Advertisement

ZDM

, Volume 38, Issue 3, pp 226–246 | Cite as

Mathematical modelling as a tool for the connection of school mathematics

  • Francisco Javier Garcia
  • Josep Gascón Pérez
  • Luisa Ruiz Higueras
  • Marianna Bosch Casabó
Analyses

Abstract

We start introducing some aspects of the theoretical framework: the Anthropological Theory of Didactics (ATD). Then, we consider on the research domain commonly known as “modelling and applications” and briefly describe its evolution using the ATD as an analytical tool. We propose a reformulation of the modelling processes from the point of view of the ATD, which is useful to identify new educational phenomena and to propose and tackle new research problems. Finally, we focus on the problem of the connection of school mathematics. The reformulation of the modelling processes emerges as a didactic tool to tackle this research problem. We work on the problem of the articulation of the study of functional relationships in Secondary Education and present a teaching proposal designed to reduce the disconnection in the study of functional relationships in Spanish Secondary Education.

ZDM-Classification

D20 D30 F80 I24 M14 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barbé, Q., Bosch, M., Espinoza, L. & Gascón, J. (2005). Didactic restrictions on the teacher's practice. The case of limits of functions in Spanish high school.Educational Studies in Mathematics, 59, 235–268.CrossRefGoogle Scholar
  2. Blum, W. (2002). ICMI study 14: Applications and modelling in mathematics education— Discussion document.Educational Studies in Mathematics, 51, 149–171.CrossRefGoogle Scholar
  3. Blum, W. (2006). “Filling Up”—the problem of independence-preserving teacher interventions in lessons with demanding modelling task. In M. Bosch,Proceedings of the 4th Conference of the European Society for Research in Mathematics Education.Google Scholar
  4. Blum, W. & Niss, M. (1991). Applied mathematical problem solving, modelling, applications and links to other subjects—State, trends and issues in mathematics instruction.Educational Studies in Mathematics, 22, 37–68.CrossRefGoogle Scholar
  5. Bolea, P. (2002).El proceso de algebrización de organizaciones matemáticas escolares. (Doctoral dissertation). Universidad de Zaragoza.Google Scholar
  6. Bolea, P., Bosch, M. & Gascón, J. (1999). The role of algebraization in the study of a mathematical organization. InProceedings of the 1st Conference of the European society for Research in Mathematics Education.Google Scholar
  7. Bolea, P., Bosch, M. & Gascón, J. (2001a). Cómo se construyen problemas en didáctica de las matemáticas?Educación Matemática, 13(3), 23–63.Google Scholar
  8. Bolea, P., Bosch, M. & Gascón J. (2001b). La transposición didáctica de organizaciones matemáticas en proceso de algebrización.Recherches en Didactique des Mathématiques, 21(3), 247–304.Google Scholar
  9. Bolea, P., Bosch, M. & Gascón J. (2003). Why is modelling not included in the teaching of algebra at secondary school? In A. Mariotti,Proceedings of the 3rd Conference of the European society for Research in Mathematics Education.Google Scholar
  10. Bosch, M. (1994).La dimensión ostensiva de la actividad matemática. (Doctoral dissertation). Universidad Autónoma de Barcelona.Google Scholar
  11. Bosch, M., Chevallard, Y. & Gascón, J. (2006). Science of Magic? The use of models and theories in didactics of mathematics. In Bosch, M.Proceedings of the 4th Conference of the European Research in Mathematics Education.Google Scholar
  12. Bosch, M., Fonseca, C. & Gascón, J. (2004). Incompletud de las organizaciones matemáticas locales en las instituciones escolares.Recherches en Didactique des Mathématiques, 24/2–3, 205–250.Google Scholar
