Benefits of "concreteness fading" for children's mathematics understanding *

Children often struggle to gain understanding from instruction on a procedure, particularly when it is taught in the context of abstract mathematical symbols. We tested whether a “concreteness fading” technique, which begins with concrete materials and fades to abstract symbols, can help children extend their knowledge beyond a simple instructed procedure. In Experiment 1, children with low prior knowledge received instruction in one of four conditions: (a) concrete, (b) abstract, (c) concreteness fading, or (d) concreteness introduction. Experiment 2 was designed to rule out an alternative hypothesis that concreteness fading works merely by “warming up” children for abstract instruction. Experiment 3 tested whether the benefits of concreteness fading extend to children with high prior knowledge. In all three experiments, children in the concreteness fading condition exhibited better transfer than children in the other conditions. Children's understanding benefits when problems are presented with concrete materials that are faded into abstract representations.