Teaching is a process for which its plan should contain reflection onto previous experience. With that in mind, teaching situations should be continuously researched and improved in accordance with the research results. Led by this thought and the fact that students are uncritically using visualization to solve mathematical problems, we defined the aim of this research–determine the attitude of students about the visualization of mathematical content (VMC). The subject of this research are the attitudes of students towards VMC. By analyzing our research subject, we have discovered the research problem–students use visual aid to solve problems uncritically. Based on this problem, we have set the aim of our research. Our aim was to determine the students’ attitudes (and their opinions) about VMC. Based on the aim of our research, we have set four research tasks. Based on these research tasks, we have established the main (leading) research question–What is the attitude of high school students towards the application of VMC? We divided the main research question into five questions: Do high school students consider that they understand the term ‘VMC’? Who considers they use more methods of solving mathematical problems using visual aid–high school male students or female students? Students of which grades consider that they use solving problems using visual aid more? What is the attitude of high school students about the relationship between the substantiality of the picture (the amount of data it encompasses) and the difficulty of solving the problem? What is the attitude of students about the use of software to solve mathematical problems? The research has been conducted with 1,240 high school students from Sarajevo, Bosnia & Herzegovina. For the purposes of this article, we employed a survey, questionnaire-based research. The research was created as part of a larger study conducted in the context of preparing a doctoral dissertation related to VMC. It is one fundamental research. An essential aspect of this research involves students’ attitudes toward VMC. After obtaining all necessary approvals from relevant institutions and parents, students proceeded to testing and surveying in their school classrooms, under the supervision of designated individuals who facilitated the conduct of the research. The distribution of the data was not normal, so we used the Pearson Chi-square, likelihood ratio Chi-square, and linear-by-linear association test to examine the association between student attitudes and categorical variables (gender and grade). In addition, we used frequencies and percentages. It has been concluded that the students are mostly positive towards applying visualization in their process of solving mathematical problems and these should be used in the direction of improving the students’ success, their confidence and their level of contentment in their mathematics class, as well as in other life situations that encompass mathematical content. In future research, it could be examined why students expressed such attitudes about the presented situations. Additionally, it would be significant to explore why students do not consider themselves successful in applying VMC, despite claiming to understand the term. The analysis could extend to the content presented in textbooks or instructional materials students use–how visualized the content is or whether students are required to visualize it themselves. It would also be worthwhile to investigate the extent to which teachers encourage students to visualize specific tasks or do so on their behalf. Given the fluctuation in results (we observe affirmative answers–partially or completely) observed across grades–initial decrease, subsequent increase, followed by another decrease–it might be explored whether this is related to the curriculum taught in each grade (such as content, volume, number of class hours, etc.). Regarding images leading to incorrect conclusions, it would be interesting to investigate the types of images students have in mind, how frequently they encounter such situations, where they use these images, who creates them, and similar aspects. These are just some questions for future research.

The object of the research are fuzzy functional dependencies on given relation scheme, and the question of their obtaining using the classical and innovated techniques. The attributes of the universal set are associated to the elements of the unit interval, and are turned into fuzzy formulas in this way. We prove that the dependency (which is treated as a fuzzy formula with respect to appropriately chosen valuation) is valid whenever it agrees with the attached two-elements fuzzy relation instance. The opposite direction of the claim is proven to be incorrect in this setting. Generalizing things to sets of attributes, we prove that particular fuzzy functional dependency follows form a set of fuzzy dependencies (in both, the world of two-element and the world of arbitrary fuzzy relation instances) if and only if the dependency is valid with respect to valuation anytime the set of fuzzy formulas agrees with the valuation. The results derived in paper show that the classical techniques in the procedure for generating new fuzzy dependencies may be replaced by the resolution ones, and hence automated. The research is conducted with respect to Willmott fuzzy implication operator

