Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-04T19:47:23.171Z Has data issue: false hasContentIssue false
This chapter is part of a book that is no longer available to purchase from Cambridge Core

32 - Mathematical Expertise

from PART V.C - GAMES AND OTHER TYPES OF EXPERTISE

Brian Butterworth
Affiliation:
Institute of Cognitive Neuroscience, University College London
K. Anders Ericsson
Affiliation:
Florida State University
Neil Charness
Affiliation:
Florida State University
Paul J. Feltovich
Affiliation:
University of West Florida
Robert R. Hoffman
Affiliation:
University of West Florida
Get access

Summary

Competence in mathematics is a basic requirement for effective citizenship in a modern numerate society (Cockcroft, 1982). Poor numeracy skills are known to be a serious handicap for paid employment in the US (Rivera-Batiz, 1992) and the UK (Bynner & Parsons, 1997). Indeed, the UK Basic Skills Agency has published a report suggesting that numeracy is more important even than literacy in terms of career prospects in the UK (Bynner & Parsons, 1997). And the trend is toward an even greater emphasis on numeracy: recent research for the British Science, Technology and Mathematics Council shows that “mathematical skills in the workplace are changing, with increasing numbers of people engaged in mathematics-related work, and with such work involving increasingly sophisticated mathematical activities” (Hoyles, Wolf, Molyneux-Hodgson, & Kent, 2002).