  13. Brousseau, G. (1997).Theory of Didactical Situations in Mathematics. Dordrecht: Kluwer Academic Publishers.Google Scholar
  14. CECJA (2002). Decreto 148/2002 por el que se establecen las enseñanzas correspondientes a la Educación Secundaria Obligatoria en Andalucía.Boletín Oficial de la Junta de Andalucía, 75. June 27, 2002.Google Scholar
  15. Chevallard, Y. (1999). L'analyse des pratiques enseignantes en théorie anthropologique du didactique,Recherches en Didactique des Mathématiques 19/2, 221–226.Google Scholar
  16. Chevallard, Y. (2001). Aspectos problemáticos de la formación docente.Boletin del Seminario Interuniversitario de Investigación en Didáctica de las Matemáticas, no 12.Google Scholar
  17. Chevallard, Y. (2002a). Organiser l'étude 1. Structures et fonctions. In J. Dorier, M. Artaud, M. Artigue, R. Berthelot et R. Floris (eds.),Actes de la 11e École d'Été de didactique des mathématiques—Corps—21–30 Août 2001 (pp. 3–22). Grenoble: La Pensée Sauvage.Google Scholar
  18. Chevallard, Y. (2002b). Organiser l'étude. 3. Écologie & régulation. In J. Dorier, M. Artaud, M. Artigue, R. Berthelot et R. Floris (eds.),Actes de la 11e École d'Été de didactique des mathématiques—Corps—21–30 Août 2001 (pp. 41–56) Grenoble: La Pensée Sauvage.Google Scholar
  19. Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In M. Bosch,Proceedings of the IV Conference of the European Society for Research in Mathematics Education.Google Scholar
  20. Chevallard, Y., Bosch, M. & Gascón, J. (1997).Estudiar matemáticas. El eslabón perdido entre la enseñanza y el aprendizaje. Barcelona: ICE/Horsori.Google Scholar
  21. De Lange, J. (1996). Using and Applying Mathematics in Education. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick and C. Laborde (Eds.),International Handbook of Mathematics Education (pp. 49–98). Dordrecht: Kluwer Academic Publishers.Google Scholar
  22. Fonseca, C. (2004).Discontinuidades matemáticas y didácticas entre la Enseñaza Secundaria y la Enseñaza Universitaria (Doctoral dissertation). Universidad de Vigo.Google Scholar
  23. Freudenthal, H. (1973).Mathematics as an Educational Task: Dordrecht: Reidel.Google Scholar
  24. Freudenthal, H. (1991).Revisiting Mathematics Education. Dordrecht: Kluwer Academic Publishers.Google Scholar
  25. García, F.J. (2005).La modelización como herramienta de articulación de la matemática escolar. De la proporcionalidad a las relaciones funcionales (Doctoral dissertation). Universidad de Jaén.Google Scholar
  26. García, F.J. & Ruiz Higueras, L. (2002). Reconstrucción y evolución de organizaciones matemáticas en el ámbito de los sistemas de variación de magnitudes.Boletín del Seminario Interuniversitario de Investigación en Didáctica de las Matemáticas, no 12.Google Scholar
  27. García, F.J. & Ruiz Higueras, L. (2006). Mathematical praxeologies of increasing complexity. In M. Bosch,Proceedings of the 4th Conference of the European Research in Mathematics Education.Google Scholar
  28. Gascón, J. (1999). Fenómenos y problemas en didáctica de las matemáticas. In T. Ortega, (Ed.),Actas del III Simposio de la SEIEM (pp. 129–150) Valladolid: SEIEM.Google Scholar
  29. Gascón, J. (2001a). Incidencia del modelo epistemológico de las matemáticas sobre las prácticas docentes.Revista Latinoamericana de Investigación en Matemática Educativa, 4(2), 129–159.Google Scholar
  30. Gascón, J. (2001b). Evolución de la controversia entre geometría sintética y geometría analítica. Un punto de vista didáctico-matemático.Disertaciones del Seminario de Matemáticas Fundamentales (UNED, Madrid), 28, 1–20.Google Scholar
  31. Gascón, J. (2003). From the cognitive to the epistemological programme in the didactics of mathematics: two incommensurable scientific research programmes.For the learning of mathematics 23(2), 44–55.Google Scholar
