Why is 9+7 harder than 2+3? Strength and interference as explanations of the problem-size effect

Abstract

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 ofthe 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 I 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. 689One ofthe most fundamental findings in the research on simple arithmetic is that people take longer and make more errors when they solve large problems (i.e., problems with large digits) than when they solve small problems (i.e., problems with small digits). This is most commonly referred to as the problem-size effect. It occurs with children and with adults, with addition, subtraction, multiplication, and division problems, and with production and verification tasks. Much effort has gone into trying to explain the problem-size effect. In effect, all theories of simple arithmetic try to account for it, and how well they are able to account for it determines how good they are as theories.There are four types of theories that may explain the problem-size effect. Two are reconstructive theories and two are reproductive theories. Reconstructive theories propose an algorithm that computes the answer; reproductive theories propose a retrieval process that retrieves previous answers. The time taken by the algorithm or the retrieval process depends on the number or the difficulty of the steps to be executed. The theories argue that larger problems require more steps or more difficult steps and therefore take longer to solve and are more prone to errors. The question is, why does the number or the difficulty of steps depend on the magnitude of the problem? The answers to this question reveal the fundamental assumptions that distinguish the theories. Reconstructive Theories Number of StepsThe primary reconstructive theories are counting theories. They assume that answers are computed by setting an internal counter to a specified value and then successively incrementing it a specified number of times. Five counting models have been investigated (Groen & Parkman, 1972). In the sum model, the counter is set to zero and then incremented by both digits in the problem; in the right model, the counter is set to the left d...

Fast automated counting procedures in addition problem solving: When are they used and why are they mistaken for retrieval?

“…In the linguistic domain, dramatic differences in word frequency (from about 3000 to 60 per million) result in rather small differences in RTs (i.e., 15 ms in a lexical decision task, Ferrand et al, 2011), whereas the observed difference in RTs between probably very frequent additions such as 2 + 1 and 2 + 4 was higher than 90 ms. Finally, the size effect we observed does not seem to result from interferences in a memory network as Zbrodoff, 1995) suggested. An index we called Overlap reflecting the number of problems sharing the same answer as the problem under study was the worst among the eight predictors of RTs that we entered in our analyses (r = .10, n.s.).…”

“…Assuming a process of spreading activation through the memory network, the time needed to reach a given intersection (i.e., the correct sum) would be proportional to the area of the network to be traversed, hence the predictive power of the product of the two addends. Zbrodoff (1995) and Zbrodoff and Logan (2005) proposed a network interference model in which problem-answer associations take longer to retrieve for larger problems because they suffer from more interference created by overlap of their operands or answers.…”

Effects of problem size, operation, and working-memory span on simple-arithmetic strategies: differences between children and adults?

“…The problem-size eVect, which refers to slower and more error-prone performance on large problems (e.g., 8 £ 9) than on small problems (e.g., 2 £ 3), is one of the most robust eVects observed in mental-arithmetic research (Ashcraft 1992;ZbrodoV 1995). According to Campbell and Xue (2001), there are three strategy-related sources of the problem-size eVect in adults: less frequent retrieval use for large than for small problems, lower retrieval eYciency for large than for small problems, and lower procedural eYciency for large than for small problems.…”

Storage and retrieval of addition facts: The role of number comparison

“…LeFevre et al's (1996) college subjects frequently reported using a counting strategy. The different status of the N+1 problems compared to problems with addends larger than 1 has implicitly been assumed by researchers who did not include these items in testing a model of arithmetical fact retrieval (e.g., Campbell, 1994Campbell, , 1995Zbrodoff, 1995).…”