Children’s Knowledge of Simple Arithmetic: A Developmental Model and Simulation

  • Mark H. Ashcraft
Part of the Springer Series in Cognitive Development book series (SSCOG)


This chapter is about children’s mental arithmetic, the knowledge that is acquired across the school years, the early representation of that knowledge in memory, and the evolution of the mental representation and processes across childhood. The largest portion of the chapter is devoted to a model of children’s knowledge and performance in a simple addition task. I propose that knowledge in the domain of arithmetic is, in principle, similar to other long-term memory knowledge, both in its representational format and in the processes used to access the knowledge. The computer simulation based on the model successfully predicts the major empirical effects found in the literature and generates new predictions about the nature of memory retrieval across the developmental span.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard.Google Scholar
  2. Anderson, J. R., & Bower, G. H. (1973). Human associative memory. Washington:Winston-Wiley.Google Scholar
  3. Ashcraft, M. H. (1976). Priming and property dominance effects in semantic memory. Memory & Cognition, 4, 490–500.CrossRefGoogle Scholar
  4. Ashcraft, M. H. (1978). Property dominance and typicality effects in property statement verification. Journal of Verbal Learning and Verbal Behavior, 17, 155–164.CrossRefGoogle Scholar
  5. Ashcraft, M. H. (1982). The development of mental arithmetic: A chronometric approach. Developmental Review, 2,213–236.CrossRefGoogle Scholar
  6. Ashcraft, M. H. (1983). Simulating network retrieval of arithmetic facts. Learning Research and Development Center Publication Series, University of Pittsburgh, 1983/10.Google Scholar
  7. Ashcraft, M. H. (1985a, April). Children’s mental arithmetic: Toward a model of retrieval and problem solving. Paper presented at the meetings of the Society for Research in Child Development, Toronto.Google Scholar
  8. Ashcraft, M. H. (1985b). Is it farfetched that some of us remember our arithmetic facts? Journal for Research in Mathematics Education, 16, 99–105.CrossRefGoogle Scholar
  9. Ashcraft, M. H., & Battaglia, J. (1977, November). Cognitive arithmetic: Evidence for two-stage decision and search. Paper presented at the meeting of the Psychonomic Society, Washington.Google Scholar
  10. Ashcraft, M. H., & Battaglia, J. (1978). Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition. Journal of Experimental Psychology: Human Learning and Memory, 4,527–538.CrossRefGoogle Scholar
  11. Ashcraft, M. H., & Fierman, B. A. (1982). Mental addition in third, fourth, and sixth graders. Journal of Experimental Child Psychology, 33, 216–234.CrossRefGoogle Scholar
  12. Ashcraft, M. H., Fierman, B. A., & Bartolotta, R. (1984). The production and verification tasks in mental addition: An empirical comparison. Developmental Review, 4, 157–170.CrossRefGoogle Scholar
  13. Ashcraft, M. H., & Stazyk, E. H. (1981). Mental addition: A test of three verification models. Memory & Cognition, 9, 185–196.CrossRefGoogle Scholar
  14. Banks, W. P. (1977). Encoding and processing of symbolic information in comparativejudgments. In G. H. Bower (Ed.), The psychology of learning and motivation (Vol. 11, pp. 101–159). New York: Academic.Google Scholar
  15. Baroody, A. J. (1983). The development of procedural knowledge: An alternative explanation for chronometric trends of mental arithmetic. Developmental Review, 3, 225–230.CrossRefGoogle Scholar
  16. Baroody, A. J. (1984). A reexamination of mental arithmetic models and data: A reply to Ashcraft. Developmental Review, 4, 148–156.CrossRefGoogle Scholar
  17. Baroody, A. J. (1985). Mastery of basic number combinations: Internalization of relationships or facts? Journalfor Research in Mathematics Education, 16, 83–98.CrossRefGoogle Scholar
  18. Campbell, J. I. D. (1985). Associative interference in mental multiplication. Unpublished doctoral dissertation, University of Waterloo, Ontario, Canada.Google Scholar
  19. Campbell, J.I. D., & Graham, D. J. (1985). Mental multiplication skill: Structure, process, and acquisition. Canadian Journal of Psychology, 39, 338–366.CrossRefGoogle Scholar
  20. Collins, A. M., & Loftus, E. F. (1975). A spreading-activation theory of semantic procesing. Psychological Review, 82, 407–428.CrossRefGoogle Scholar
  21. Collins, A. M., & Quillian, M. R. (1972). How to make a language user. In E. Tulving & W. Donaldson (Eds.), Organization of memory. New York: Academic.Google Scholar
  22. Duffy, S. A., & Fisher, D. L. (1980, May). The organization and processing of multiplication facts. Paper presented at the meetings of the Midwestern Psychological Association, St. Louis.Google Scholar
  23. Estes, W. K. (1964). All-or-none processes in learning and retention. American Psychologist, 19, 16–25.CrossRefGoogle Scholar
  24. Fierman, B. A. (1980). Developmental mental addition: A test of two models and two methods. Unpublished master’s thesis, Cleveland State University, Cleveland, OH.Google Scholar
  25. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard.Google Scholar
  26. Ginsburg, H. (1977). Children’s arithmetic: The learning process. New York: Van Nostrand.Google Scholar
