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The aim of this study was to test the hypothesis of a complex encoding of numbers according to which each numerical processing requires a specific representational format for input. In three experiments, adult participants were given two numbers presented successively on screen through a self-presentation procedure after being asked to add, to subtract, or to compare them. We considered the self-presentation time of the first number as reflecting the complexity of the encoding for a given planned processing. In line with Dehaene's triple-code model, self-presentation times were longer for additions and subtractions than for comparisons with two-digit numbers but longer for subtractions than for additions and comparisons with one-digit numbers. The implications of these results for different theories of number processing are discussed.

The identical elements (IE) model of arithmetic fact representation (Rickard, 2005; Rickard & Bourne, 1996) was developed and tested with multiplication and division. In Experiment 1, we demonstrated that the model also applies to addition and subtraction by examining transfer of response time (RT) savings between prime and probe problems tested in the same block of trials. As is predicted by the IE model, there were equivalent probe RT savings for addition with identical repetition (prime 6 + 9 [right arrow] probe 6 + 9) or an order change (9 + 6 [right arrow] 6 + 9), but much greater savings for subtraction with identical repetition (15 - 6 [right arrow] 15 - 6) than with an order change (15 - 9 [right arrow] 15 - 6), and no savings with an operation change (15 - 9 [right arrow] 6 + 9 or 6 + 9 [right arrow] 15 - 6). In Experiment 2, we examined transfer in simple multiplication and division and demonstrated symmetrical transfer between operations. Cross-operation RT savings were eliminated, however, when the RT analysis included only trials on which both the prime and the probe problems were reportedly solved by direct retrieval. An IE model extended to accommodate savings associated with procedural strategies provides a coherent account of facilitative transfer effects in simple arithmetic.

The size congruity effect is the interesting result that comparisons of the sizes of the physical formats in which numerals appear are affected by the numerical magnitudes of the respective numerals. We demonstrated that separating the physical and the numerical attributes in space leaves the effect unchanged. We then applied the spatially separated version to two-digit numerals and showed the effect to be comparable to that obtained with single numerals. We showed further that the effect is sensitive to the relative salience of the numeric and physical dimensions, to the extent that when the latter is the more salient dimension, a reverse effect obtains by which physical size interferes with number comparison. The results can be explained by a relative speed of processing account, but they are also compatible with an attention account that does not appeal to the notion of automaticity.

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 [micro] (reflecting shift) and [tau] (reflecting skew) confirmed that retrieval trials contributed predominantly to [micro], whereas procedure trials contributed predominantly to [tau]. The format X problem size RT interaction (i.e., greater word-format RT costs for large problems than for small problems) was associated entirely with [micro] and not with [tau]. Reported use of procedures presented a corresponding format X 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.

We used the choice/no-choice methodology in two experiments to examine patterns of strategy selection and execution in groups of undergraduates. Comparisons between choice and no-choice trials revealed three groups. Some participants (good retrievers) were consistently able to use retrieval to solve almost all arithmetic problems. Other participants (perfectionists) successfully used retrieval substantially less often in choice-allowed trials than when strategy choices were prohibited. Not-sogood retrievers retrieved correct answers less often than the other participants in both the choiceallowed and no-choice conditions. No group differences emerged with respect to time needed to search and access answers from long-term memory; however, not-so-good retrievers were consistently slower than the other subgroups at executing fact-retrieval processes that are peripheral to memory search and access. Theoretical models of simple arithmetic, such as the Strategy Choice and Discovery Simulation (Shrager & Siegler, 1998), should be updated to include the existence of both perfectionist and not-so-good retriever adults.

Participants practiced a set of complex multiplication problems (e.g., 3 x 18) in a pre-/postpractice design. Before, during, and after practice, the participants gave self-reports of problem-solving strategies. At prepractice, the most common strategy was a mental version of the standard multidigit algorithm, and dual tasks revealed that working memory load was high and heavier for problems solved via nonretrieval strategies. After practice, retrieval was used almost exclusively, and participant variability, automaticity level of problems (proportion of trials on which retrieval was used over the entire experiment), and error rates were significant predictors of problem-solving latencies. Practice reduced working memory involvement to minimal levels, and there was no relationship between automaticity level and working memory load. The commonalities between the present findings and findings related to automaticity development in simple arithmetic are discussed.

The study explored the contribution of working memory (WM) to mathematical problem solving in younger (8-year-old) and older (11-year-old) children. The results showed that (1) significant age-related differences in WM performance were maintained when measures of phonological processing (i.e., digit naming speed, short-term memory, phonological deletion) were partialed from the analysis; (2) WM predicted solution accuracy of word problems independently of measures of problem representation, knowledge of operations and algorithms, phonological processing, fluid intelligence, reading, and math skill; and (3) a second-order WM factor was correlated with problem solving, suggesting that a general or executive system underlies age-related performance. The results were interpreted as support for the notion that the executive system was an important predictor of age-related changes in problem solving beyond the contribution of math and reading skills, and this system operates independently of the phonological system and domain-specific knowledge in predicting solution accuracy.

In four experiments, the problem-size effect was investigated, using an alphabet-arithmetic task in which subjects verified such problems as A + 2 = C. Problem size was manipulated by varying the magnitude of the digit addend (e.g., A + 2, A + 3, and A + 4). The frequency and similarity of problems was also manipulated to determine the contribution of strength and interference, respectively. Experiment 1 manipulated frequency at low levels of practice and found that strength could account for the problem-size effect. Experiment 2 manipulated frequency at higher levels of practice, and found that strength alone could not account for the problem-size effect at asymptote. Experiment 3 manipulated frequency and similarity and found a substantial problem-size effect at asymptote, suggesting that both strength and interference contribute to the problem-size effect. Experiment 4 manipulated similarity, keeping frequency constant, and found no problem-size effect at asymptote, suggesting that interference alone is not responsible for the problem-size effect. The results are related to findings with number arithmetic.

The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people's strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.