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Tools, Human Development and Mathematics

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Tools and Mathematics

Part of the book series: Mathematics Education Library ((MELI,volume 110))

Abstract

This chapter raises a number of issues from pre-history and history that one mathematics educator considers ‘worthy of mention’ with regard to tools and mathematics. These issues are: tool use in the development of the human species (phylogenesis); tool use in a mathematical culture, ancient Greek mathematics that goes beyond the obvious tools; an example from ancient Indian mathematics that bears some resemblances to Jon’s experimental mathematics described in Chap. 3; the mutual support of hand, mind and artefact in expert use of an abacus; a consideration of a period (sixteenth-century Europe) where there was a rapid advance in the development of mathematical tools.

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Notes

  1. 1.

    Homo sapiens is a specie in the genus Homo in the family Hominidae.

  2. 2.

    Before common era, a term preferred by scholars to BC (but virtually identical in terms of dates).

  3. 3.

    Almost all the mathematicians of ancient Greece were upper class males.

  4. 4.

    Thirteen strictly sequenced books of definitions, postulates, common notions and propositions.

  5. 5.

    Evidence suggests that ancient Greeks also ‘used pebbles for calculations on abaci … but in a marginal role … never at the centre of mathematical activity’ (Netz, 1999, pp. 63–64).

  6. 6.

    Without detracting from the wonder of Greek mathematics, there are mathematical problems with its definitions and proofs. We do not consider these here. The interested reader may consult Netz (1999).

  7. 7.

    My reference here could be (Heath, 1926, p. 382) but I shall use Netz (1999) to describe the proof of this proposition.

  8. 8.

    This makes implicit reference to Proposition I.46. We return to implicit references and expected knowledge in our discussion of the ‘the tool box’ after the proof.

  9. 9.

    This makes implicit reference to Proposition I.31.

  10. 10.

    The complement of a parallelogram would be expected to be known to readers. Line <1> makes implicit reference to Proposition I.43.

  11. 11.

    Implicit reference to Proposition I.36.

  12. 12.

    The gnomon is defined in Definition II.2.

  13. 13.

    ‘Netz’ in the following pages refers to ‘Netz (1999)’.

  14. 14.

    I have represented this using a symbol for ‘therefore’ instead of representing this as ‘<1> & <2> → <3>’ as I am far from certain that implication in terms of mathematical logic (suggested by the ‘→’ sign) is how the Greeks understood the relationship between ‘<1>, <2> and <3>’.

  15. 15.

    I use the term Saito coined and Netz followed but do not scrutinise the term with regard to Chap. 1 definitions of tools. It can be assumed that the terms ‘tool box’ in this chapter and ‘tool-box’ in Chap. 1 refer to different things.

  16. 16.

    The mathematical community considered also constructed algorithm for computing cube roots but I restrict my focus to square roots in this section.

  17. 17.

    I drop the prefix ‘European’ for the remainder of this section.

  18. 18.

    \( a={ \log}_bc\iff c={b}^a,b>0\ \mathrm{and}\ b\ne 1. \)

References

  • Bailey, D. H., & Borwein, J. M. (2012). Ancient Indian square roots: An exercise in forensic paleo-mathematics. The American Mathematical Monthly, 119(8), 646–657.

    Article  Google Scholar 

  • Beck, B. B. (1980). Animal tool behavior: The use and manufacture of tools by animals. New York: Garland STPM Press.

    Google Scholar 

  • Boesch, C. (1993). Aspect of transmission of tool use in wild chimpanzees. In K. R. Gibson & T. Ingold (Eds.), Tools, language and cognition in human evolution (pp. 171–183). Cambridge: Cambridge University Press.

    Google Scholar 

  • Bosch, M., & Chevallard, Y. (1999). La sensibilité de l’activité mathématique aux ostensifs. Objet d’étude et problématique. Recherches en Didactique des Mathématiques, 19(1), 79–124.

    Google Scholar 

  • Clarke, G. (1969). World history: A new outline (2nd ed.). Cambridge: Cambridge University Press.

    Google Scholar 

  • Darwin, C. (1879). The descent of man (2nd ed.). St Ives: Penguin Classics (2004).

    Google Scholar 

  • Dunbar, R. I. M. (1993). Coevolution of neocortical size, group size and language in humans. Behavioral and Brain Sciences, 16, 661–736.

