Designing and evaluating multi-representational learning environments for primary mathematics
This thesis reports the design and evaluation of multi-representational learning environments that teach aspects of number sense. COPPERS is concerned with children's belief that mathematical problems can have only a single correct answer. CENTS addresses the skills and knowledge required for successful computational estimation. Although, there is much multi-representational software and a significant body of research which suggests that learning with multiple external representations (MERs) is beneficial, little is known about the conditions under which MERs promote effective learning. To address this, a framework was proposed for considering MERs. It consists of a set of dimensions along which multi-representational software can be described and specifies learning demands of MERs. This framework was used to generate predictions about the effectiveness of different multi-representational systems. Experiments investigated children's performance in multiple solutions and computational estimation before they received direct teaching and tested whether the learning environments could help children develop these skills. Each experiment examined how specific aspects of the learning environments contributed to learning outcomes. Experiments with COPPERS showed that children's pre-test performance was generally poor. Improved post-test performance on multiple solutions tasks occurred when children gave substantially more answers on the computer than their pre-test base-line. They rarely chose this strategy for themselves. It was found that providing a tabular representation of solutions in addition to the familiar row and column representation improved learning. Estimation is difficult for primary school children, but limited teaching led to substantial improvements in strategies and accuracy of estimates. Three experiments with CENTS addressed the effects of MERs on learning. When representations were too difficult to co-ordinate, then either children did not improve at understanding the accuracy of estimates, or focused their attention upon a single representation. Additionally, varying how information was distributed across representations influenced how representations were used. These experiments show that when considering learning with MERs, it is not sufficient to consider the effects of each representation in isolation. Behaviour with representations changes depending on how they are combined. These findings are discussed in terms of their implications for the design of multi-representational learning environments.