The level of competence routinely demanded in numerate cultures today would have been considered quite exceptional 200 years ago. How then does one distinguish today's expert from the normally competent school-leaver who can handle numbers of arbitrary size, fractions and decimals, logarithms, equations with unknowns and negative roots, and some differentiation and integration? One could arbitrarily take the top n% of a standard test (like the SAT-M), but what should n be? Francis Galton, in Hereditary Genius, used obituaries from The Times of London and a biographical dictionary, Men of our Time, as the criteria of “eminence.” This gave him an estimated proportion of 0.025% of the population. Really exceptional individuals, his class G, were about one-twentieth of these.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alarcon, M., Defries, J., Light, Gillis J., & Pennington, B. (1997). A twin study of mathematics disability. Journal of Learning Disabilities, 30, 617–623.CrossRefGoogle ScholarPubMed
Alexander, J. E., O'Boyle, M. W., & Benbow, C. P. (1996). Developmentally advanced EEG alpha power in gifted male and female adolescents. International Journal of Psychophysiology, 23, 25–31.CrossRefGoogle ScholarPubMed
Amunts, K., Schlaug, G., Jancke, L., Steinmetz, H., Schleicher, A., Dabringhaus, A., et al. (1997). Motor cortex and hand motor skills: Structural compliance in the human brain. Human Brain Mapping, 5, 206–215.3.0.CO;2-7>CrossRefGoogle ScholarPubMed
Antell, S. E., & Keating, D. P. (1983). Perception of numerical invariance in neonates. Child Development, 54, 695–701.CrossRefGoogle ScholarPubMed
Ashcraft, M. (1995). Cognitive psychology and simple arithmetic: A review and summary of new directions. Mathematical Cognition, 1, 3–34.Google Scholar
Barlow, F. (1952). Mental Prodigies. New York: Greenwood Press.Google Scholar
Becker, B. J., & Hedges, L. V. (1988). The effects of selection and variability in studies of gender differences. Commentary on Benbow (1988). Behavioral and Brain Sciences, 11(2), 183–184.CrossRefGoogle Scholar
Benbow, C. P. (1988). Sex differences in mathematical reasoning ability in intellectually talented preadolescents: Their nature, effects, and possible causes. Behavioral and Brain Sciences, 11(2), 169–183.CrossRefGoogle Scholar
Binet, A. (1894). Psychologie des grands calculateurs et joueurs d'échecs. Paris: Hachette.Google Scholar
Butterworth, B. (1999). The Mathematical Brain. London: Macmillan.
Butterworth, B. (2001). What makes a prodigy? Nature Neuroscience, 4(1), 11–12.CrossRefGoogle ScholarPubMed
Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology & Psychiatry, 46(1), 3–18.CrossRefGoogle ScholarPubMed
Butterworth, B., Cipolotti, L., & Warrington, E. K. (1996). Short-term memory impairments and arithmetical ability. Quarterly Journal of Experimental Psychology, 49A, 251–262.CrossRefGoogle Scholar
Butterworth, B., Shallice, T., & Watson, F. (1990). Short-term retention of sentences without “short-term memory.” In Vallar, G. & Shallice, T. (Eds.), Neuropsychological Impairments of Short-Term Memory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Bynner, J., & Parsons, S. (1997). Does Numeracy Matter? London: The Basic Skills Agency.Google Scholar
Cappelletti, M., Kopelman, M., & Butterworth, B. (2002). Why semantic dementia drives you to the dogs (but not to the horses): A theoretical account. Cognitive Neuropsychology, 19(6), 483–503.CrossRefGoogle ScholarPubMed
Charness, N., & Campbell, J. I. D. (1988). Acquiring skill at mental calculation in adulthood: A task decomposition. Journal of Experimental Psychology: General, 117(2), 115–129.CrossRefGoogle Scholar
Cipolotti, L., & van Harskamp, N. (2001). Disturbances of number processing and calculation. In Berndt, R. S. (Ed.), Handbook of Neuropsychology (2nd. ed., Vol. 3, pp. 305–334). Amsterdam: Elsevier Science.Google Scholar
Cockcroft, W. H. (1982). Mathematics Counts: Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr. W. H. Cockcroft. London: HMSO.Google Scholar
Dehaene, S. (1997). The Number Sense: How the Mind creates Mathematics. New York: Oxford University Press.Google Scholar
Dehaene, S., & Cohen, L. (1995). Towards and anatomical and functional model of number processing. Mathematical Cognition, 1, 83–120.Google Scholar
Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506.CrossRefGoogle ScholarPubMed
