From intrinsic to non-intrinsic geometry : a study of children's understandings in Logo-based microworlds
The aim of the present study was to investigate the potential for children to use the turtle methaphor to develop understandings of intrinsic, euclidean and cartesian geometrical ideas. Four aspects of the problem were investigated. a) the nature of the schema children form when they identify with the turtle in order to change its state on the screen; b) whether it is possible for them to use the schema to gain insights into certain basic geometrical principles of the cartesian geometrical system; c) how they might use the schema to form understandings of euclidean geometry developed inductively from specific experiences; d) the criteria they develop for choosing between intrinsic and euclidean ideas. Ten 11 to 12 year - old children participated in the research, previously having had 40 to 50 hours of experience with Turtle geometry. The research involved three case - studies of pairs of children engaging in cooperative activities, each case - study within a geometrical Logo microworld. The data included hard copies of everything that was said, typed and written. Issues a) and b) were investigated by means of the first case - study which involved three pairs of children and a microworld embedding intrinsic and coordinate ideas. A model of the children's intrinsic schema and a model of the coordinate schema which they formed during the study were devised. The analysis shows that the two schemas remained separate in the children's minds with the exception of a limited number of occasions of context specific links between the two. Issue c) was investigated in the second case - study involving one pair of children and a microworld where the turtle was equipped with distance and turn measuring instruments and a facility to mark positions. The analysis illustrates how a turtle geometric environment of a dynamic mathematical nature was generated by the children, who used their intrinsic schema and predominantly engaged in inductive thinking. The geometrical content available to the children within this environment was extended from intrinsic to both intrinsic and euclidean geometry. Issue d) was investigated by means of the third case - study involving a pair of children and a microworld where the children could choose among circle procedures embedding intrinsic and/or euclidean notions in order to construct figures of circle compositions. The analysis shows that the children employed their turtle schema in using both kinds of notions and did not seem to perceive qualitative differences between them. Their decisions on which type of notion to use were influenced by certain broader aspects of the mathematical situations generated in the study.