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Automatic Fact Retrieval in Simple Arithmetic: A Study of Problem Size Effects.

Researcher Kataryzna Rogozinski

Supervisor Dr Jeff Coney

Date: 18th May, 2010

The research in mental arithmetic suggests that reaction times to simple calculation problems are, in general, slower and more error prone if the operands and their correct solutions become numerically larger. This phenomenon, called problem – size effect is often explained by the differences in automaticity of retrieving the answers to arithmetic problems. In this view, basic arithmetic facts of addition and multiplication are stored in the form of the interrelated long- term memory networks (Ashcraft, 1992; Graham & Campbell, 1992). The retrieval of facts is thought to be mainly automatic with reaction times varying as a function of strength and accessibility, where strength of connections depends on the frequency of exposure. As small problems are encountered and practiced in natural settings more often than large problems, they exhibit much stronger associations with their correct solution (Ashcraft, 1992; Campbell, 1987).

In contrast, other theories argue that prolonged reaction times for the problems with large operands are largely due to non-retrieval procedures. In that sense, the prompt reaction times for small problems results from the use of automatic and direct retrieval of facts from the memory, whereas prolonged reaction times for the problems with large operands are largely due to non-retrieval procedures (LeFevre at al., 1996).

The current study replicated the methodology of simple arithmetic priming paradigm developed by Jackson and Coney (2005, 2007) in order to investigate automatic and strategic processing on the effects of problem size. In each trial, participants were presented with a small or large arithmetic problem (e.g. 2+3) and were then presented with a target that was either congruent (e.g. 5) or incongruent (e.g. 7) with this prime. The reaction times for congruent and incongruent condition were than compared with a neutral condition (A + C). Moreover, three levels of SOA (120ms, 240ms, 1500ms) were employed to distinguish between automatic and strategic processing, i.e. ˃ 250ms SOA represented automatic processing and ˂ 800ms represented strategic processing. Further, to enable the analysis of skill on arithmetic performance, participants were tested using the arithmetic section of the ACER Short Clerical Test (ACER, 1984). Based on participant’s fluency score, subjects were assigned to one of two groups: low ang high skilled.

Consistent with previous research, it was expected that facilitation will occur for small and large multiplication problems and small addition problems only suggesting access via automatic processing. Moreover, facilitation was expected to be observed at the 240 ms SOA for the high skill group and at the 1000ms SOA for the low skill group. Such results will confirm the assumption that automatic and procedural processes are involved in the arithmetic problem size effect.

As expected, the results indicated significant facilitation in naming congruent targets for the small and large multiplication problems together with small addition problems. No priming effect was observed for large addition problems. Moreover, for the high skill group, significant facilitation was found at the 240 ms SOA and 1000 ms SOA, whereas for the low skill group facilitation was only observed at the 1000 ms SOA. Finally, the pattern of inhibition observed in the current study was stable over time and reaching significance at the 120 ms SOA and 240 ms SOA.

The findings of the current study imply several causes for the problem size effect in adult arithmetic performance. Firstly, because of the significant facilitation effects found for the high skill group at the 240 ms SOA together with the differences in access to solutions for small and large problems and between the fluency groups, it is assumed that automatic activation of answers is apparent in simple arithmetic processing. This agrees with the strength of activation and interference theory of problem size effect as supposed by Campbell (1987, 1991). Secondly, the lack of priming effect for large addition problems coupled together with priming effect for large multiplication problems at the 1000 ms SOA might suggest that large problems may invoke additional strategic processing that is not trigger by small problems.

Moreover, the pattern of inhibition found in the current study suggests that facilitation and inhibition effects occur due to different functional mechanisms as the levels of inhibition indicate that strategic processing was employed after presentation of the target. According to Jackson and Coney (2005, 2007) such findings might be explained in terms of a self- regulatory validity check mechanism, where participants after the exposure to the target and before responding may have quickly compared the target to the solution evoked from the memory. In the incongruent condition, where the incorrect solutions to the problems were presented as a target, this may have led to an uncertainty in responding.

In summary, it is apparent that results from the studies that employ arithmetic priming paradigm provide valuable findings that need to be incorporated into the current theoretical framework of arithmetic problems size effect, as it seems that the existing models are incomplete and thus unable to account for the existing outcomes on their own. Specifically, the current findings suggest that the differences in processing of different problem size may originate not only in the strength of association but also in strategic access to solutions as revealed by significant pattern of facilitation and inhibition effects.

Reference:

Ashcraft, M.H. (1992). Cognitive arithmetic: A review of data and theory. Cognition, 44, 75, 106
Campbell, J.I.D. (1987). Production, verification, and priming of multiplication facts.

Memory and Cognition, 15(4), 349-364
Graham, D.G. & Campbell, J.I.D. (1992). Network interference and number- fact retrieval:
Evidence from children’s alphaplication. Canadian Journal of Psychology, 46(2), 65-91
Jackson, N. & Coney, J.R. (2005). Simple arithmetic processing: The question of
automaticity. Acta Psychologica, 119, 41-66
Jackson, N. & Coney, J.R. (2007). Simple arithmetic processing: Individual differences in
automaticity. European Journal of Cognitive Psychology. 19(1), 141-160
LeFevre, J.; Sadesky, G.S. & Bisanz, J. (1996). Selection of procedures in mental addition:
Reassessing the problem size effect in adults. Journal of Experimental Psychology: Learning, Memory and Cognition, 22, 216- 230