Abstract
The aim of this chapter is to present some issues concerning secondary teacher education, drawing on the activity of the Laboratory of Mathematical Machines at the Department of Mathematics of the University of Modena and Reggio Emilia (MMLab: http://www.mmlab.unimore.it). The name comes from the most important collection of the Laboratory, containing more than two hundred working reconstructions (based on the original sources) of mathematical artefacts taken from the history of geometry. In this chapter we intend to discuss, in the setting of teacher education and within a suitable theoretical framework, a single case, i.e., an ellipse drawing device, from different perspectives (historic-epistemological, manipulative and virtual), to develop expertise in selecting and adjusting appropriate tools for the mathematics classroom.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For the Italian version see http://www.mmlab.unimore.it/on-line/Home/VisitealLaboratorio/Materiale.html
- 2.
See Goos’ contribution in this volume for other elements concerning the socio-cultural perspective and cultural tools.
- 3.
This construction problem is taken from the First Book of Euclid’s elements (Proposition 10, see Heath 1956, p. 267). The solution we propose is a bit different from Euclid’s one.
- 4.
http://www.tpub.com/engbas/4.htm. Accessed February 2010.
- 5.
Béguin and Rabardel (2000) define instrumentation as follows:
Utilization schemes have both a private and a social dimension. The private dimension is specific to each individual. The social dimension, i.e., the fact that it is shared by many members of a social group, results from the fact that schemes develop during a process involving individuals who are not isolated. Other users as well as the artefact’s designers contribute to the elaboration of the scheme. (Bèguin and Rabardel 2000, p. 182)
- 6.
We refer in a short way to the Cabri commands. Legend:
-
compass: to transport the given segment with a vertex in a given point (the software draws a circle);
-
intersection: to find the intersection point of two objects on the screen;
-
intersection (after compass command): to intersect the circle with another object on the screen;
-
segment: to draw a segment joining two points;
The others (axis, locus, symmetrical point) hint at geometrical meanings, and are realized by means of the available commands.
-
References
Anichini, G., Arzarello, F., Ciarrapico, L., & Robutti, O. (Eds.). (2004). Matematica 2003. La matematica per il cittadino. Attività didattiche e prove di verifica per un nuovo curricolo di Matematica (Ciclo secondario). Lucca: Matteoni stampatore.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.
Bartolini Bussi, M. G. (2001). The geometry of drawing instruments: Arguments for a didactical use of real and virtual copies. Cubo, 3(2), 27–54.
Bartolini Bussi, M. G. (in press). Challenges: The laboratory of mathematics. Proceeding conference of the future of mathematics education in Europe. Lisboa, 2007, see http://www.fmee2007.org/.
Bartolini Bussi, M. G., & Mariotti M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. English & M. G. Bartolini Bussi (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 746–783). New York: Routledge.
Bartolini Bussi, M. G., & Maschietto, M. (2006). Macchine Matematiche: Dalla storia alla scuola. Milano: Springer.
Bartolini Bussi, M. G., & Maschietto, M. (2008). Machines as tools in teacher education. In D. Tirosh & T. Wood (Eds.), The international handbook of mathematics teacher education: Vol. 2. Tools and processes in mathematics teacher education (pp. 183–208). Rotterdam: Sense.
Béguin, P., & Rabardel, P. (2000). Designing for instrument-mediated activity. Scandinavian Journal of Information Systems, 12, 173–190.
Bos, H. J. M. (2001). Redefining geometrical exactness: Descartes’ transformation of the early modern concept of construction. New York: Springer.
Dennis, D., & Confrey, J. (1995). Functions of a curve: Leibniz’s original notion of functions and it’s meaning for the parabola. The College Mathematics Journal, 26(2), 124–131.
Descartes, R. (1637). La géomètrie (nouvelle édition 1886). Paris: Hermann.
Heath, T. L. (1956). Euclid: The thirteen books of the elements. New York: Dover.
Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1–2), 151–161.
Lebesgue, H. (1950). Leçons sur les constructions géométriques. Paris: Gauthier-Villars.
Maschietto, M. (2005). The laboratory of mathematical machines of modena. Newsletter of the European Mathematical Society, 57, 34–37.
Maschietto, M., & Martignone, F. (2008). Activities with the mathematical machines: Pantographs and curve drawers. In E. Barbin, N. Stehlikova, & C. Tzanakis (Eds.), History and epistemology in mathematics education: Proceedings of the fifth European Summer University (pp. 285–296). Prague: Vydavatelsky.
Rabardel, P. (1995). Les hommes et les technologies: Approche cognitive des instruments contemporains. Paris: A. Colin.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(Feb), 4–14.
van Schooten, F. (1657). Exercitationum mathematicarum liber IV, sive de organica conicarum sectionum in plano descriptione. Lugd. Batav ex officina J. Elsevirii.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press.
Wartofsky, M. (1979). Perception, representation, and the forms of action: towards an historical epistemology. In M. Wartofsky (Ed.), Models: Representation and the scientific understanding (pp. 188–209). Dordrecht: D. Reidel.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer US
About this chapter
Cite this chapter
Maschietto, M., Bartolini Bussi, M.G. (2011). Mathematical Machines: From History to Mathematics Classroom. In: Zaslavsky, O., Sullivan, P. (eds) Constructing Knowledge for Teaching Secondary Mathematics. Mathematics Teacher Education, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09812-8_14
Download citation
DOI: https://doi.org/10.1007/978-0-387-09812-8_14
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-09811-1
Online ISBN: 978-0-387-09812-8
eBook Packages: Humanities, Social Sciences and LawEducation (R0)