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., 6 × 8 = 48 and 8 × 6 = 48) share a common representation in memory, whereas facts with nonidentical elements (e.g., 6 × 8 = 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.