| Three models of mathematical learning were evaluated using a cognitive framework based on the ACT-R general cognitive architecture. Two models were based on the concrete-representational-abstract (CRA) learning theory proposed by Bruner and others. Two models were designed with lesson activities to prompt the recognize, build-with, construct/consolidate (RBC) actions Hershkowitz, Schwarz and Dreyfus found to develop mathematical abstractions. The three models were implemented and evaluated in an urban middle school mathematics classroom learning beginning algebra equation transformations. The results indicate the RBC activities had a positive effect on learning. Future research on the RBC activities is proposed to identify the characteristics of effective activities for each phase, to develop alternative activities when a student does not recognize the need for new knowledge, to identify effective teacher interventions to prompt construction and to develop criteria for determining how many build-with activities are need to achieve a threshold activation level. The computer simulation of a balance beam used in one model to provide the concrete experience did not increase learning as expected. The cognitive framework analysis suggests this could be due to the trade-offs between features that benefit learning (e.g. a structure for initial learning actions, immediate feedback on learning actions and multi-modality knowledge construction) and that hinder learning (e.g. learning an additional metaphor for the target knowledge, focusing on surface features of the simulation and reducing the need for meta-cognition). As with most prior studies, the models based on the concrete-representational-abstract (CRA) theory did not prove superior. The cognitive framework analysis of these models suggests explanations for this finding. Three areas are proposed for future research to integrate cognitive science and mathematics education learning theory. |
