Trigonometry is an important part of secondary school mathematics, but it is usually challenging for students to understand and learn. Since trigonometry is learned and used at a university level in many fields, like physics or geodesy, it is important to have an insight into students’ trigonometry knowledge before the beginning of the university courses. This research aimed to develop a test in triangle and circle trigonometry, which can be used to test students’ prior knowledge of basic trigonometric concepts. A test with multiple-choice questions was developed based on content and learning outcomes from the secondary school curriculum. After a group of three experts evaluated the content validity of the test, the corrected version was given to 58 first-year students at the Department of Mathematics and Computer Science. After the test and item analysis, a final version of the test was made which can be used as a short initial test to measure students’ prior knowledge of trigonometry at the beginning of a university course.

According to Mitchelmore [1], generalisations are the cornerstone of school mathematics, covering various aspects like numerical generalisation in algebra, spatial generalisation in geometry and measurement, as well as logical generalisations in diverse contexts. The process of generalising lies at the heart of mathematical activity, serving as the fundamental method for constructing new knowledge [2, 3]. In this paper we will generalise an interesting geometry problem that appeared in the 1995 edition of the International Mathematical Olympiad (IMO) [4].

ABSTRACT

The inclination is the most important criteria for determining stability of the object for objects for which the ratio of height and width is high (industrial chimneys, towers, tall buildings, etc.). In this paper, two methods for testing the verticality of objects with a circular base will be presented: principal components analysis (PCA), and a new method based on fitting circles and analytical geometry in space presented in the paper "Rapid assessment of verticality of structural objects with a circular base" (Hamzić, Kamber Hamzić i Avdagić, 2021). The cases when the verticality of the object is calculated from the cloud of measuring points of all or part of the object and when the verticality is calculated from the measurement of the circular arc of the object in two levels are analyzed. The results showed that both methods give quality results when the point cloud covers the entire circular arc of the object. In cases where only a part of the circular arc was observed, the method based on the circle fitting gave better results. In the case when the measurement is performed only in two levels on the object, the method based on circle fitting gave good results while the PCA method had large errors in estimating the inclination of the object.

Structural health monitoring of the large infrastructural objects (high buildings, bridges, tunnels, dams, etc.) is in the domain of civil and geodetic engineers who use different methods and instruments for this task. Dam movement is influenced by various factors among which the most important are: thermal variations, hydrostatic pressure and dam ageing. This research investigates influence of thermal variations on dam crest movement by using statistical methods: autoregressive integrated moving average (ARIMA) and multiple linear regression. Dam crest movement data is obtained by using optical alignement method on the concrete gravity dam HP Salakovac. In the first part of this research correlation between dam crest movement and concrete temperature is determined, the second part deals with short term concrete temperature prediction and in the final part of this research previously fitted statistical models are used for dam movement prediction. The results showed that proposed model based on statistical methods can provide quality prediction of dam crest movement.

Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in geometric problems turns out to be particularly difficult, even for high attaining students [2]. Sometimes, students do not even know where to start when trying to solve these [3].

Cooperative learning is a modern teaching strategy in which team work and cooperation become the most important activities of the entire teaching process. The quality of interaction between students and teacher, as main participants in teaching process, is important for successful application of cooperative learning. Beside faster and longer lasting knowledge acquiring, cooperative learning develops critical and creative thinking, communication and social skills and it strengthens self-confidence. Modern methods of teaching mathematics focus on didactical principle of conscious activity above other principles. This means students are major, active factors of mathematics teaching, and not only they participate in the process of teaching, but they also participate in the selection of methods of teaching. This enhances their motivation for work during classes. This means, what is learned through cooperative learning is better used in new situations, knowledge transfer is greater and new knowledge is acquired easier and lasts longer. Specific and abstract contents of mathematics lead to different ways of applying cooperative learning in this subject. That is why we chose this subject, i.e. to explore and point out the possibilities and ways of applying cooperative learning in mathematics.

Some zeta functions which are naturally attached to the locally homogeneous vector bundles over compact locally symmetric spaces of rank one are investigated. We prove that such functions can be expressed in terms of entire functions whose order is not larger than the dimension of the corresponding compact, even-dimensional, locally symmetric space.