Transfer and Learning • Fractions

Often times, instruction provides scaffolds that help students learn to perform some task. The canonical example might be the use of training wheels that help children complete the task of riding a bicycle without having to master all the elements of balance. A different issue is how can we help people to self-scaffold when they leave the confines of our instruction. People have to continue learning and the ability to use the social and physical environment to scaffold this learning is relatively unexplored. In our work on children learning about fractions, we have been exploring the relation between initial scaffolding and subsequent abilities to self-scaffold. Students learned to do fraction addition with two kinds of manipulatives, pie pieces and tile pieces. We found that while both groups of students looked good on initial learning, the tile students were much more prepared to learn in a new situation. In the new situation, they received new manipulatives (e.g., beans and cups) to help learn to solve new problems. We knew they had to learn with the new materials, because we separated those who could solve the problems in their head and those who could not. The tile students were able to use the new materials to help them learn. In contrast, students who had learned with pie pieces reverted back to whole number interpretations of new materials. (In the figure, the accuracy drops over the days, because the children were doing increasingly complicated problems; the important comparison is between the Tile and Pie conditions.) This finding highlights the importance of assessing the effects of instruction on people’s abilities to continue to learn. The differences between instruction with tiles and pies would not have been evident had we only looked at the children’s immediate problem solving and had not assessed ability to use new resources to help themselves learn.