# A multidimensional scaling approach to mental multiplication

- 240 Downloads
- 1 Citations

## Abstract

Adults consistently make errors in solving simple multiplication problems. These errors have been explained with reference to the interference between similar problems. In this paper, we apply multidimensional scaling (MDS) to the domain of multiplication problems, to uncover their underlying similarity structure. A tree-sorting task was used to obtain perceived dissimilarity ratings. The derived representation shows greater similarity between problems containing larger operands and suggests that*tie* problems (e.g., 7 x 7) hold special status. A version of the generalized context model (Nosofsky, 1986) was used to explore the derived MDS solution. The similarity of multiplication problems made an important contribution to producing a model consistent with human performance, as did the frequency with which such problems arise in textbooks, suggesting that both factors may be involved in the explanation of errors.

## Keywords

Multidimensional Scaling Problem Frequency Mental Multiplication Arithmetic Fact Generalize Context Model## References

- Ashcraft, M. H. (1985). Is it farfetched that some of us remember our arithmetic facts?
*Journal for Research in Mathematics Education*,**16**, 99–105.CrossRefGoogle Scholar - Ashcraft, M. H. (1992). Cognitive arithmetic: A review of theory and data.
*Cognition*,**44**, 75–106.CrossRefPubMedGoogle Scholar - Ashcraft, M. H., &Christy, K. S. (1995). The frequency of arithmetic facts in elementary texts: Addition and multiplication in grades 16.
*Journal for Research in Mathematics Education*,**26**, 396–421.CrossRefGoogle Scholar - Benford, F. (1938). The law of anomalous numbers.
*Proceedings of the American Philosophical Society*,**78**, 551–572.Google Scholar - Campbell, J. I. D. (1987). Network interference and mental multiplication.
*Journal of Experimental Psychology: Learning, Memory, & Cognition*,**13**, 109–123.CrossRefGoogle Scholar - Campbell, J. I. D. (1994). Architectures for numerical cognition.
*Cognition*,**53**, 1–44.CrossRefPubMedGoogle Scholar - Campbell, J. I. D. (1995). Mechanisms of simple addition and multiplication: A modified network interference model and simulation.
*Mathematical Cognition*,**1**, 121–164.Google Scholar - Campbell, J. I. D. (1997). Reading-based interference in cognitive arithmetic.
*Canadian Journal of Experimental Psychology*,**51**, 74–81.PubMedGoogle Scholar - Campbell, J. I. D., &Graham, D. J. (1985). Mental multiplication skill: Structure, process and acquisition.
*Canadian Journal of Psychology*,**39**, 338–366.CrossRefGoogle Scholar - Campbell, J. I. D., &Oliphant, M. (1992). Representation and retrieval of arithmetic facts: A network interference model and simulation. In J. I. D. Campbell (Ed.),
*The nature and origins of mathematical skills*(pp. 331–364). Amsterdam: Elsevier.CrossRefGoogle Scholar - Campbell, J. I. D., &Tarling, D. P. M. (1996). Retrieval processes in arithmetic production and verification.
*Memory & Cognition*,**24**, 156–172.CrossRefGoogle Scholar - Coxon, A. P. M. (1982).
*The users guide to multidimensional scaling: With special reference to the MDS(X) library of computer programs*. Exeter, NH: Heinemann.Google Scholar - Dehaene, S. (1992). Varieties of numerical abilities.
*Cognition*,**44**, 1–42.CrossRefPubMedGoogle Scholar - Fendrich, D. W., Healy, A. F., &Bourne, L. E. (1993). Mental arithmetic: Training and retention of multiplication skill. In C. Izawa (Ed.),
*Cognitive psychology applied*(pp. 111–133). Hillsdale, NJ: Erlbaum.Google Scholar - Fillenbaum, S., &Rapoport, A. (1971).
*Structures in the subjective lexicon*. New York: Academic Press.Google Scholar - Geary, D. C., Widaman, K. F., &Little, T. D. (1986). Cognitive addition and multiplication: Evidence for a single memory network.
*Memory & Cognition*,**14**, 478–487.CrossRefGoogle Scholar - Graham, D. J., &Campbell, J. I. D. (1992). Network interference and number-fact retrieval: Evidence from childrens alphaplication.
*Canadian Journal of Psychology*,**46**, 65–91.CrossRefPubMedGoogle Scholar - Koshmider, J. W., &Ashcraft, M. H. (1991). The development of childrens mental multiplication skills.
*Journal of Experimental Child Psychology*,**51**, 53–89.CrossRefGoogle Scholar - Kruschke, J. K. (1992). ALCOVE: An exemplar-based connectionist model of category learning.
