Abstract
The present study reports on curriculum design implemented in eight parallel classes of high school, with students aged 14–15, by means of work-sheets dealing with properties of the principal plane geometric figures. Van Hiele’s theory was used to facilitate teachers’ discussion of student performances on work-sheets in order to enhance their didactical awareness. We hypothesize that van Hiele’s theory is a proper means to provide encouragement and support for teachers in rethinking their teaching goals and practices. This environment enables teachers to manage their students’ transition from the perception of a figure to its definition. In this paper, we will recount a teachers’ discussion as it unfolded in a school-university work group. The analysis of students’ work on work-sheets interweaves with an analysis of the teaching actions that foster the process of understanding definitions.
Résumé
La présente étude rend compte d’un programme mis en pratique dans 8 classes parallèles de niveau secondaire, auprès d’étudiants de 14 et 15 ans, grâce à des fiches d’exercices traitant des propriétés qui caractérisent les principales figures géométriques planes. La théorie de van Hiele a été utilisée pour faciliter la discussion chez les enseignants quant au rendement des élèves dans ces fiches d’exercices, afin d’affiner leur niveau de conscience pédagogique. Nous posons comme hypothèse que la théorie de van Hiele constitue un instrument adéquat pour encourager et soutenir les enseignants qui souhaitent revoir leurs objectifs et leurs pratiques d’enseignement. Un tel environnement permet aux enseignants d’aider leurs élèves à faire la transition qui consiste à passer de la perception d’une figure à sa définition. Dans cet article, nous rapportons certaines discussions entre enseignants, telles qu’elles ont eu lieu au sein d’un groupe de travail mixte scolaire et universitaire. L’analyse du travail fait par les étudiants sur les fiches d’exercices est ensuite mise en rapport avec l’analyse des techniques d’enseignement servant à favoriser le processus de compréhension des définitions.
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References
Cannizzaro L., & Menghini, M. (2001). From geometrical figures to linguistic rigor: Van Hiele’s model and the growing of teachers awareness. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of PME (pp. 1–291). Utrecht, The Netherlands: Freudenthal Institute.
Castelnuovo, E., & Barra, M. (2000). Matematica nella realtà. Torino: Boringhieri. (Original work published 1976)
Clements, D.H., & Battista, M.T. (1992). Geometry and spatial reasoning. In D. Grouws (Ed.), Handbook of research on teaching and learning mathematics (pp. 420–464). New York: Macmillan.
DeMarois, P., & Tall, D. (1999). Function: Organizing principle or cognitive root? In O. Zaslavsky (Ed.), Proceedings of the twenty-third conference of PME (Vol. 2, pp. 257–264). Haifa, Israel: Haifa Institute of Technology.
De Santis, C., & Percario, Z. (1997). Dalle costruzioni geometriche al rigore linguistico: un’esperienza didattica all’inizio del biennio. In B. Micale & S. Pluchino (Eds.), 18° Convegno Nazionale UMI-CIIM (Notiziario U.M.I, supplemento al n.7, pp. 129–132).
de Villiers, M. (1994). The role and function of a hierarchical classification of quadrilaterals. For the Learning of Mathematics, 14(1), 11–18.
de Villiers, M. (1998). To teach definitions in geometry or teach to define? In A. Olivier & K. Newstead (Eds.), Proceedings of 22nd PME Conference (Vol. 2, pp. 248–255). Stellenbosch, South Africa: University of Stellenbosch.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In David Tall (Ed.), Advanced mathematical thinking (pp. 95–126). Dordrecht, The Netherlands: Reidel.
Duval, R. (1988). Approche cognitive des problemes de geometrie en terms de congruence. Annales de Didactique et de Sciences Cognitives 1, 57–74.
Freudenthal, H. (1973). The case of geometry. In Mathematics as an educational task. Dordrecht, The Netherlands: Reidel.
Heath, T.L. (1908). The thirteen books of Euclid’s Elements. Cambridge, UK: Cambridge University Press.
Krainer, K. (1994). Integrating research and teacher in-service education as a means of mediating theory and practice in mathematics education. In Luciana Bazzini (Ed.), Theory and practice in mathematical education (pp. 121–132). Pavia, Italy: ISDAF.
Jacobson, C., & Lehrer, R. (2000). Teacher appropriation and student learning of geometry through design. Journal for Research in Mathematics Education, 31, 71–88.
Love, E. (1994). Mathematics teachers’ accounts seen as narratives. In L. Bazzini (Ed.), Theory and practice in mathematical education (pp. 143–155). Pavia, Italy: ISDAF.
Malara, N., Menghini, M., & Reggiani, M. (1996). General aspects of the Italian research in mathematics education. In N. Malara, M. Menghini, & M. Reggiani (Eds.), Italian research in mathematics education 1988-1995 (pp. 7–23). Roma: Seminario Nazionale di didattica della Matematica e CNR.
Mariotti, M.A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34, 220–248.
Monaghan, F. (2000). What difference does it make? Children’s views of the differences between some quadrilaterals. Educational Studies in Mathematics, 42, 179–196.
Pinto, M.M.F., & Tall, D. (1999). Student construction of formal theory: Giving and extracting meaning. In O. Zaslavsky (Ed.), Proceedings of the twenty-third conference of PME (Vol. 4, pp. 65–72). Haifa, Israel.
Pirie, S., Martin, L., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 165–190.
Skemp R.R. (1979). Intelligence, learning, and action. Chichester, UK: John Wiley.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with special reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.
Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. Chicago: University of Chicago. (ERIC Document Reproduction Service No. ED 220 288)
van Hiele, P.M. (1986). Structure and insight. New York: Academy Press.
van Hiele, P.M., & van Hiele-Geldof, D. (1958). A method of initiation into geometry at secondary schools. In H. Freudenthal (Ed.), Report on methods of initiation into geometry (pp. 67–80). Groningen, The Netherlands: J.B. Wolters.
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Cannizzaro, L., Menghini, M. From Geometrical Figures to Definitional Rigour: Teachers’ Analysis of Teaching Units Mediated through Van Hiele’s Theory. Can J Sci Math Techn 6, 369–386 (2006). https://doi.org/10.1080/14926150609556711
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DOI: https://doi.org/10.1080/14926150609556711