Academic journal article Memory & Cognition

Calculation Latency: The [Mu] of Memory and the [Tau] of Transformation

Academic journal article Memory & Cognition

Calculation Latency: The [Mu] of Memory and the [Tau] of Transformation

Article excerpt

Does numeral format (e.g., 4 + 8 vs. four + eight) affect calculation per se? University students (N = 47) solved single-digit addition problems presented as Arabic digits or English words and reported their strategies (memory retrieval or procedures such as counting or transformation). Decomposition of the response time (RT) distributions into μ (reflecting shift) and τ (reflecting skew) confirmed that retrieval trials contributed predominantly to μ, whereas procedure trials contributed predominantly to τ. The format × problem size RT interaction (i.e., greater word-format RT costs for large problems than for small problems) was associated entirely with μ and not with τ. Reported use of procedures presented a corresponding format × size interaction. Together, these results indicate that, relative to the well-practiced digit format, the unfamiliar word format disrupts number-fact retrieval and promotes use of procedural strategies.

The effect of numeral format (e.g., Arabic digits, written number words, Roman numerals) on performance of simple arithmetic continues to be a controversial issue in numerical cognition research. One view is that effects of format are restricted to systems that encode numerical stimuli, and do not penetrate downstream to affect calculation (Dehaene & Cohen, 1995; McCloskey, 1992; Noël, Fias, & Brysbaert, 1997). Others have reported evidence that format can directly affect the efficiency of calculation per se (Blankenberger & Vorberg, 1997; Campbell, 1994; Campbell, Kanz, & Xue, 1999; Campbell, Parker, & Doetzel, 2004; McNeiI & Warrington, 1994; Sciama, Semenza, & Butterworth, 1999). Here, we pursue this issue by analyzing parameters of the latency distributions produced by digit and word problems. Specifically, we estimated μ, which is sensitive to a uniform shift in the entire response time (RT) distribution, and τ, which is sensitive to shifts in skew (Smith & Mewhort, 1998). We will begin by briefly reviewing a key phenomenon in the debate, and then explain our predictions regarding effects of format on RT distributions.

Format × Problem Size Interaction in Cognitive Arithmetic

Numerous experimental studies of simple addition and multiplication have compared performance with problem operands presented as Arabic digits to performance with operands as written number words. A wide range of languages have been examined, including French, Dutch, English, German, Chinese, and Filipino (see Campbell & Epp, 2005, for a review). One general finding is that performance of simple arithmetic with written number words (e.g., seven × six) is much more difficult than with Arabic numerals (e.g., 7 × 6). Simple arithmetic with written words is as much as 30% slower and 30% more error prone than with Arabic digits (Campbell, 1994). Thus, arithmetic performance with the less-typical written word format is substantially impaired relative to the typical Arabic format.

Evidence that format affects calculation per se, rather than only more peripheral encoding or response processes, comes from the problem-size effect (PSE). The PSE is the common finding that the difficulty of simple arithmetic problems generally increases with the numerical size of the operands. Small-number problems have greater memory strength because they are encountered more frequently and may be less susceptible to retrieval interference (see Zbrodoff& Logon, 2005, for a detailed discussion of the PSE). Several studies have demonstrated, however, that the PSE in both simple addition and multiplication is larger with problems in written word format (e.g., three + eight) than in digit format (3 + 8; Campbell, 1994; Campbell & Clark, 1992; Campbell et al., 1999; Noël et al., 1997). For example, Campbell (1994) contrasted performance on small addition problems (both operands ≤5) to performance on large addition problems (at least one operand >5), and found that the RT PSE was much larger with English number words (192 msec) than with Arabic digits (108 msec). …

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