Abstract
Two current models of arithmetic fact retrieval, the network interference theory (NIT; Campbell, 1995) and the interacting neighbors (IN) model (Verguts & Fias, 2005a), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are “consistent,” compared with “inconsistent” errors). In a reanalysis of error data originally reported by Campbell (1997), we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001). However, the NIT and IN‐model differ in their explanations of the problem‐size effect, a hallmark finding robustly observed in arithmetic fact retrieval. Only the IN‐model predicts that a correct answer's neighborhood consistency can fully account for the problem‐size effect, which was confirmed in our analysis.
References
(1987). Children's knowledge of simple arithmetic: A developmental model and simulation. In , Formal methods in developmental research (pp. 302–338). New York: Springer‐Verlag
(1995). The frequency of arithmetic facts in elementary texts: Addition and multiplication in grades 1‐6. Journal for Research in Mathematics Education, 5, 396–421.
(2001). The arithmetic tie effect is mainly encoding‐based. Cognition, 82, B15–B24.
(1991). Conditions of error priming in number‐ fact retrieval. Memory & Cognition, 19, 197–209.
(1995). Mechanisms of simple addition and multiplication: A modified network‐interference theory and simulation. Mathematical Cognition, 1, 121–165.
(1997). On the relation between skilled performance of simple division and multiplication. Journal of Experimental Psychology: Learning, Memory, & Cognition, 23, 1140–1159.
(1996). Inhibitory processes in sequential retrieval: Evidence from variable‐lag repetition priming. Brain & Cognition, 30, 59–80.
(1992). Cognitive number processing: An encoding‐complex perspective. In J. I. D. Campbell (Ed.). The nature and origins of mathematical skills (pp. 457–491). Amsterdam: Elsevier Science
(1985). Mental multiplication skills: Structures, process and acquisition. Canadian Journal of Psychology, 39, 338–366.
(2003). The neural basis of the Weber‐Fechner law: A logarithmic mental number line. Trends in Cognitive Science, 7, 145–147.
(2005). The addition of two‐digit numbers: Exploring carry versus no‐carry problems. Psychology Science, 47, 74–83.
(1996). Multiple routes to solution of single‐digit multiplication problems. Journal of Experimental Psychology: General, 125, 284–306.
(1996). Selection of procedures in mental addition: Reassessing the problem size effect in adults. Journal of Experimental Psychology: Learning, Memory, & Cognition, 22, 216–230.
(1999). What affects strategy selection in arithmetic? The example of parity and five effects on product verification. Memory & Cognition, 27, 364–382.
(2000). The odd‐even effect in multiplication: Parity rule or familiarity with even numbers? Memory & Cognition, 28, 358–365.
(1992). Mathnet: Preliminary results from a distributed model of arithmetic fact retrieval. In , The nature and origins of mathematical skills (pp. 365–409). Amsterdam: North Holland/Elsevier
(2001). Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition, 82, B25–B33.
(2005). On the magnitude representation of two‐digit numbers. Psychology Science, 47, 52–72.
(2005). A revised identical elements model of arithmetic fact representation. Journal of Experimental Psychology: Learning, Memory, & Cognition, 31, 250–257.
(1990). Priming word recognition with orthographic neighbors: Effects of relative prime‐target frequency. Journal of Experimental Psychology: Human Perception & Performance, 16, 65–76.
(1998). SCADS: A model of children's strategy choices and strategy discoveries. Psychological Science, 9, 405–410.
(1988). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology: General, 117, 258–275.
(1982). A network approach to mental multiplication. Journal of Experimental Psychology: Learning, Memory, & Cognition, 8, 320–335.
(2005a). Interacting neighbours: A connectionist model of retrieval in single‐digit multiplication. Memory & Cognition, 33, 1–16.
(2005b). Neighborhood effects in mental arithmetic. Psychology Science, 47, 132–140.
(2004). What everyone finds: The problem size effect. In , Handbook of mathematical cognition (pp. 331–346). New York: Psychology Press
