Title: Promoting the Development of an Integrated Numerical Representation through the Coordination of Physical Materials
Description: How do children use physical and virtual tools to develop new numerical knowledge? While concrete instructional materials may support the delivery of novel information to learners, they may also over-simplify the task, unintentionally reducing learners' performance in recall and transfer tasks. This reduction in testing performance may be mitigated by embedding physical incongruencies in the design of instructional materials. The effort of resolving this incongruency can foster a richer understanding of the underlying concept. In two experiments children were trained on a computerized number line estimation task, with a novel scale (0-180), and then asked to perform a series of posttest number line estimation tasks that varied spatial features of the training number line. In experiment 1, during training with feedback, children either received a ruler depicting endpoint and quartile magnitudes (i.e., 0, 45, 90, 135, 180) that physically matched the on-screen number line (congruent ruler), a proportionally-similar ruler scaled 33% larger than the on-screen number line (incongruent ruler), or no ruler.

Children were trained to criterion before proceeding to posttest. Results indicated that while children who used the congruent ruler performed well during training, their performance at posttest was less accurate than the other two conditions. On the other hand, by increasing the difficulty of the learning task, while providing relevant landmark information, children in the incongruent ruler condition produced the highest accuracy at posttest. In experiment 2, controlling for learning task duration, the incongruent ruler and congruent ruler conditions were compared directly. Posttest results confirmed an advantage for children in the more complex, incongruent ruler condition. These results are interpreted to suggest that landmarks representations are an important and accessible means of developing a mature numerical representation of the number line. Furthermore, the results confirm that desirable difficulties are an essential component of the learning process. Potential implications for the design of learning activities that balance instructional support with conceptual challenge are discussed.