Abstract
Mathematics begins in human experience thousands of years ago as empirical and intuitive experiences. It takes the deliberate naming of concepts to help crystallize and secure those observations and intuitions as abstract concepts, and to begin separating the concept of number from specific instances of objects. It takes the creation of compact symbols to enable efficient calculation and to begin raising a consciousness of this activity we call mathematics. And it takes the sustained development and discussion of mathematical conventions and practices to create entire domains of mathematical thought, such as we find in geometry. The major innovations and conceptual reformulations are few in number, but these represent perhaps the greatest challenges to learners. Historically significant transformative events have their counterpart in the cognitive growth of the individual. This article examines the interplay between these big ideas in cultural history and the deliberate processes of cognitive change that are their counterpart in the educational process.
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References
Ball W.W.R. (1960). A Short Account of the History of Mathematics. Dover, New York
Bodrova E., Leong D. (1996). Tools of the Mind: The Vygotskian Approach to Early Childhood Education. Merill Englewood Cliffs, NJ
Carson, R.: 2003, ‘Pivotal Events as the Basis for Narrative Coherence in Science Education’, in E.D. Metz, Proceedings of The Seventh International History and Philosophy of Science and Science Teaching Conference, Winnipeg
Carson R. (2001). The Epic Narrative of Intellectual Culture as a Framework for Curricular Coherence. In: Bevilacqua F., Giannetto E., Matthews M. (eds), Science Education and Culture. Kluwer, Dordrecht
Cooke R. (1997). The History of Mathematic: A Brief Course. John Wiley and Sons, New York
Devlin K. (2000). The Language of Mathematics – Making the Invisible Visible. W.H. Freeman, New York
Egan K. (1997) The Educated Mind: How Cognitive Tools Shape Our Understanding. University of Chicago Press, Chicago
Fauvel J., Van Maanen J. (2000). History in Mathematics Education. The ICMI Study, Kluwer Dordrecht
Geertz C. (1973). The Interpretation of Cultures. Basic Books, New York
Goody J., Watt I. (1977). The Consequences of Literacy. In: Karabel J., Halsey A. (eds), Power and Ideology in Education. Oxford University Press, New York
Gould S.J. (1977) Ontogeny and Phylogeny. Harvard University Press, Cambridge
Greenspan S., Shanker S. (2004) The First Idea – How Symbols, Language, and Intelligence Evolved From Our Primate Ancestors to Modern Humans. Da Capo Press, Cambridge, MA
Gulikers I., Blom K. (2001). A Historical Angle, A Survey of Recent Literature on the Use and Value of History in Geometrical Education. Educational Studies in Mathematics 47:223–258
Guthrie K.S. (1987). The Pythagorean Sourcebook and Library. Phanes Press, Grand Rapids MI
Havelock, E.A.: 1963, Preface to Plato. Harvard University Press, Cambridge, MA
Heath, T.L.: 1981a, A History of Greek Mathematics (vol. I From Thales to Euclid), Dover, New York
Heath, T.L.: 1981b, A History of Greek Mathematics (vol. II From Aristarchus to Diaphantus), Dover, New York
Heath, T.L.: 1956, Euclid, The Thirteen Books of the Elements (vol 1) (second edition), Dover, New York
Hirst P.H. (1973). Liberal Education and the Nature of Knowledge. In: Peters R.S. (eds), The Philosophy of Education. Oxford University Press, Oxford, pp. 87–111
Jaegar, W.: 1967, Paideia: The Ideals of Greek Culture (vol 1. Archaic Greece: The Mind of Athens) (trans. G. Highet), Oxford University Press, New York
Jaeger, W.: 1986, Paideia: The Ideals of Greek Culture (vol 2. In Search of the Divine Center) (trans. G. Highet), Oxford University Press, New York
Klein H. (1974). The Science of Measurement: A Historical Survey. Dover, New York
Kline, M.: 1982, Mathematics: The Loss of Certainity. Oxford University Press, Oxford
Plato, 1986: The Republic (trans. B. Jowett), Prometheus Books, New York
Ronan, C.:1982, Science: Its History and Development Among the World’s Cultures, Facts on File, New York
Sambursky S. (1956). The Physical World of the Greeks. Princeton University Press, Princeton, NJ
Scribner S. (1995). Vygotsky’s Uses of History. In: Wertsch J. (eds), Culture, Communication, and Cognition: Vygotskian Perspectives. Cambridge University Press, New York
Senk S. (1989). van Hiele Levels and Achievement in Writing Geometry Proofs. Journal for Research in Mathematics Education 20(3):309–321
Shweder R.A. (1991). Thinking Through Cultures – Expeditions in Cultural Psychology. Harvard University Press, Cambridge, MA
Simson R. (1834). The Elements of Euclid. Desilver & Thomas, Philadelphia
Sklar, L.: 1977, Space, Time, and Spacetime. University of California Press, Los Angeles
Skemp R. (1987). The Psychology of Learning Mathematics. Laurence Erlbaum, New York
Sloan W. (1983). The Craft of Writing. W.W. Norton, New York
Smith D.E. (1958). History of Mathematics. Dover, New York
Van Doren C. (1991). A History of Knowledge: The Pivotal Events, People, and Achievements of World History. Ballantine Books, New York
Van der Veer R., Valsiner J. (1998). The Vygotsky Reader. Blackwell, Cambridge, UK
Van Hiele, P. M.: 1957, De Problematiek van Het Inzicht Gedomonstreed van Het Inzicht von Schoolkintren in Meerkundefeerstof [The Problem of Insight in Connection with Schoolchildren’s Insight into the Subject Matter of Geometry]. Unpublished Doctoral Dissertation, University Of Utrecht
Van Hiele P.M. (1959). Development and Learning Process: A Study of Some Aspects of Piaget’s Psychology in Relation with the Didactics of Mathematics. J.B. Wolters, Groningen, The Netherlands
van Hiele P.M., van Hiele-Geldof D. (1958). A Method of Initiation into Geometry at Secondary School. In Freudenthal H. (eds), Report on Methods of Initiation into Geometry. J. B. Wolters, Groningen, The Netherlands, pp. 67–80
Vygotsky L. (1986). Thought and Language. MIT Press, Cambridge MA
Vygotsky L. (1978). Mind in Society: The Development of Higher Psychological Processes. Harvard University Press, Cambridge MA
Watson P. (2005). Ideas: A History from Fire to Freud. Weidenfeld & Nicolson, London
Wenke R. (1990). Patterns in Prehistory – Humankind’s First Three Million Years. Oxford University Press, New York
Wertsch J. (eds) (1995). Culture, Communication, and Cognition: Vygotskian Perspectives. Cambridge University Press, New York
Wertsch J. (1985). Vygotsky and the Social Formation of Mind. Harvard University Press, Cambridge, MA
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Based on a Presentation to the Eighth International History, Philosophy, & Science Teaching Conference, Leeds, England. July 15–18, 2005
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Carson, R.N., Rowlands, S. Teaching the Conceptual Revolutions in Geometry. Sci & Educ 16, 921–954 (2007). https://doi.org/10.1007/s11191-006-9041-y
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DOI: https://doi.org/10.1007/s11191-006-9041-y