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.

[1]  A. Paivio.  Mental Representations: A Dual Coding Approach , 1986 .

[2]  Stanislas Dehaene,et al.  Two mental calculation systems: A case study of severe acalculia with preserved approximation , 1991, Neuropsychologia.

[3]  Jamie I. D. Campbell.  Retrieval inhibition and interference in cognitive arithmetic , 1990 .

[4]  X. Seron,et al.  Cognitive Models of Simple Mental Arithmetic - a Critical-review , 1988 .

[5]  W. Banks,et al.  Semantic congruity effects in comparative judgments of magnitudes of digits. , 1976 .

[6]  S Dehaene,et al.  The psychophysics of numerical comparison: A reexamination of apparently incompatible data , 1989, Perception & psychophysics.

[7]  Michael McCloskey,et al.  Models of arithmetic fact retrieval: an evaluation in light of findings from normal and brain-damaged subjects. , 1991, Journal of experimental psychology. Learning, memory, and cognition.

[8]  Alfonso Caramazza,et al.  Cognitive mechanisms in number processing and calculation: Evidence from dyscalculia , 1985, Brain and Cognition.

[9]  Donald G. MacKay,et al.  Self-Inhibition and the Disruptive Effects of Internal and External Feedback in Skilled Behavior* , 1986 .

[10]  J. I. Campbell,et al.  Network interference and number-fact retrieval: evidence from children's alphaplication. , 1992, Canadian journal of psychology.

[11]  David R. Paredes,et al.  Starting to add worse: Effects of learning to multiply on children's addition , 1990, Cognition.

[12]  M. McCloskey,et al.  Cognitive processes in verbal-number production: inferences from the performance of brain-damaged subjects. , 1986, Journal of experimental psychology. General.

[13]  Jamie I. D. Campbell,et al.  Conditions of error priming in number-fact retrieval , 1991, Memory & cognition.

[14]  O. Svenson.  Memory retrieval of answers of simple additions as reflected in response latencies , 1985 .

[15]  Mark H. Ashcraft,et al.  Children’s Knowledge of Simple Arithmetic: A Developmental Model and Simulation , 1987 .

[16]  L E Krueger,et al.  Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification , 1984, Memory & cognition.

[17]  Emmanuel Dupoux,et al.  Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. , 1990 .

[18]  R. Bjork.  Retrieval inhibition as an adaptive mechanism in human memory. , 1989 .

[19]  D. Geary,et al.  Cognitive addition and multiplication: Evidence for a single memory network , 1986, Memory & cognition.

[20]  Lee Ryan,et al.  Transfer-appropriate processing for implicit and explicit memory. , 1990 .

[21]  F. B. Knight,et al.  The Learning of the One Hundred Multiplication Combinations , 1930, Teachers College Record: The Voice of Scholarship in Education.

[22]  R. Siegler,et al.  Strategy choice procedures and the development of multiplication skill. , 1988, Journal of experimental psychology. General.

[23]  Mark H. Ashcraft,et al.  A network approach to mental multiplication. , 1982 .

[24]  Gordon D. Logan,et al.  On the relation between production and verification tasks in the psychology of simple arithmetic , 1990 .

[25]  F. Restle.  Speed of Adding and Comparing Numbers. , 1970 .

[26]  Jamie I. D. Campbell,et al.  An encoding-complex view of cognitive number processing: Comment on McCloskey, Sokol, and Goodman (1986). , 1988 .

[27]  Guy J. Groen,et al.  Temporal aspects of simple addition and comparison , 1971 .

[28]  G. R. Potts,et al.  Mental comparison of size and magnitude: size congruity effects. , 1984, Journal of experimental psychology. Learning, memory, and cognition.

[29]  Jamie I. D. Campbell,et al.  Integrated versus modular theories of number skills and acalculia , 1991, Brain and Cognition.

[30]  M. Ashcraft,et al.  The development of children's mental multiplication skills , 1991 .

[31]  Lewis R. Aiken,et al.  Response Times in Adding and Multiplying Single-Digit Numbers , 1973 .

[32]  Mark H. Ashcraft,et al.  Textbook Presentations of the Basic Addition Facts , 1986 .

[33]  Jamie I. D. Campbell,et al.  Time course of error priming in number-fact retrieval: evidence for excitatory and inhibitory mechanisms , 1989 .

[34]  A. T. Welford,et al.  THE MEASUREMENT OF SENSORY-MOTOR PERFORMANCE : SURVEY AND REAPPRAISAL OF TWELVE YEARS' PROGRESS , 1960 .

[35]  R. Shepard,et al.  The internal representation of numbers , 1975, Cognitive Psychology.

[36]  T. Blaxton.  Investigating dissociations among memory measures: Support for a transfer-appropriate processing framework. , 1989 .

[37]  Jamie I. D. Campbell,et al.  Mental multiplication skill: Structure, process, and acquisition. , 1985 .

[38]  M. Perlmutter,et al.  Cognitive arithmetic: comparison of operations. , 1984, Journal of experimental psychology. Learning, memory, and cognition.

[39]  M. McCloskey,et al.  In defense of a modular architecture for the number-processing system: reply to Campbell and Clark. , 1989, Journal of experimental psychology. General.

[40]  D L Schacter,et al.  Modality specificity of implicit memory for new associations. , 1989, Journal of experimental psychology. Learning, memory, and cognition.

[41]  H. Heuer,et al.  Generation and modulation of action patterns , 1986 .

[42]  Jamie I. D. Campbell.  Network interference and mental multiplication. , 1987 .