*Psychological Review*,**99**, 22–44.CrossRefPubMedGoogle Scholar - LeFevre, J., Bisanz, J., Daley, K. E., Buffone, L., Greenham, S.L., &Sadesky, G. S. (1996). Multiple routes to solution of single-digit multiplication problems.
*Journal of Experimental Psychology: General*,**125**, 284–306.CrossRefGoogle Scholar - Lewandowsky, S., &Newman, D. A. (1993). Chronometric scaling of numbers. In R. Steyer, K. F. Wender, & K. F. Widaman (Eds.),
*Psychometric methodology: Proceedings of the 7th European Meeting of the Psychometric Society in Trier*(pp. 272–277). Stuttgart: Gustav Fischer Verlag.Google Scholar - Luce, R. D. (1963). Detection and recognition. In R.D. Luce, R. R. Bush, & E. Galanter (Eds.),
*Handbook of mathematical psychology*(pp. 103–189). New York: Wiley.Google Scholar - McCloskey, M., Harley, W., &Sokol, S. M. (1991). Models of arithmetic fact retrieval: An evaluation in light of findings from normal and brain-damaged subjects.
*Journal of Experimental Psychology: Learning, Memory, & Cognition*,**17**, 377–397.CrossRefGoogle Scholar - Meagher, P. D., &Campbell, J. I. D. (1995). Effects of prime type and delay on multiplication priming: Evidence for a dual process model.
*Quarterly Journal of Experimental Psychology*,**48A**, 801–821.Google Scholar - Miller, K. F. (1992). What a number is: Mathematical foundations and developing number concepts. In J. I. D. Campbell (Ed.),
*The nature and origins of mathematical skills*(pp. 3–38). Amsterdam: Elsevier.CrossRefGoogle Scholar - Miller, K. [F.], &Gelman, R. (1983). The childs representation of number: A multidimensional scaling analysis.
*Child Development*,**54**, 1470–1479.Google Scholar - Miller, K. [F.], Perlmutter, M., &Keating, D. (1984). Cognitive arithmetic: Comparison of operations.
*Journal of Experimental Psychology: Learning, Memory, & Cognition*,**10**, 46–60.CrossRefGoogle Scholar - Nosofsky, R. M. (1986). Attention, similarity, and the identification categorization relationship.
*Journal of Experimental Psychology: General*,**115**, 39–57.CrossRefGoogle Scholar - Nosofsky, R. M. (1987). Attention and learning processes in the identification and categorization of integral stimuli.
*Journal of Experimental Psychology: Learning, Memory, & Cognition*,**13**, 87–109.CrossRefGoogle Scholar - Nosofsky, R. M. (1991a). Exemplar-based approach to relating categorization, identification, and recognition. In F. G. Ashby (Ed.),
*Multidimensional models of perception and cognition*(pp. 363–393). Hillsdale, NJ: Erlbaum.Google Scholar - Nosofsky, R. M. (1991b). Tests of an exemplar model for relating perceptual classification and recognition memory.
*Journal of Experimental Psychology: Human Perception & Performance*,**17**, 3–27.CrossRefGoogle Scholar - Parkman, J. M. (1972). Temporal aspects of simple multiplication and comparison.
*Journal of Experimental Psychology*,**95**, 437–444.CrossRefGoogle Scholar - Parkman, J. M., &Groen, G. J. (1971). Temporal aspects of simple addition and comparison.
*Journal of Experimental Psychology*,**89**, 335–342.CrossRefGoogle Scholar - Rhodes, G. (1985). Looking at faces: First-order and second-order features as determinants of facial appearance.
*Perception*,**17**, 43–63.CrossRefGoogle Scholar - Rickard, T. C., Healy, A. F., &Bourne, L. E., Jr. (1994). On the representation of arithmetic facts: Operand order, symbol, and operation transfer effects.
*Journal of Experimental Psychology: Learning, Memory, & Cognition*,**20**, 1139–1153.CrossRefGoogle Scholar - Shepard, R. N. (1980) Multidimensional scaling, tree-fitting, and clustering.
*Science*,**210**, 390–398.CrossRefPubMedGoogle Scholar - Shepard, R. N. (1987) Toward a universal law of generalization for psychological science.
*Science*,**237**, 1317–1323.CrossRefPubMedGoogle Scholar - Shepard, R. N., Kilpatric, D. W., &Cunningham, J. P. (1975). The internal representation of numbers.
*Cognitive Psychology*,**7**, 82–138.CrossRefGoogle Scholar - Stazyk, E. H., Ashcraft, M. H., &Hamann, M. S. (1982). A network approach to mental multiplication.
*Journal of Experimental Psychology: Learning, Memory, & Cognition*,**8**, 320–335.CrossRefGoogle Scholar - Wickens, T. D. (1989).
*Multiway contingency tables analysis for the social sciences*. Hillsdale, NJ: Erlbaum.Google Scholar - Zbrodoff, N. J. (1995). Why is 9 + 7 harder than 2 + 3? Strength and interference as explanations for the problem-size effect.
*Memory & Cognition*,**23**, 689–700.CrossRefGoogle Scholar