the context of contemporary mathematics teaching, the author uses a critical approach to assess the way and content of teacher education as well as the required competencies in quality mathematics teachers. Taking into account the requirements and specifics of teaching mathematics that are in the domain of pedagogical theory and practice, the author emphasises teachers’ pedagogical competencies and their definition from the pedagogical perspective. A review of relevant research has led the author to state that a stimulating classroom environment is one of the key assumptions of students’ success in mathematics and that the didactic-methodological guidelines of teaching mathematics should be based on the individuality of students and the demands and difficulties they encounter in the classroom.

The article discusses the experiences with the use of concept maps in teaching mathematics in the first semester at the Faculty of Teacher Education of the University of Zagreb. We present and analyze the results of a survey conducted among students that reflects their attitudes toward the use of concept maps. The results are compared with the results of previously conducted survey among future teacher in the United Arab Emirates. Finally, we give directions for our future work and research.

Work with Roma children raises special challenges for educators and teachers who during their pre-service training have not been prepared for the specific work within the culture of the Roma community and the Roma language and in particular for the flexibility in work with Roma pupils. Interest of Roma pupils in educational process is a challenge to which we have tried to respond by introducing elements of Vedic mathematics in tuition. This article provides a methodological proposal which shows a connection of an active approach of Roma students' mathematical learning with the introduction of the Vedic way of calculating in additional mathematics classes and extracurricular activities.

In the monograph Mathematics Teaching for the Future we use the term teacher to denote a person who teaches mathematics, and the context of each paper will reveal whether this refers to a teacher in a pre- school institution, a school teacher or a university instructor. In their papers the authors critically assess the results of many years of research within PISA and TIMSS projects regarding learning outcomes of ten-year-olds and fifteen-year-olds in the neighbouring countries of Bosnia and Herzegovina, Croatia, Hungary, Germany, Romania, Slovakia and Slovenia. Ten years ago the most successful examples of mathematics teaching practice were found in the Anglo-Saxon countries: Great Britain, the United States, New Zealand, Australia, and Canada. Today the best learning outcomes are achieved by the pupils of primary schools in Finland and in several Asian countries. At a global level there is great support for the popularization of mathematics. This is visible, among other, via examples of numerous popular mathematical contests organised at national levels, with special emphasis on attendance. This book contains reports from authors who critically observe all the events, as well as scientific, popular science and professional contributions in mathematics, and propose new approaches for the improvement of pupils’ achievements in the neighbouring countries of Bosnia and Herzegovina, Croatia, Hungary, Germany, Romania, Slovakia and Slovenia. The first chapter titled Fostering the development of mathematical competences in children of pre-school age and during compulsory education presents comparative analyses on learning outcomes in the subject of mathematics for pupils from the neighbouring countries during their compulsory education. The authors point to the continuous necessity for popularizing mathematics from the lower primary inclusion, in other words laws that govern the inclusion of children with disabilities into formal instruction, should in our sorroundings move beyond words towards deeds, which is primarily the task of governments, not necessarily of mathematics teachers. What is more closely considered is the need to improve learning outcomes in mathematics among economically disadvantaged groups, such as the Roma people, with this section of the book providing certain guidelines for teachers of mathematics in these particular cases. In the second chapter titled The role of information and communication technologies in other approaches to teaching high school mathematics the authors analyse the pre-existing conditions connected to using ICT in high school instruction, but also discuss goals and visions for the future. During the past few years there has been a great increase in the application of artificial intelligence methods, foremostly of tutor systems as modern teaching aids, but also for the purpose of detecting mathematically gifted pupils. From the papers included in this chapter it is obvious that in the sorrounding countries ICT is recognised as an important factor in high school instruction. Yet, the quality and availability of technical equipment, as well as the professional competence of teachers regarding its application within mathematics instruction, differ among the neighbouring countries. The authors highlight the need to encourage active, and subsequently creative approach of pupils/students to computers and computer software, the use of which, however, can cause resistance in pupils. By supporting pupils during specific activities, individually or as a team, using ICT can significantly contribute to the development of their creativity. In the third chapter titled Encouraging the development of students’ cognitive domain within mathematics instruction the authors underline the importance of such type of support at faculties and universities of teacher studies and technical studies. While observing the pre-existing levels of knowledge and the understanding of specific mathematical concepts of students from the sorrounding countries, we detected similar situations and issues. Some scholars are convinced that improvements in mathematics instruction at faculties of teacher studies and technical studies could be achieved if instructors managed to master various approaches to learning mathematics with their students. With regards to teacher studies, including students in research on the levels of knowledge of their future pupils, as well as various methods for the teaching of mathematical concepts which appear in the latest curricula (probability, data analysis, reading and presentation of data) has proven to be most effective. Furthermore, the emphasis should be placed on encouraging the understanding of and courageous arriving at “different paths to solutions”, which presupposes teachers’ skills of communicating on different levels of their pupils’ knowledge. In the fourth chapter titled The influence of convictions, views, norms, emotions and attitudes of mathematics teachers on the efficiency of mathematics instruction the authors emphasize the importance of a teacher’s personality in solving problems and designing instruction as a problem-based environment. Moreover, during task solving in mathematics instruction different attitudes of teachers, as well as those of pupils (joy, motivation, interest) can cause the participants of the teaching process to experience a variety of emotions (frustration, anxiety, pleasure, joy, impatience, rage). The authors suggest that we take into consideration teachers’ attitudes, as they influence the efficiency of different segments of mathematics instruction. The fifth chapter is titled Examples of good teaching practice. Teaching practice is the source of scientific research. The task of scientists in education, as well as related sciences, is to improve the quality of teaching practice by means of research and with the aid of their respective governments. The link between science and practice is created through efforts of benevolent scientists and critical practitioners. Only through costant interaction of scientific activities by noted mathematicians, psychologists, computer scientists, educationists, educational specialists and the critical work of teachers in the field can we, in the socially acceptable atmosphere, count on the achieved mathematical potential in pupils and students.