The presented study analyzes the usage of the didactic efficiency of multiple representations in a computer environment in learning absolute value functions and equations. It is known that the axis of symmetry of the graph of the absolute value function is the y-axis. The research was applied at the University of Novi Sad, Serbia. The data were collected by testing a group of 226 students: major chemistry and physics students at the beginning of their common calculus course. The students worked individually in two groups: the experimental and control group. The experimental group of students practiced using GeoGebra software, and the control group of students practiced using paper and pencil. At the end of the experiment, which lasted for two weeks (six school classes), both groups were tested with a post-test of knowledge without using a computer. It can be concluded that GeoGebra software had a positive influence on the students’ achievements in solving absolute value equations.

In this paper, we describe the influence of the implementation of methodology for teaching and learning multiple integrals based on the constructivist approach and the interconnectedness of multiple representations (with an emphasis on dynamic representations) for different mathematical objects using GeoGebra’s applications Graphing Calculator and 3D Calculator. The empirical research was conducted at the University of Kragujevac, Serbia, with two groups (the experimental and the control groups) of second-year students. During the learning process of the experimental group, students created digital materials to visualize the domain of the integration for the concrete tasks, to set the integration bounds, to recognize in which situation they needed to introduce the switch of variables, and all that to solve double or triple integral problems. Students in the control group, on the other hand, didn't use technology of any kind, including the programs Graphing Calculator and 3D Calculator. The findings of the empirical study showed that students' use of these tools to create digital materials improved their academic achievement and their practical and theoretical knowledge of multiple integrals.

Introduction In our research the collaborative learning with and without the use of the computer is analyzed. The process of collaborative learning is applied to the first year calculus students at the University of Novi Sad, Serbia, for examining functions and drawing their graphs. The collaborative learning process was analyzed, compared and described, for experimental and control group regarding their way of learning. First part of research was conducted in 2011 and 2012, were both experimental and control group were divided to small four member groups by using Kagan’s (1994) principles. In 2013 the new experimental group was formed by using mathematical model for the group making. The students in both experimental groups learned with the help of GeoGebra, and the students in the control group learned without using GeoGebra. After the collaborative learning, the students were tested with a test and post test, and the results of the experimental groups were significantly better than the results of students in the control group. The results are of the first part of research is published in the paper Takači et al (2015). Methods The research had started in 2011 year with the control group with 180 students, and continued in 2012 with the experimental group. In 2012 we had 210 students, but 180 students were chosen for experimental group having a corresponding student from 2011, same major, according to the pre-test results. Both the experimental and the control group were structured according to Kagan’s, instructions: The results of the pre-tests considering their pre-calculus knowledge were used to form a list order of all the students (ranging from the worst to the best one). Each student had to make two lists of persons whom he likes or dislikes, particularizing up to three names. The research was continued in 2013 when the second experimental group is formed analogously as in 2012, but the small four members groups are formed by using variable neighborhood search algorithm, (metaheuristic for solving the mathematical optimization problems) based on Kagan’s (1994) principles. The students also worked with the help of Geogebra. The task for all students was to examine twenty 20 appropriate functions and draw their graphs. The calculations for chosen functions could be done easily even without computer. The control group finished their task in two weeks (2 times 5 classes a week), but both experimental groups finished the whole task in 5 hours. The students worked on the task at the university, with the help of the authors. All the members in the group were given 5 points (of 100) for the successfully finished task, both in experimental and control group. After collaborative learning the students got test (colloquium) and post test (exam) to be done without the computer. Results and Discussion The analysis of the students work (examining 20 functions) had confirmed that learning calculus with the use of GeoGebra is more “constructivist” than without computer. Students used the opportunities to consider together algebraic and graphic view in order to examine, analyze and test the adequacy of their knowledge of the properties of functions. The students used the characteristics of the package GeoGebra to analyze both the formula and its graph at the same time in different ways. Both experimental groups finished all 20 tasks with the precision of 18.02% (2012), 18.25%, (2013), but the control group finished all 19.64 tasks with the precision of 12.84%. The statistical analysis (t-test, effect size) proves that the students’ learning achievement in examining functions and drawing their graphs is better when they use GeoGebra. The difference between the results of the test and post tests was statistically significant at the level of significance of 0.05 and 0.01, regarding control and both experimental groups. Conclusions Taking into account the results of the students’ achievement both at the test and the post-test, the results of the interviews, and the average number of solved tasks (of 20) per one group, we can say that the learning of examining functions and drawing their graphs by GeoGebra in CSCL groups is more efficient than the learning without it in collaborative groups. Since the difference between the tested experimental groups, of pre test, and post test was not statistically significant at the level of significance of 0.05 05 and 0.01, it can be concluded that the application of variable neighborhood search algorithm is very useful for team building procedure when we have lot of students to work with. References and Notes 1. Kagan, S. (1994). Cooperative Learning. San Clemente, CA: Resources for Teachers,Inc. 2. O'Malley, C. (ed.) (1995). Computer-supported cooperative learning. Berlin: Springer- Verlag. 3. Tall, D. (2011). A Sensible approach to the Calculus. Handbook Calculus and its teaching. Ed. F. Pluminage and A. Cuevas. 4. Takači, Dj., Stankov, G., Milanovic, I., (2015) Efficiency of learning environment using GeoGebra when calculus contents are learned in collaborative groups, Computers & Education 82: 421-431.