Abstract
This chapter contains a teaching proposal based on the history of matrices inspired by the conceptual and methodological framework introduced by Kjeldsen (2011) to integrate history into the teaching of mathematics. Kjeldsen’s conceptual framework is based on Sfard’s (2008) theory of thinking as communicating. Our goal is to create conflicting situations in which students are encouraged to reflect upon the metadiscursive rules related to matrices and determinants, comparing them with those found in some historical writings. Two teaching modules were created, dealing with two episodes in the history of matrices, based on the works of the mathematicians Sylvester and Cayley, and on the historical interpretation of Brechenmacher (2006). Two field studies were conducted with undergraduate mathematics students, from two universities in Rio de Janeiro. In this chapter we also explain how some historical metadiscursive rules were identified.
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Notes
- 1.
The term “multiplicity,” used in reference to the points of intersection of the conics, refers to the algebraic concept of the index of intersection, which generalizes the intuitive notion of counting the number of times that two algebraic curves intersect at a point.
- 2.
It is worthwhile to note that the expression “linear transformations” was often used by Cayley, despite his not having used it in the 1858 memoir. Some articles are dedicated entirely to linear transformations; for example, Cayley (1845).
- 3.
To keep the presentation simpler, Caley’s original notation for a matrix (a combination of parentheses and vertical lines) is not used here (though his notation for determinants using vertical lines is identical to the modern one).
- 4.
In the original: “Ces publications successives permettent de suivre l’élaboration progressive d’une méthode qui se caractérise par une traduction de propriétés géométriques ou analytiques dans le cadre du calcul des déterminants.”
- 5.
References
Bernardes, A. (2016). História e Ensino de matrizes: Promovendo reflexões sobre o discurso matemático (Unpublished doctoral dissertation). Federal University of Rio de Janeiro, Brazil. http://www.cos.ufrj.br/index.php/pt-BR/publicacoes-pesquisa/details/15/2606. Accessed August 9, 2017.
Bernardes, A., & Roque, T. (2014). Reflecting on metadiscursive rules through episodes from the history of matrices. In É. Barbin, U. T. Jankvist, & T. H. Kjeldsen (Eds.), History and epistemology in mathematics education: Proceedings of the 7th ESU (pp. 153–167). Copenhagen: Danish School of Education, Aarhus University.
Brechenmacher, F. (2006). Les matrices: Formes de représentation et pratiques opératoires (1850–1930). Site expert des Ecoles Normales Supérieures et du Ministère de l’Education Nationale. Retrieved from http://www.math.ens.fr/culturemath/histoire%20des%20maths/index-auteur.htm#B. Accessed August 9, 2017.
Cayley, A. (1845). On the theory of linear transformations, In Cambridge Mathematical Journal (Vol. IV, pp. 193–209). (reprinted in A. Cayley, The Collected Mathematical Papers of Arthur Cayley, vol. I, pp. 80–94, Cambridge University Press 1889).
Cayley, A. (1855). Remarque sur le notation des fonctions algébriques, In Journal für die reine und angewandte Mathematik (Crelles Journal), tom. I (pp. 282–285). (reprinted in A. Cayley, The Collected Mathematical Papers of Arthur Cayley, vol.II, pp.185–188, Cambridge University Press 1889).
Cayley, A. (1858). A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148, 17–37.
Kjeldsen, T. H. (2011). Does history have a significant role to play for the learning of mathematics? Multiple perspective approach to history, and the learning of meta level rules of mathematical discourse. In É. Barbin, M. Kronfellner, & C. Tzanakis (Eds.), History and epistemology in mathematics education: Proceedings of the 6th ESU (pp. 51–62). Wien: Verlag Holzhausen GmbH.
Kjeldsen, T. H., & Blomhøj, M. (2012). Beyond motivation: History as a method for the learning of metadiscursive rules in mathematics. Educational Studies in Mathematics, 80, 327–349.
Kjeldsen, T. H., & Petersen, P. H. (2014). Bridging history of the concept of a function with learning of mathematics: Students’ metadiscursive rules, concept formation and historical awareness. Science & Education, 23, 29–45.
Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint. The Journal of the Learning Sciences, 16(4), 567–615.
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.
Sylvester, J. J. (1850a). Additions to the articles “on a new class of theorems,” and “on Pascal’s theorems.” In D. Brewster, R. Taylor, R. Phillips, & R. Kane (Eds.), Philosophical Magazine and Journal of Science (Vol. XXXVII, pp. 213–218). London: Printers and Publishers of the University of London.
Sylvester, J. J. (1850b). On the intersections, contacts, and other correlations of two conics expressed by indeterminate coordinates. In W. Thomson (Ed.), Cambridge and Dublin Mathematical Journal, V (pp. 262–282). Cambridge: Macmillan and Co.
Sylvester, J. J. (1851a). An enumeration of the contacts of lines and surfaces of the second order. In H. F. Baker (Ed.), The collected mathematical papers of James Joseph Sylvester (Vol. 1, 1904, pp. 219–240). Cambridge: Cambridge University Press.
Sylvester, J. J. (1851b). On the relation between the minor determinants of linearly equivalent quadratic functions. In H. F. Baker (Ed.), The collected mathematical papers of James Joseph Sylvester (Vol. 1, 1904, pp. 241–250). Cambridge: Cambridge University Press.
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Bernardes, A., Roque, T. (2018). History of Matrices. In: Clark, K., Kjeldsen, T., Schorcht, S., Tzanakis, C. (eds) Mathematics, Education and History . ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-73924-3_11
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