Abstract
This paper summarizes our analysis of the complexity of ratio problems at Grades 6 and 7, and reports a two-year experiment related to the teaching and learning of rational numbers and proportionality in these grades. Two classes were followed and observed. Part of the teaching material was common to both classes, mainly the objectives and the corpus of ratio problems in a physical context. But in one class, here called “Partial-experiment”, the learning environment was exclusively a paper-pencil one and the teacher followed his usual method in designing and conducting teaching sequences. In the other class, here called “Full-experiment”, the teaching was based on a framework, emerging from our analysis of complexity of ratio problems, involving precise guidelines and a specific computer environment. Using a pre-test and a post-test, we observed clear progress in both classes compared to a sample of “standard” pupils. Our comparative pupil-oriented study indicates more complete improvement in the “full-experiment” class, i.e., a better acquisition of fractions and their use for solving usual proportionality problems. The average pupil’s progress is greater in the “full experiment”, with the pupils who were initially high- or low-level attainers benefiting the most from the “full-experiment”.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adjiage, R. and Heideier, A.: 1998, Didacticiels de la série Oratio, éditions Pierron, 57206 Sarreguemines.
Adjiage, R.: 1999, L'expression des nombres rationnels et leur enseignement initial, thése, IREM, ULP Strasbourg 1.
Adjiage R, et Pluvinage F.: 2000, ‘Un registre géométrique unidimensionnel pour l'expression des rationnels’, Recherches en Didactique des Mathématiques 20(1), 41–88.
Adjiage, R.: 2001, ‘Maturations du fonctionnement rationnel. Fractions et décimaux: Acquisitions d'une classe’, Annales de didactique et de sciences cognitives, Vol. 7, IREM de Strasbourg, pp. 7–48.
Adjiage, R.: 2005, ‘Diversité et invariants des problémes mettant en jeu des rapports’, Annales de didactique et sciences cognitives, Vol. 10, IREM de Strasbourg, pp. 95–129.
Alatorre, S. and Figueras O. 2005, ‘Routine and adaptive experts in proportional reasoning’, in G.M. Lloyd, M. Wilson, J.L.M. Wilkins and S.L. Behm (eds.), Proceeding_24718 of the 27th Annual Meeting of PME-NA, Virginia Tech, October 2005, 1–3.
Behr, M. et al.: 1979–2002, http://education.umn.edu/rationalnumberproject/default.html.
Bloch, I.: 2003. ‘Teaching functions in a graphic milieu: What forms of knowledge enable students to conjecture and prove’, Educational Studies in Mathematics, 52(1), 3–28.
Brousseau, G.: 1986, Théorisation des phénoménes d'enseignement des mathématiques, thése, Université de Bordeaux 1.
Clark, M.: 2005, ‘Using multiple-missing-values problems to promote the development of middle-school students’ proportional reasoning’, in G.M. Lloyd, M. Wilson, J.L.M. Wilkins, and S.L. Behm (eds.), Proceeding_24824 of the 27th Annual Meeting of PME-NA, Virginia Tech, October 2005, 1–7.
Comin, E.: 2000, Proportionnalité et fonction linéaire. Caractéres, causes et effets didactiques des évolutions et des réformes dans la scolarité obligatoire, thése, Université de Bordeaux 1.
Douady, R.: 1984, Jeux de cadres et dialectique outil-objet dans l'enseignement des mathématiques, une réalisation de tout le cursus primaire, Doctorat détat, Université de Paris VII, Paris.
Dupé, C., Robin, I. et Vugdalic, S.: 1998, ‘Profil et compétences en lecture, calcul et géométrie des éléves à l'entrée en sixiéme, évaluation septembre 1997’, Note d'information no. 98-24 (6 pages), Paris, Ministére de l'Education –DPD D1.
Duval, R.: 1995, Sémiosis et pensée humaine, Peter Lang, Bern.
Duval, R.: 2000, ‘Basic issues for research in mathematics education’, Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (PME), Vol. 1, Hiroshima, Japan, pp. 55–69.
Harel, G., Behr, M., Post, T. and Lesh, R.: 1991, ‘Variables affecting proportionality: Understanding of physical principles, formation of quantitative relations, and multiplicative invariance’, in F. Furinghetti (ed.), Proceedings of PME XV Conference, Assisi, Italy, 125–133.
Hart, K. and the CSMS Mathematics Team: 1981, Children's Understanding of Mathematics: 11–16, John Murray, London.
Hart, K.: 1984, Ratio: Children's Strategies and Errors, NFER-NELSON, Windsor.
Hart, K. and Sinkinson, A.: 1989, ‘They're useful —children's view of concrete materials’, PME XIII Vol. 2, Paris, pp. 60–66.
Karplus, R., Pulos, S. and Stage, E.K.: 1983. ‘Early adolescents' proportional reasoning on ‘rate’ Problems’, Educational Studies in Mathematics 14(3), 219–233.
Kieren, T.E.: 1980, ‘The rational number construct —its elements and mechanism’, in T.E. Kieren (ed.), Recent Research on Number Learning, ERIC/SMEAC, Columbus, pp. 125–150.
Legrand, M.: 2000, ‘Sciences, Enseignement, Démocratie et Humanisme’, Actes du XXVII Colloque Inter-IREM des formateurs chargés de la formation des maitres, IREM de Grenoble, pp. 9–28.
Lesh, R., Post, T. and Behr, M.: 1988, ‘Proportional reasoning’, in Hiebert, J. and Behr, M. (eds.), Number Concepts and Operations in the Middle Grades, Reston, Va: NCTM, Erlbaum, pp. 93–118.
Noelting, G.: 1980, ‘The development of proportional reasoning and the ratio concept’, Educational Studies in Mathematics, 11, 217–253.
Piaget, J., Grize, S.A. and Bang, V.: 1968. Epistemologie et psychologie de la fonction. Paris: Paris University Press.
Pluvinage, F. and Mallier, A.: 1998, ‘Le repérage des difficultés de lecture à l'aide du franccais et des mathématiques’, Annales de didactique et de sciences cognitives, IREM de Strasbourg, Vol. 6, pp. 117–124.
Post, T., Cramer, K., Harel, G., Kiernen, T. and Lesh, R.: 1998, ‘Research on rational number, ratio and proportionality’, Proceedings of PME-NA XX, Vol. I. Raleigh, North Carolina, pp. 89–93. (http://education.umn.edu/rationalnumberproject/98_1.html).
Rouche, N.: 1997, ‘Faut-il enseigner les grandeurs?’, in ‘Grandeurs physiques et grandeurs mathématiques’, Centre de Recherche sur l'Enseignement des Mathématiques, Nivelles, Belgique.
Streefland, L.: 1993, ‘The design of a mathematics course, a theoretical reflection’, Educational Studies in Mathematics 25, 109–135.
Tourniaire, F. and Pulos, S.: 1985, ‘Proportional reasoning: A review of the literature’, Educational Studies in Mathematics 16, 181–204.
Vergnaud, G.: 1983, ‘Multiplicative structures’, in R. Lesh and M. Landau (eds.), Acquisition of Mathematics Concepts and Processes, Academic Press, New York, pp. 127–174.
Vergnaud, G.: 1988, ‘Multiplicative structures’, in J. Hiebert and M. Behr (eds.), Number Concepts and Operations in the Middle Grades, NJ: Lawrence Erlbaum Association, pp. 141–161.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Adjiage, R., Pluvinage, F. An Experiment in Teaching Ratio and Proportion. Educ Stud Math 65, 149–175 (2007). https://doi.org/10.1007/s10649-006-9049-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-006-9049-x