Retrieval savings with nonidentical elements: the case of simple addition and subtraction

The identical elements (IE) theory of fact representation (Rickard, 2005) proposes that memorized facts that are composed of identical elements (e.g., 6x8=48 and 8x6=48) share a common representation in memory, whereas facts with nonidentical elements (e.g., 6x8=48 and 48/8=6) are represented separately in memory. The IE model has been successfully applied to the transfer of practice in simple multiplication and division, in transition from procedure-based to retrieval-based performance, and in cued episodic recall. In the present article, we examined the effects of practicing simple addition problems (e.g., 3+6=9) on the performance of corresponding subtraction problems (9-6=3), and vice versa. According to IE theory, there should be no transfer of retrieval savings between addition and subtraction facts if performance is based on discrete IE fact representations. Cross-operation response time savings were observed, however, for both small, well-memorized problems (e.g., practice 3+2, test 5-2) and larger problems (6+8, 14-6), and they were statistically robust when trials that were self-reported as direct retrieval were analyzed. The transfer of retrieval practice savings between facts with nonidentical elements challenges IE theory as a comprehensive model of transfer in memory retrieval.