Bidirectional associations in multiplication memory: conditions of negative and positive transfer

A variety of experimental evidence indicates that the memory representation for multiplication facts (e.g., 6 x 9 = 54) incorporates bidirectional links with a forward association from factors to product and a reverse association from product to factors. Surprisingly, the authors did not find evidence in Experiment 1 of facilitative transfer-of-practice from multiplication (6 x 9 = ?) to factoring (54 = ? x ?); in fact, multiplication practice produced item-specific interference with factoring. Similarly, the authors found no evidence in Experiment 2 that repetition of specific factoring problems (54 = ? x ?) facilitated performance of corresponding multiplication problems (6 x 9 = ?). In Experiment 3, participants practiced both multiplication and factoring and presented facilitative transfer in both directions. Thus, bidirectional facilitation occurred if both operations were practiced, but interference occurred when only one operation was practiced. We propose that this seemingly paradoxical behavior occurs because it is adaptive for the bidirectional retrieval structure to retain operational flexibility in the context of practicing both operations, whereas it is adaptive to specialize the memory representation for the practiced operation (i.e., factoring or multiplication) when only one operation is practiced.