The kite family and other animals : Does a dragging utilisation scheme generate only shapes or can it also generate mathematical meanings?
thesisposted on 16.06.2014, 10:32 by Susan Kathleen Forsythe
This thesis is about development of students' geometrical reasoning, in particular of inclusive relations between 2 dimensional shapes, e.g. the rhombus as a special case of the kite. Students in the study worked with a dynamic figure constructed using Dynamic Geometry Software. The figure is a quadrilateral whose diagonals are constructed so that they are of fixed length and perpendicular. All students in the study were observed to use a strategy of 'dragging' to keep one diagonal as the perpendicular bisector of the other. This generated a 'family of shapes' which was comprised of an infinite number of kites, arrowheads (i.e. concave kites), one rhombus and two isosceles triangles. I have called this strategy 'Dragging Maintaining Symmetry' (DMS) and I claim it has the potential to mediate the understanding of the rhombus as a special case of the kites (and the isosceles triangle in the context of dynamic geometry). However students in the study typically perceived the shapes, generated using DMS, according to a partitional view i.e. as different shapes, albeit with common properties such as line symmetry. When asked how many kites it would be possible to make by dragging the figure some students reported that there were four kites (one typical kite in each of four relative positions). It appears that they perceived the dragging activity as a journey to a discrete end position rather than as an action that resulted in a continuously changing figure. To address this problem I showed the students an animation of the figure under DMS. This proved to be the catalyst which moved their reasoning towards perceiving inclusive relations between the rhombus and kite.