The Representation of Multiplication and Division Facts in Memory
Evidence for Cross-Operation Transfer Without Mediation
Abstract
Recently, using a training paradigm, Campbell and Agnew (2009) observed cross-operation response time savings with nonidentical elements (e.g., practice 3 + 2, test 5 − 2) for addition and subtraction, showing that a single memory representation underlies addition and subtraction performance. Evidence for cross-operation savings between multiplication and division have been described frequently (e.g., Campbell, Fuchs-Lacelle, & Phenix, 2006) but they have always been attributed to a mediation strategy (reformulating a division problem as a multiplication problem, e.g., Campbell et al., 2006). Campbell and Agnew (2009) therefore concluded that there exists a fundamental difference between addition and subtraction on the one hand and multiplication and division on the other hand. However, our results suggest that retrieval savings between inverse multiplication and division problems can be observed. Even for small problems (solved by direct retrieval) practicing a division problem facilitated the corresponding multiplication problem and vice versa. These findings indicate that shared memory representations underlie multiplication and division retrieval. Hence, memory and learning processes do not seem to differ fundamentally between addition-subtraction and multiplication-division.
References
(1997). The role of examples and rules in the acquisition of a cognitive skill. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 923–945.
(1999). Division by multiplication. Memory & Cognition, 27, 791–802.
(2009). Retrieval savings with nonidentical elements: The case of simple addition and subtraction. Psychonomic Bulletin & Review, 16, 938–944.
(2006). Identical elements model of arithmetic memory: Extension to addition and subtraction. Memory & Cognition, 34, 633–647.
(2008). Bidirectional associations in multiplication memory: Conditions of negative and positive transfer. Journal of Experimental Psychology: Learning, Memory, and Cognition, 34, 546–555.
(2001). Cognitive arithmetic across cultures. Journal of Experimental Psychology: General, 130, 299–315.
(2009). A longitudinal study of children’s performance on simple multiplication and division problems. Developmental Psychology, 45, 1480–1496.
(2008). Improving accuracy in detecting acoustic onsets. Journal of Experimental Psychology: Human Perception and Performance, 34, 1317–1326.
(2007). Do multiplication and division strategies rely on executive and phonological working-memory resources?. Memory & Cognition, 35, 1759–1771.
(1999). More on the relation between division and multiplication in simple arithmetic: Evidence for mediation of division solutions via multiplication. Memory & Cognition, 27, 803–812.
. (2002). E-Prime V1.1.
(2005). Revised identical elements model of arithmetic fact representation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 31, 750–767.
(2006). Cued recall from image and sentence memory: A shift from episodic to identical elements representation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 32, 734–748.
(1996). Some tests of an identical elements model of basic arithmetic skills. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 1281–1295.
(1994). On the cognitive structure of basic arithmetic skills – operation, order and symbol transfer effects. Journal of Experimental Psychology: Learning, Memory, and Cognition, 20, 1139–1153.
(1901). The influence of improvement in one mental function upon the efficiency of other functions. Psychological Review, 8, 247–261.
