Connecting Calculus with Reality through a Virtual Environment

The overarching ambition of this research is to utilize Virtual Environment (VE) computer technology to connect students’ calculus knowledge with a corresponding reality. The fundamental assumption of the study is that VE simulations are perceived by students as a reality. The study explores how students, who had completed an AP calculus course, find the optimal path in a VE empirically and, after that, mathematically. The basis for the experimental design and examination of the data is Realistic Mathematics Education (RME) theory. The students’ activity is analysed through the perspective of RME vertical and horizontal mathematizing and modeling principles. Fischbein’s theory of intuition is used for studying the influence of intuition on VE empirical and mathematical activities. In addition, students’ horizontal and vertical mathematizing are analysed with the help of theoretical constructs of ‘cognitive map’ and ‘intellectual schemata’ respectively. An interactive setting for the empirical real-life optimal path-finding problem is programmed in the Second Life VE. Ten students from Vancouver’s Templeton Secondary School, ranging in age from 17 to 18 years, participated in the study. The data were collected from 3 sources: screen-capture of students’ VE activities, video recordings of students’ mathematizing, and specially designed guiding-reflecting journals. The data recorded from the activities of five out of ten students were selected for detailed analysis of different ways of mathematizing. To accurately capture the early stages of mathematizing during their empirical activity, a new term, ‘empirical mathematizing’, is introduced and utilised in this research. The results presented in this study demonstrate that all five students constructed their models-of the situational problem on the basis of their empirical mathematizing. The results also show that new empirical knowledge obtained from empirical mathematizing prevails over intuitions. Another finding of this study is the connection between the way of mathematizing and the stage of epistemological empowerment (as determined by confidence and personal power over the use of knowledge). Particularly, the study conjectures that the way of mathematizing depends on the stage of epistemological empowerment, which in turn can be developed by engaging in empirical and mathematical exploration of real-life problems through VE simulations in mathematics classrooms.