  27. Glass, A. L., Holyoak, K. J., & O’Dell, C. (1974). Production frequency and the verification of quantified statements. Journal of Verbal Learning and Verbal Behavior, 13, 237–254.CrossRefGoogle Scholar
  28. Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329–343.CrossRefGoogle Scholar
  29. Hamann, M. S. (1981). Cognitive processes in simple and complex addition. Unpublished master’s thesis, Cleveland State University, Cleveland, OH.Google Scholar
  30. Hamann, M. S. (1983, May). Acquisition and practice of addition and quasiaddition facts. Paper presented at the Pittsburgh-Carnegie-Mellon Conference on Cognition, Pittsburgh.Google Scholar
  31. Hamann, M. S., & Ashcraft, M. H. (1985). Simple and complex mental addition across development. Journal of Experimental Child Psychology, 40, 49–72.CrossRefGoogle Scholar
  32. Hamann, M. S., & Ashcraft, M. H. (in press). Textbook presentations ofthe basic addition facts. Cognition and Instruction Google Scholar
  33. Hempel, C. G., &Oppenheim, P. (1953). The logic of explanation, In H. Feigl & M. Brodbeck (Eds.), Readings in the philosophy of science. New York: Appleton Century Crofts.Google Scholar
  34. Kieras, D. (1984, November). The why, when, and how of cognitive simulation: A tutorial. Paper presented at the meetings of the Society for Computers in Psychology, San Antonio, TX.Google Scholar
  35. Kintsch, W. (1974). The representation of meaning in memory. Hillsdale, NJ: Erlbaum.Google Scholar
  36. Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92, 109–129.PubMedCrossRefGoogle Scholar
  37. Koshmider, J. W. III (1986). Development of children’s multiplication skills. Unpublished master’s thesis. Cleveland State University, Cleveland, OH.Google Scholar
  38. Loftus, G. (1985). Johannes Kepler’s computer simulation of the universe: Some remarks about theory in psychology. Behavior Research Methods, Instruments, & Computers, 17, 149–156.Google Scholar
  39. Miller, K. F., Perlmutter, M., & Keating, D. (1984). Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 10, 46–60.PubMedCrossRefGoogle Scholar
  40. Norman, D. A., & Rumelhart, D. E. (1975). Explorations in cognition. San Francisco: Freeman.Google Scholar
  41. Parkman, J. M. (1972). Temporal aspects of simple multiplication and comparison. Journal of Experimental Psychology, 95, 437–444.CrossRefGoogle Scholar
  42. Posner, M. I. (1978). Chronometric explorations of mind. Hillsdale, NJ: Erlbaum.Google Scholar
  43. Posner, M. I., & Snyder, C. R. R. (1975). Facilitation and inhibition in the processing of signals. In P. M. A. Rabbitt & S. Domic (Eds.), Attention and performance V (pp. 669–682). New York: Academic.Google Scholar
  44. Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale, NJ: Erlbaum.Google Scholar
  45. Rosch, E. (1975). Cognitive representations of semantic categories. Journal of Experimental Psychology: General, 104, 192–233.CrossRefGoogle Scholar
  46. Rumelhart, D. E., & McClelland, J. L. (1982). An interactive activation model of context effects in letter perception: Part 2. The contextual enhancement effect and some tests and extensions of the model. Psychological Review, 89, 60–94.PubMedCrossRefGoogle Scholar
  47. Schank, R., & Abelson, R. (1977). Scripts, plans, goals and understanding-An inquiry into human knowledge structures. Hillsdale, NJ: Erlbaum.Google Scholar
  48. Schank, R. C., & Riesbeck, C. K. (Eds.) (1981). Inside computer understanding Hillsdale, NJ: Erlbaum.Google Scholar
  49. Selfridge, O. G. (1959). Pandemonium: A paradigm for learning. In The mechanization of thought processes. London: H. M. Stationery Office.Google Scholar
  50. Shiffrin, R. M., & Schneider, W. (1977). Controlled and automatic human information processing: II. Perceptual learning, automatic attending, and a general theory. Psychological Review, 84, 127–190.CrossRefGoogle Scholar
  51. Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), Origins of cognitive skills. Hillsdale, NJ: Erlbaum.Google Scholar
  52. Simpson, G. B., & Lorsbach, T. C. (1983). The development of automatic and conscious components of contextual facilitation. Child Development, 54, 760–772.CrossRefGoogle Scholar
  53. Stazyk, E. H. (1980). A network approach to mental multiplication. Unpublished master’s thesis, Cleveland State University, Cleveland, OH.Google Scholar
  54. Stazyk, E. H., Ashcraft, M. H., & Hamann, M. S. (1982). A network approach to mental multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8, 320–335.CrossRefGoogle Scholar
  55. Svenson, O. (1975). Analysis of time required by children for simple additions. Acta Psychologica, 39, 289–302.CrossRefGoogle Scholar
  56. Whaley, C. P. (1978). Word-nonword classification time. Journal of Verbal Learning and Verbal Behavior, 17, 143–154.CrossRefGoogle Scholar
  57. Wheeler, L. R. (1939). A comparative study of the difficulty of the 100 addition combinations. Journal of Genetic Psychology, 54,295–312.Google Scholar
  58. Winkelman, H. J., & Schmidt, J. (1974). Associative confusions in mental arithmetic. Journal of Experimental Psychology, 102, 734–736.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Mark H. Ashcraft

There are no affiliations available

Personalised recommendations