    Article  Google Scholar 

  • Fauvel, J., & Gray, J. (1987). The history of mathematics: A reader. Milton Keynes: The Open University.

    Google Scholar 

  • Gibson, K. R. (1993a). General introduction: Animal minds, human minds. In K. R. Gibson & T. Ingold (Eds.), Tools, language and cognition in human evolution (pp. 3–19). Cambridge: Cambridge University Press.

    Google Scholar 

  • Gibson, K. R. (1993b). Tool use, language and social behavior in relationship to information processing capacities. In K. R. Gibson & T. Ingold (Eds.), Tools, language and cognition in human evolution (pp. 251–269). Cambridge: Cambridge University Press.

    Google Scholar 

  • Gibson, K. R., & Ingold, T. (Eds.). (1993). Tools, language and cognition in human evolution. Cambridge: Cambridge University Press.

    Google Scholar 

  • Heath, T. L. (1926). The thirteen books of Euclid’s Element (Vol. 1). London: Cambridge University Press.

    Google Scholar 

  • Joseph, G. G. (2010). The crest of the peacock: Non-European roots of mathematics. Princeton: Princeton University Press.

    Google Scholar 

  • Kanes, C., & Lerman, S. (2008). Analysing concepts of communities of practice. In A. Watson & P. Winbourne (Eds.), New directions for situated cognition in mathematics education (pp. 30–328). New York: Springer.

    Google Scholar 

  • Kenward, B., Rutz, C., Weir, A. A. S., & Kacelnik, A. (2006). Development of tool use in New Caledonian crows: Inherited action patterns and social influences. Animal Behaviour, 72, 1329–1343.

    Article  Google Scholar 

  • Lave, J., & Wenger, E. (1991). Situated learning: Learning as situated peripheral participation. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Miller, J. M. A. (2000). Craniofacial variation in Homo habilis: An analysis of the evidence for multiple species. American Journal of Physical Anthropology, 112(1), 103–128.

    Article  Google Scholar 

  • Netz, R. (1999). The shaping of deduction in Greek mathematics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Pierce, R. C. (1977). A brief history of logarithms. Two-Year College Mathematics Journal, 22–26.

    Google Scholar 

  • Pruetz, J. D., & Bertolani, P. (2007). Savanna chimpanzees, Pan troglodytes verus, hunt with tools. Current Biology, 17, 412–417.

    Article  Google Scholar 

  • Rosińska, G. (1987). Tables of decimal trigonometric functions from ca. 1450 to ca. 1550. Annals of the New York Academy of Sciences, 500(1), 419–426.

    Article  Google Scholar 

  • Smith, D. E. (1958). History of Mathematics, Volume II. New York: Dover Publications.

    Google Scholar 

  • Solla Price, D. (1984). A history of calculating machines. Micro, IEEE, 4(1), 22–52.

    Google Scholar 

  • Tattersall, I. (2002). The case for saltational events in human evolution. In T. J. Crow (Ed.), The speciation of modern Homo sapiens: Proceedings of the British Academy (Vol. 106, pp. 49–59). Oxford: Oxford University Press.

    Google Scholar 

  • Tattersall, I., & Schwartz, J. H. (2000). Extinct humans. New York: Perseus books.

    Google Scholar 

  • Thoren, V. E. (1988). Prosthaphaeresis revisited. Historia Mathematica, 15, 32–39.

    Article  Google Scholar 

  • Vygotsky, L. S. (1978). Mind in Society. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Washburn, S. L. (1959). Speculations on the interrelations of the history of tools and biological evolution. Human Biology, 31(1), 21–31.

    Google Scholar 

  • Washburn, S. L. (1960). Tools and human evolution. Scientific American, 203, 63–75.

    Article  Google Scholar 

  • Weir, A. S., Chappell, J., & Kacelnik, A. (2002). Shaping of hooks in New Caledonian crows. Science, 297, 981.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Agder, Kristiansand, Norway

    John Monaghan

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  1. John Monaghan
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Monaghan, J. (2016). Tools, Human Development and Mathematics. In: Tools and Mathematics. Mathematics Education Library, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-02396-0_4

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  • DOI: https://doi.org/10.1007/978-3-319-02396-0_4

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