Analysing mathematical modelling problems solving processes in the technology-aided environment: An explanation on approaches and thought processes

The purpose of this study is to explain the approach and thinking processes occurring during the solution process of mathematical modeling problems in technology- supported environment. In the research, by revealing the basic components of the aforesaid complex modeling process, the basic characteristics of sub steps and the connections of each other was examined in detail. The technology- supported media was provided through the Geogebra software, videos, animations, pictures and using Screenhunter program.The research was conducted by using the grounded theory approach one of the qualitative research methods. The participants of the research consisted of 19 voluntary senior mathematics student teachers having training in Secondary Mathematics Teaching Department in 2011-2012 academic years. The participants were given a Mathematical Modeling Course by the master's thesis advisor during one term and the researcher also followed this course. At the same time the participants took courses regarding computer based and mathematics based software during six terms (including GeoGebra for two terms) in their undergraduate education. In addition, some implementations about the GeoGebra and modeling were made by being made the mathematics student teachers to remember the basic structure of GeoGebra. Therefore, the acquisition of a substantial solution process related to mathematical modeling problems during the data collection was aimed by promoting the development of the participants? skills in mathematical modeling and technology. In this study in which the modeling processes of the mathematics student teachers were examined in the collaboration groups, three mathematical modeling problems designed by the researchers, the video analysis including the solution processes of the groups and their thinking aloud, GeoGebra solution files, the papers given by the groups for the problems? solutions and brief reminder observation notes were used as data collection tools. These three problems used for the data collection tools were designed considering the structures and qualities of the mathematical modeling problems in literature, and the students? pre-knowledge. The problems in question were called as ?Swing Problem?, ?Height-Foot Length Problem? and ?Stadium Problem?. These problems were solved by the collaboration groups of 2, 3 or 4 students made in accordance with their wishes. The students were asked to think aloud, and to explain all their approaches and thoughts during solving the problems. In the study, each group met in an empty class for each problem, and the researcher also participated in all the problem solutions. The students benefited from the GeoGebra Software and the ScreenHunter program actively during the solution processes. In addition, the animations, videos and pictures were given to the students. The solution process of the students was recorded using a camera and a computer. The constant comparative analysis based on grounded theory method was used for the data analysis. After the qualitative detailed data analysis process including open coding, axial coding and selective coding based on grounded theory approach, a theory concerning approaches and thought processes supported technology by creating categories and emphasizing the relations between these categories and the model explaining this theory were put forward in accordance with the analyses carried out by three people.It was seen that the technology usage contributed to mathematical modeling process through the data obtained in the study. The solutions of the designed technology-aided mathematical modeling problems put forward a rich cognitive process for the participants. The students demonstrated some different approaches for the solution using their technology and mathematics knowledge for during the process. The animations, videos and pictures given with the problems were some important items of an appropriate solution strategy during the process. With the result of the data analysis, eight basic components (complex real-world situation, real-life problem situation, a model of real-life problem situation, helping mathematical models, the main mathematical model/s, mathematical solution, real-world solution, a brief solution report or a real-world problem situation), seven basic steps (analyzing the problem, establishing the systematic structure, mathematization, top mathematization, mathematical analysis, interpretation/evaluation, the model verification) and 47 sub-steps revealing these seven basic steps, and the relations between them were put forward by the researchers. It is thought that the thesis study will bring an original and differen aspect considering the studies about modeling process. The effects of technology were observed in each step during the process. The effects of technology were observed especially when a general solution strategy was being composed, the real- life solutions were being obtained and the approaches on behalf of the model verification were being displayed. The operations? complexity stemmed from real- life solutions was minimized thanks to technology. Thus, this operation prevented the mathematics student teachers to get drowned in mathematical operations, and presented a more efficient mental process report to the researcher. It was seen that GeoGebra software could create an ideal environment for a rich cognitive process in mathematical modeling. It was observed that the regular comparison of the relations between algebraic and geometric representations in GeoGebra during the process, being Turkish of the software?s language and being easy of its use prepared a ground in order to take on a more effective role. Accordingly, some rich environments which improve their modeling and technology skills can be provided by being provided the students to participate in a mathematical modeling process enriched through technology for classroom and extracurricular activities. It can be considered for under graduate lessons that the lesson for technology- supported mathematical modeling should be taken into account, or a richer learning environment can be created by integrated technology into the lesson for mathematical modeling. The modeling process can be observed by created some problems in a different classification framework for mathematical modeling for further studies. A more intensive qualitative data analysis can be performed for metacognitive process like for cognitive process. In addition, the effects of the software can be explained by using GeoGebra 3D software.Key Words: Mathematics Education, Mathematical Modeling, Mathematical Modeling Process, Technology-aided Mathematical Modeling, GeoGebra, Mathematics Student Teacher, Grounded Theory