The Future of Education

In this paper, I report on the design and implementation of two lessons aimed at introducing junior high school mathematics students (age 13-14) to the idea of shearing in a multitouch dynamic geometry technology. By using shearing, I hoped to shift students’ attention away from a formula-driven, computational conception of area towards a more geometric conception of area. Furthermore, I propose a dynamic geometry “learnable system” for the concept of area that transforms not only how it is learned but also the nature of the concept itself. My hypothesis is that this system will shift students’ attention away from formulae that compute the area of polygons, which they had already encountered in previous years of schooling, and towards geometric ways of comparing and reasoning about those polygons.  I found that the introduction of the notion of shearing did indeed change the students’ thinking about area. The students were also able to solve several problems involving the comparison of polygons based on verbal and diagrammatic explanations. These findings are encouraging because of the evidence that students could move back and forth between the medium of the dynamic, multi-touch screen, in which objects moved continuously over time, and the medium of paper, where the students were able to devise various strategies to communicate the temporal nature of shearing as well as the inferential steps from one diagram to another. The ease with which students could go back and forth between the screen and the paper is also partly due to the mobility and size of the iPad in contrast to a computer monitor or even a laptop. Using the theoretical lens of Semitic Mediation, I describe the features of the multi-touch dynamic geometry technology that supported their learning, as well as the specific role that the teacher played in modelling students’ reasoning about area.