Article
Effects of problem format on division and multiplication performance: division facts are mediated via multiplicationbased representations.
Institute of Cognitive Science, Carleton University, Ottawa, Ontario, Canada.
Journal of Experimental Psychology Learning Memory and Cognition (Impact Factor: 3.1). 04/2003; 29(2):16370. DOI: 10.1037/02787393.29.2.163 Source: PubMed

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ABSTRACT: Recent imaging studies could show that fact acquisition in arithmetic is associated with decreasing activation in several frontal and parietal areas, and relatively increasing activation within the angular gyrus, indicating a switch from direct calculation to retrieval of a learned fact from memory. So far, however, little is known about the transfer of learned facts between arithmetic operations. The aim of the present fMRI study was to investigate whether and how newly acquired arithmetic knowledge might transfer from trained multiplication problems to related division problems. On the day before scanning, ten complex multiplication problems were trained. Within the scanner, trained multiplication problems were compared with untrained multiplication problems, and division problems related to multiplication (transfer condition) were compared with unrelated division problems (notransfer condition). Replicating earlier results, untrained multiplication problems activated several frontal and parietal brain areas more strongly than trained multiplication problems, while trained multiplication problems showed relatively stronger activation in the left angular gyrus than untrained multiplication problems. Concerning division, an ROI analysis indicated that activation in the left angular gyrus was relatively stronger for the transfer condition than for the notransfer condition. We also observed distinct interindividual differences with regard to transfer that modulated activation within the left angular gyrus. Activation within the left angular gyrus was generally higher for participants who showed a transfer effect for division problems. In conclusion, the present study yielded some evidence that successful transfer of knowledge between arithmetic operations is accompanied by modifications of brain activation patterns. The left angular gyrus seems not only to be involved in the retrieval of stored arithmetic facts, but also in the transfer between arithmetic operations.NeuroImage 01/2009; · 6.25 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Researchers have speculated that children find it more difficult to acquire conceptual understanding of the inverse relation between multiplication and division than that between addition and subtraction. We reviewed research on children and adults’ use of shortcut procedures that make use of the inverse relation on two kinds of problems: inversion problems (e.g., 9 24 ¸24 {9} \times {24} \div {24} ) and associativity problems (e.g., 9 24 ¸8 {9} \times {24} \div {8} ). Both can be solved more easily if the division of the second and third numbers is performed before the multiplication of the first and second numbers. The findings we reviewed suggest that understanding and use of the inverse relation between multiplication and division develops relatively slowly and is difficult for both children and adults to implement in shortcut procedures if they are not flexible problem solvers. We use the findings to expand an existing model, highlight some similarities and differences in solvers’ use of conceptual knowledge across operations, and discuss educational implications of the findings. KeywordsMathematical inversion–Multiplication and division–Concepts–StrategiesEducational Studies in Mathematics 01/2012; 79(3):409428. · 0.55 Impact Factor
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