On the GeoGebra supported collaborative learning

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.


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.