Iterative design and comparison of learning systems for reflection in two dimensions
This research involved the iterative design and comparison of learning systems aiming to mediate between students' personal knowledge of reflection and conventional mathematical knowledge. The study comprised a two-phase teaching experiment. In the first phase, four learning systems were developed using an iterative methodology that cycled between design of computer-based tools and observation of students interacting with them. Each learning system was structured according to a filling-outwards or a fillinginward instructional approach and incorporated a computer microworld based on either dynamic Euclidean geometry (DEG) or multiple-turtle geometry (MFO). Learning systems were intended (i) to help students build from views of reflection based on internal (intra) properties of two-dimensional figures to views taking account of external (inter) relationships between figures and (ii) to emphasise a functional perspective on the transformation. In the second phase, analysis focused on how meanings for reflection evolved as six students' interacted in each system. Results suggested that, in all four systems, students developed meanings by coordinating intra and interfigural analyses while they built computational models of reflection. Microworld tools had a role in mediating all aspects of students' activities: with DEG tools, reflection tended to be represented as a correspondence relationship based on perpendicular distances; MTG tools led to expression of reflection as a mapping of one set of turtles onto another and emphasised equal turns and distances. In all learning systems, mathematical meaning-making involved forging connections between general models of reflection and physical movements of screen objects. Additionally~ students using MTG tools gave more meaning to their models by connecting feedback to imaginary social practices. The impact of instructional approach on learning trajectories was also mediated by microworld tools. Specific effects associated with filling-inwards and filling-outwards interventions were identifie~ but were limited to particular tasks and the tools used to negotiate them.