Article
Effects of Problem Format on Division and Multiplication Performance: Division Facts Are Mediated via MultiplicationBased Representations
Department of Psychology, Carleton University, Ottawa, Ontario, Canada
Journal of Experimental Psychology Learning Memory and Cognition (Impact Factor: 2.86). 04/2003; 29(2):16370. DOI: 10.1037/02787393.29.2.163 Source: PubMed
ABSTRACT
In 2 experiments participants solved division problems presented in multiplicationbased formats (e.g., 8 x _ = 72) more quickly than division problems presented in divisionbased formats (e.g., 72 / 8 = _). In contrast, participants solved multiplication problems presented in a divisionbased format (e.g., _ / 8 = 9) slowly and made many errors. In both experiments, the advantage for multiplicationbased formats on division problems was found only for large problems (i.e., those with products or dividends greater than 25). These findings provide support for the view that large singledigit division facts are mediated via multiplicationbased representations and that multiplication is the primary mode of representation for both division and multiplication facts.

 "The current evidence for an intuitive understanding of wholenumber scaling stands in stark contrast to children's often reported difficulty in learning symbolic division in school (Burns, 2007; Campbell, 1997; Mauro, LeFevre, & Morris, 2003; Rich & Schmidt, 1997; Siegler, 1988). Children were above chance on initial training trials and readily applied the scaling factors of onehalf and onequarter to amounts that extended well beyond those on which they were initially trained. "
Article: Nonsymbolic division in childhood
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ABSTRACT: The approximate number system (ANS) underlies representations of large numbers of objects as well as the additive, subtractive, and multiplicative relationships between them. In this set of studies, 5 and 6yearold children were shown a series of videobased events that conveyed a transformation of a large number of objects into onehalf or onequarter of the original number. Children were able to estimate correctly the outcomes to these halving and quartering problems, and they based their responses on scaling by number, not on continuous quantities or guessing strategies. Children's performance exhibited the ratio signature of the ANS. Moreover, children performed above chance on relatively early trials, suggesting that this scaling operation is easily conveyed and readily performed. The results support the existence of a flexible and substantially untrained capacity to scale numerical amounts. 
 "When direct retrieval for division facts fails, people may recast the division problem (e.g., 36 ÷ 4 = __) to multiplication form (e.g., 4 × __ = 36). The solution to the multiplication fact is then accessed and retrieved as the solution to the original division problem (Mauro et al., 2003). These studies in numerical computation suggest that the estimation of percentage savings will be easy when the cash equivalent of the points is a simple, welllearned fraction of the price. "
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ABSTRACT: Many consumers today hold loyalty program points which function as a currency, but are not cash. This paper examines factors that influence consumers' decisions to keep or spend their accumulated points. We found that consumers are more likely to spend points when they can easily anticipate the benefits they can enjoy with the points. Specifically, the decision to spend points is facilitated when it is easier to compute the percentage savings one can get by using the points. This computational ease has effects on point spending beyond that of saving magnitude. 
 "The first component is influenced by factors such as familiarity of the surface notation (Metcalfe & Campbell, 2007), problem layout (Campbell, 2008; Mauro et al., 2003), taskswitching costs (e.g., switching from addition to multiplication) and speedaccuracy criteria (Campbell & Austen, 2002). The second component is influenced by factors including problem retrieval strength (which varies both across arithmetic operation and problem size) and performance context (e.g., different presentation formats activate distinct retrieval structures; interference or facilitation from recent primes or problems; Bassok, 2001; Campbell, 1994; Campbell & Metcalfe, 2007). "
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ABSTRACT: Educated adults solve simple addition problems primarily by direct memory retrieval, as opposed to by counting or other procedural strategies, but they report using retrieval substantially less often with problems in writtenword format (four + eight) compared with digit format (4 + 8). It was hypothesized that retrieval efficiency is relatively low with word operands compared with digits and that this promotes a shift to procedural backup strategies. Consistent with this hypothesis, Experiment 1 demonstrated greater wordformat costs on retrieval usage for addition than subtraction, which was due to increased counting for addition but not subtraction. Experiment 2 demonstrated greater wordformat costs on retrieval for division than multiplication, which was due to increased use of multiplicationfact reference to solve division problems. Formatrelated strategy shifts away from retrieval reflected both the efficiency of retrieval for a given operation and the availability of viable alternative strategies. The results demonstrate that calculation processes are not abstracted away from problem surface form. The authors propose that retrieval efficiency for arithmetic connects diverse performance and strategyrelated effects across key arithmetic factors, including arithmetic operation, numerical size, and numeral format.
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