Supervisors info:
Χρυσαυγή Τριανταφύλλου, Επικ. Καθηγήτρια, Τμήμα Μαθηματικών, ΕΚΠΑ.
Summary:
The present study attempted to study the modeling processes of a group of five high school students (grade 11) in an experimental teaching when they were given an open-ended problem. The problem given to the students was to calculate the total length of a Formula 1 circuit taking into account the data from the tracking of a Formula 1 racing car. The video recorded the instantaneous speed (in Km/h) every moment and there was an indication for acceleration (throttle) or deceleration (brake). In an attempt to solve the problem, students intuitively developed the modelling process of numerical integration. In the present work, we study the processes used and developed by the students in their efforts to solve the problem, their evolution, and the factors that influenced this development. The research data were the audio and video recordings of a two-hour teaching session. The data analysis was qualitative and based on the modeling cycle and with the assistance of Atlas.ti program. The results of the study revealed four modeling processes that were either rejected or subsequently developed by the participants. These processes were (1st) visualization of motion as linear with smooth acceleration as a whole and use of the corresponding physics types (rejected), (2nd) visualization of motion as linear with smooth acceleration per intervals of accelerated and decelerated motion (rejected), (3rd) Connection of the mean speed to the mean of the speeds for different partitions of time (applied but then rejected) and (4th) Finding the numerical value of the area under the velocity – time graph. In the development of the 4th modeling process we recognized five stages: the importance of finding the area under the velocity-time graph, designing different types of graphs, how to calculate the area under graph, searching for the best approach, and calculating the numerical value of the area using a digital tool. Students in the above trajectory “reinvented” (Van den Heuvel-Panhuizen, 2000) the numerical integration method which they applied to the final solution of the problem. The results revealed that the evolution factors of the of students' knowledge were (a) communication, (b) the realistic context of the problem, (c) the search for accuracy, (d) teacher interventions, and (e) the use of representations and digital tools.
Keywords:
modeling cycle, open-ended problems, realistic mathematics education, numerical integration methods
