Transfer of training in alphabet arithmetic.

In recent years, several researchers have proposed that skilled adults may solve single-digit addition problems (e.g., 3 + 1 = 4, 4 + 3 = 7) using a fast counting procedure. Practicing a procedure, however, often leads to transfer of learning to unpracticed items; consequently, the fast counting theory was potentially challenged by subsequent studies that found no generalization of practice for simple addition. In two experiments reported here (Ns = 48), we examined generalization in an alphabet arithmetic task (e.g., B + 5 = C D E F G) to determine that counting-based procedures do produce generalization. Both experiments showed robust generalization (i.e., faster response times relative to control problems) when a test problem's letter augend and answer letter sequence overlapped with practiced problems (e.g., practice B + 5 = C D E F G, test B + 3 = C D E ). In Experiment 2, test items with an unpracticed letter but whose answer was in a practiced letter sequence (e.g., practice C + 3 = DEF, test D + 2 = E F) also displayed generalization. Reanalysis of previously published addition generalization experiments (combined n = 172) found no evidence of facilitation when problems were preceded by problems with a matching augend and counting sequence. The clear presence of generalization in counting-based alphabet arithmetic, and the absence of generalization of practice effects in genuine addition, represent a challenge to fast counting theories of skilled adults' simple addition.

No generalization of practice for nonzero simple addition.

Several types of converging evidence have suggested recently that skilled adults solve very simple addition problems (e.g., 2 + 1, 4 + 2) using a fast, unconscious counting algorithm. These results stand in opposition to the long-held assumption in the cognitive arithmetic literature that such simple addition problems normally are solved by fact retrieval from declarative memory. Here we tested a large sample of diversely skilled and culturally diverse men and women at the University of Saskatchewan and examined multiple categories of simple (1 digit plus 1 digit) addition problems for evidence of generalization of practice, a signature of procedure use. The procedure-based 0 + N = N problems presented clear evidence of generalization (i.e., practicing a subset of 0 + N problems lead to speed-up for a different subset of 0 + N problems), but there was no evidence of such generalization of practice for the nonzero problems, although the experiment had good power to detect small effects. Given that generalization of practice is a basic marker of procedure-based processing, its absence for the nonzero addition problems casts doubt on the compacted counting theory.