Delazer, M., & Benke, T. (1997). Arithmetic facts without meaning. Cortex, 33, 697–710.CrossRefGoogle ScholarPubMed
DfES. (2002). GCSE/GNVQ National Summary Results, from http://www.standards.dfes.gov.uk/performance/2002gcseresults.pdf?version=1.
Donlan, C. (2003). The early numeracy of children with specific language impairments. In Baroody, A. J. & Dowker, A. D. (Eds.), The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise (pp. 337–358). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
Edwards, C. J., Alder, T. B., & Rose, G. J. (2002). Auditory midbrain neurons that count. Nature Neuroscience, 5(10), 934–936.CrossRefGoogle ScholarPubMed
Ericsson, K. A., & Charness, N. (1994). Expert performance: Its structure and acquisition. American Psychologist, 49(8), 725–747.CrossRefGoogle Scholar
Ericsson, K. A., & Kintsch, W. (1995). Long-term working memory. Psychological Review, 102, 211–245.CrossRefGoogle ScholarPubMed
Ericsson, K. A., Krampe, R. T., & Tesch-Römer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100, 363–406.CrossRefGoogle Scholar
Galton, F. (1979). Hereditary Genius: An Inquiry into its Laws and Consequences (Originally published in 1869). London: Julian Friedman Publishers.Google Scholar
Gardner, H. (1983). Frames of Mind: The Theory of Multiple Intelligences. New York: Basic Books.Google Scholar
Geary, D. C. (1996). Sexual selection and sex differences in mathematical abilities. Behavioral and Brain Sciences, 19, 229 et seq.CrossRefGoogle Scholar
Gelman, R., & Gallistel, C. R. (1978). The Child's Understanding of Number (1986 ed.). Cambridge, MA: Harvard University Press.Google Scholar
Girelli, L., & Delazer, M. (1996). Subtraction bugs in an acalculic patient. Cortex, 32, 547–555.CrossRefGoogle Scholar
Goel, V., & Dolan, R. J. (2004). Differential involvement of left prefrontal cortexin inductive and deductive reasoning. Cognition, 93(3), B109–B121.CrossRefGoogle Scholar
Gruber, O., Indefrey, P., Steinmetz, H., & Kleinschmidt, A. (2001). Dissociating neural correlates of cognitive components in mental calculation. Cerebral Cortex, 11, 350–359.CrossRefGoogle ScholarPubMed
Hardy, G. H. (1969). A Mathematician's Apology (Originally published in 1940). Cambridge: Cambridge University Press.Google Scholar
Hauser, M., MacNeilage, P., & Ware, M. (1996). Numerical representations in primates. Proceedings of the National Academy of Sciences, USA, 93, 1514–1517.CrossRefGoogle ScholarPubMed
Hécaen, H., Angelergues, R., & Houillier, S. (1961). Les variétés cliniques des acalculies au cours des lésions rétro-rolandiques: Approche statistique du problème. Revue Neurologique, 105, 85–103.Google Scholar
Hermelin, B., & O'Connor, N. (1990). Factors and primes: A specific numerical ability. Psychological Medicine, 20, 163–169.CrossRefGoogle ScholarPubMed
Hittmair-Delazer, M., Semenza, C., & Denes, G. (1994). Concepts and facts in calculation. Brain, 117, 715–728.CrossRefGoogle ScholarPubMed
Horwitz, W. A., Deming, W. E., & Winter, R. F. (1969). A further account of the idiot savants: Experts with the calendar. American Journal of Psychiatry, 126, 160–163.CrossRefGoogle Scholar
Hoyles, C., Wolf, A., Molyneux-Hodgson, S., & Kent, P. (2002). Mathematical Skills in the Workplace. London: Institute of Education.Google Scholar
Hunter, I. M. L. (1962). An exceptional talent for calculative thinking. British Journal of Psychology, 53, 243–280.CrossRefGoogle ScholarPubMed
Jensen, A. R. (1990). Speed of information-processing in a calculating prodigy. Intelligence, 14, 3.CrossRefGoogle Scholar
Keys, W., Harris, S., & Fernandes, C. (1996). Third International Mathematics and Science Study. First national Report. Part 1. Slough: National Foundation for Educational Research.Google Scholar
Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A Study of 8–9 Year Old Students. Cognition, 93, 99–125.CrossRefGoogle ScholarPubMed
Logie, R. H., Gilhooly, K. J., & Wynn, V. (1994). Counting on working memory in arithmetic problem solving. Memory & Cognition, 22, 395–410.CrossRefGoogle ScholarPubMed
McCloskey, M., & Caramazza, A. (1987). Dissociations of calculation processes. In Deloche, G. & Seron, X. (Eds.), Mathematical Disabilities: A Cognitive Neuropsychological Perspective. Hillsdale, NJ: Lowrence Erlbaum Associetes.Google Scholar
McComb, K., Packer, C., & Pusey, A. (1994). Roaring and numerical assessment in contests between groups of female lions, Panthera leo . Animal Behaviour, 47, 379–387.CrossRefGoogle Scholar