In this paper we consider all possible dependencies that can be built upon similarity-based fuzzy relations, that is, fuzzy functional and fuzzy multivalued dependencies. Motivated by the fact that the classical obtaining of new dependencies via inference rules may be tedious and uncertain, we replace it by the automated one, where the key role is played by the resolution principle techniques and the fuzzy formulas in place of fuzzy dependencies. We prove that some fuzzy multivalued dependency is actively correct with respect to given fuzzy relation instance if and only if the corresponding fuzzy formula is in line with the attached interpretation. Additionally, we require the tuples of the instance to be conformant (up to some extent) on the leading set of attributes. The equivalence as well as the conclusion are generalized to sets of attributes. The research is conducted by representing the attributes and fuzzy dependencies in the form of fuzzy formulas, and the application of fuzzy implication operators derived from carefully selected Frank’s classes of additive generators

There is a view that playing sports is positively related to pupils' academic achievement. Results of studies worldwide indicate this correlation, while few studies have been done in Bosnia and Herzegovina on this problem. The current study aims to expand on findings from previous studies by examining associations between (1) mode: active vs. recreational); (2) type: individual vs. team based and (3) a particular kind of sports and academic achievement particularly math achievement among middle school children. Population involves middle school pupils in two urban areas in Bosnia and Herzegovina and research sample consists of 1036 female and 1055 male pupils wanting to take part in study voluntarily. Kruskal-Wallis and Wilcoxon test were conduced to obtain results. Results show that middle school pupils who are actively involved in sports have better overall midterm academic success as well as better mathematics achievement compared to those who are involved in sports recreationally or not at all (p < 0.001). On the other hand, differences do not occur between pupils who are engaged in team or individual sports. Also, pupils who practice football were found to have lower school performance compared to pupils who practice some of the other sports.

In this paper we apply the h-generated fuzzy implications to prove a number of results which are of fundamental importance to the theory of fuzzy and vague functional and multivalued dependencies defined on given scheme. Our research is motivated by the fact that some analogous results already hold true for the families of f- and g-generated fuzzy implications, and the fact that these three collections of implications share many similar mutual properties. While some of the aforementioned implications are introduced in order to be applied in approximate reasoning, the results derived in this paper represent the main tool in the process of automation and are also used to complement the resolution principle. More precisely, the main result of this research states that the fact that some fuzzy (vague) relation instance r, |r| = 2, satisfies some fuzzy (vague) functional or fuzzy (vague) multivalued dependency c /∈ C (under assumption that r satisfies some set C of fuzzy (vague) functional and fuzzy (vague) multivalued dependencies), yields that the fuzzy formula attached to c is valid whenever all of the fuzzy formulas attached to the elements of C are valid. What is more important is that the opposite claim is also proven. Its importance stems from the fact that the verification by hand, which means purely theoretical verification, that C implies c is not required anymore. Now, in order to prove that some C yields some c, it is enough to make the use of the resolution principle, and automatically verify whether or not the set of the attached fuzzy formulas yields the fuzzy formula attached to c. In the case of affirmative answer, the desired dependency follows. The research conducted in this paper represent a natural generalization of our previous research since it includes and considers both, fuzzy and vague theories.

Klir-Yuan fuzzy implication, as fuzzy implication generated from the standard strong fuzzy negation, the probabilistic sum t-conorm, and the product t-norm, represents a classical example of QL-implication, where QL-implications are the short for quantum logic fuzzy implications. In this paper we prove that the recent results on equivalences between fuzzy formulas and fuzzy dependencies remain invariant with respect to QL-implications when considered through Klir-Yuan fuzzy implication.

In this paper we complement the most recent results on soundness of inference rules for new vague multivalued dependencies. Motivated by the fact that the inclusive and the augmentation rules are sound, we prove that: complementation, transitivity, replication, coalescence, union, pseudo-transitivity, decomposition, and mixed pseudo-transitivity rules are also sound. Our research relies on definitions of vague functional and vague multivalued dependencies based on appropriately selected similarity measures between vague values, vague sets, and tuples on sets of attributes.

This paper represents a natural continuation of our previous study. In our earlier research we proved that the inclusive inference rule and the union inference rule for new vague functional dependencies are sound, and sketched a proof of the fact that the set of the main inference rules is a complete set. In the present paper we rigorously prove that: reflexive, augmentation, transitivity, pseudo-transitivity, and decomposition inference rules are also sound. Some additional insights in completeness of the main inference rules are also provided.