An investigation of 7th grade students' mathematical modeling processes

Mathematical modeling problems are problem solving activities in which students discover, extend and refine their mathematical structures by making sense of realistic problem situations. Mathematical modeling problems are powerful tools that reveal ways of students thinking. Mathematical modeling process provides a rich environment for teachers to understand how students express their mathematical ideas, how they interpret the information presented in different forms, and what level of meaningful solutions they can produce. The aim of the study is to examine the mathematical modeling processes of six 7th grade students while working on mathematical modeling problems. A qualitative case research design was utilized in this study. This study was conducted in an elective mathematics course entitled "Mathematics Applications for Grades 7" at a public middle school. 24 students were enrolled in this course and the students worked in eight groups of three throughout the study. In order to make in-depth investigation, two groups were selected as the focus group according to a purposeful sampling method. The participants in this study were six 7th grade students worked in two groups of three. Three modeling problems "The Weather Problem, The Car Problem and The Lawnmower Problem" were used in order to examine students' modeling processes. The groups had nearly 90 minutes (two class periods) to work on these problems. After students worked on each modeling problem, the researcher has conducted a focus group interview with both groups in order to obtain detailed information about their solution processes. The data sources for this study were audiotapes of the students' responses; their written worksheets on modeling problems, focus group interviews and observation notes. Qualitative data analyses were carried out to analyze data. The findings of this study revealed that in the three modeling problems, the students in both groups developed their own mathematical ideas through a nonlinear cyclic process and modelled real-life problem situations using these ideas. It was also seen that the students in both groups tried to understand the problem statements, offered several mathematical ideas and developed their models as they were working on the modeling problems. Furthermore, it was also observed that the students in both groups frequently came back the problem situations to understand the problems and evaluated the ideas they created. However, the findings also showed that in the three modeling problems, both groups of the students presented their mathematical ideas without internalizing the problem situations. Findings also showed that the students in the two groups expressed their desire to scoring as a mathematical idea for three modeling problems, and they created systems that include selection, classification, the ranking of factors, quantitative and qualitative data quantification and transformation of quantities. On the other hand, it was seen that the students expressed and tried many different mathematical ideas in the first modeling problem that they worked on. However, it was seen that the number of the different ideas presented by students decreased while working on the second modeling problem and they did not present the various ideas and used only "Scoring" while working on the third modeling problem. When the modeling processes of the students in Group 1 and 2 were compared, the findings showed that the modeling processes of students in Group 1 in terms of examination of the data given in tables, deciding which data is important in the light of the given situation, associating the data with each other and the quantification of the data are more robust and valid than students in Group 2. However, when the modeling processes of these two groups of students were evaluated, it was observed that they were able to provide acceptable systematic solutions and nearly able to deal successfully with three complex mathematical modeling problems. Findings of the study displayed that throughout the study the students were able to work collaboratively, make occasionally assumptions and interpreted the problem situation and data with basing upon the real-life situations. However, it was observed that students were mostly inadequate to explain and justify what and why they did in their processes and they mostly focused on the mathematical operations. On the other hand, it was understood that students in both groups just examined the correctness of their mathematical operations in order to understand if there were any mistakes in their modeling processes and they did not interpret their developed models with real life. Findings of this study suggest that mathematics teachers should regularly apply mathematical modeling problems in their mathematics classes. The applications of the modeling problems can be maintained within the scope of a planned teaching, by including mathematical modeling problems in mathematics courses or in elective mathematics courses as it was done in this study. Keywords: Mathematical Modeling, Mathematical Modeling Problems, Middle School Students