Retrieval processes in arithmetic production and verification

JAMIEI. D. CAMPBELL ) 0 1 DEREKP. M. TARLING 0 1 0 We thank M. Ashcraft, 1. A. Lefevre, M. Masson, P Meagher, and 1. Zbrodoff for very helpful comments on a previous draft. This re search was supported by Grant OPGOOO 1980 from the Natural Sci should be addressed to 1. I. D. Campbell, Department of Psychology, University of Saskatchewan , Saskatoon, SK, Canada S7N 5A5 (e mail: 1 University ofSaskatcheuian ; Saskatoon, Saskatcheuxm; Canada To investigate whether arithmetic production and verification involve the same retrieval processes, we alternated multiplication production trials (e.g., 9 x 6 =?) with verification trials (4 x 9 =36, true or false?) and analyzed positive error priming. Positive error priming is the phenomenon in which errors frequently match correct answers from preceding problems. Production errors were strongly primed by previous production trials (the error-answer matching rate was about 900A! greater than expected by chance), but production errors were not strongly primed by previous verification trials ("'" 13% above chance). Conversely, false-verification errors were primed by previous verification trials (""'25% above chance), but not by production trials. The results indicated that arithmetic production and verification were mediated by different memory processes and suggest a familiaritybased over a retrieval-based model of arithmetic verification. - skill (Ashcraft, 1982; Koshmider & Ashcraft, 1991; Lemaire, Barrett, & Fayol, 1994; Lemaire & Fayol, 1995), although, in everyday practice, arithmetic production is probably much more common than arithmetic verifica tion. Furthermore, it is common for researchers to pre suppose a specific model ofverification. For example, the validity of the retrieve-compare model was assumed in several recent componential analyses of both simple and complex arithmetic (Frensch & Geary, 1993; Geary, Wida man, & Little, 1986; Widaman & Little, 1992). Similarly, in analyses of arithmetic deficits due to brain injury, arithmetic verification has been used as a control condi tion to assess the integrity ofretrieval processes assumed to operate in arithmetic production (e.g., McCloskey, Sokol, & Goodman, 1986). This is appropriate only if arithmetic production and verification involve essen tially identical computational processes. We begin with a review ofretrieval-based and familiarity based approaches to arithmetic verification. We then de scribe error priming in arithmetic production and explain how error priming may be used to discriminate retrieval based from familiarity-based verification. Retrieve-Compare Model ofVerification Four stages are hypothesized in the retrieve-compare model of arithmetic verification introduced by Ashcraft and Battaglia (1978; see also Ashcraft, 1982, 1987, 1992): (1) encode the problem and presented answer, (2) com pute the answer to the problem, (3) decide if the com puted and presented answers are the same or different, (4) execute the "true" or "false" response. It is assumed in the model that computation by adults and older chil dren usually involves answer retrieval and that retrieval is mediated by automatic spreading activation through a network of related problems and answers. For example, the problem 2 X 7 activates the memorized multiples of 2 and 7, and answers that are numerically near to the correct answer are more strongly activated. The correct an swer is identified by the intersection of activation from both operands. In Ashcraft's model, speed of retrieval is determined by the associative strength linking the prob lem and its correct answer, and strength is assumed to depend primarily on problem frequency. In general, prob lem frequency decreases with problem size (Hamann & Ashcraft, 1986). Also, consistent with the relations among frequency, strength, and response time (RT) hypothe sized in the model, it is commonly observed that RT tends to increase with problem size (see Ashcraft, 1992; Campbell & Oliphant, 1992; Gallistel & Gelman, 1992; LeFevre, Sadesky, & Bisanz, in press; Siegler, 1988). Furthermore, in keeping with the assumption that veri fication and production involve the same retrieval pro cess, Ashcraft, Fierman, and Bartolotta (1984) tested first graders, fifth graders, and college students on ver ification and production of simple addition facts and found no task X problem size interactions beyond the first grade. Interactions between the retrieval and comparison stages in Ashcraft's model account for relatedness ef Jects observed in verification of false-answer equations: RTs to false equations are longer when the split (i.e., nu merical difference) between the presented and correct answer is small (e.g., 4 + 8 = 13) compared to larger splits (4 + 8 = 17) (Ashcraft & Stazyk, 1981; Stazyk, Ashcraft, & Hamann, 1982). Similarly, RTs to false equations are longer and errors are more common when the incorrect answer is arithmetically related to the prob lem. For example, RT is relatively slow when a presented answer is the correct answer to another problem in the same times table (e.g., 4 X 7 = 24) (Campbell, 1987; Stazyk et aI., 1982) or the correct answer for a different arithmetic operation (e.g., 4 X 7 = 11) (Winkelman & Schmidt, 1974; Zbrodoff & Logan, 1986). It is assumed in Ashcraft's model that the verification decision is based on a comparison of the activation levels of the computed and presented answers in the retrieval network (see, e.g., Ashcraft, 1992, p. 88). Because problems pro duce strong activation of numerically near and related answers, the activation levels of these answers may be relatively close to the activation level of the correct an swer compared to the weak activation ofnumerically dis tant or unrelated answers. Given that the same-different decision is more difficult when activation levels are sim ilar, it follows that the comparison stage should require longer RTs and be more error prone when a presented false answer is near or related to the correct answer. The retrieve-compare model is plausible and provides straightforward explanations for the major phenomena of verification, but it is likely that use of a retrieve-compare strategy depends on a number of factors. For example, Ashcraft and Stazyk (1981) found that whereas false verification RTs are generally longer than true RTs, splits of 13 in addition verification resulted in shorter RTs than did true equations. This suggests that if the split is large enough, subjects quickly recognize that the equation is implausible without necessarily retrieving the answer to the problem. Krueger (1986; Krueger & Hallford, 1984) similarly proposed that subjects can make plausibility judgments in verification on the basis of numerical relationships (e.g., a product is odd only if both operands are odd), thereby bypassing retrieval of the correct answer (see also Lemaire & Fayol, 1995). Thus, when there are salient characteristics of false equations that reliably distingui (...truncated)