Transfer of training in alphabet arithmetic

Abstract

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.

Transfer of training in simple addition

“…There was some evidence, mainly in connection with Experiment 2, that transfer problems were slightly (26 ms) faster than control problems in Test Block 2. A generalization effect in Block 2, but not Block 1, is opposite to the pattern observed in the alphabet arithmetic studies (Campbell et al, 2016) and contrary to the prediction that generalization effects (i.e., the RT advantage for transfer relative to control problems) ought to diminish with repeated testing. Indeed, the 0 + N problems in Experiment 1 confirmed diminished RT gains in Block 2 after the generalization effect seen in Block 1.…”

“…The critical items with respect to the counting hypothesis were the augend-matched transfer problems and controls analogous to the augend-sequence overlap alphabet arithmetic problems used by Campbell et al (2016). These were small, non-tie additions with sums ≤10 (including n + 1 problems), which are more likely candidates for fast counting-based procedures than larger simple additions.…”

“…Thus, there is no clear converging evidence for automated procedures for addition. Campbell et al (2016) used alphabet arithmetic to demonstrate that counting-based procedures can produce substantial generalization of learning to unpracticed items that would be implicitly practiced during training (e.g., solving B + 3 by counting implicitly practices B + 1 and B + 2).…”

“…Campbell, Chen, Allen, and Beech (2016) sought direct evidence that practice of counting-based procedures can produce generalization that leads to the speed up of unpracticed problems. They examined generalization in an alphabet arithmetic task (e.g., B + 1 = C, D + 3 = G).…”

“Compacted” procedures for adults’ simple addition: A review and critique of the evidence

“…Generalization for n + 0 problems, but no generalization for nonzero simple addition problems, has been repeatedly replicated (Campbell & Beech, 2014;Campbell, Dufour, & Chen, 2015;Campbell & Therriault, 2013;Chen & Campbell, 2014, 2017. Campbell, Chen, Allen, and Beech (2016) demonstrated robust generalization of practice in a countingbased alphabet addition task (e.g., B + 5 = G), thereby confirming generalization of counting-based procedures. Furthermore, one cannot argue that simple addition skills for small problems are so overlearned that they would not benefit from generalization or practice.…”