Jamie I. D. Campbell

Architectures for numerical cognition.

Current theories of numerical cognition differ in assumptions about the componential architecture of number processing and about the extent of notation-specific processes. To investigate these issues, 64 adult subjects were tested on simple addition and multiplication problems presented in Arabic digit or English number-word format. Overall, response times and error rates were much higher with the word format, but more importantly, presentation format interacted with arithmetic operation and problem size. Operation errors (2 + 4 = 8), operand-naming errors (2 + 8 = 8), and operand-intrusion errors (9 x 6 = 36) were each characterized by a different format x operation interaction, and analysis of inter-trial error priming showed selective interference from preceding trials as a function of number format. These types of format-specific retrieval interference and operation-specific effects of format are problematic for models that hypothesize notation-independent memory processes for arithmetic. Furthermore, analyses of operand-naming errors, operand-intrusion errors, and other operand-priming effects, revealed strong interactions of number reading and number-fact retrieval processes; processes that are typically posited to be functionally independent. The results suggest a complex encoding architecture that incorporates notation-dependent activation of addition and multiplication facts, as well as interpenetration of number reading and number-fact retrieval processes.

Integrated versus modular theories of number skills and acalculia.

This paper contrasts two views of the cognitive architecture underlying numerical skills and acalculia. According to the abstract-modular theory (e.g., McCloskey, Caramazza, & Basili, 1985), number processing is comprised of independent comprehension, calculation, and production subsystems that communicate via a single type of abstract quantity code. The alternative, specific-integrated theory (e.g., Campbell & Clark, 1988), proposes that visuospatial, verbal, and other modality-specific number codes are associatively connected as an encoding complex and that different facets of number processing generally involve common, rather than independent, processes. The hypothesis of specific number codes is supported by conceptual inadequacies of abstract codes, format-specific phenomena in calculation, the diversity of acalculias and individual differences in number processing, lateralization issues, and the role of format-specific codes in working memory. The integrated, associative view of number processing is supported by the dependence of modular views on abstract codes and other conceptual inadequacies, evidence for integrated associative networks in calculation tasks, acalculia phenomena, shortcomings in modular architectures for number-processing dissociations, close ties between semantic and verbal aspects of numbers, and continuities between number and nonnumber processing. These numerous logical and empirical considerations challenge the abstract-modular theory and support the encoding-complex view that number processing is effected by integrated associative networks of modality-specific number codes.

Production, verification, and priming of multiplication facts

In the arithmetic-verification procedure, subjects are presented with a simple equation (e.g., 4 × 8 = 24) and must decide quickly whether it is true or false. The prevailing model of arithmetic verification holds that the presented answer (e.g., 24) has no direct effect on the speed and accuracy of retrieving an answer to the problem. It follows that models of the retrieval stage based on verification are also valid models of retrieval in the production task, in which subjects simply retrieve and state the answer to a given problem. Results of two experiments using singledigit multiplication problems challenge these assumptions. It is argued that the presented answer in verification functions as a priming stimulus and that on “true” verification trials the effects of priming are sufficient to distort estimates of problem difficulty and to mask important evidence about the nature of the retrieval process. It is also argued that the priming of false answers that have associative links to a presented problem induces interference that disrupts both speed and accuracy of retrieval. The results raise questions about the interpretation of verification data and offer support for a network-interference theory of the mental processes underlying simple multiplication.

Numerical order and quantity processing in number comparison.

We investigated processing of numerical order information and its relation to mechanisms of numerical quantity processing. In two experiments, performance on a quantity-comparison task (e.g. 2 5; which is larger?) was compared with performance on a relative-order judgment task (e.g. 2 5; ascending or descending order?). The comparison task consistently produced the standard distance effect (faster judgments for far relative to close number pairs), but the distance effect was smaller for ascending (e.g. 2 5) compared to descending pairs (e.g. 5 2). The order task produced a pair-order effect (faster judgments for ascending pairs) and a reverse distance effect for consecutive pairs in ascending order. The reverse effect implies an order-specific process, such as serial search or direct recognition of order for successive numbers. Thus, numerical quantity and order judgments recruited different cognitive mechanisms. Nonetheless, the reduced distance effect for ascending pairs in the quantity task implies involvement of order-related processes in magnitude comparison. Accordingly, distance effects in the quantity-comparison task are not necessarily a process-pure measure of magnitude representation.