Mitchell, F. D. (1907). Mathematical prodigies. Amercian Journal of Psychology, 18(1), 61–143.CrossRefGoogle Scholar
O'Boyle, M. W., Benbow, C. P., & Alexander, J. E. (1995). Sex differences, hemispheric laterality, and associated brain activity in the intellectually gifted. Developmental Neuropsychology, 11(4), 415–443.CrossRefGoogle Scholar
O'Boyle, M. W., Gill, H. S., Benbow, C. P., & Alexander, J. E. (1994). Concurrent finger-tapping in mathematically gifted males – evidence for enhanced right-hemisphere involvement during linguistic processing. Cortex, 30(3), 519–526.CrossRefGoogle ScholarPubMed
Pascual-Leone, A., & Torres, F. (1993). Plasticity of the sensorimotor cortex representation of the reading finger in Braille readers. Brain, 116, 39–52.CrossRefGoogle ScholarPubMed
Paulesu, E., Frith, C. D., & Frackowiak, R. S. J. (1993). The neural correlates of the verbal component of working memory. Nature, 362, 342–345.CrossRefGoogle ScholarPubMed
Pesenti, M. (2005). Calculation abilities in expert calculators. In Campbell, J. I. D. (Ed.), Handbook of Mathematical Cognition (pp. 413–430). Hove: Psychology Press.Google Scholar
Pesenti, M., Seron, X., Samson, D., & Duroux, B. (1999). Basic and exceptional calculation abilities in a calculating prodigy: A case study. Mathematical Cognition, 5, 97–148.CrossRefGoogle Scholar
Pesenti, M., Thioux, M., Seron, X., & Volder, A. (2000). Neuroanatomical substrates of Arabic number processing, numerical comparison and simple addition: A PET study. Journal of Cognitive Neuroscience, 12, 461–479.CrossRefGoogle ScholarPubMed
Pesenti, M., Zago, L., Crivello, F., Mellet, E., Samson, D., Duroux, B., et al. (2001). Mental calculation expertise in a prodigy is sustained by right prefrontal and medial-temporal areas. Nature Neuroscience, 4(1), 103–107.CrossRefGoogle Scholar
Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of employment among young adults in the United States. The Journal of Human Resources, 27(2), 313–328.CrossRefGoogle Scholar
Schlaug, G., Jancke, L., Huang, Y. X., Staiger, J. F., & Steinmetz, H. (1995). Increased corpus callosum size in musicians. Neuropsychologia, 33, 1047–1055.CrossRefGoogle ScholarPubMed
Schlaug, G., Jancke, L., Huang, Y. X., & Steinmetz, H. (1995). In-vivo evidence of structural brain asymmetry in musicians. Science, 267, 699–701.CrossRefGoogle ScholarPubMed
Scripture, E. W. (1891). Arithmetical prodigies. Amercian Journal of Psychology, 4(1), 1–59.CrossRefGoogle Scholar
Shalev, R. S., & Gross-Tsur, V. (2001). Developmental dyscalculia. Review article. Pediatric Neurology, 24, 337–342.CrossRefGoogle Scholar
Singh, H., & O'Boyle, M. W. (2004). Interhemispheric interaction during global-local processing in mathematically gifted adolescents, average-ability youth, and college students. Neuropsychology, 18(2), 371–377.CrossRefGoogle Scholar
Smith, S. B. (1983). The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies. New York: Columbia University Press.Google Scholar
Starkey, P., & Cooper, R. G. Jr. (1980). Perception of numbers by human infants. Science, 210, 1033–1035.CrossRefGoogle ScholarPubMed
Harskamp, N. J., & Cipolotti, L. (2001). Selective impairments for addition, subtraction and multiplication. Implications for the organisation of arithmetical facts. Cortex, 37, 363–388.CrossRefGoogle ScholarPubMed
Weinland, J. D., & Schlauch, W. S. (1937). An examination of the computing ability of Mr. Salo Finkelstein. Journal of Experimental Psychology, 21, 382–402.CrossRefGoogle Scholar
Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358, 749–751.CrossRefGoogle ScholarPubMed
Wynn, K. (2000). Findings of addition and subtraction in infants are robust and consistent: Reply to Wakeley, Rivera, and Langer. Child Development, 71(6), 1535–1536.CrossRefGoogle ScholarPubMed
Wynn, K. (2002). Do infants have numerical expectations or just perceptual preferences? Commentary. Developmental Science, 5(2), 207–209.CrossRefGoogle Scholar
Wynn, K., Bloom, P., & Chiang, W. C. (2002). Enumeration of collective entities by 5-month-old infants. Cognition, 83(3), B55–B62.CrossRefGoogle ScholarPubMed
Zago, L., Pesenti, M., Mellet, E., Crivello, F., Mazoyer, B., & Tzourio-Mazoyer, N. (2001). Neural correlates of simple and complex mental calculation. Neuroimage, 13(2), 314–327.CrossRefGoogle ScholarPubMed

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×