  32. Harel, G. y Behr, M. (1989). Structure and Hierarchy of Missing-Value Proportion Problems and Their Representations.Journal of mathematical behaviour, 8, 77–119.Google Scholar
  33. Hart, K. M. (1988). Ratio and proportion. In J. Hiebert and M. Behr (Eds.),Number Concepts and operations in the middle grades (Vol 2) (pp. 198–219). Lawrence Erlbaum Associates.Google Scholar
  34. Karplus, R., Pulos, S. & Stage, E. (1981).Proportional Reasoning in Early Adolescents: Comparison and Missing Value Problems in Three Schools. Paper presented at the Third Annual Conference for the Psychology of Mathematics Education, Minneapolis, Minnesota.Google Scholar
  35. Karplus, R., Pulos, S. & Stage, E. (1983a). Proportional reasoning of early adolescents. In R. Lesh and M. Landau (Eds.),Acquisition of Mathematics Concepts and Processes. New York: Academic Press.Google Scholar
  36. Karplus, R., Pulos, S. & Stage, E. (1983b). Early adolescents' proportional reasoning on “rate” problems.Educational Studies in Mathematics, 14, 219–233.CrossRefGoogle Scholar
  37. Lamon, S. (1991). Ratio and proportion: connecting content and children's thinking.Journal of Research in Mathematics Education, 24 (1), 41–61.CrossRefGoogle Scholar
  38. Maaß, K. (2005). Barriers and opportunities for the integration of modelling in mathematics classes: results of an empirical study.Teaching Mathematics and its Applications, 24, 61–74.CrossRefGoogle Scholar
  39. Niss, M. (1999). Aspects of the nature and state of research in mathematics education.Educational Studies in Mathematics, 40(1), 1–24.CrossRefGoogle Scholar
  40. Niss, M. (2003). Mathematical Competencies and the Learning of Mathematics. Retrieved May 4, 2006, fromhttp://www7.nationalacademies.org/mseb/Mat hematical_Competencies_and_the_Learning_o f Mathematics.pdf Google Scholar
  41. Noelting, G. (1980a). The development of proportional reasoning and the ratio concept: Part I—Differentiation of stages.Educational Studies in Mathematics, 11, 217–253.CrossRefGoogle Scholar
  42. Noelting, G. (1980b). The development of proportional reasoning and the ratio concept: Part II—Problem structure at successive stages; problem solving strategies and the mechanism of adaptive restructuring.Educational Studies in Mathematics 11, 331–363.CrossRefGoogle Scholar
  43. OECD (2003).The PISA 2003 assessment framework: mathematics, readings, science and problem solving knowledge and skills. Paris: OECD.Google Scholar
  44. Singer, J. & Resnick, L. (1992). Representations of proportional relationships: are children part-part or part-whole reasoners?Educational Studies in Mathematics, 23, 231–246.CrossRefGoogle Scholar
  45. Tourniaire, F. (1986). Proportions in elementary school.Educational Studies in Mathematics, 17, 401–412.CrossRefGoogle Scholar
  46. Tourniaire, F. & Pulos, S. (1985). Proportional reasoning: a review of the literature.Educational Studies in Mathematics, 16, 181–204.CrossRefGoogle Scholar
  47. Treffers, A. & Goffree, F. (1985). Rational Analysis of Realistic Mathematics Education. In L. Streefland (Ed.)Proceedings of PME-9 (pp. 97–123). Utrecht: OW & OC.Google Scholar

Copyright information

© ZDM 2006

Authors and Affiliations

  • Francisco Javier Garcia
    • 1
  • Josep Gascón Pérez
    • 2
  • Luisa Ruiz Higueras
    • 1
  • Marianna Bosch Casabó
    • 3
  1. 1.Department of Didactic of SciencesUniversity of JaénJaénSpain
  2. 2.Department of MathematicsAutonomic University of Barcelona UAB Campus-Edifici CBellatera (Cerdanyola del Vallès)Spain
  3. 3.Department of Applied StatisticsFundemi IQS-Ramon Llull UniversityBarcelonaSpain

Personalised recommendations