Chapter 12 Cognitive Number Processing: An Encoding-Complex Perspective

Summary According to the encoding-complex approach (Campbell & Clark, 1988; Clark & Campbell, 1991), numerical skills are based on a variety of modality-specific representations (e.g., visuo-spatial and verbal-auditory codes), and diverse number-processing tasks (e.g., numerical comparisons, calculation, reading numbers, etc.) generally involve common, rather than independent, cognitive mechanisms. In contrast, the abstract-modular theory (e.g., McCloskey, Caramazza, & Basili, 1985) assumes that number processing is comprised of separate comprehension, calculation, and production subsystems that communicate via a single type of abstract quantity code. We review evidence supporting the specific-integrated (encoding-complex) view of number processing over the abstract-modular view, and report new experimental evidence that one aspect of number processing, retrieval of simple multiplication facts, involves non-abstract, format-specific representations and processes. We also consider implications of the encoding-complex hypothesis for the modularity of number skills.

On the relation between skilled performance of simple division and multiplication

Are corresponding multiplication and division facts (e.g., 7 x 8, 56 divided by 7) based on common or on independent memory processes? University students received division problems alternated with multiplication problems under instructions for speeded responses. Response times were highly correlated for corresponding division and multiplication problems, and error characteristics indicated parallel retrieval structures. Specifically, division errors were constrained by the distance between the dividend and the product implied by the error, rather than by distance from the correct quotient. This suggests that division memory is organized in terms of multiplicative relationships. Multiplication errors (e.g., 7 x 9 = 56) were primed by previous division trials (56 divided by 7 = 8), but division errors were not primed by previous multiplications. The error priming results suggest that multiplication is often used at least to check division.

RETRIEVAL PROCESSES IN ARITHMETIC PRODUCTION AND VERIFICATION

To investigate whether arithmetic production and verification involve the same retrieval processes, we alternated multiplication production trials (e.g., 9 × 6 = ?) with verification trials (4 × 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 90% 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 familiarity-based over a retrieval-based model of arithmetic verification.

Conditions of error priming in number-fact retrieval

Analysis of errors in simple multiplication has shown that answers retrieved on previous trials are initially inhibited (negative error priming) but later are promoted as errors to subsequent problems (positive error priming). Two experiments investigated whether error priming is associated either with problem-specific retrieval processes or with representations of answers that can be manipulated independently of problems. In Experiment 1, answers were primed by visually presenting products for 200 msec prior to problems. Correct-answer primes facilitated retrieval, related-incorrect primes interfered with retrieval more than unrelated primes, and both effects were greater for more difficult problems. Primes affected only the trial on which they were presented, however, whereas both negative and positive error priming from previous problems were observed across trials. In Experiment 2, subjects named and retrieved multiplication products on alternating trials. Just-named products were inhibited as errors to the following multiplication problem (i.e., negative error priming), but, compared to positive priming from previous retrieved products, positive error priming from previously named numbers was weak. The results indicate that positive error priming is due mainly to an encoding or retrieval bias produced by previous problems, whereas negative error priming entails suppression, or de-selection, of answer representations.

Syllogistic reasoning time: Disconfirmation disconfirmed

University of Saskatchewan, Saskatoon, Saskatchewan, Canada Models of deductive reasoning typically assume that reasoners dedicate more logical analysis to unbelievable conclusions than to believable ones (e.g., Evans, Newstead, Allen, & Pollard, 1994; Newstead, Pollard, Evans, & Allen, 1992). When the conclusion is believable, reasoners are assumed to accept it without much further thought, but when it is unbelievable, they are assumed to analyze the conclusion, presumably in an attempt to disconfirm it. This disconfirmation hypothesis leads to two predictions, which were tested in the present experiment: Reasoners should take longer to reason about problems leading to unbelievable conclusions, and reasoners should consider more models or representations of premise information for unbelievable conclusions than for believable ones. Neither prediction was supported by our data. Indeed, we observed that reasoners took significantly longer to reason about believable conclusions than about unbelievable ones and generated the same number of representations regardless of the believability of the premises. We propose a model, based on a modified version of verbal reasoning theory (Polk & Newell, 1995), that does not depend on the disconfirmation assumption.

Chapter 9 Representation And Retrieval Of Arithmetic Facts: A Network-Interference Model And Simulation

Summary We present a computer model of a network-interference theory of memory for single-digit multiplication and addition facts. According to the model, a presented problem activates representations for a large number of related arithmetic facts, with strength of activation of specific facts determined by similarity to the presented problem. Similarity is assumed to be based on both physical codes (e.g., common visual or phonological features) and visuo-spatial magnitude codes. Nodes representing numerical facts that are related to the presented problem are continuously activated and compete by way of mutual inhibition until one reaches the critical activation threshold and triggers a response. The counteracting processes of excitation and inhibition in the model reproduce a large number of response time and error phenomena observed in skilled memory for number facts. The general form of the representational structures proposed in the simulation provide for a natural extension of the model to other areas of cognitive arithmetic and associated research.